A Presentation of the Absolute Galois Group of ℚ₂

A Presentation of the Absolute Galois Group of $\mathbb{Q}_2$

David Roe

Department of Mathematics
Massachusetts Institute of Technology
roed@mit.edu

David Turturean

^* Incoming, Physical Superintelligence PBC

Abstract

We give an explicit profinite presentation of the absolute Galois group $\notn{GQ}{G_{\mathbb{Q}_2}}\text{,}$ filling a gap left by Jannsen and Wingberg [JW, Thm. 2] (who found a presentation for all $p$-adic fields with odd $p$) and Diekert [Die, Theorem, p. 152] (who found a presentation for $2$-adic fields where adjoining $i$ is unramified). The presentation has four generators $\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notn{x1}{x_1}\text{;}$ the closed normal subgroup generated by $\notn{x0}{x_0},\notn{x1}{x_1}$ is required to be pro-$2\text{;}$ and there are two defining relations. The first is the tame relation $\notn{tau}{\tau}^\notn{sigma}{\sigma}=\notn{tau}{\tau}^2$ and the second is the explicit relation stated in Theorem 1.2.

Section 1 Introduction and main theorem

For odd primes $p\text{,}$ the absolute Galois group of $\notn{Qp}{\mathbb{Q}_p}$ admits an explicit four-generator presentation of Jannsen–Wingberg type; see [NSW, Thm. 7.5.14]. Yet both the statement and proof break down for $p=2\text{.}$ Jarden and Shusterman [JS, Thms. 2.1, 3.4, and 4.4] have since shown that the absolute Galois group of every $p$-adic field, including the dyadic ones, is finitely presented, and they computed the minimal numbers of generators and relations; their argument is cohomological and does not produce explicit generators or relations. We give an explicit presentation for $p=2$ and a new approach that works in the dyadic setting.

Throughout, exponentiation by an element of $\notn{Zhat}{\Zhat}$ has its usual profinite meaning. Let

\begin{equation*} \notn{w}{\w}=(1\text{ in }\mathbb{Z}_2,\;0\text{ in }\mathbb{Z}_\ell\text{ for every odd prime }\ell). \end{equation*}

Thus $\notn{w}{\w}^2=\notn{w}{\w}\text{.}$

We use the conventions

\begin{equation*} x^g=g^{-1}xg, \qquad [x,y]=x^{-1}y^{-1}xy. \end{equation*}

The unramified coordinate is normalized geometrically throughout: if $\Phi_{\mathrm{ar}}$ is arithmetic Frobenius and $\Phi_{\mathrm{geom}}=\Phi_{\mathrm{ar}}^{-1}\text{,}$ then $\notn{nuur}{\nu_{\mathrm{ur}}}(\Phi_{\mathrm{geom}})=1\text{.}$ With the right-conjugation convention above, a standard tame generator satisfies

\begin{equation*} \notn{tau}{\tau}^{\Phi_{\mathrm{geom}}}=\notn{tau}{\tau}^q. \end{equation*}

Thus the displayed relation $\notn{tau}{\tau}^\notn{sigma}{\sigma}=\notn{tau}{\tau}^2$ means that $\notn{sigma}{\sigma}$ maps to geometric Frobenius. Put

\begin{equation} \sigma_2=\notn{sigma}{\sigma}^{\notn{w}{\w}},\qquad u_i=(x_i\notn{tau}{\tau})^{\notn{w}{\w}}\;(i=0,1),\qquad d_0=u_0\notn{x0}{x_0}^{-1},\tag{1.1} \end{equation}

\begin{equation} z_0=\notn{x0}{x_0}^{\sigma_2}, \qquad c_0=[d_0,z_0], \qquad g_0=\sigma_2^2, \qquad d_g=d_0^{g_0},\tag{1.2} \end{equation}

\begin{equation} h_c=[d_g,d_0], \qquad h_0=\notn{x0}{x_0}^{g_0}\notn{x0}{x_0}\,d_gd_0d_0^2h_c.\tag{1.3} \end{equation}

Proposition 1.1. Marked dyadic Demushkin normalization.

Let $\notn{Dloc}{D_{\mathrm{loc}}}=G_{\Qtwo}(2)$ and let $\notn{nuur}{\nu_{\mathrm{ur}}}:\notn{Dloc}{D_{\mathrm{loc}}}\twoheadrightarrow\mathbb{Z}_2$ be normalized by sending geometric Frobenius to $1\text{.}$ There exist topological generators $a,s,y$ such that

\begin{equation} \notn{Dloc}{D_{\mathrm{loc}}}\cong \left\langle a,s,y\;\middle|\;a^2s^4[s,y]=1\right\rangle_{\mathrm{pro}-2}, \qquad \notn{nuur}{\nu_{\mathrm{ur}}}(a)=-2,\quad \notn{nuur}{\nu_{\mathrm{ur}}}(s)=1,\quad \notn{nuur}{\nu_{\mathrm{ur}}}(y)=0.\tag{1.4} \end{equation}

The proof is given in Sections 3.1–3.3.

Lean

With the marked Demushkin normalization available, the main theorem takes the following form.

Theorem 1.2. Presentation theorem.

The absolute Galois group $\notn{GQ}{\GQ}$ is isomorphic to the profinite group topologically generated by

\begin{equation*} \notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notn{x1}{x_1}, \end{equation*}

for which the closed normal subgroup generated by $\notn{x0}{x_0},\notn{x1}{x_1}$ is pro-$2\text{,}$ and which is subject to the two relations

\begin{equation} \notn{tau}{\tau}^\notn{sigma}{\sigma}=\notn{tau}{\tau}^2\tag{1.5} \end{equation}

and

\begin{equation} h_0u_1^{-1}\notn{x1}{x_1}^\notn{sigma}{\sigma} c_0=1,\tag{1.6} \end{equation}

where the auxiliary words are defined in (1.1)–(1.3).

Lean (literal relators)Lean (counting form)Lean (main)Lean

Explicit presentations of this kind have a substantial history. For odd $p\text{,}$ the structure of the full group $G_{\mathbb{Q}_p}$ by generators and relations is due to Jannsen and Wingberg [JW]; see [NSW, Chapter VII] for a systematic account. In residue characteristic two, Zel’venskiĭ [Zel, Theorem 4] described the Galois group of the maximal extension without simple ramification for $2$-adic base fields of odd degree, and Diekert [Die, Theorem, p. 152] obtained structure results for absolute Galois groups of dyadic fields under additional hypotheses. More recently, Jarden and Shusterman [JS, Thms. 2.1, 3.4, 4.4] proved that $G_K$ is finitely presented as a profinite group for every finite extension $K/\mathbb{Q}_p\text{,}$ with no restriction at $p=2\text{,}$ and computed the minimal numbers of generators and relations ($[K:\mathbb{Q}_p]+2$ and $[K:\mathbb{Q}_p]+1$); their proof, via Lubotzky’s cohomological criterion for finite presentability, deliberately produces no explicit generators or relations. The present paper’s contribution is therefore not finite presentability, which is theirs, but the explicit marked presentation: concrete generators tied to the tame–wild structure, concrete word relations, and the finite-quotient characterization on which the counting proof runs. Presentations of local Galois groups also feed anabelian questions about their automorphisms; see for instance [Nis].

Theorem 1.2 is not the only presentation of $G_{\mathbb{Q}_2}$ that has been proposed. A second four-generator, two-relator candidate, the square-commutator presentation, was written down independently of it and checked against finite-quotient counts; Appendix C proves that it, too, presents $G_{\mathbb{Q}_2}\text{.}$ It is not obtained from Theorem 1.2 by a change of generators (no triple of words identifies the two maximal pro-$2$ quotients) and is proved instead by running the argument of this paper over a second source: the interface of Corollary 6.19 is re-verified for the new relator, and the instance of the Demushkin classification needed to identify its maximal pro-$2$ quotient is proved there rather than quoted. That appendix therefore rests on the same classical inputs as Theorem 1.2 and on no others, and the chain it completes is formalized without gaps.

During preparation of this manuscript, we also learned of independently proposed candidate presentations for $G_{\mathbb{Q}_2}$ by other researchers. Some may be isomorphic, after changes of generators, to the presentation in Theorem 1.2. At the time of writing, those candidate presentations and the proposed isomorphisms have not been fully validated.

Write $\notn{GA}{\GA}$ for the profinite group in Theorem 1.2. The condition that the normal closure of $\notn{x0}{x_0},\notn{x1}{x_1}$ be pro-$2$ is part of its presentation. For an epimorphism to a finite group $G\text{,}$ this condition is equivalent to requiring the images of $\notn{x0}{x_0},\notn{x1}{x_1}$ to lie in the $2$-core $O_2(G)\text{.}$

We use presentation in the following marked sense. The group $\notn{GA}{\GA}$ is the quotient of the free profinite group on $\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notn{x1}{x_1}$ by the intersection of the kernels of all its admissible finite quotients: those in which the images of the four generators generate, satisfy (1.5) and (1.6), and have $2$-group normal closure of the images of $\notn{x0}{x_0},\notn{x1}{x_1}\text{;}$ the formal construction is (2.1). The pro-$2$ requirement is thus a condition imposed on every finite quotient alongside the two word relations. It is not itself a word relation, and we do not claim that it follows from the two displayed relators; in particular, $\notn{GA}{\GA}$ is not asserted to coincide with the quotient of the free profinite group by the closed normal subgroup generated by the two relators alone. This enriched format is the classical one for absolute Galois groups of $p$-adic fields: for odd $p\text{,}$ the Jannsen–Wingberg presentation likewise imposes, alongside its word relations, the condition that the normal closure of the wild generators be pro-$p$ [JW, Thm. 2]. In the unmarked sense, finite presentability of $\notn{GQ}{\GQ}\text{,}$ with minimal generator and relation numbers three and two, is the theorem of Jarden and Shusterman [JS, Thms. 2.1, 3.4, and 4.4] recalled above; Theorem 1.2 provides the explicit marked presentation.

Subsection 1.1 Strategy of the proof

The proof has six stages.

The candidate $\notn{GA}{\GA}$ is constructed from its admissible finite quotients. A reconstruction lemma reduces the theorem to equality of epimorphism counts from $\notn{GA}{\GA}$ and $\notn{GQ}{\GQ}$ to every finite group.
The tame quotient of $\notn{GA}{\GA}$ is identified with the tame quotient of $\notn{GQ}{\GQ}\text{.}$ Its maximal pro-$2$ quotient is identified with $G_{\Qtwo}(2)$ in a form preserving the full unramified character to $\mathbb{Z}_2\text{,}$ not merely its reduction modulo $2\text{.}$
The tame and maximal pro-$2$ maps are combined over their common unramified quotient. Fixing this fibre-product datum turns the comparison into a finite lifting problem with a prescribed tame quotient and a prescribed map to an elementary abelian $2$-group.
For extensions with elementary abelian $2$-group kernel, lifts from the candidate are governed by the two defining relators, while lifts from $\notn{GQ}{\GQ}$ are governed by continuous cohomology. The two theories are shown to have the same obstruction dimensions, lift multiplicities, duality pairings, and connecting maps.
When a simple coefficient module is self-dual, a scalar central pushout produces a quadratic obstruction. An exact word calculation on the candidate side and the dyadic square-class filtration on the local side give the same quadratic Gauss sum.
For a finite target, the normal $2$-subgroup is reduced through an elementary quotient and a central Frattini layer. Fourier inversion and the partition of homomorphisms according to their image subgroup reduce the count to strictly smaller targets. Strong induction gives equality of all finite epimorphism counts, and profinite reconstruction gives the required isomorphism.

The dependence of the argument is therefore

\begin{equation*} \begin{gathered} \text{marked tame and maximal pro-$2$ quotients} \\ \Downarrow \\ \text{common tame--pro-$2$ data over $\mathbb{Z}_2$} \\ \Downarrow \\ \text{linear lifting comparison and quadratic obstruction comparison} \\ \Downarrow \\ \text{induction on the normal $2$-subgroup of a finite target} \\ \Downarrow \\ |\Sur(\notn{GA}{\GA},G)|=|\Sur(\notn{GQ}{\GQ},G)|\quad\text{for every finite group $G$} \\ \Downarrow \\ \notn{GA}{\GA}\cong\notn{GQ}{\GQ}. \end{gathered} \end{equation*}

In order, the five arrows are carried by: the marked identifications of Section 3 (Proposition 1.1, Proposition 3.14); the fibre-product boundary of Section 4; the two lifting theories and their numerical comparison in Section 5, Section 6, summarized in the source interface of Corollary 6.19, together with the minimal-block analysis and closed recursion of Section 7, Section 8 (Theorem 8.17) driving the induction of Section 9 (Theorem 4.2); the exhaustion over tame frames of Section 10 (Lemma 10.1, Proposition 2.3); and the one-sided reconstruction lemma (Lemma 2.5). This chain agrees with the formalization’s dependency graph: in Lean, the main theorem consumes exactly Lemma 2.5, Lemma 10.1, Proposition 2.3, Theorem 4.2, with Theorem 8.17 and the source interface feeding Theorem 4.2.

The paper is organized as follows. In Section 2 we construct the candidate $\notn{GA}{\GA}$ as a profinite group and prove the one-sided reconstruction lemma, reducing Theorem 1.2 to the equality of finite epimorphism counts. In Section 3 we identify the tame and maximal pro-$2$ quotients of $\notn{GA}{\GA}$ with those of $\notn{GQ}{\GQ}$ in marked form over the full unramified quotient $\mathbb{Z}_2\text{,}$ proving Proposition 1.1. In Section 4 we combine the two quotients over their common unramified data and reduce the comparison to a boundary-framed lifting problem. In Section 5 we develop the linear lifting theory on both sides: the evaluated Fox Jacobian complex for $\notn{GA}{\GA}\text{,}$ and continuous cohomology with local Tate duality for $\notn{GQ}{\GQ}\text{.}$ In Section 6 we compute the quadratic obstruction that appears for self-dual simple $\F_2$-modules, comparing the candidate word calculation with the dyadic Gauss sum. In Section 7 we isolate a minimal non-scalar module layer in the wild kernel, and in Section 8 we invert the resulting lifting-count recursion by finite Fourier analysis. In Section 9 we assemble these ingredients into the inductive proof of the boundary-framed theorem, and in Section 10 we sum over tame boundary frames to complete the proof of Theorem 1.2. A closing section records further questions. The normalized cochain and transfer conventions used throughout are fixed in Appendix A, and the finite Fox–Heisenberg evaluation rules and the word ledger behind the Jacobian and quadratic calculations are recorded in Appendix B. In Appendix C we prove that a second, independently proposed presentation also presents $\notn{GQ}{\GQ}\text{.}$ A final appendix collects background material.

Acknowledgements.

David Roe was supported by Simons Foundation grant SFI-MPS-SSRFA-0000542 and by the AI for Math fund of the Renaissance Charitable Foundation. David Turturean used computing resources provided by Fulcrum to develop the solution and produce his Lean formalization.

David Turturean used ChatGPT-5.6-Pro to generate the presentation, develop and repair its proof, and draft the first 60-page manuscript. He directed much of this work by voice, using voice mode in Claude Code to operate the surrounding research workflow and speech-to-text in ChatGPT.

Roe and Turturean developed separate Lean 4 formalizations with coding agents. Roe used Claude Opus 4.8 and Fable 5; Turturean used ChatGPT-5.5-Pro, Codex (GPT-5.5-xhigh), Claude Code, and Fable 5. Turturean’s completed release later incorporated a documented adaptation from Roe’s code. Fable 5 also assisted Roe in preparing the PaperForge edition of this paper. The authors reviewed and revised the material included here and take responsibility for the final mathematical claims and exposition. A detailed development record accompanies the paper.

Subsection 1.2 What is new

We record the aspects of the argument that we believe are new.

The route to the presentation is enumerative: we match $|\Sur(\GA,G)|=|\Sur(\GQ,G)|$ for every finite group $G$ and conclude by a one-sided profinite reconstruction. To our knowledge, this counting comparison has not previously been used to establish a presentation of a local absolute Galois group; the known proofs for odd $p$ [JW] and the dyadic partial results [Zel], [Die] proceed through embedding-problem solvability and the Demushkin classification.

The reconstruction lemma (Lemma 2.5) is stated in a one-sided form: $P$ is assumed topologically finitely generated while $Q$ is an arbitrary profinite group, with equality of the surjection counts $|\Sur(P,H)|=|\Sur(Q,H)|$ the only hypothesis required for $P \cong Q\text{.}$ Classical finite-quotient rigidity [DFPR, Theorem 1] (see also [RZ, Prop. 2.5.2]) assumes finite generation on both sides, while for small profinite groups the set of finite quotients is already determining [FJ, Proposition 16.10.7]. Although both sides are known to be finitely generated in the present application, the one-sided form is the natural strength of the argument (it matches the counting data the proof actually produces) and it is the shape required in settings where finite generation of the Galois side is open, such as the restricted-ramification groups $G_{\mathbb{Q},S}\text{,}$ which are small by Hermite–Minkowski yet not known to be topologically finitely generated.

Lifting problems are controlled by an evaluated Fox–Jacobian three-term complex, with cup products computed in an explicit Heisenberg model. Free differential calculus is an established tool in pro-$p$ Galois theory [Koc]; the device of evaluating the Jacobian on each finite target to control lifting counts appears to be new here. Finite Fourier inversion over character groups of finite modules is used to invert the lifting-count recursion: a classical enumerative device in the Frobenius tradition, applied inside a Galois-presentation argument.

For self-dual simple $\F_2$-modules, the lifting count acquires a quadratic obstruction, computed through Arf invariants and the dyadic filtration of square classes: the count is not determined by the linear duality data alone, in contrast with the odd-$p$ pattern of [NSW, Chapter VII]. The dyadic obstruction is computed with tools imported from quadratic-form theory and topology: Arf invariants, Stiefel–Whitney classes [Kah], [Gui], the Evens norm from finite-group cohomology [Eve], [Koz], and Gauss-sign computations.

The boundary-framed marked target $\mathcal Y=(Y,L_Y,\pi_Y,\theta_Y)$ of Definition 4.1 and the exact-image count $e_\Gamma^{\beta}$ are defined for an arbitrary profinite source with a tame–pro-$2$ boundary; nothing in the construction is specific to $\Qtwo\text{.}$ The framework is a candidate approach to presentations of $G_K$ for other dyadic fields, where only partial results are known [Die], [Sch].

The normalized-cochain ledger of Appendix A carries Shapiro maps, corestriction, and the Evens norm at the level of explicit normalized cochains with transversal bookkeeping, making transfer-level cohomology computations finite and machine-checkable; standard treatments [Bro], [Eve] work at the level of classes.

Subsection 1.3 The formalizations and their trust base

The theorem has been formalized in Lean 4 twice, in separately initiated projects, and the statements in the web edition of this paper link directly to the corresponding Lean declarations. The first author’s formalization proves the literal continuous isomorphism GQ2.main_presentation_literal with no sorry, together with the equivalent finite-counting form; the second author’s formalization was developed separately from the same manuscript, and its completed version incorporates a documented later adaptation of several steps from the first. Beyond checking the argument, the formalization effort sharpened the paper: a number of standing hypotheses, cochain normalizations, and citations were made explicit only when the proofs were machine-checked, and several statements in this revision repeat hypotheses whose necessity the formalization exposed.

Neither formalization is unconditional. Lean verifies the entire argument of this paper down to nine named interfaces with the classical literature, stated as axioms with a citation recorded for each; the kernel then checks everything above them. The nine interfaces are not of equal character, and we classify them accordingly: a direct classical theorem is transcribed from a single published statement; a classical theorem with encoding choices is classical mathematics whose Lean form passes through project normalizations that must be checked against the paper’s conventions; and a composite interface bundles several classical inputs, or a classical input plus a project convention, into one statement. The composite interfaces (B3c, B8, B11a below) are where independent expert attention is most valuable. The list below is generated from the same census as the formalization itself; the literal Lean statements, with links into the API documentation and this paper, are collected on the project’s axioms page.

B1 (direct classical theorem): $\GQ$ is topologically finitely generated. Lean name: absGalQ2_isTopologicallyFinitelyGenerated. Source: [NSW, Ch. VII §7.4, Theorem (7.4.1)]; the weaker bound of [JW] also suffices. Encoding: Stated verbatim for the fixed model of $\GQ\text{;}$ no normalization enters. Used at: Lemma 2.5.
B3c (composite interface): $G_{\Qtwo}(2)$ is the rank-$3$ Demushkin group with $q=2$ in Labute’s classification, its dualizing character is the cyclotomic character through this quotient, and a normalized marked identification exists. Lean name: dyadicOrientation. Source: [Lab, Théorème 4(2), Théorème 8]; [NSW, (7.5.11)–(7.5.12)]; [Ser63]. Encoding: Bundles the classification, the dualizing-character identification, and the project’s normalized unramified marking into one statement; the marking normalization is the paper’s (geometric Frobenius to $1$). Used at: Lemma 3.4, Proposition 1.1.
B5 (classical theorem with encoding choices): Local class field theory for $\Qtwo\text{:}$ the reciprocity map with its unramified normalization and the compatibilities used by the marked-generator calculations. Lean name: localReciprocity. Source: [NSW, (7.1.1), (7.1.5)]; [Ser79, Ch. XI–XIII]. Encoding: The reciprocity normalization (uniformizer to geometric Frobenius) and the interface to the marked generators are project conventions to be checked against the sources. Used at: Lemma 3.5.
B6 (direct classical theorem): Local Tate duality at $\GQ$ and at $G_K$ for every finite $K/\Qtwo\text{:}$ the invariant isomorphism and the perfect cup pairing, with the induced nondegenerate mod-$2$ pairing. Lean name: tateDualityAt. Source: [NSW, Theorem (7.2.6)]; [Mil, I.2.3]; [FV, Ch. IV §5]. Encoding: Direct, restricted to the open subgroups actually used; the mod-$2$ specialization is spelled out in the Lean statement. Used at: Subsection 6.3.
B7 (direct classical theorem): Finiteness of $H^i(\GQ,A)$ for finite $A$ and the local Euler–Poincaré characteristic formula. Lean name: absGalQ2_localEulerCharacteristic. Source: [NSW, Theorem (7.3.1)]; [Mil, I.2.8]. Encoding: Direct; the retained $H^0$-finiteness clause is redundant and kept only for faithfulness to the source. Used at: (8.21), Subsection 9.2.
B8 (composite interface): For every $u\in\mathbb Z_2^\times$ there is a continuous automorphism of the pro-$2$ fundamental group of $\mathbb P^1\smallsetminus\{0,1,\infty\}$ acting on the peripheral generators by $u$-th powers up to conjugacy. Lean name: peripheralCyclotomicAction. Source: [Sti, §3.3, Definition 37] (classical origin: Deligne, in [Iha]). Encoding: Combines the cited geometric action with the realizability of the required $2$-adic-unit actions and with the project’s chosen peripheral marking; the cyclotomic surjectivity input comes from B5. Used at: Lemma 3.7.
B9 (classical theorem with encoding choices): The relative Stiefel–Whitney identity of (6.53) in degrees $1$ and $2$ for the twisted trace forms. Lean name: relativeStiefelWhitney_dyadic. Source: [Kah, Théorèmes 1–2]; [Koz, Theorem 1.1]; [Eve, Theorem 1]. Encoding: The theorem is classical; the exact relative Stiefel–Whitney normalization must match the convention fixed at (6.53). Used at: (6.53), Section 6.
B10 (classical theorem with encoding choices): A closed normal pro-$2$ wild inertia subgroup $W\le\GQ$ with $\GQ/W$ the tame group on $\sigma,\tau$ with $\tau^\sigma=\tau^2\text{,}$ marked by geometric Frobenius. Lean name: tameQuotient. Source: [NSW, Theorem (7.5.3), (7.5.2)]; [Ser79, Ch. IV; Ch. XIII §4, Proposition 13]. Encoding: The theorem (Iwasawa) is classical; the reciprocity-oriented choice of tame frame and the geometric-Frobenius marking are project conventions. Used at: Proposition 3.2.
B11a (composite interface): Over any finite $k/\Qtwo\text{,}$ in Kummer-cup form: the class $[a]\cup[b]$ vanishes in $H^2(G_k,\F_2)$ if and only if $b$ is a norm from $k(\sqrt a)\text{.}$ Lean name: hilbertSymbol_normCriterion_finiteDyadic. Source: [Ser79, Ch. XIV §2, Propositions 4(iii), 5, 7(iii)]; [Ser73, Ch. III §1.1, Proposition 1]. Encoding: The norm criterion is standard; it is stated through the repository’s continuous-cohomology and Kummer interfaces, which must be matched to the classical symbol. Used at: Subsection 6.3.

A finite-quotient verifier provides a complementary empirical check: the presentation predicts the correct number of Galois extensions for every group in a battery of 5,402 finite test groups. Passing every test is strong evidence but is not a proof, and it is independent of the Lean verification above.

Section 2 The candidate as a profinite group

Starting from the four generators, the two relators, and the requirement on the normal closure of $\notn{x0}{x_0},\notnfar{x1}{x_1}\text{,}$ we construct the candidate as an inverse limit of finite quotients. The section ends with an exact finite-quotient semantics for this construction and the reconstruction criterion used in Section 10.

Subsection 2.1 The marked quotient construction

Let $F$ be the free profinite group on $\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1}\text{.}$ The finite quotients relevant to the presentation are precisely those in which the four distinguished elements generate, satisfy both relators, and have $2$-group wild normal closure. Formally, call an epimorphism

\begin{equation*} \varphi:F\twoheadrightarrow G \end{equation*}

to a finite group admissible if the distinguished tuple generates $G\text{,}$ satisfies (1.5) and (1.6), and the normal closure of $\varphi(\notn{x0}{x_0}),\varphi(\notnfar{x1}{x_1})$ is a $2$-group. Let $\mathcal Q_A$ be the class of all admissible finite quotients and set

\begin{equation} \notn{NA}{N_A}=\bigcap_{\varphi\in\mathcal Q_A}\ker\varphi, \qquad \notn{GA}{\GA}=F/\notn{NA}{N_A}.\tag{2.1} \end{equation}

Let $\notn{WA}{\WA}$ denote the closed normal subgroup of $\notn{GA}{\GA}$ generated by $\notn{x0}{x_0},\notnfar{x1}{x_1}\text{.}$

Lemma 2.1. Finite subdirect closure.

If $F\twoheadrightarrow G_i$ are finitely many admissible quotients and $Y$ is the image of $F$ in $\prod_iG_i\text{,}$ then $F\twoheadrightarrow Y$ is admissible and its kernel is the intersection of the given kernels.

Lean

Proof.

The two relations hold coordinatewise. The distinguished tuple generates $Y\text{.}$ The normal closure of $\notn{x0}{x_0},\notnfar{x1}{x_1}$ in $Y$ lies in

\begin{equation*} Y\cap\prod_i O_2(G_i), \end{equation*}

which is a finite $2$-group. Since $Y$ is the image of the diagonal map $F\to\prod_iG_i\text{,}$ one has

\begin{equation*} \ker(F\to Y)=\bigcap_i\ker(F\to G_i), \end{equation*}

which proves the assertion about kernels.

Two features make the construction usable in counting arguments: every finite quotient of the candidate is itself admissible, and epimorphisms to a finite group admit a purely equational description. We record them in turn.

Lemma 2.2. Cofinality.

Every finite quotient of $\notn{GA}{\GA}$ is an admissible marked quotient. In particular, $\notn{WA}{\WA}$ is pro-$2\text{.}$

Lean

Proof.

Let $K\triangleleft_{\mathrm{open}}F$ contain $\notn{NA}{N_A}\text{.}$ For every $g\notin K\text{,}$ the definition of $\notn{NA}{N_A}$ gives an admissible quotient whose kernel does not contain $g\text{.}$ Compactness of $F\setminus K$ gives finitely many such kernels whose intersection is contained in $K\text{.}$ By Lemma 2.1, this intersection is itself the kernel of an admissible quotient. Therefore $F/K$ is a quotient of an admissible quotient, and admissibility passes to quotients. The last assertion follows because every finite image of $\notn{WA}{\WA}$ is a $2$-group.

Proposition 2.3. Epimorphic finite-quotient semantics.

For every finite group $G\text{,}$ epimorphisms $\notn{GA}{\GA}\twoheadrightarrow G$ are in natural bijection with generating quadruples $(\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1})\in G^4$ satisfying (1.5), (1.6), and $\notn{x0}{x_0},\notnfar{x1}{x_1}\in O_2(G)\text{.}$

Lean (equivalence)Lean (main)Lean

Proof.

An epimorphism sends $\notn{WA}{\WA}$ to a normal $2$-subgroup of $G\text{,}$ hence into $O_2(G)\text{.}$ Conversely, a generating quadruple with $\notn{x0}{x_0},\notnfar{x1}{x_1}\in O_2(G)$ has normal $2$-group closure and defines an admissible quotient in $\mathcal Q_A\text{.}$

Remark 2.4.

The corresponding statement for arbitrary, non-surjective homomorphisms into a fixed ambient group is false: the image of $\notn{WA}{\WA}$ is normal in the image subgroup, but need not lie in the $2$-core of the ambient group. This is why the proof below keeps track of exact images and their induced markings.

Subsection 2.2 A reconstruction lemma

Lemma 2.5. One-sided profinite reconstruction.

Let $P$ be a topologically finitely generated profinite group and let $Q$ be any profinite group. If

\begin{equation*} |\Sur(P,H)|=|\Sur(Q,H)| \end{equation*}

for every finite group $H\text{,}$ then $P\cong Q\text{.}$

Lean (from equal counts)Lean (main)

Proof.

We construct an epimorphism in each direction from compatible finite quotients and then use Hopficity to identify both maps as isomorphisms.

Since $P$ is finitely generated, $\Sur(P,H)$ and hence $\Sur(Q,H)$ are finite for every finite $H\text{.}$ The two groups have exactly the same finite quotients.

For $n\ge1\text{,}$ let $R_n(P)$ be the intersection of the kernels of all maps from $P$ to groups of order at most $n\text{,}$ and put $P_n=P/R_n(P)\text{.}$ The $P_n$ form a cofinal inverse system of finite quotients of $P\text{.}$ Every $P_n$ is a quotient of $Q\text{.}$ For any finite set of indices, a sufficiently large $P_m$ maps onto all corresponding $P_n\text{;}$ composing an epimorphism $Q\twoheadrightarrow P_m$ gives a compatible family on that finite set. Compactness of the inverse limit of the finite nonempty sets $\Sur(Q,P_n)$ yields a compatible family and therefore an epimorphism $Q\twoheadrightarrow P\text{.}$

Interchanging the two groups, use the directed system of all finite quotients of $Q\text{.}$ The sets of epimorphisms from $P$ to those quotients are finite and have the finite intersection property, so there is an epimorphism $P\twoheadrightarrow Q\text{.}$ The composite $P\twoheadrightarrow Q\twoheadrightarrow P$ is a surjective endomorphism of a finitely generated profinite group. Such groups are Hopfian; see, for example, [RZ, Prop. 2.5.2]. Hence the composite is an isomorphism, and so is $P\twoheadrightarrow Q\text{.}$

In more detail: the counting hypothesis is used only through finite data. Summing surjection counts over the subgroups of a finite $H$ converts the hypothesis into $|\Hom(P,H)|=|\Hom(Q,H)|$ for every finite $H\text{,}$ and these counts are finite because $P$ is topologically finitely generated. The formalization also assumes $Q$ topologically finitely generated: with counts encoded as natural numbers, an infinite homomorphism set would make the count hypothesis vacuous rather than false, so finiteness on both sides is part of the statement’s content. In the application this is harmless: the candidate is finitely generated by construction, and the finite generation of $\GQ$ is exactly the foundational input B1.

The topological finite generation of $\notn{GQ}{\GQ}$ required in this argument is classical; the direct dyadic reference is [NSW, Ch. VII §7.4, Theorem (7.4.1)], which bounds the number of generators of $G_k$ for every $p$-adic $k$ with no parity hypothesis; see also [JW]. Reconstruction of profinite groups from their finite quotients goes back to [DFPR, Theorem 1]; the counting refinement used here is a one-sided variant.

Section 3 The tame and maximal pro-$2$ quotients

The candidate construction of Section 2 is now compared with the corresponding tame and maximal pro-$2$ quotients of $\notn{GQ}{G_{\Qtwo}}\text{.}$ Besides the abstract quotient identifications, we determine compatible maps to the full unramified quotient $\mathbb{Z}_2\text{;}$ those marked maps are the input to the fibre product in Section 4.

Let

\begin{equation*} \notn{TA}{\TA}=\langle \notn{sigma}{\sigma},\notn{tau}{\tau}\mid \notn{tau}{\tau}^\notn{sigma}{\sigma}=\notn{tau}{\tau}^2\rangle_{\mathrm{prof}} \end{equation*}

denote the finite-quotient tame group.

Lemma 3.1. Finite tame quotients.

In every finite quotient generated by $s,t$ with $t^s=t^2\text{,}$ the element $t$ has odd order, and the quotient has the form

\begin{equation*} C_e\rtimes C_n, \qquad e\text{ odd}, \qquad s^{-1}ts=t^2. \end{equation*}

Every normal $2$-subgroup of such a quotient is central and lies in the unramified cyclic direction.

Lean (odd order)Lean (central 2-subgroup)Lean

Proof.

Conjugation preserves order, whereas $|t^2|=|t|/\gcd(2,|t|)\text{,}$ so $|t|$ is odd. Let $I=\langle t\rangle$ and suppose $sI$ has order $n\text{.}$ Then $s^n=t^m\text{.}$ Since $s^n$ commutes with $s\text{,}$ the relation $t^s=t^2$ gives $t^m=t^{2m}\text{,}$ hence $m\equiv0\pmod{|t|}\text{.}$ Thus the extension splits.

If $N$ is a normal $2$-subgroup, then $N\cap I=1$ and $[N,I]\le N\cap I=1\text{.}$ Hence $N$ centralizes $I\text{.}$ It is therefore contained in the $2$-primary part of the kernel of the Frobenius action on $I\text{,}$ which is central and unramified.

The first comparison between the two sources happens on tame quotients, where both collapse onto the same explicit group.

Proposition 3.2. Common tame quotient.

There are canonical isomorphisms

\begin{equation*} \notn{GA}{\GA}/\notn{WA}{\WA}\cong \notn{TA}{\TA}\cong \notn{GQ}{\GQ}/\notn{WF}{\WF}, \end{equation*}

where $\notn{WF}{\WF}$ is wild inertia.

Lean (Γ_A)Lean (local)Lean (structure)Lean

Proof.

In the quotient by $\notn{WA}{\WA}\text{,}$ the elements $\notn{x0}{x_0},\notnfar{x1}{x_1}$ vanish. By Lemma 3.1, $\notn{tau}{\tau}$ has odd order in every finite quotient, so $\notn{tau}{\tau}^\notn{w}{\w}=1\text{.}$ Hence $u_i=d_0=z_0=c_0=d_g=h_c=h_0=1\text{;}$ the words $\sigma_2$ and $g_0$ may remain nontrivial, but they only conjugate the identity. The wild relation therefore disappears. For the local group, the tame quotient has pro-prime-to-$2$ cyclic inertia and procyclic unramified quotient; geometric Frobenius acts on tame inertia by $x\mapsto x^2\text{.}$ With $x^g=g^{-1}xg\text{,}$ this is precisely the relation $\notn{tau}{\tau}^\notn{sigma}{\sigma}=\notn{tau}{\tau}^2\text{.}$

The description of the tame quotient is Iwasawa’s; see [NSW, Theorem (7.5.3)] and [Ser79, Chapter IV].

Lemma 3.3. The wild subgroup is the characteristic $2$-core.

The tame group $\notn{TA}{\TA}$ has no nontrivial closed normal pro-$2$ subgroup. Consequently

\begin{equation*} O_2(\notn{GA}{\GA})=\notn{WA}{\WA}, \qquad O_2(\notn{GQ}{\GQ})=\notn{WF}{\WF}. \end{equation*}

In particular, the marked wild subgroups of the two sources are characteristic.

Lean (trivial images)Lean (maximality)Lean

Proof.

Let $I=\overline{\langle\notn{tau}{\tau}\rangle}\triangleleft\notn{TA}{\TA}$ be tame inertia; it is pro-odd. If $N\triangleleft\notn{TA}{\TA}$ is pro-$2\text{,}$ then

\begin{equation*} N\cap I=1, \qquad [N,I]\le N\cap I=1, \end{equation*}

so $N$ centralizes $I\text{.}$ Its image in $\notn{TA}{\TA}/I\cong\widehat{\mathbb{Z}}$ lies in the $2$-primary factor and has a $2$-adic geometric Frobenius coordinate $a\in\mathbb{Z}_2\text{.}$ For every $m\ge1\text{,}$ use the finite inertia quotient of order

\begin{equation*} e_m=2^{2^m}-1. \end{equation*}

Centrality forces $2^a\equiv1\pmod {e_m}\text{.}$ The order of $2$ modulo $e_m$ is exactly $2^m\text{,}$ and hence $a\in2^m\mathbb{Z}_2$ for every $m\text{.}$ Thus $a=0\text{,}$ so $N\le I\text{,}$ and therefore $N=1\text{.}$

The subgroup $\notn{WA}{\WA}$ is normal pro-$2$ and $\notn{GA}{\GA}/\notn{WA}{\WA}\cong\notn{TA}{\TA}\text{.}$ Every normal pro-$2$ subgroup of $\notn{GA}{\GA}$ maps trivially to $\notn{TA}{\TA}$ and is contained in $\notn{WA}{\WA}\text{.}$ This proves $O_2(\notn{GA}{\GA})=\notn{WA}{\WA}\text{.}$ The identical argument applied to $1\to\notn{WF}{\WF}\to\notn{GQ}{\GQ}\to\notn{TA}{\TA}\to1$ gives the local assertion.

Subsection 3.1 The standard rank-three dyadic Demushkin group

Labute’s classification identifies $G_{\Qtwo}(2)$ with the group below as an abstract pro-$2$ group, but it does not choose an isomorphism preserving the full unramified character. An abstract identification is insufficient because the tame and pro-$2$ quotients must agree over their common quotient $\mathbb{Z}_2\text{.}$ The canonical Demushkin orientation supplies the additional rigidity: together with the unramified character, it separates the three summands of the abelianization.

Put

\begin{equation} \notn{D0}{D_0}= \left\langle \notn{Agen}{A},\notn{Sgen}{S},\notn{Ygen}{Y}\;\middle|\;\notn{Agen}{A}^2\notn{Sgen}{S}^4[\notn{Sgen}{S},\notn{Ygen}{Y}]=1 \right\rangle_{\mathrm{pro}-2}\tag{3.1} \end{equation}

and

\begin{equation} \notn{Bab}{B_{\mathrm{ab}}}=\notn{D0}{D_0}^{\mathrm{ab}} =\left\langle \notn{barA}{\bar A},\notn{barS}{\bar S},\notn{barY}{\bar Y}\;\middle|\; 2\notn{barA}{\bar A}+4\notn{barS}{\bar S}=0\right\rangle_{\mathbb{Z}_2}.\tag{3.2} \end{equation}

Let

\begin{equation} \notn{tTors}{t}=\notn{barA}{\bar A}+2\notn{barS}{\bar S}, \qquad \notn{eta}{\eta}=(-3)^{-1}\in\mathbb{Z}_2^\times.\tag{3.3} \end{equation}

Then

\begin{equation} \notn{Bab}{B_{\mathrm{ab}}}=C_2\notn{tTors}{t}\oplus\mathbb{Z}_2\notn{barS}{\bar S}\oplus\mathbb{Z}_2\notn{barY}{\bar Y}.\tag{3.4} \end{equation}

Lemma 3.4. Standard dyadic group and its orientation.

The group $\notn{D0}{D_0}$ is the standard rank-three, $q=2$ Demushkin group, and $G_{\Qtwo}(2)$ is abstractly isomorphic to $\notn{D0}{D_0}\text{.}$ The canonical Demushkin orientation of $\notn{D0}{D_0}$ factors through $\notn{Bab}{B_{\mathrm{ab}}}$ and is the character

\begin{equation} \notn{chi0}{\chi_0}(\notn{tTors}{t})=-1, \qquad \notn{chi0}{\chi_0}(\notn{barS}{\bar S})=1, \qquad \notn{chi0}{\chi_0}(\notn{barY}{\bar Y})=\notn{eta}{\eta}.\tag{3.5} \end{equation}

Equivalently,

\begin{equation*} \notn{chi0}{\chi_0}(\notn{barA}{\bar A})=-1, \qquad \notn{chi0}{\chi_0}(\notn{barS}{\bar S})=1, \qquad \notn{chi0}{\chi_0}(\notn{barY}{\bar Y})=(-3)^{-1}. \end{equation*}

Proof.

The presentation (3.1) is the odd-rank dyadic case of Labute’s classification, with parameter $q=2$ and secondary exponent $4\text{;}$ see [Lab, Theorem 8]. The canonical character for the odd-rank normal form is computed in [Lab, Theorem 4, case (2)]: for the relation $\notnfar{x1}{x_1}^2x_2^{2^f}[x_2,x_3]$ it takes the values $-1,1,(1-2^f)^{-1}$ on $\notnfar{x1}{x_1},x_2,x_3\text{.}$ With $f=2$ and the standing commutator convention, this gives $-1,1,(1-4)^{-1}=(-3)^{-1}$ on $\notn{Agen}{A},\notn{Sgen}{S},\notn{Ygen}{Y}\text{,}$ respectively. The maximal pro-$2$ Galois group of $\notn{Qtwo}{\Qtwo}$ is the rank-three local Demushkin group with these parameters, so the abstract isomorphism follows from the same classification.

Lemma 3.5. Marked abelianization, orientation, and initial form.

Use arithmetic local reciprocity $\notn{rec}{\rec}\text{,}$ normalized so that $\notn{rec}{\rec}(2)$ is arithmetic Frobenius, and normalize $\notn{nuur}{\nu_{\mathrm{ur}}}$ geometrically. Put

\begin{equation*} \notn{bars}{\bar s}=\notn{rec}{\rec}(2)^{-1}=\notn{rec}{\rec}(1/2),\qquad \bar\epsilon=\notn{rec}{\rec}(-1),\qquad \notn{bary}{\bar y}=\notn{rec}{\rec}(-3),\qquad \notn{bara}{\bar a}=\bar\epsilon\notn{bars}{\bar s}^{-2}=\notn{rec}{\rec}(-4). \end{equation*}

Then the pro-$2$ abelianization of $\notnfar{Dloc}{D_{\mathrm{loc}}}=G_{\Qtwo}(2)$ is identified with $\notn{Bab}{B_{\mathrm{ab}}}$ by

\begin{equation*} \notn{barA}{\bar A}\longmapsto\notn{bara}{\bar a}, \qquad \notn{barS}{\bar S}\longmapsto\notn{bars}{\bar s}, \qquad \notn{barY}{\bar Y}\longmapsto\notn{bary}{\bar y}. \end{equation*}

Under this identification,

\begin{equation} \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{bara}{\bar a},\notn{bars}{\bar s},\notn{bary}{\bar y})=(-2,1,0), \qquad \notn{chiloc}{\chi_{\mathrm{loc}}}(\notn{bara}{\bar a},\notn{bars}{\bar s},\notn{bary}{\bar y})=(-1,1,(-3)^{-1}),\tag{3.6} \end{equation}

where $\notn{chiloc}{\chi_{\mathrm{loc}}}$ is the canonical Demushkin orientation. In the basis of $H^1(\notnfar{Dloc}{D_{\mathrm{loc}}},\F_2)$ dual to $(\notn{bara}{\bar a},\notn{bars}{\bar s},\notn{bary}{\bar y})\text{,}$ the Demushkin quadratic initial form is

\begin{equation*} \alpha^2+\beta\gamma+\gamma\beta. \end{equation*}

Consequently a Demushkin relator for lifts of these classes has the same quadratic initial form as

\begin{equation*} \notn{r0}{r_0}=\notn{Agen}{A}^2\notn{Sgen}{S}^4[\notn{Sgen}{S},\notn{Ygen}{Y}], \end{equation*}

namely $\notn{Agen}{A}^2+[\notn{Sgen}{S},\notn{Ygen}{Y}]$ in degree two. Moreover the pair

\begin{equation*} (\notn{nuur}{\nu_{\mathrm{ur}}},\notn{chiloc}{\chi_{\mathrm{loc}}}):\notn{Bab}{B_{\mathrm{ab}}}\longrightarrow \mathbb{Z}_2\times\mathbb{Z}_2^\times \end{equation*}

is injective.

Lean (injectivity)Lean (abelianization)Lean (Hilbert ledger)

Proof.

We compute successively the unramified character, the cyclotomic orientation, the quadratic initial form, and the injectivity of the two-character marking.

Since $\notn{nuur}{\nu_{\mathrm{ur}}}$ is normalized by geometric Frobenius while $\notn{rec}{\rec}(2)$ is arithmetic Frobenius, one has $\notn{nuur}{\nu_{\mathrm{ur}}}(\notn{rec}{\rec}(2))=-1\text{.}$ Hence

\begin{equation*} \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{bars}{\bar s})=1, \qquad \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{bary}{\bar y})=0, \qquad \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{bara}{\bar a})=-2. \end{equation*}

The pro-$2$ completion of $\notn{Qtwo}{\Qtwo}^\times$ is

\begin{equation*} \widehat{\notn{Qtwo}{\Qtwo}^\times}^{(2)} \cong \langle 2\rangle_{\mathbb{Z}_2} \oplus \langle -1\rangle_{C_2} \oplus \langle -3\rangle_{\mathbb{Z}_2}. \end{equation*}

The single relation among $\notn{barA}{\bar A},\notn{barS}{\bar S},\notn{barY}{\bar Y}$ is

\begin{equation*} \notn{bara}{\bar a}^2\notn{bars}{\bar s}^4 =\notn{rec}{\rec}((-4)^2(1/2)^4)=\notn{rec}{\rec}(1)=1, \end{equation*}

that is, additively, $2\notn{barA}{\bar A}+4\notn{barS}{\bar S}=0\text{.}$

For a local Demushkin group the canonical orientation is the $2$-adic cyclotomic character. With the reciprocity convention above, local reciprocity gives

\begin{equation*} \notn{chiloc}{\chi_{\mathrm{loc}}}(\notn{rec}{\rec}(u))=u^{-1}\qquad(u\in\mathbb{Z}_2^\times), \end{equation*}

while the uniformizer component acts trivially on the totally ramified extension $\notn{Qtwo}{\Qtwo}(\mu_{2^\infty})/\notn{Qtwo}{\Qtwo}\text{.}$ This proves the orientation row in (3.6).

Modulo squares, the classes are

\begin{equation*} \notn{bara}{\bar a}\equiv[-1],\qquad \notn{bars}{\bar s}\equiv[2],\qquad \notn{bary}{\bar y}\equiv[-3]. \end{equation*}

For odd units $u,v\text{,}$ write

\begin{equation*} \notn{epsu}{\epsilon}(u)=\frac{u-1}{2}\bmod 2, \qquad \notn{omgu}{\omega}(u)=\frac{u^2-1}{8}\bmod 2. \end{equation*}

The dyadic Hilbert-symbol formula

\begin{equation*} (2^\alpha u,2^\beta v)_2 =(-1)^{\notn{epsu}{\epsilon}(u)\notn{epsu}{\epsilon}(v)+\alpha\notn{omgu}{\omega}(v)+\beta\notn{omgu}{\omega}(u)} \end{equation*}

gives, in the square-class basis $(-1,2,-3)\text{,}$

\begin{equation*} (-1,-1)_2=-1, \qquad (2,-3)_2=-1, \end{equation*}

with all other pairings among these basis elements trivial, up to symmetry. Under Kummer theory, cup product is the Hilbert symbol. Thus the dual basis has Bockstein/cup initial form $\alpha^2+\beta\gamma+\gamma\beta\text{,}$ which is exactly the degree-two initial form of $\notn{Agen}{A}^2\notn{Sgen}{S}^4[\notn{Sgen}{S},\notn{Ygen}{Y}]\text{.}$

Finally, in the decomposition (3.4),

\begin{equation*} \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{tTors}{t},\notn{barS}{\bar S},\notn{barY}{\bar Y})=(0,1,0), \qquad \notn{chiloc}{\chi_{\mathrm{loc}}}(\notn{tTors}{t},\notn{barS}{\bar S},\notn{barY}{\bar Y})=(-1,1,\notn{eta}{\eta}). \end{equation*}

The element $\notn{eta}{\eta}$ topologically generates $1+4\mathbb{Z}_2\text{,}$ because $v_2(\notn{eta}{\eta}-1)=2\text{.}$ The two displayed characters therefore detect the three direct summands of (3.4), proving injectivity.

The Demushkin classification underlying this normalization goes back to [Ser63] and [Lab]; the local reciprocity inputs are standard, see [NSW, Chapter VII §7.1] and [Ser79, Chapters XI–XIII].

Remark 3.6. Why the orientation cannot be omitted.

An un-oriented marked-relator normalization with only the abelianized relation and quadratic initial form is false. Let $F=\langle \notn{Agen}{A},\notn{Sgen}{S},\notn{Ygen}{Y}\rangle_{\mathrm{pro}-2}$ and define

\begin{equation*} \psi(\notn{Agen}{A})=AY^{-4}, \qquad \psi(\notn{Sgen}{S})=SY^2, \qquad \psi(\notn{Ygen}{Y})=\notn{Ygen}{Y}. \end{equation*}

This is an automorphism, with inverse $\notn{Agen}{A}\mapsto AY^4\text{,}$ $\notn{Sgen}{S}\mapsto SY^{-2}\text{,}$ $\notn{Ygen}{Y}\mapsto \notn{Ygen}{Y}\text{.}$ For $\notn{r0}{r_0}=\notn{Agen}{A}^2\notn{Sgen}{S}^4[\notn{Sgen}{S},\notn{Ygen}{Y}]\text{,}$ put $r=\psi^{-1}(\notn{r0}{r_0})\text{.}$ On the free abelianization,

\begin{equation*} \psi_{\mathrm{ab}}(2\notn{barA}{\bar A}+4\notn{barS}{\bar S}) =2(\notn{barA}{\bar A}-4\notn{barY}{\bar Y})+4(\notn{barS}{\bar S}+2\notn{barY}{\bar Y}) =2\notn{barA}{\bar A}+4\notn{barS}{\bar S}. \end{equation*}

Moreover

\begin{equation*} r=(AY^4)^2(SY^{-2})^4[SY^{-2},\notn{Ygen}{Y}], \end{equation*}

so, modulo the third Zassenhaus term, its initial form is $\notn{Agen}{A}^2+[\notn{Sgen}{S},\notn{Ygen}{Y}]\text{.}$ Thus $r$ has the same abelianized relation and quadratic initial form as $\notn{r0}{r_0}\text{,}$ and its quotient is abstractly $\notn{D0}{D_0}\text{.}$ However, under the isomorphism induced by $\psi\text{,}$ the transported canonical orientation takes $\notn{Sgen}{S}$ to $\notn{eta}{\eta}^2\ne1\text{.}$ Therefore no isomorphism from this quotient to $\notn{D0}{D_0}$ which is the identity on $\notn{Bab}{B_{\mathrm{ab}}}$ can exist. The orientation-preserving lifting statement Proposition 3.9, rather than the false un-oriented normalization, is the exact structural input needed below.

Lean

Subsection 3.2 Automorphisms preserving the orientation

To correct the action of an abstract isomorphism on $\notn{Bab}{B_{\mathrm{ab}}}\text{,}$ it is enough to show that every automorphism of $\notn{Bab}{B_{\mathrm{ab}}}$ preserving $\notn{chi0}{\chi_0}$ is induced by an automorphism of $\notn{D0}{D_0}\text{.}$

Lemma 3.7. Cyclotomic powering of the three peripheral classes.

Let

\begin{equation*} \notn{Deltatame}{\Delta}=\langle \notn{Pperi}{P},\notn{Tperi}{T}\rangle_{\mathrm{pro}-2}, \qquad \notn{Cperi}{C}=(PT)^{-1}. \end{equation*}

For every $u\in\mathbb{Z}_2^\times$ there are an automorphism $\notn{varphiu}{\varphi_u}\in\Aut(\notn{Deltatame}{\Delta})$ and elements $\notn{cP}{c_P},\notn{cT}{c_T},\notn{cC}{c_C}\in\Delta$ such that

\begin{equation} \notn{varphiu}{\varphi_u}(\notn{Pperi}{P})=\notn{cP}{c_P}^{-1}\notn{Pperi}{P}^u \notn{cP}{c_P}, \qquad \notn{varphiu}{\varphi_u}(\notn{Tperi}{T})=\notn{cT}{c_T}^{-1}\notn{Tperi}{T}^u \notn{cT}{c_T}, \qquad \notn{varphiu}{\varphi_u}(\notn{Cperi}{C})=\notn{cC}{c_C}^{-1}\notn{Cperi}{C}^u \notn{cC}{c_C}.\tag{3.7} \end{equation}

Lean

Proof.

We obtain the required outer automorphism from the arithmetic Galois action and then choose a representative that realizes the three peripheral conjugacy classes simultaneously.

Identify $\notn{Deltatame}{\Delta}$ with the maximal pro-$2$ geometric fundamental group of

\begin{equation*} X=\mathbb{P}^1_{\mathbb{Q}}\setminus\{0,1,\infty\}, \end{equation*}

with $\notn{Pperi}{P},\notn{Tperi}{T},\notn{Cperi}{C}$ the three compatible peripheral inertia generators. The Galois action on the geometric fundamental group gives

\begin{equation*} G_{\mathbb{Q}}\longrightarrow\operatorname{Out}(\notn{Deltatame}{\Delta}). \end{equation*}

The $2$-adic cyclotomic character is surjective, since

\begin{equation*} \operatorname{Gal}(\mathbb{Q}(\mu_{2^\infty})/\mathbb{Q}) \cong\mathbb{Z}_2^\times. \end{equation*}

Each of the three cusps is rational, so its peripheral conjugacy class is preserved. The corresponding decomposition group acts on the procyclic inertia group through the cyclotomic character; see [Sti, Section 3.3 and Definition 37]. Concretely, if $z$ is a local parameter and a chosen inertia generator sends $z^{1/2^n}$ to $\zeta_{2^n}z^{1/2^n}\text{,}$ then the arithmetic action of $g$ replaces $\zeta_{2^n}$ by $\zeta_{2^n}^{\chi_2(g)}\text{.}$ With the opposite convention for representing the outer action, the exponent is $\chi_2(g)^{-1}\text{.}$ Accordingly choose $g$ with $\chi_2(g)=u$ in the first convention and with $\chi_2(g)=u^{-1}$ in the second. In either convention the resulting outer automorphism sends each peripheral generator to a conjugate of its $u$-th power. Choosing a representative of the outer class gives (3.7).

Lemma 3.8. Square-root and HNN lifting.

For every $u\in\mathbb{Z}_2^\times$ there is an automorphism $\notn{Psiu}{\Psi_u}\in\Aut(\notn{D0}{D_0})$ such that, on $\notn{Bab}{B_{\mathrm{ab}}}\text{,}$

\begin{equation} \notn{barA}{\bar A}\longmapsto u\notn{barA}{\bar A}, \qquad \notn{barS}{\bar S}\longmapsto u\notn{barS}{\bar S}.\tag{3.8} \end{equation}

Lean (classification)Lean (lift)

Proof.

The proof first lifts peripheral powering across the square-root relation, then transports the lift to the free pro-$2$ group on $\notnfar{Sgen}{S},\notn{Agen}{A}\text{,}$ and finally extends it over the HNN stable letter $\notn{Ygen}{Y}\text{.}$

Start with $\notn{varphiu}{\varphi_u}$ and the conjugators from Lemma 3.7. Consider

\begin{equation*} \notn{Esq}{E_\square}= \left\langle \notn{Pperi}{P},\notn{Tperi}{T},\notn{Agen}{A}\;\middle|\;PTA^2=1\right\rangle_{\mathrm{pro}-2} \cong\langle \notn{Pperi}{P},\notn{Agen}{A}\rangle_{\mathrm{pro}-2}, \qquad \notn{Cperi}{C}=\notn{Agen}{A}^2. \end{equation*}

View the words $\notn{cP}{c_P},\notn{cT}{c_T},\notn{cC}{c_C}$ in $\notn{Esq}{E_\square}$ via $\notn{Pperi}{P},\notn{Tperi}{T}\text{,}$ and define

\begin{equation*} \notn{Pperi}{P}\longmapsto\notn{varphiu}{\varphi_u}(\notn{Pperi}{P}), \qquad \notn{Tperi}{T}\longmapsto\notn{varphiu}{\varphi_u}(\notn{Tperi}{T}), \qquad \notn{Agen}{A}\longmapsto \notn{cC}{c_C}^{-1}\notn{Agen}{A}^u \notn{cC}{c_C}. \end{equation*}

The defining relation is preserved, because

\begin{equation*} (\notn{cC}{c_C}^{-1}\notn{Agen}{A}^u \notn{cC}{c_C})^2 =\notn{cC}{c_C}^{-1}\notn{Cperi}{C}^u \notn{cC}{c_C} =\notn{varphiu}{\varphi_u}(\notn{Cperi}{C}) =(\notn{varphiu}{\varphi_u}(\notn{Pperi}{P})\notn{varphiu}{\varphi_u}(\notn{Tperi}{T}))^{-1}. \end{equation*}

Modulo the Frattini subgroup, the free basis $\notn{Pperi}{P},\notn{Agen}{A}$ is sent to $u\bar P,u\notn{barA}{\bar A}\text{,}$ hence to $\bar P,\notn{barA}{\bar A}\text{.}$ The pro-$2$ Burnside basis theorem and Hopficity therefore show that this endomorphism is an automorphism; call it $\notn{thetau}{\theta_u}\text{.}$

In the free pro-$2$ group $\notn{FSA}{F_{S,A}}=\langle \notnfar{Sgen}{S},\notn{Agen}{A}\rangle_{\mathrm{pro}-2}$ put

\begin{equation*} \notn{Pperi}{P}=\notnfar{Sgen}{S}^3, \qquad \notn{Tperi}{T}=\notnfar{Sgen}{S}^{-3}\notn{Agen}{A}^{-2}. \end{equation*}

Since $3\in\mathbb{Z}_2^\times\text{,}$ the pair $\notn{Pperi}{P},\notn{Agen}{A}$ is again a free basis of $\notn{FSA}{F_{S,A}}\text{,}$ and $PTA^2=1\text{.}$ For every odd integer $m\text{,}$ the power map $x\mapsto x^m$ is bijective on a pro-$2$ group, with inverse $x\mapsto x^{m^{-1}}$ for $m^{-1}\in\mathbb{Z}_2^\times\text{.}$ Comparing the unique cube roots of the two sides of (3.7) therefore gives

\begin{equation*} \notn{thetau}{\theta_u}(\notnfar{Sgen}{S})=\notn{cP}{c_P}^{-1}\notnfar{Sgen}{S}^u \notn{cP}{c_P}, \qquad \notn{thetau}{\theta_u}(\notn{Tperi}{T})=\notn{cT}{c_T}^{-1}\notn{Tperi}{T}^u \notn{cT}{c_T}. \end{equation*}

The relation of $\notn{D0}{D_0}$ is equivalently

\begin{equation} \notn{Ygen}{Y}^{-1}SY=\notn{Tperi}{T}, \qquad \notn{Tperi}{T}=\notnfar{Sgen}{S}^{-3}\notn{Agen}{A}^{-2}.\tag{3.9} \end{equation}

Define an endomorphism of the presentation (3.9) by

\begin{equation*} \notn{Psiu}{\Psi_u}|_E=\notn{thetau}{\theta_u}, \qquad \notn{Psiu}{\Psi_u}(\notn{Ygen}{Y})=\notn{cP}{c_P}^{-1}\notn{Ygen}{Y}\notn{cT}{c_T}. \end{equation*}

Then

\begin{equation*} \notn{Psiu}{\Psi_u}(\notn{Ygen}{Y})^{-1}\notn{Psiu}{\Psi_u}(\notnfar{Sgen}{S})\notn{Psiu}{\Psi_u}(\notn{Ygen}{Y}) =\notn{cT}{c_T}^{-1}\notn{Ygen}{Y}^{-1}\notnfar{Sgen}{S}^uY\notn{cT}{c_T} =\notn{cT}{c_T}^{-1}\notn{Tperi}{T}^u \notn{cT}{c_T} =\notn{Psiu}{\Psi_u}(\notn{Tperi}{T}), \end{equation*}

so the relation is respected. On the Frattini quotient of $\notn{D0}{D_0}\text{,}$ the images of $\notn{Agen}{A},\notnfar{Sgen}{S},\notn{Ygen}{Y}$ form a triangular basis with diagonal entries $1,1,1\text{.}$ Consequently $\notn{Psiu}{\Psi_u}$ is surjective and hence an automorphism. Its action on $\notn{barA}{\bar A},\notn{barS}{\bar S}$ is (3.8).

Orientation-preserving automorphisms of the abelianization must now be lifted to the full pro-$2$ model.

Proposition 3.9. Lifting the orientation-preserving abelian automorphisms.

The natural map

\begin{equation} \Aut(\notn{D0}{D_0})\longrightarrow\Aut(\notn{Bab}{B_{\mathrm{ab}}},\notn{chi0}{\chi_0})\tag{3.10} \end{equation}

is surjective. More explicitly, every element of $\Aut(\notn{Bab}{B_{\mathrm{ab}}},\notn{chi0}{\chi_0})$ has the unique form

\begin{equation} \notn{alphaub}{\alpha_{u,b}}: \quad \notn{tTors}{t}\longmapsto \notn{tTors}{t}, \qquad \notn{barS}{\bar S}\longmapsto u\notn{barS}{\bar S}, \qquad \notn{barY}{\bar Y}\longmapsto\notn{barY}{\bar Y}+b\notn{barS}{\bar S},\tag{3.11} \end{equation}

with $u\in\mathbb{Z}_2^\times$ and $b\in\mathbb{Z}_2\text{,}$ and every $\notn{alphaub}{\alpha_{u,b}}$ lifts to $\Aut(\notn{D0}{D_0})\text{.}$

Proof.

We first determine the orientation-preserving automorphisms of the abelianization, then realize their unit-scaling part, and finally realize the remaining shear.

Because $v_2(\notn{eta}{\eta}-1)=2\text{,}$ the element $\notn{eta}{\eta}$ topologically generates $1+4\mathbb{Z}_2\text{.}$ From (3.4) and (3.5) one obtains

\begin{equation*} \ker\notn{chi0}{\chi_0}=\mathbb{Z}_2\notn{barS}{\bar S}. \end{equation*}

The torsion subgroup of $\notn{Bab}{B_{\mathrm{ab}}}$ is the order-two group generated by $\notn{tTors}{t}\text{.}$ Hence an orientation-preserving automorphism fixes $\notn{tTors}{t}\text{,}$ sends $\notn{barS}{\bar S}$ to $u\notn{barS}{\bar S}$ for a unique unit $u\text{,}$ and sends $\notn{barY}{\bar Y}$ to $\notn{barY}{\bar Y}+b\notn{barS}{\bar S}$ for a unique $b\text{.}$ Conversely, every map (3.11) is an orientation-preserving automorphism of $\notn{Bab}{B_{\mathrm{ab}}}\text{.}$

In more detail: the fact that the coefficient $u$ is a unit is not an extra assumption in the classification. Applying the same row extraction to an automorphism and to its inverse produces coefficients $u$ and $u'$ with $uu'=1\text{.}$ Preservation of the order-two element $t=\bar A+2\bar S$ then forces the $\bar A$-row once the $\bar S$-row is known.

The automorphism $\notn{Psiu}{\Psi_u}$ of Lemma 3.8 sends $\notn{barA}{\bar A},\notn{barS}{\bar S}$ to $u\notn{barA}{\bar A},u\notn{barS}{\bar S}\text{,}$ so it fixes $\notn{tTors}{t}=\notn{barA}{\bar A}+2\notn{barS}{\bar S}$ and sends $\notn{barS}{\bar S}$ to $u\notn{barS}{\bar S}\text{.}$ The canonical orientation is the action on the dualizing module and is therefore preserved by every automorphism of $\notn{D0}{D_0}\text{.}$ Thus $\notn{Psiu}{\Psi_u}$ induces $\alpha_{u,b(u)}$ for some $b(u)\in\mathbb{Z}_2\text{.}$

For the lift, the formal proof makes a small coordinate correction explicit. If $\Psi_u$ sends $\bar Y$ to $\bar Y+c(u)\bar S\text{,}$ then composing with a shear of parameter $b'$ contributes $ub'\text{,}$ not $b'\text{,}$ after the unit scaling. Thus one takes $b'=(b-c(u))u^{-1}\text{.}$ The inverse shear has parameter $-b'\text{,}$ so this construction gives an automorphism rather than only an endomorphism.

For $b\in\mathbb{Z}_2\text{,}$ define

\begin{equation} \notn{Thetab}{\Theta_b}(\notn{Agen}{A})=\notn{Agen}{A}^{\notnfar{Sgen}{S}^b}, \qquad \notn{Thetab}{\Theta_b}(\notnfar{Sgen}{S})=\notnfar{Sgen}{S}, \qquad \notn{Thetab}{\Theta_b}(\notn{Ygen}{Y})=YS^b,\tag{3.12} \end{equation}

Here $\notn{Agen}{A}^{\notnfar{Sgen}{S}^b}=(\notnfar{Sgen}{S}^b)^{-1}AS^b$ is conjugation, not the product $AS^b\text{.}$ This is an automorphism of the free pro-$2$ group on $\notn{Agen}{A},\notnfar{Sgen}{S},\notn{Ygen}{Y}\text{,}$ with inverse $\Theta_{-b}\text{.}$ Moreover

\begin{equation*} [\notnfar{Sgen}{S},YS^b]=[\notnfar{Sgen}{S},\notn{Ygen}{Y}]^{\notnfar{Sgen}{S}^b}, \end{equation*}

and therefore

\begin{equation*} \notn{Thetab}{\Theta_b}(\notn{Agen}{A}^2\notnfar{Sgen}{S}^4[\notnfar{Sgen}{S},\notn{Ygen}{Y}]) =(\notn{Agen}{A}^2)^{\notnfar{Sgen}{S}^b}\notnfar{Sgen}{S}^4[\notnfar{Sgen}{S},\notn{Ygen}{Y}]^{\notnfar{Sgen}{S}^b} =(\notn{Agen}{A}^2\notnfar{Sgen}{S}^4[\notnfar{Sgen}{S},\notn{Ygen}{Y}])^{\notnfar{Sgen}{S}^b}. \end{equation*}

It therefore descends to an automorphism of $\notn{D0}{D_0}\text{,}$ inducing $\alpha_{1,b}\text{.}$ The composite $\notn{Thetab}{\Theta_{b-b(u)}}\circ\notn{Psiu}{\Psi_u}$ induces $\notn{alphaub}{\alpha_{u,b}}\text{,}$ proving surjectivity.

Subsection 3.3 Matching the unramified marking

The preceding surjectivity permits an arbitrary abstract isomorphism $G_{\Qtwo}(2)\cong \notn{D0}{D_0}$ to be corrected so that it induces the identity on the marked abelianization $\notn{Bab}{B_{\mathrm{ab}}}\text{.}$ The unramified character is then preserved integrally, not only after reduction modulo $2\text{.}$

Proof of Proposition 1.1.

By Lemma 3.4, choose an abstract isomorphism

\begin{equation*} f:\notnfar{Dloc}{D_{\mathrm{loc}}}\xrightarrow{\sim}\notn{D0}{D_0}. \end{equation*}

Use Lemma 3.5 to identify $\notnfar{Dloc}{D_{\mathrm{loc}}}^{\mathrm{ab}}$ with $\notn{Bab}{B_{\mathrm{ab}}}\text{,}$ and use (3.2) for $\notn{D0}{D_0}^{\mathrm{ab}}\text{.}$ Canonical orientations are functorial under group isomorphisms. The orientation row (3.6) and (3.5) therefore imply

\begin{equation*} \notn{chi0}{\chi_0}\circ f_{\mathrm{ab}}=\notn{chiloc}{\chi_{\mathrm{loc}}}=\notn{chi0}{\chi_0}. \end{equation*}

Thus $f_{\mathrm{ab}}\in\Aut(\notn{Bab}{B_{\mathrm{ab}}},\notn{chi0}{\chi_0})\text{.}$ By Proposition 3.9, choose $\gamma\in\Aut(\notn{D0}{D_0})$ with

\begin{equation*} \gamma_{\mathrm{ab}}=f_{\mathrm{ab}}^{-1}. \end{equation*}

Then

\begin{equation*} \beta=\gamma\circ f:\notnfar{Dloc}{D_{\mathrm{loc}}}\xrightarrow{\sim}\notn{D0}{D_0} \end{equation*}

induces the identity on the marked abelianization $\notn{Bab}{B_{\mathrm{ab}}}\text{.}$

Set

\begin{equation*} a=\beta^{-1}(\notn{Agen}{A}), \qquad s=\beta^{-1}(\notnfar{Sgen}{S}), \qquad y=\beta^{-1}(\notn{Ygen}{Y}). \end{equation*}

These elements topologically generate $\notnfar{Dloc}{D_{\mathrm{loc}}}\text{,}$ satisfy $a^2s^4[s,y]=1\text{,}$ and have the prescribed abelianized classes. Hence

\begin{equation*} \notn{nuur}{\nu_{\mathrm{ur}}}(a,s,y)=(-2,1,0), \end{equation*}

which proves Proposition 1.1.

Subsection 3.4 The maximal pro-$2$ quotient of the candidate

Proposition 3.10. Maximal pro-$2$ quotient.

The maximal pro-$2$ quotient of $\notn{GA}{\GA}$ is

\begin{equation} \notn{Pi}{\Pi}= \left\langle \notn{sigma}{\sigma},\notn{x0}{x_0},\notnfar{x1}{x_1}\;\middle|\; \notn{x0}{x_0}^{\notn{sigma}{\sigma}^2}\notn{x0}{x_0}[\notnfar{x1}{x_1},\notn{sigma}{\sigma}]=1 \right\rangle_{\mathrm{pro-}2}.\tag{3.13} \end{equation}

The group $\notn{Pi}{\Pi}$ is abstractly isomorphic to $G_{\Qtwo}(2)\text{.}$ Equipped with

\begin{equation} \notn{nu2}{\nu_2}:\notn{Pi}{\Pi}\twoheadrightarrow\mathbb{Z}_2, \qquad \notn{nu2}{\nu_2}(\notn{sigma}{\sigma})=1,\quad \notn{nu2}{\nu_2}(\notn{x0}{x_0})=\notn{nu2}{\nu_2}(\notnfar{x1}{x_1})=0,\tag{3.14} \end{equation}

it is isomorphic to the fully unramified marked pair $(G_{\Qtwo}(2),\notn{nuur}{\nu_{\mathrm{ur}}})\text{.}$

Lean (Γ_A, proved)Lean (local, proved)Lean (Γ_A)Lean (local)Lean

Proof.

In a finite $2$-group quotient, Lemma 3.1 forces $\notn{tau}{\tau}=1\text{,}$ and $\notn{w}{\w}$ acts as the identity. Hence

\begin{equation*} u_i=x_i,\qquad d_0=1,\qquad c_0=1,\qquad g_0=\notn{sigma}{\sigma}^2,\qquad d_g=h_c=1, \end{equation*}

and

\begin{equation*} h_0=\notn{x0}{x_0}^{\notn{sigma}{\sigma}^2}\notn{x0}{x_0}. \end{equation*}

Moreover

\begin{equation*} u_1^{-1}\notnfar{x1}{x_1}^\notn{sigma}{\sigma}=\notnfar{x1}{x_1}^{-1}\notnfar{x1}{x_1}^\notn{sigma}{\sigma}=[\notnfar{x1}{x_1},\notn{sigma}{\sigma}]. \end{equation*}

Thus (1.6) becomes precisely the relator in (3.13). Conversely, every finite quotient of (3.13) is admissible after setting $\notn{tau}{\tau}=1\text{.}$ The fully marked local identification is given by Corollary 3.12.

Proposition 3.11. Nielsen transform to the pro-$2$ boundary.

For any pro-$2$ group with topological generators $a,s,y$ and unique defining relation

\begin{equation} a^2s^4[s,y]=1,\tag{3.15} \end{equation}

the Nielsen change

\begin{equation} \notn{sigma}{\sigma}=s,\qquad \notnfar{x1}{x_1}=y,\qquad \notn{x0}{x_0}=s^{-2}a^{-1}\tag{3.16} \end{equation}

changes the closed normal closure of the relator to that of

\begin{equation} \notn{x0}{x_0}^{\notn{sigma}{\sigma}^2}\notn{x0}{x_0}[\notnfar{x1}{x_1},\notn{sigma}{\sigma}].\tag{3.17} \end{equation}

Consequently Proposition 1.1 identifies $\notn{Dpro2}{D}=G_{\Qtwo}(2)$ abstractly with $\notn{Pi}{\Pi}\text{.}$

Lean

Proof.

Under (3.16), one has $a=\notn{x0}{x_0}^{-1}\notn{sigma}{\sigma}^{-2}$ and therefore

\begin{equation*} a^2s^4=\notn{x0}{x_0}^{-1}(\notn{x0}{x_0}^{-1})^{\notn{sigma}{\sigma}^2}. \end{equation*}

Writing $P=\notn{x0}{x_0}^{\notn{sigma}{\sigma}^2}\notn{x0}{x_0}$ and $Q=[\notnfar{x1}{x_1},\notn{sigma}{\sigma}]\text{,}$ the inverse of the standard relator is $QP\text{.}$ The words $QP$ and $PQ$ are cyclic conjugates, so they have the same closed normal closure. This proves the assertion.

Corollary 3.12. Fully marked form.

The generators furnished by Proposition 1.1 and transformed by (3.16) satisfy

\begin{equation*} \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{sigma}{\sigma})=1, \qquad \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{x0}{x_0})=\notn{nuur}{\nu_{\mathrm{ur}}}(\notnfar{x1}{x_1})=0, \end{equation*}

and (3.17) is the unique defining relation. Hence $(\notn{Dpro2}{D},\notn{nuur}{\nu_{\mathrm{ur}}})\cong(\notn{Pi}{\Pi},\notn{nu2}{\nu_2})\text{.}$

Lean

Proof.

The relation is Proposition 3.11. The values follow directly from (1.4):

\begin{equation*} \notn{nuur}{\nu_{\mathrm{ur}}}(\notn{x0}{x_0}) =-2\notn{nuur}{\nu_{\mathrm{ur}}}(s)-\notn{nuur}{\nu_{\mathrm{ur}}}(a)=0. \end{equation*}

Remark 3.13. The integral marking is not detected by the mod-$2$ cup form.

The more familiar unmarked Labute relator

\begin{equation*} \notn{x0}{x_0}^2\notnfar{x1}{x_1}^4[\notnfar{x1}{x_1},\notn{sigma}{\sigma}] \end{equation*}

has the same quadratic initial form $b^2+ac+ca$ as (3.17). It does not, however, admit the marking

\begin{equation*} \nu(\notn{sigma}{\sigma})=1, \qquad \nu(\notn{x0}{x_0})=\nu(\notnfar{x1}{x_1})=0 \end{equation*}

required by the tame boundary. In those Labute variables the canonical orientation has values

\begin{equation*} \chi(\notn{x0}{x_0})=-1, \qquad \chi(\notnfar{x1}{x_1})=1, \qquad \chi(\notn{sigma}{\sigma})=(-3)^{-1}. \end{equation*}

The abelianization is $\mathbb{Z}_2^2\oplus C_2\text{,}$ and $\bar x_1$ has infinite order. The asserted marking would nevertheless put $\notnfar{x1}{x_1}$ in $\ker\nu\cap\ker\chi\text{,}$ contradicting the injectivity of $(\notn{nuur}{\nu_{\mathrm{ur}}},\chi)$ on $\notn{Dpro2}{D}^{\mathrm{ab}}\text{.}$ Thus agreement of the maximal pro-$2$ quotient as an unmarked Demushkin group is insufficient. The Nielsen transformation (3.16) replaces that relator by (3.17). It has the same mod-$2$ two-jet; its full unramified marking follows from Proposition 1.1.

For the boundary construction it is not enough that the tame and pro-$2$ quotients exist separately: they must carry one compatible unramified character. We arrange this next.

Proposition 3.14. Fully marked tame and pro-$2$ quotients.

Let

\begin{equation*} \nu_{\mathrm t}:\notnfar{TA}{\TA}\twoheadrightarrow\mathbb{Z}_2, \qquad \nu_{\mathrm t}(\notn{sigma}{\sigma})=1, \quad \nu_{\mathrm t}(\notn{tau}{\tau})=0, \end{equation*}

and equip $\notn{Pi}{\Pi}$ with (3.14). For each $\notn{GammaSrc}{\Gamma}\in\{\notn{GA}{\GA},\notn{GQ}{\GQ}\}\text{,}$ the tame quotient map and the maximal pro-$2$ quotient map may be chosen so that their composites with $\nu_{\mathrm t}$ and $\notn{nu2}{\nu_2}$ are the same natural unramified character

\begin{equation*} \nu_{\Gamma}:\notn{GammaSrc}{\Gamma}\twoheadrightarrow\mathbb{Z}_2. \end{equation*}

The compatibility therefore holds after passage to every finite quotient $C_{2^m}\text{,}$ not only modulo $2\text{.}$

Lean (proved)Lean (statement)Lean

Proof.

For the candidate, the assignment

\begin{equation*} \notn{sigma}{\sigma}\longmapsto1,\qquad \notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1}\longmapsto0 \end{equation*}

respects both defining relations and induces the displayed maps on the tame and pro-$2$ quotients. For the local source, use Corollary 3.12. The tame geometric Frobenius and the pro-$2$ generator $\notn{sigma}{\sigma}$ have the same full unramified image, while $\notn{x0}{x_0},\notnfar{x1}{x_1}$ lie in its kernel.

Remark 3.15. Why the full marking is load-bearing.

The construction

\begin{equation*} \notn{bdry}{\bdry}=\notnfar{TA}{\TA}\times_{\mathbb{Z}_2}\notn{Pi}{\Pi} \end{equation*}

uses the full marked isomorphism $(\notn{Pi}{\Pi},\notn{nu2}{\nu_2})\cong(G_{\Qtwo}(2),\notn{nuur}{\nu_{\mathrm{ur}}})\text{,}$ not merely the abstract pro-$2$ group of Remark 3.13: it is the marking that fixes the terminal comparison after passage to the finite quotients $C_{2^m}$ of the unramified coordinate.

Let $(a,b,c)$ be the basis of $H^1(\notn{Pi}{\Pi},\F_2)$ dual to $(\notn{sigma}{\sigma},\notn{x0}{x_0},\notnfar{x1}{x_1})\text{.}$ The quadratic initial form of the relator in (3.13) is

\begin{equation*} b^2+ac+ca. \end{equation*}

Consequently the scalar cup/Bockstein table is

\begin{equation} \mathcal H((a,b,c),(a',b',c'))=ac'+ca'+bb'.\tag{3.18} \end{equation}

This is the local Hilbert pairing on $\notn{Qtwo}{\Qtwo}^\times/\notn{Qtwo}{\Qtwo}^{\times2}$ under the corresponding basis.

Section 4 The common tame–pro-$2$ boundary and the global induction

The compatible quotient maps obtained in Section 3 are assembled here into a single boundary condition. This produces the epimorphisms $\notn{bGamma}{b_\Gamma}:\notn{GammaSrc}{\Gamma}\twoheadrightarrow\notn{bdry}{\bdry}$ and the finite targets on which the induction of Section 9 is carried out.

The tame quotient and the maximal pro-$2$ quotient share the full unramified pro-$2$ quotient:

\begin{equation} \notnfar{TA}{\TA}\xrightarrow{\,\nu_{\mathrm t}\,}\mathbb{Z}_2 \xleftarrow{\,\notn{nu2}{\nu_2}\,}\notn{Pi}{\Pi}, \qquad \notn{bdry}{\bdry}=\notnfar{TA}{\TA}\times_{\mathbb{Z}_2}\notn{Pi}{\Pi}.\tag{4.1} \end{equation}

The fibre product records the requirement that the two quotient maps have the same unramified coordinate. Thus their markings cannot be chosen independently, or merely matched after reduction modulo $2\text{.}$ For $\notn{GammaSrc}{\Gamma}\in\{\notn{GA}{\GA},\notn{GQ}{\GQ}\}\text{,}$ the compatible tame and maximal pro-$2$ maps of Proposition 3.14 combine to an epimorphism

\begin{equation} \notn{bGamma}{b_\Gamma}:\notn{GammaSrc}{\Gamma}\twoheadrightarrow\notn{bdry}{\bdry}.\tag{4.2} \end{equation}

Surjectivity follows from profinite Goursat. Any common quotient of $\notnfar{TA}{\TA}$ and the pro-$2$ group $\notn{Pi}{\Pi}$ is a pro-$2$ quotient of $\notnfar{TA}{\TA}\text{,}$ hence a quotient of its maximal pro-$2$ quotient $\mathbb{Z}_2\text{;}$ Proposition 3.14 shows that the full $\mathbb{Z}_2$ quotient occurs.

Fix a finite tame quotient $\notn{Htarget}{H}\text{,}$ an epimorphism $\alpha:\notnfar{TA}{\TA}\twoheadrightarrow \notn{Htarget}{H}\text{,}$ an elementary abelian $2$-group $\notn{Etarget}{E}\text{,}$ and a homomorphism $\bar\psi:\notn{Pi}{\Pi}\to \notn{Etarget}{E}\text{.}$ They define

\begin{equation} \beta:\notn{bdry}{\bdry}\longrightarrow \notn{Htarget}{H}\times \notn{Etarget}{E}, \qquad \beta(t,p)=(\alpha(t),\bar\psi(p)).\tag{4.3} \end{equation}

We compare only surjections that induce the fixed map to $\notn{Htarget}{H}$ and the fixed elementary decoration in $\notn{Etarget}{E}\text{.}$ A finite target carrying these two prescribed pieces of data will be called boundary-framed.

Definition 4.1. Boundary-framed marked target.

Keep the frame data fixed above: the finite tame quotient $\notn{Htarget}{H}$ with $\alpha:\notnfar{TA}{\TA}\twoheadrightarrow \notn{Htarget}{H}\text{,}$ and the decoration group $\notn{Etarget}{E}\text{,}$ an elementary abelian $2$-group (abelian of exponent dividing $2$), with $\bar\psi:\notn{Pi}{\Pi}\to \notn{Etarget}{E}\text{.}$ A boundary-framed marked target is a tuple

\begin{equation*} \notn{calY}{\mathcal Y}=(\notn{Ytarget}{Y},\notn{LY}{L_Y},\notn{piY}{\pi_Y},\notn{thetaY}{\theta_Y}) \end{equation*}

where $\notn{LY}{L_Y}\triangleleft \notn{Ytarget}{Y}$ is a finite $2$-group, $\notn{piY}{\pi_Y}:\notn{Ytarget}{Y}\twoheadrightarrow \notn{Htarget}{H}$ has kernel $\notn{LY}{L_Y}\text{,}$ and $\notn{thetaY}{\theta_Y}:\notn{Ytarget}{Y}\to \notn{Etarget}{E}$ is a homomorphism. Put

\begin{equation*} \notn{qY}{q_Y}=(\notn{piY}{\pi_Y},\notn{thetaY}{\theta_Y}):\notn{Ytarget}{Y}\longrightarrow \notn{Htarget}{H}\times \notn{Etarget}{E}. \end{equation*}

For $\notn{GammaSrc}{\Gamma}\in\{\notn{GA}{\GA},\notn{GQ}{\GQ}\}$ define

\begin{equation} \notn{eGamma}{e_\Gamma}^\beta(\notn{calY}{\mathcal Y}) =\#\{f:\notn{GammaSrc}{\Gamma}\twoheadrightarrow \notn{Ytarget}{Y}:q_Yf=\beta \notn{bGamma}{b_\Gamma}\}.\tag{4.4} \end{equation}

Lean

The tuple $\notn{calY}{\mathcal Y}$ therefore fixes, on the same finite target, both the tame quotient $\notn{Ytarget}{Y}/\notn{LY}{L_Y}\cong \notn{Htarget}{H}$ and the homomorphism $\notn{Ytarget}{Y}\to \notn{Etarget}{E}\text{.}$ The equation $q_Yf=\beta \notn{bGamma}{b_\Gamma}$ counts surjections inducing both prescribed maps. Since the wild subgroup is the kernel of the tame quotient, it also forces $\notnfar{WA}{\WA}$ or $\notnfar{WF}{\WF}$ into $\notn{LY}{L_Y}\text{.}$

An exact-image stratum consists of maps whose image is a specified subgroup $J\text{,}$ rather than maps that merely land in $J\text{.}$ If $J\le \notn{Ytarget}{Y}$ projects onto $\notn{Htarget}{H}\text{,}$ then

\begin{equation*} \mathcal J=(J,J\cap \notn{LY}{L_Y},\notn{piY}{\pi_Y}|_J,\notn{thetaY}{\theta_Y}|_J) \end{equation*}

inherits the same structure. Consequently, partitioning maps according to their image subgroup produces targets of the same form.

Theorem 4.2. Boundary-framed exact-image theorem.

For every boundary-framed marked target $\notn{calY}{\mathcal Y}\text{,}$

\begin{equation*} e_{\notn{GA}{\GA}}^\beta(\notn{calY}{\mathcal Y})=e_{\notn{GQ}{\GQ}}^\beta(\notn{calY}{\mathcal Y}). \end{equation*}

The same equality holds for every exact-image target $\mathcal J$ defined above.

Lean (main)Lean (stratum)Lean

The proof is a global induction over all boundary-framed marked targets, ordered by $|\notn{LY}{L_Y}|\text{.}$ Auxiliary central covers need not lie inside the original ambient group, which is why the induction is stated globally.

Section 5 Linear lifting theory and duality

This section starts with a lower map and an elementary kernel and extracts the linear information needed by the target induction. The candidate and local cochain theories remain distinct; their common numerical and duality content is stated in Corollary 5.17.

Fix a finite group $\notn{Cfin}{C}\text{,}$ a boundary-compatible epimorphism $\notn{rhoEpi}{\rho}:\notn{GammaSrc}{\Gamma}\twoheadrightarrow \notn{Cfin}{C}\text{,}$ and a finite elementary $\F_2[\notn{Cfin}{C}]$-module $\notn{Amod}{A}\text{.}$ Here boundary-compatible means that $\notn{Cfin}{C}$ is the underlying group of a boundary-framed marked target $(\notn{Cfin}{C},L_C,\pi_C,\theta_C)$ as in Definition 4.1 and that $\notn{rhoEpi}{\rho}$ induces the fixed frame: $(\pi_C,\theta_C)\circ\rho=\beta\circ \notn{bGamma}{b_\Gamma}$ with $\beta$ as in (4.3). The marking of $\notn{Cfin}{C}$ is thus compatible with the fixed boundary frame, and all comparisons in this section are between the two sources over this common frame. Here elementary means that the additive group of $\notn{Amod}{A}$ is a finite-dimensional $\F_2$-vector space; no semisimplicity assumption is made. A lifting datum over $\notn{rhoEpi}{\rho}$ is an extension

\begin{equation*} 1\longrightarrow \notn{Amod}{A}\longrightarrow \widetilde C \xrightarrow{\,\pi\,} \notn{Cfin}{C}\longrightarrow1 \end{equation*}

inducing the prescribed $\notn{Cfin}{C}$-action on $\notn{Amod}{A}\text{.}$ In either source theory it has a degree-two obstruction

\begin{equation*} o_{\Gamma,\notn{rhoEpi}{\rho}}(\widetilde C)\in \notn{HGr}{H^2_{\Gamma,\rho}}(\notn{Amod}{A}). \end{equation*}

The obstruction vanishes exactly when $\notn{rhoEpi}{\rho}$ lifts to $\widetilde C\text{;}$ after one lift is fixed, every other lift is obtained uniquely by multiplication by an element of $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Amod}{A})\text{.}$ Thus a nonempty lift set is a principal homogeneous space under $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Amod}{A})\text{.}$ Only the obstruction dimensions, torsor cardinalities, perfect pairings, and connecting homomorphisms will be compared; no map between the two cochain complexes or their degree-one cohomology spaces is used.

For $\notn{GammaSrc}{\Gamma}=\notn{GA}{\GA}\text{,}$ varying the four distinguished generators through an extension by $\notn{Amod}{A}$ gives the finite word complex

\begin{equation} C_{\rho,\notn{Amod}{A}}^\bullet: 0\longrightarrow \notn{Amod}{A}\xrightarrow{d^0_{\rho,\notn{Amod}{A}}}\notn{Amod}{A}^4 \xrightarrow{d^1_{\rho,\notn{Amod}{A}}}\notn{Amod}{A}^2\longrightarrow0.\tag{5.1} \end{equation}

Write $\notn{opS}{\mathsf S},\notn{opT}{\mathsf T},\notn{opX0}{\mathsf X_0},\notn{opX1}{\mathsf X_1}$ for the actions of the four lower values on $\notn{Amod}{A}\text{.}$ With the lift convention $(a,g)$ in a semidirect product, simultaneous infinitesimal conjugation gives

\begin{equation} d^0_{\rho,\notn{Amod}{A}}(v)=((\notn{opS}{\mathsf S}-1)v,(\notn{opT}{\mathsf T}-1)v,(\notn{opX0}{\mathsf X_0}-1)v,(\notn{opX1}{\mathsf X_1}-1)v).\tag{5.2} \end{equation}

The map $d^1_{\rho,\notn{Amod}{A}}$ is the ordered pair of the exact tame- and wild-relator defects. Concretely, if the lower map has values in $\notn{Cfin}{C}$ and a lift to $\notn{Cfin}{C}\ltimes \notn{Amod}{A}$ is written

\begin{equation*} (\notn{rhoEpi}{\rho}(\notn{sigma}{\sigma}),a),\quad (\notn{rhoEpi}{\rho}(\notn{tau}{\tau}),b),\quad (\notn{rhoEpi}{\rho}(\notn{x0}{x_0}),c),\quad (\notn{rhoEpi}{\rho}(\notnfar{x1}{x_1}),d), \end{equation*}

then $d^1_{\rho,\notn{Amod}{A}}(a,b,c,d)$ is the coefficient of $\notn{Amod}{A}$ in the evaluated tame and wild relators. This construction is functorial in the $\notn{Cfin}{C}$-module $\notn{Amod}{A}$ because it is obtained by evaluating the same two finite words in semidirect products. Its degree-one kernel is denoted $Z^1_{A,\notn{rhoEpi}{\rho}}(\notn{Amod}{A})$ and its cohomology by $H^i_{A,\notn{rhoEpi}{\rho}}(\notn{Amod}{A})\text{.}$

The pro-$2$ normal-closure condition contributes no additional linear equation. The elementary kernel $\notn{Amod}{A}$ lies in the marked normal $2$-subgroup of the target, so multiplying any of the four chosen lifts by an element of $\notn{Amod}{A}$ preserves the marked condition; the only defects are the two displayed relators.

For $\notnfar{GammaSrc}{\Gamma}=\notn{GQ}{\GQ}\text{,}$ put

\begin{equation*} H^i_{F,\notn{rhoEpi}{\rho}}(\notn{Amod}{A})=H^i_{\mathrm{cont}}(\notn{GQ}{\GQ},A_\rho), \qquad Z^1_{F,\notn{rhoEpi}{\rho}}(\notn{Amod}{A})=Z^1_{\mathrm{cont}}(\notn{GQ}{\GQ},A_\rho), \end{equation*}

where the action is induced by $\notn{rhoEpi}{\rho}\text{.}$ These are the ordinary continuous Galois cohomology groups. No two-relator complex is assigned to $\notn{GQ}{\GQ}\text{.}$

Subsection 5.1 The evaluated Fox Jacobian

Let $\notn{opS}{\mathsf S},\notn{opT}{\mathsf T}$ denote the actions of $\notn{rhoEpi}{\rho}(\notn{sigma}{\sigma}),\notn{rhoEpi}{\rho}(\notn{tau}{\tau})$ on $\notn{Amod}{A}\text{.}$ If $e=|\notn{opT}{\mathsf T}|$ is odd, put

\begin{equation*} \notn{Pnorm}{P}=1+\notn{opT}{\mathsf T}+\cdots+\notn{opT}{\mathsf T}^{e-1}. \end{equation*}

The following calculation is valid for any finite coefficient module after replacing $\notn{w}{\w}$ by an integer representative modulo the exponent of the finite semidirect products used in the calculation.

Lemma 5.1. Finite-exponent independence.

Suppose $\nu$ and $\nu'$ represent $\notn{w}{\w}$ modulo the exponent of every finite semidirect and Heisenberg group occurring in a fixed first- or second-order calculation. Replacing every $\notn{w}{\w}$-power in the relators by $\nu$ or by $\nu'$ gives the same first derivative and the same mixed second derivative.

Lean

Proof.

The two ordinary powers define the same element in each finite group in which the relator is evaluated. First and mixed derivatives are, by definition, the noncentral and central coordinates of those finite evaluations. They are therefore identical. The representative may be chosen separately for each finite coefficient calculation; no global coherent integer is required.

Lemma 5.2. Exact class-two identity.

Let

\begin{equation*} 1\longrightarrow Z\longrightarrow\widetilde V\longrightarrow V\longrightarrow1 \end{equation*}

be a central extension in which $V$ and $Z$ are elementary abelian $2$-groups, and let $\phi$ be an automorphism of $\widetilde V$ fixing $Z\text{.}$ For $X,D\in\widetilde V\text{,}$ put

\begin{equation*} \mathfrak h_\phi(X,D) =\phi(X)X\,\phi(D)D\,D^2[\phi(D),D]. \end{equation*}

Then

\begin{equation} \mathfrak h_\phi(X,D)=\phi(X)X D^{-1}\phi(D).\tag{5.3} \end{equation}

Consequently:

if $D$ and $X$ have the same image in $V\text{,}$ then $\mathfrak h_\phi(X,D)=X^2\text{;}$
if $\phi$ is the identity, then $\mathfrak h_\phi(X,D)=X^2$ for every $D\text{.}$

Lean (core)Lean (identity)Lean (identity, id case)Lean

Proof.

The group $\widetilde V$ has class at most two, every commutator has order at most two, and every element has fourth power one. Put $\notn{Amod}{A}=\phi(\notn{Dfox}{D})\text{,}$ $B=\notn{Dfox}{D}\text{,}$ and $k=[\notn{Amod}{A},B]\text{.}$ With our commutator convention $AB=BAk\text{;}$ since $k$ is central of order at most two and $B^2$ is central,

\begin{equation*} ABB^2k=BAB^2=B^3\notn{Amod}{A}=B^{-1}\notn{Amod}{A}. \end{equation*}

This proves (5.3).

If $\notn{Dfox}{D}=Xz$ with $z\in Z\text{,}$ then $\phi(z)=z$ and $\notn{Dfox}{D}^{-1}\phi(\notn{Dfox}{D})=X^{-1}\phi(X)\text{.}$ Hence $\mathfrak h_\phi(X,\notn{Dfox}{D})=\phi(X)^2=X^2\text{,}$ because $X^2\in Z$ and $\phi$ fixes $Z\text{.}$ If $\phi=1\text{,}$ (5.3) gives the second assertion directly.

Lemma 5.3. The elementary and class-two shadow of $h_0$.

Let $\notn{Vhead}{V}$ be either the trivial module or a nontrivial simple tame $\F_2[\notn{Cfin}{C}]$-module. For an arbitrary lower map, the marked wild subgroup acts trivially on every simple factor, so the evaluated simple-factor derivatives are the same as those of the induced tame lower map. At that tame map:

the first derivative of $h_0$ is the first derivative of $\notn{x0}{x_0}^2\text{,}$ namely zero;
on the normalized degree-one representatives supported on $\notn{x0}{x_0}\text{,}$ the mixed Heisenberg and extraspecial central coordinates of $h_0$ are respectively those of $\notn{x0}{x_0}^2\text{.}$

Thus $h_0$ may replace $\notn{x0}{x_0}^2$ in every linear, cup-product, and determinant calculation used below.

Lean

Proof.

We first compute the linear variation of $h_0$ and then treat its class-two central coordinate separately in the cases $\notn{Pnorm}{P}=0$ and $\notn{Pnorm}{P}=1\text{.}$

Let $\notn{opU}{\mathsf U}=\notn{opS}{\mathsf S}^\notn{w}{\w}\text{.}$ If the first-order variations of $\notn{tau}{\tau}$ and $\notn{x0}{x_0}$ are $b$ and $c\text{,}$ then

\begin{equation*} Dd_0=\notn{Pnorm}{P} b+(\notn{Pnorm}{P}+1)c, \qquad \notn{Dfox}{D}\notn{x0}{x_0}=c. \end{equation*}

The variation of $g_0$ affects only terms of order at least two, because the base values of $\notn{x0}{x_0}$ and $d_0$ at the tame map are both the identity. If $\notn{Pnorm}{P}=0\text{,}$ then $Dd_0=c=\notn{Dfox}{D}\notn{x0}{x_0}\text{,}$ and the four linear terms in $\mathfrak h_{g_0}(\notn{x0}{x_0},d_0)$ cancel. If $\notn{Pnorm}{P}=1\text{,}$ then $\notn{opT}{\mathsf T}=1$ and the $2$-primary part $\notn{opU}{\mathsf U}$ acts trivially on a simple characteristic-$2$ module; again the four linear terms cancel. This proves (i).

For the normalized cohomology representatives used in the mixed and quadratic calculations, only $\notn{x0}{x_0}$ varies. If $\notn{Pnorm}{P}=0\text{,}$ the two elements $d_0$ and $\notn{x0}{x_0}$ have the same image $c$ in $\notn{Vhead}{V}\text{,}$ so Lemma 5.2(i) applies. If $\notn{Pnorm}{P}=1\text{,}$ then $Dd_0=0$ and $\notn{opT}{\mathsf T}=1\text{.}$ The $2$-primary operator $\notn{opU}{\mathsf U}$ acts trivially on the simple quotient $\notn{Vhead}{V}\text{.}$ Its lift to a Heisenberg or extraspecial coefficient group is therefore a central automorphism. Such an automorphism has order at most $2\text{,}$ because it is determined by a homomorphism from the elementary quotient to the central involution. Hence $g_0=\sigma_2^2$ acts trivially on the full coefficient group, and Lemma 5.2(ii) applies. For the trivial module, $\notn{opT}{\mathsf T}=\notn{opU}{\mathsf U}=1\text{,}$ so the second case applies without modification. Taking the mixed Heisenberg central kernel gives the square diagonal $\lambda(c)\text{;}$ taking the extraspecial central kernel gives $q(c)\text{.}$ This proves (ii).

Lemma 5.4. Full finite-word ledger.

At a tame lower map, write first-order variations of $(\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1})$ as $(a,b,c,d)\text{.}$ Let $\notn{opS}{\mathsf S},\notn{opT}{\mathsf T}$ be the lower actions of $\notn{sigma}{\sigma},\notn{tau}{\tau}\text{,}$ put $\notn{opU}{\mathsf U}=\notn{opS}{\mathsf S}^\notn{w}{\w}\text{,}$ and let $\notn{Pnorm}{P}=1+\notn{opT}{\mathsf T}+\cdots+\notn{opT}{\mathsf T}^{e-1}$ for the odd order $e$ of $\notn{opT}{\mathsf T}\text{.}$ The raw finite Fox rules give

\begin{equation*} \notn{Dfox}{D}(u_i)=\notn{Pnorm}{P}(Dx_i+\notn{Dfox}{D}\notn{tau}{\tau}), \qquad \notn{Dfox}{D}(d_0)=\notn{Pnorm}{P} b+(\notn{Pnorm}{P}+1)c, \qquad \notn{Dfox}{D}(z_0)=\notn{opU}{\mathsf U}^{-1}c. \end{equation*}

The first-order contributions of the wild factors are

\begin{equation*} \begin{array}{c|cccc} \text{factor}&a&b&c&d\\ \hline h_0&0&0&0&0\\ u_1^{-1}&0&\notn{Pnorm}{P}&0&\notn{Pnorm}{P}\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0&0&\notn{opS}{\mathsf S}^{-1}\\ {[d_0,z_0]}&0&0&0&0. \end{array} \end{equation*}

Thus the wild row is $L_{\mathrm w}=Pb+(\notn{Pnorm}{P}+\notn{opS}{\mathsf S}^{-1})d\text{.}$

Write $b_{\mathrm{cl}}$ for the bilinear commutator pairing of the chosen class-two coefficient group. For the mixed Heisenberg or extraspecial calculation on the normalized cohomology representative supported on $\notn{x0}{x_0}\text{,}$ the complete central ledger is

\begin{equation*} \begin{array}{c|c|c} \text{factor}&\text{$\notn{Vhead}{V}$-coordinate}&\text{central coordinate}\\ \hline h_0&0&\lambda(c)\text{ in }\mathcal H(\notn{Vhead}{V}),\; q(c)\text{ for }\notn{kappaq0}{\kappa_q^0}\\ u_1^{-1}&0&0\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0\\ d_0=(\notn{x0}{x_0}\notn{tau}{\tau})^\notn{w}{\w} \notn{x0}{x_0}^{-1}&(\notn{Pnorm}{P}+1)c&\ast\\ z_0=\notn{x0}{x_0}^{\sigma_2}&\notn{opU}{\mathsf U}^{-1}c&\ast\\ {[d_0,z_0]}&0&b_{\mathrm{cl}}((\notn{Pnorm}{P}+1)c,\notn{opU}{\mathsf U}^{-1}c). \end{array} \end{equation*}

The symbols $\ast$ are irrelevant because $d_0$ and $z_0$ enter the relator only through their commutator. The safety of replacing $h_0$ by $\notn{x0}{x_0}^2$ in these ledgers is exactly Lemma 5.3: the variation of $g_0=\sigma_2^2$ starts in order at least two; when $\notn{Pnorm}{P}=0$ one has $Dd_0=\notn{Dfox}{D}\notn{x0}{x_0}\text{,}$ so Lemma 5.2(i) applies; when $\notn{Pnorm}{P}=1\text{,}$ the lift of $\notn{opU}{\mathsf U}$ to the class-two coefficient group is a central automorphism of order at most two, so $g_0$ acts trivially and Lemma 5.2(ii) applies.

Differentiating the relator words in these ledgers linearizes the lifting problem; the resulting Jacobian can be evaluated on each simple tame module.

Lemma 5.5. Evaluated Jacobian.

Write the lift variables on $(\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1})$ as $(a,b,c,d)\in \notn{Amod}{A}^4\text{.}$ On a simple tame module the actual relation differential is

\begin{equation} d^1_{\rho,\notn{Amod}{A}}= \begin{pmatrix} \notn{opS}{\mathsf S}^{-1}(1+\notn{opT}{\mathsf T})&\notn{opS}{\mathsf S}^{-1}+1+\notn{opT}{\mathsf T}&0&0\\ 0&\notn{Pnorm}{P}&0&\notn{Pnorm}{P}+\notn{opS}{\mathsf S}^{-1} \end{pmatrix}.\tag{5.4} \end{equation}

Equivalently,

\begin{align} L_{\mathrm t} &=\notn{opS}{\mathsf S}^{-1}(1+\notn{opT}{\mathsf T})a+(\notn{opS}{\mathsf S}^{-1}+1+\notn{opT}{\mathsf T})b,\tag{5.5}\\ L_{\mathrm w} &=Pb+(\notn{Pnorm}{P}+\notn{opS}{\mathsf S}^{-1})d.\tag{5.6} \end{align}

Postcomposing the tame defect coordinate by $\notn{opS}{\mathsf S}$ gives the row-normalized expression

\begin{equation*} (1+\notn{opT}{\mathsf T})a+\bigl(1+\notn{opS}{\mathsf S}(1+\notn{opT}{\mathsf T})\bigr)b, \end{equation*}

which may be used for rank calculations only. The word complex, the mixed derivative, and the endpoint maps below use the unnormalized row (5.5).

Lean

Proof.

Use $(u,g)(v,h)=(u+gv,gh)$ in $\notn{Amod}{A}\rtimes \notn{Cfin}{C}\text{.}$ A direct calculation gives

\begin{equation*} \notn{Dfox}{D}(\notn{tau}{\tau}^\notn{sigma}{\sigma}\notn{tau}{\tau}^{-2})(a,b) =\notn{opS}{\mathsf S}^{-1}(\notn{opT}{\mathsf T}-1)a+\bigl(\notn{opS}{\mathsf S}^{-1}-(1+\notn{opT}{\mathsf T})\bigr)b. \end{equation*}

Over $\F_2$ this is exactly (5.5).

For an odd-order $\notn{opT}{\mathsf T}$-orbit, differentiation of a $\notn{w}{\w}$-power gives the norm projector $\notn{Pnorm}{P}\text{.}$ By Lemma 5.3, the word $h_0$ has the first-order value of $\notn{x0}{x_0}^2\text{,}$ hence zero. The remaining wild factors give

\begin{equation*} \begin{array}{c|cccc} &a&b&c&d\\ \hline h_0&0&0&0&0\\ u_1^{-1}&0&\notn{Pnorm}{P}&0&\notn{Pnorm}{P}\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0&0&\notn{opS}{\mathsf S}^{-1}\\ {[d_0,z_0]}&0&0&0&0. \end{array} \end{equation*}

Their sum is (5.6). Without the simplicity assumption, the same finite Fox rules define the word complex; the displayed reduction is the simple-factor calculation used in the devissage.

Here and below, the duality inputs on the Galois side are local Tate duality and the Euler–Poincaré characteristic; see [NSW, Theorems (7.2.6) and (7.3.1)] and [Mil, Chapter I].

Subsection 5.2 The Fox–Heisenberg cup calculation

For an elementary module $\notn{Amod}{A}\text{,}$ set

\begin{equation*} \mathcal H(\notn{Amod}{A})=\notn{Amod}{A}\times \notn{Amod}{A}^\vee\times\F_2, \end{equation*}

with multiplication

\begin{equation*} (a,\lambda,z)(a',\lambda',z') =(a+a',\lambda+\lambda',z+z'+\lambda(a')). \end{equation*}

The $\notn{Cfin}{C}$-action is diagonal. If a finite word has evaluated coordinates

\begin{equation*} (\bar u,D_u,D_u^\vee,\beta_u), \end{equation*}

then multiplication and inversion obey

\begin{align} D_{uv}&=D_u+\bar uD_v,& D_{uv}^\vee&=D_u^\vee+\bar uD_v^\vee,\tag{5.7}\\ \beta_{uv}&=\beta_u+\beta_v+(D_u^\vee)(\bar uD_v),& \beta_{u^{-1}}&=\beta_u+(\bar u^{-1}D_u^\vee)(\bar u^{-1}D_u).\tag{5.8} \end{align}

These are the finite Fox recursion formulas. The Stokes identities below are proved directly by conjugation in the finite Heisenberg group; no cellular diagonal approximation is used. All identities in this subsection are identities of finite word evaluations before cohomology. In particular, coefficient functoriality and the later passage to nonsplit elementary modules are proved at the level of complexes and mapping cones; no semisimple decomposition of the $\notn{Cfin}{C}$-module $\notn{Amod}{A}$ is used.

Lemma 5.6. Strict coefficient naturality.

For every $\notn{Cfin}{C}$-module map $f:\notn{Amod}{A}\to A'\text{,}$ the mixed derivatives satisfy

\begin{equation*} \beta^{A'}_{\rho,r}(f_*a,\lambda') =\beta^A_{\rho,r}(a,f^\vee\lambda'). \end{equation*}

Lean Lean

Proof.

Evaluate the same finite relator in the mixed Heisenberg group

\begin{equation*} \notn{Amod}{A}\times A'^\vee\times\F_2, \qquad (a,\lambda,z)(a',\lambda',z') =(a+a',\lambda+\lambda',z+z'+\lambda(f(a'))). \end{equation*}

Its two natural images in $\mathcal H(A')$ and $\mathcal H(\notn{Amod}{A})$ have the same central coordinate. This gives the identity before passage to cohomology.

Lemma 5.7. Finite-word Stokes formula.

Let $F_n=\langle g_1,\dots,g_n\rangle$ be a free group, let $\rho:F_n\to \notn{Cfin}{C}$ be a homomorphism to a finite group, and write $c_i=\rho(g_i)\text{.}$ Let $r\in F_n$ be an ordinary finite word with $\rho(r)=1\text{,}$ and let $\epsilon_i(r)\in\F_2$ be the total exponent of $g_i$ in $r\text{,}$ reduced modulo $2\text{.}$

For $x=(x_i)\in \notn{Amod}{A}^n$ and $y=(y_i)\in(\notn{Amod}{A}^\vee)^n\text{,}$ evaluate $r$ after replacing $g_i$ by $(x_i,y_i,0;c_i)\in\mathcal H(\notn{Amod}{A})\rtimes \notn{Cfin}{C}\text{.}$ Write the result as

\begin{equation*} \bigl(L_r^\notn{Amod}{A}(x),L_r^{\notn{Amod}{A}^\vee}(y),\beta_r(x,y);1\bigr). \end{equation*}

Then, for $a\in \notn{Amod}{A}$ and $\lambda\in \notn{Amod}{A}^\vee\text{,}$

\begin{align} \beta_r(d^0a,y) &=\langle a,L_r^{\notn{Amod}{A}^\vee}(y)\rangle +\sum_{i=1}^n\epsilon_i(r)\,y_i(c_i a),\tag{5.9}\\ \beta_r(x,d^0\lambda) &=\langle L_r^\notn{Amod}{A}(x),\lambda\rangle +\sum_{i=1}^n\epsilon_i(r)\,\lambda(x_i),\tag{5.10} \end{align}

where

\begin{equation*} d^0a=((c_i-1)a)_i, \qquad d^0\lambda=((c_i-1)\lambda)_i. \end{equation*}

Consequently, for words $r_1,\dots,r_m$ and coefficients $t_j\in\F_2$ satisfying

\begin{equation} \sum_{j=1}^m t_j\epsilon_i(r_j)=0 \qquad(1\le i\le n),\tag{5.11} \end{equation}

the corresponding traced mixed coordinate satisfies the two Stokes identities without correction terms.

Lean (left)Lean (right)

Proof.

The two identities follow from conjugating by a primal and a dual coefficient, after which the total exponent vector records the central correction contributed by each generator.

Put

\begin{equation*} p_a=(a,0,0;1),\qquad p_\lambda=(0,\lambda,0;1), \qquad \mathbf z(t)=(0,0,t;1). \end{equation*}

A direct multiplication in $\mathcal H(\notn{Amod}{A})\rtimes \notn{Cfin}{C}$ gives

\begin{align*} p_a^{-1}(0,y_i,0;c_i)p_a &=((c_i-1)a,y_i,0;c_i)\,\mathbf z\bigl(y_i(c_i a)\bigr),\\ p_\lambda^{-1}(x_i,0,0;c_i)p_\lambda &=(x_i,(c_i-1)\lambda,0;c_i)\,\mathbf z\bigl(\lambda(x_i)\bigr). \end{align*}

The elements $\mathbf z(t)$ are central of order at most two. Therefore, on evaluating an ordinary word, the extra central factor attached to the $i$th generator occurs exactly $\epsilon_i(r)$ times.

If only the dual variables $y_i$ are present, the value of $r$ is $(0,L_r^{\notn{Amod}{A}^\vee}(y),0;1)\text{.}$ Conjugating this value by $p_a$ contributes the central coordinate $L_r^{\notn{Amod}{A}^\vee}(y)(a)\text{.}$ Adding the generatorwise central corrections gives (5.9). Repeating this conjugation calculation with $p_\lambda$ starts from $(L_r^\notn{Amod}{A}(x),0,0;1)$ and gives (5.10). Condition (5.11) cancels the correction terms. Every step is functorial in $\notn{Amod}{A}\text{.}$

The finite-word identity acquires its useful form after tracing.

Proposition 5.8. Traced finite-word Stokes identity.

Let $\beta_{\mathrm t}$ and $\beta_{\mathrm w}$ be the mixed central coordinates obtained from the tame and wild relators, and set

\begin{equation*} \mathcal B_{\rho,\notn{Amod}{A}}=\beta_{\mathrm t}+\beta_{\mathrm w}. \end{equation*}

Let $L_{\mathrm t}$ and $L_{\mathrm w}$ denote the two unnormalized first relation differentials. Then, before passage to cohomology,

\begin{align} \mathcal B_{\rho,\notn{Amod}{A}}(d^0_{\rho,\notn{Amod}{A}}a,y) &=\langle a, L_{\mathrm t}^{\notn{Amod}{A}^\vee}(y)+L_{\mathrm w}^{\notn{Amod}{A}^\vee}(y)\rangle,\tag{5.12}\\ \mathcal B_{\rho,\notn{Amod}{A}}(x,d^0_{\rho,\notn{Amod}{A}^\vee}\lambda) &=\langle L_{\mathrm t}^{\notn{Amod}{A}}(x)+L_{\mathrm w}^{\notn{Amod}{A}}(x),\lambda\rangle.\tag{5.13} \end{align}

The identities are natural in $\notn{Amod}{A}\text{.}$ They are assertions about the sum of the two relator coordinates; in general they are false for either relator separately.

Lean (left)Lean (right)Lean

Proof.

We compute the two relators’ mod-$2$ exponent vectors, apply the finite-word Stokes formula to their sum, and then invoke finite-exponent independence and coefficient naturality.

Choose an odd ordinary integer representative of $\notn{w}{\w}$ which is valid in every finite semidirect-Heisenberg group in the calculation, and replace the two profinite relators by the resulting ordinary words $r_{\mathrm t}$ and $r_{\mathrm w}\text{.}$ In the ordered generator list $(\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1})$ their mod-$2$ exponent vectors are

\begin{equation*} \notn{epsu}{\epsilon}(r_{\mathrm t})=(0,1,0,0), \qquad \notn{epsu}{\epsilon}(r_{\mathrm w})=(0,1,0,0). \end{equation*}

Indeed, $r_{\mathrm t}=\notn{tau}{\tau}^\notn{sigma}{\sigma}\notn{tau}{\tau}^{-2}$ has odd total $\notn{tau}{\tau}$-exponent. For $r_{\mathrm w}=h_0u_1^{-1}\notnfar{x1}{x_1}^\notn{sigma}{\sigma} c_0\text{,}$ the commutator $c_0$ has zero exponent vector. The word $h_0$ also has zero exponent vector: the two $\notn{x0}{x_0}$-terms cancel modulo $2\text{,}$ the two occurrences of $d_0$ cancel, $d_0^2$ has even exponent vector, and $h_c$ is a commutator. Finally, $u_1^{-1}$ contributes $\notnfar{x1}{x_1}+\notn{tau}{\tau}$ and $\notnfar{x1}{x_1}^\notn{sigma}{\sigma}$ contributes $\notnfar{x1}{x_1}\text{.}$ Thus the sum of the two exponent vectors is zero.

Apply Lemma 5.7 with trace coefficients $(1,1)\text{.}$ The correction terms in (5.9) cancel and give (5.12); the correction terms in (5.10) cancel and give (5.13). Independence of the chosen integer representative is Lemma 5.1, and strict coefficient naturality is Lemma 5.6.

Remark 5.9. The trace is essential.

There is a small explicit regression test. Take $\notn{Cfin}{C}=\GL_2(\F_2)$ with

\begin{equation*} \notn{opS}{\mathsf S}=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \notn{opT}{\mathsf T}=\begin{pmatrix}0&1\\1&1\end{pmatrix}, \qquad \notn{opS}{\mathsf S}^{-1}\notn{opT}{\mathsf T}\notn{opS}{\mathsf S}=\notn{opT}{\mathsf T}^2, \end{equation*}

and let $\notn{Amod}{A}=\F_2^2$ be the natural module. For $a=e_2$ and the dual variation supported on $\notn{tau}{\tau}$ with value $e_2^\vee\text{,}$ the tame relator alone has mixed central value $0\text{,}$ whereas pairing its dual first derivative with $a$ gives $1\text{.}$ The wild relator supplies the compensating value. Thus a relator-by-relator Stokes formula would assert $0=1\text{,}$ while the traced formula above is exact.

For $x\in \notn{Amod}{A}^4$ and $y\in(\notn{Amod}{A}^\vee)^4\text{,}$ let

\begin{equation*} \mathcal B_{\rho,\notn{Amod}{A}}(x,y) =\beta_{\rho,r_{\mathrm t}}(x,y) +\beta_{\rho,r_{\mathrm w}}(x,y) \end{equation*}

be the sum of the two mixed central coordinates. The following proposition constructs the chain map used below, including its degree-zero and degree-two components.

Proposition 5.10. The Fox–Heisenberg chain map.

Under the canonical identifications

\begin{equation*} \Hom((\notn{Amod}{A}^\vee)^2,\F_2)\cong \notn{Amod}{A}^2, \qquad \Hom((\notn{Amod}{A}^\vee)^4,\F_2)\cong \notn{Amod}{A}^4, \qquad \Hom(\notn{Amod}{A}^\vee,\F_2)\cong \notn{Amod}{A}, \end{equation*}

define

\begin{align} \notn{Dfox}{D}^0_{\rho,\notn{Amod}{A}}(a)&=(a,a),\tag{5.14}\\ \langle \notn{Dfox}{D}^1_{\rho,\notn{Amod}{A}}(x),y\rangle&=\mathcal B_{\rho,\notn{Amod}{A}}(x,y),\tag{5.15}\\ \notn{Dfox}{D}^2_{\rho,\notn{Amod}{A}}(u_{\mathrm t},u_{\mathrm w})&=u_{\mathrm t}+u_{\mathrm w}.\tag{5.16} \end{align}

Then $D_{\rho,\notn{Amod}{A}}=(\notn{Dfox}{D}^0,\notn{Dfox}{D}^1,\notn{Dfox}{D}^2)$ is a natural chain map

\begin{equation} D_{\rho,\notn{Amod}{A}}:C_{\rho,\notn{Amod}{A}}^\bullet \longrightarrow \Hom_{\F_2}(C_{\rho,\notn{Amod}{A}^\vee}^\bullet,\F_2)[-2].\tag{5.17} \end{equation}

More explicitly,

\begin{align} \notn{Dfox}{D}^1_{\rho,\notn{Amod}{A}}d^0_{\rho,\notn{Amod}{A}} &=(d^1_{\rho,\notn{Amod}{A}^\vee})^\vee \notn{Dfox}{D}^0_{\rho,\notn{Amod}{A}},\tag{5.18}\\ \notn{Dfox}{D}^2_{\rho,\notn{Amod}{A}}d^1_{\rho,\notn{Amod}{A}} &=(d^0_{\rho,\notn{Amod}{A}^\vee})^\vee \notn{Dfox}{D}^1_{\rho,\notn{Amod}{A}}.\tag{5.19} \end{align}

Lean

Proof.

The endpoint maps are induced by the scalar trace $\operatorname{tr}(y_{\mathrm t},y_{\mathrm w})=y_{\mathrm t}+y_{\mathrm w}\text{.}$ Equations (5.12) and (5.13), proved by the direct finite-word calculation of Lemma 5.7, are exactly (5.18) and (5.19), respectively. Strict naturality in $\notn{Amod}{A}$ is Lemma 5.6, Proposition 5.8. Hence (5.17) is a natural chain map in all three degrees.

Lemma 5.11. Exact cone dévissage.

For an elementary $\F_2[\notn{Cfin}{C}]$-module $\notn{Amod}{A}\text{,}$ put

\begin{equation} K(\notn{Amod}{A})=\Cone\!\left( D_{\rho,\notn{Amod}{A}}:\notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}}\longrightarrow \Hom_{\F_2}(\notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}^\vee},\F_2)[-2] \right).\tag{5.20} \end{equation}

For every short exact sequence of finite $\F_2[\notn{Cfin}{C}]$-modules

\begin{equation*} 0\longrightarrow A'\longrightarrow \notn{Amod}{A}\longrightarrow A''\longrightarrow0, \end{equation*}

there is a degreewise short exact sequence of complexes

\begin{equation} 0\longrightarrow K(A')\longrightarrow K(\notn{Amod}{A})\longrightarrow K(A'') \longrightarrow0.\tag{5.21} \end{equation}

Consequently, if two of $K(A')\text{,}$ $K(\notn{Amod}{A})\text{,}$ and $K(A'')$ are acyclic, then so is the third. In particular, acyclicity for all simple modules implies acyclicity for every elementary module, including nonsplit and nonsemisimple modules.

Lean (main)Lean (two-of-three)Lean

Proof.

We verify exactness for the word complexes, then for their duals, and finally for the associated mapping cones.

The finite-word construction is additive and exact in the coefficient module, so the source complexes form a degreewise short exact sequence

\begin{equation} 0\longrightarrow \notn{Cfin}{C}^\bullet_{\rho,A'} \longrightarrow \notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}} \longrightarrow \notn{Cfin}{C}^\bullet_{\rho,A''}\longrightarrow0.\tag{5.22} \end{equation}

Dualizing the coefficient sequence gives

\begin{equation*} 0\longrightarrow A''^\vee\longrightarrow \notn{Amod}{A}^\vee \longrightarrow A'^\vee\longrightarrow0. \end{equation*}

Applying the same finite-word complex and then $\Hom_{\F_2}(-,\F_2)$ reverses this sequence once more, and yields

\begin{equation} \begin{split} 0\longrightarrow&\ \Hom_{\F_2}(\notn{Cfin}{C}^\bullet_{\rho,A'^\vee},\F_2)[-2] \longrightarrow \Hom_{\F_2}(\notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}^\vee},\F_2)[-2]\\ \longrightarrow&\ \Hom_{\F_2}(\notn{Cfin}{C}^\bullet_{\rho,A''^\vee},\F_2)[-2] \longrightarrow0. \end{split}\tag{5.23} \end{equation}

For compactness write

\begin{equation*} \notn{Cfin}{C}^\dagger(\notn{Amod}{A})=\Hom_{\F_2}(\notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}^\vee},\F_2)[-2]. \end{equation*}

Strict coefficient naturality of the Heisenberg central coordinate gives the commutative diagram

\begin{equation*} \begin{array}{ccccc} \notn{Cfin}{C}^\bullet_{\rho,A'}&\longrightarrow&\notn{Cfin}{C}^\bullet_{\rho,\notn{Amod}{A}} &\longrightarrow&\notn{Cfin}{C}^\bullet_{\rho,A''}\\ \downarrow D_{\rho,A'}&&\downarrow D_{\rho,\notn{Amod}{A}}&&\downarrow D_{\rho,A''}\\ \notn{Cfin}{C}^\dagger(A')&\longrightarrow&\notn{Cfin}{C}^\dagger(\notn{Amod}{A})&\longrightarrow&\notn{Cfin}{C}^\dagger(A''). \end{array} \end{equation*}

Here the rows are the nonzero portions of (5.22) and (5.23), and commutativity is Lemma 5.6, Proposition 5.10. Mapping cones are formed degree by degree, so this diagram gives (5.21). The associated long exact cohomology sequence proves the two-out-of-three assertion. Applying it successively to a composition series of $\notn{Amod}{A}$ proves the final statement.

Lemma 5.12. Simple characteristic-two modules are tame.

Let $\notnfar{Vhead}{V}$ be a simple $\F_2[\notn{Cfin}{C}]$-module occurring in the induction. Then the marked normal $2$-subgroup of $\notn{Cfin}{C}$ acts trivially on $\notnfar{Vhead}{V}\text{.}$ Hence $\notnfar{Vhead}{V}$ factors through the finite tame quotient of $\notn{Cfin}{C}\text{.}$

Lean Lean

Proof.

A finite $2$-group acting on a nonzero vector space over $\F_2$ has a nonzero fixed vector. Since the $2$-subgroup is normal in $\notn{Cfin}{C}\text{,}$ the fixed subspace is a nonzero $\notn{Cfin}{C}$-submodule. Simplicity forces this fixed subspace to be all of $\notnfar{Vhead}{V}\text{.}$

Lemma 5.13. Normal forms for nontrivial simple modules.

Let $\notnfar{Vhead}{V}$ be a nontrivial simple tame $\F_2[\notn{Cfin}{C}]$-module. Every class in $H^1_{A,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})$ has a unique representative supported on $\notn{x0}{x_0}\text{:}$

\begin{equation} (a,b,c,d)\sim(0,0,c,0).\tag{5.24} \end{equation}

More precisely, the two possible inertia cases are as follows.

If $\notn{opT}{\mathsf T}=1\text{,}$ then $\notn{Pnorm}{P}=1\text{,}$ $1+\notn{opS}{\mathsf S}^{-1}$ is invertible, and

\begin{equation*} Z^1_{A,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})=\{(a,0,c,0):a,c\in \notnfar{Vhead}{V}\},\qquad B^1_{A,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})=\{((\notn{opS}{\mathsf S}-1)v,0,0,0):v\in \notnfar{Vhead}{V}\}. \end{equation*}
If $\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0\text{,}$ then $\notn{Pnorm}{P}=0$ and $\notn{opT}{\mathsf T}-1=1+\notn{opT}{\mathsf T}$ is invertible. The wild row first forces $d=0\text{.}$ Subtracting the coboundary of $v=(\notn{opT}{\mathsf T}-1)^{-1}b$ kills $b\text{,}$ and the tame row then forces $a=0\text{.}$

Under the representatives (5.24), the degree-one pairing with the corresponding representative $\lambda\in \notnfar{Vhead}{V}^\vee$ is

\begin{equation} (c,\lambda)\longmapsto \begin{cases} \lambda(c),&\notn{opT}{\mathsf T}=1,\\[1mm] \lambda((1+\notnfar{opU}{\mathsf U}+\notnfar{opU}{\mathsf U}^{-1})c),&\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0. \end{cases}\tag{5.25} \end{equation}

Both operators in (5.25) are invertible.

Lean (split)Lean (ramified)Lean (pairing, split)Lean (pairing, ramified)Lean

Proof.

We reduce separately in the cases of trivial inertia and vanishing inertia fixed space, and then evaluate the pairing on the resulting unique normal forms.

Because tame inertia is normal, $\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}$ is a $\notn{Cfin}{C}$-submodule; simplicity gives either $\notn{opT}{\mathsf T}=1$ and $\notn{Pnorm}{P}=1\text{,}$ or $\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0$ and $\notn{Pnorm}{P}=0\text{.}$

If $\notn{opT}{\mathsf T}=1\text{,}$ the two rows of (5.4) reduce to $b=0$ and $(1+\notn{opS}{\mathsf S}^{-1})d=0\text{.}$ A nonzero kernel of $1+\notn{opS}{\mathsf S}^{-1}$ would be the $\notn{opS}{\mathsf S}$-fixed subspace and hence a nonzero submodule, so $1+\notn{opS}{\mathsf S}^{-1}$ is invertible and $d=0\text{.}$ The displayed descriptions of $Z^1$ and $B^1$ follow. The same argument shows that $\notn{opS}{\mathsf S}-1$ is invertible, so each class has a unique representative $(0,0,c,0)\text{.}$

If $\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0\text{,}$ then $1+\notn{opT}{\mathsf T}$ is invertible. The wild row is $\notn{opS}{\mathsf S}^{-1}d=0\text{,}$ hence $d=0\text{.}$ For a cocycle $(a,b,c,0)\text{,}$ subtract the coboundary of $v=(\notn{opT}{\mathsf T}-1)^{-1}b\text{.}$ The new representative has $b=0\text{,}$ and its tame row is $\notn{opS}{\mathsf S}^{-1}(1+\notn{opT}{\mathsf T})a=0\text{,}$ so $a=0\text{.}$ Uniqueness follows because a coboundary with zero $\notn{opT}{\mathsf T}$-coordinate has $(\notn{opT}{\mathsf T}-1)v=0\text{,}$ hence $v=0\text{.}$

The pairing formula is Lemma 5.14. In the ramified case $\notnfar{opU}{\mathsf U}$ has $2$-power order, so its minimal polynomial divides a power of $X+1\text{.}$ Since $X^2+X+1$ is coprime to $X+1\text{,}$

\begin{equation*} 1+\notnfar{opU}{\mathsf U}+\notnfar{opU}{\mathsf U}^{-1}=\notnfar{opU}{\mathsf U}^{-1}(\notnfar{opU}{\mathsf U}^2+\notnfar{opU}{\mathsf U}+1) \end{equation*}

is invertible. The unramified operator is the identity.

Lemma 5.14. Mixed Hessian calculation.

Let $\notnfar{Vhead}{V}$ be a nontrivial simple tame module, put $\notnfar{opU}{\mathsf U}=\notn{opS}{\mathsf S}^\notn{w}{\w}\text{,}$ and represent a class in $H^1_{A,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})$ by the cocycle supported on $\notn{x0}{x_0}$ with value $c\text{.}$ Represent the dual class similarly by $\lambda\in \notnfar{Vhead}{V}^\vee\text{.}$ The mixed central contributions are

\begin{equation*} \begin{array}{c|cc} \text{word contribution}&\notn{opT}{\mathsf T}=1&\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0\\ \hline r_{\mathrm t}&0&0\\ h_0&\lambda(c)&\lambda(c)\\ {[d_0,z_0]}&0&\lambda(\mathsf Uc)+\lambda(\notnfar{opU}{\mathsf U}^{-1}c)\\ \text{all other factors of }r_{\mathrm w}&0&0. \end{array} \end{equation*}

Consequently the degree-one pairing is induced by

\begin{equation} \begin{cases} 1,&\notn{opT}{\mathsf T}=1,\\ 1+\notnfar{opU}{\mathsf U}+\notnfar{opU}{\mathsf U}^{-1},&\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0. \end{cases}\tag{5.26} \end{equation}

Lean

Proof.

The representative exists by the row reduction of (5.4). By Lemma 5.3, the word $h_0$ contributes exactly the usual Heisenberg square diagonal $\lambda(c)\text{.}$ In the ramified case the first derivatives on the $\notn{x0}{x_0}$-coordinate are

\begin{equation*} Dd_0(c)=c, \qquad Dz_0(c)=\notnfar{opU}{\mathsf U}^{-1}c, \end{equation*}

whereas on the dual coordinate they are

\begin{equation*} Dd_0(\lambda)=\lambda, \qquad Dz_0(\lambda)=\notnfar{opU}{\mathsf U}^\vee\lambda. \end{equation*}

Applying (5.7)–(5.8) to the commutator gives

\begin{equation*} \lambda(\notnfar{opU}{\mathsf U}^{-1}c)+(\notnfar{opU}{\mathsf U}^\vee\lambda)(c) =\lambda((\notnfar{opU}{\mathsf U}^{-1}+\notnfar{opU}{\mathsf U})c). \end{equation*}

The remaining factors have either zero first derivative on the normalized representatives or no mixed central term. Adding the $h_0$ contribution proves the table.

Assembled, these computations say that the candidate’s lifting complex satisfies a two-dimensional duality.

Proposition 5.15. Candidate deformation duality.

For every elementary $\F_2[\notn{Cfin}{C}]$-module $\notn{Amod}{A}\text{,}$ the chain map (5.17) is a quasi-isomorphism. Hence

\begin{equation} H^2_{A,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})^\vee\cong (\notn{Amod}{A}^\vee)^\notn{Cfin}{C}, \qquad \dim Z^1_{A,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})=2\dim \notn{Amod}{A}+\dim(\notn{Amod}{A}^\vee)^\notn{Cfin}{C}.\tag{5.27} \end{equation}

The induced pairings

\begin{equation*} H^i_{A,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})\times H^{2-i}_{A,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A}^\vee)\to\F_2 \end{equation*}

are perfect and natural in $\notn{Amod}{A}\text{.}$

Lean (simple case)Lean (split)Lean (ramified)Lean (main)Lean

Proof.

We treat the trivial simple module, then the nontrivial simple modules, and finally pass to arbitrary elementary modules by exact cone dévissage.

For $\notn{Amod}{A}=\F_2\text{,}$ all lower actions are trivial, so $d^0=0$ and the actual Jacobian is

\begin{equation*} d^1(a,b,c,d)=(b,b). \end{equation*}

Thus $\dim H^0=1\text{,}$ $\dim H^1=3\text{,}$ and $\dim H^2=1\text{.}$ On the basis of $H^1$ represented by the coordinates $(\notn{sigma}{\sigma},\notn{x0}{x_0},\notnfar{x1}{x_1})\text{,}$ the traced Fox–Heisenberg form is

\begin{equation*} ((a,c,d),(a',c',d'))\longmapsto ad'+da'+cc', \end{equation*}

with matrix

\begin{equation*} \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0 \end{pmatrix}. \end{equation*}

It is nonsingular, and the endpoint trace gives a nonzero $H^0$–$H^2$ pairing. This is exactly the scalar cup–Bockstein table (3.18).

For a nontrivial simple module, Lemma 5.12 makes the action tame, and Lemma 5.13 gives a unique degree-one representative supported on $\notn{x0}{x_0}\text{.}$ The Jacobian is surjective, so $H^0=H^2=0$ and $\dim H^1=\dim \notn{Amod}{A}\text{.}$ On the unique normal forms the degree-one pairing is exactly (5.25), and is nonsingular by the final assertion of Lemma 5.13. Hence the cone $K(\notn{Amod}{A})$ is acyclic for every simple module, including the trivial simple module treated above.

In more detail: in the split case, fixed-point-freeness of $\mathsf S-1$ makes this operator bijective and supplies the unique $x_0$-supported normal form on both $A$ and $A^\vee\text{.}$ In the ramified case the analogous argument uses $\mathsf T^{-1}-1\text{,}$ while the perfect pairing is reduced to the bijectivity of $1+\mathsf U+\mathsf U^{-1}\text{.}$ Its injectivity comes from the unipotence of the $2$-primary operator $\mathsf U\text{;}$ finiteness then gives surjectivity.

For a general elementary module, choose a composition series. The degreewise short exact cone sequence (5.21) and its long exact cohomology sequence give acyclicity at each extension step. Thus $D_{\rho,\notn{Amod}{A}}$ is a quasi-isomorphism for every elementary, possibly nonsplit and nonsemisimple, module $\notn{Amod}{A}\text{.}$ The numerical formula follows from duality and the Euler characteristic of (5.1).

The dévissage is a strong induction on $|A|\text{,}$ rather than an appeal to semisimplicity. If $A$ is neither zero nor simple, choose a proper nonzero $C$-stable subgroup $W\text{.}$ Both $W$ and $A/W$ have strictly smaller cardinality, inherit the exponent-two action, and fit into the exact cone sequence of Lemma 5.11. The two-out-of-three property for that sequence therefore propagates duality through arbitrary nonsplit extensions.

The Galois side satisfies the same duality, with the local cup product and invariant map supplying the pairing.

Proposition 5.16. Local lifting duality.

For every elementary $\F_2[\notn{Cfin}{C}]$-module $\notn{Amod}{A}\text{,}$

\begin{equation} H^2_{F,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})^\vee\cong (\notn{Amod}{A}^\vee)^\notn{Cfin}{C}, \qquad \dim Z^1_{F,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})=2\dim \notn{Amod}{A}+\dim(\notn{Amod}{A}^\vee)^\notn{Cfin}{C}.\tag{5.28} \end{equation}

The local cup product and invariant map give perfect pairings

\begin{equation*} H^i_{F,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A})\times H^{2-i}_{F,\notnfar{rhoEpi}{\rho}}(\notn{Amod}{A}^\vee)\to\F_2. \end{equation*}

Lean (bundle)Lean (main)Lean

Proof.

This is local Tate duality, with the mod-$2$ cyclotomic twist trivial, together with the local Euler–Poincaré formula for $\notn{Qtwo}{\Qtwo}\text{;}$ see [NSW, Thms. (7.2.6) and (7.3.1)]. Since the action factors through $\notn{Cfin}{C}\text{,}$ the invariant space is $(\notn{Amod}{A}^\vee)^\notn{Cfin}{C}\text{.}$ Finally

\begin{equation*} \dim Z^1=\dim H^1+\dim B^1 =\dim H^1+\dim \notn{Amod}{A}-\dim \notn{Amod}{A}^\notn{Cfin}{C}, \end{equation*}

and the Euler characteristic gives the stated expression.

Corollary 5.17. Common lift numerics and adjoint boundaries.

For either source, use the corresponding notation $\notn{HGr}{H^i_{\Gamma,\rho}}$ and $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}\text{.}$ Then the obstruction-space dimensions, unobstructed lift multiplicities, and perfect pairings agree numerically.

For every short exact sequence

\begin{equation*} 0\to \notn{Tsub}{T}\to \notn{Mmid}{M}\to \notnfar{Vhead}{V}\to0 \end{equation*}

and its dual, the connecting maps are adjoint: if

\begin{equation*} \partial_{\Gamma,\notnfar{rhoEpi}{\rho}}:\notn{HGr}{H^1_{\Gamma,\rho}}(\notnfar{Vhead}{V})\to \notn{HGr}{H^2_{\Gamma,\rho}}(\notn{Tsub}{T}) \end{equation*}

and $\chi\in(\notn{Tsub}{T}^\vee)^\notn{Cfin}{C}$ has dual boundary

\begin{equation*} \gamma_\chi=\partial_C^\vee(\chi)\in H^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee), \end{equation*}

then

\begin{equation} \langle\partial_{\Gamma,\notnfar{rhoEpi}{\rho}}c,\chi\rangle =\langle c,\notnfar{rhoEpi}{\rho}^*\gamma_\chi\rangle_\Gamma.\tag{5.29} \end{equation}

Lean Lean

Proof.

For the candidate, (5.29) is the naturality of the chain map (5.17) applied to the coefficient sequence. For the local source it is the compatibility of connecting homomorphisms with cup product and the local invariant map in the Tate duality pairing used in Proposition 5.16. The numerical assertions are Proposition 5.15, Proposition 5.16.

Section 6 Quadratic determinant obstructions

The Evens norm and the Stiefel–Whitney comparison used throughout this section follow [Eve], [Kah], and [Koz]; for the computation of Stiefel–Whitney classes see also [Gui].

The linear interface of Section 5 does not determine the scalar central pushouts attached to self-dual simple modules, because their lifting obstruction is quadratic. We construct a normalized equivariant base class, compute the resulting form on the candidate and local sides, compare their Gauss sums, and then restore the affine terms required in the finite-target recursion.

Determinant obstruction. Let

\begin{equation*} 1\longrightarrow \notn{Rcent}{R}\longrightarrow \widetilde M \longrightarrow \notn{Mmid}{M}\longrightarrow1 \end{equation*}

be a $\notn{Cfin}{C}$-equivariant central extension with $\widetilde M$ of class at most two and with $\notn{Rcent}{R}$ and $\notn{Mmid}{M}$ elementary abelian, and let $\lambda\in(\notn{Rcent}{R}^\vee)^\notn{Cfin}{C}\text{.}$ Pushing out along $\lambda:\notn{Rcent}{R}\to\F_2$ gives a central double cover of $\notn{Mmid}{M}\text{.}$ Its square map

\begin{equation*} \notn{qlambda}{q_\lambda}(m)=\lambda(\widetilde m^{\,2})\in\F_2 \end{equation*}

is quadratic, with polar form given by the commutator pairing. The scalar degree-two obstruction obtained by pulling back this cover is called the determinant obstruction. In the application, its restriction to the nontrivial simple head $\notnfar{Vhead}{V}$ is a nonzero nonsingular quadratic form; this is proved in Proposition 7.4.

The notation used for the quadratic and affine terms is as follows.

Symbol	Meaning
$q$	a $\notn{Cfin}{C}$-invariant nonsingular quadratic form on the simple self-dual module $\notnfar{Vhead}{V}$
$\notnfar{kappaq0}{\kappa_q^0}$	the zero-section-normalized equivariant central $2$-cocycle on $\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}$ whose fibre square map is $q$
$\gamma$	the filtration-one class in $Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee)\text{;}$ its contribution on $\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}$ is the cocycle $\Gamma_\gamma$
$\delta$	the scalar class in $Z^2(\notn{Cfin}{C},\F_2)$ inflated from the lower quotient $\notn{Cfin}{C}$
$e$	the class in $H^2(\notn{Cfin}{C},\notn{Tsub}{T})$ of the affine $\notn{Tsub}{T}$-extension governing which $\notnfar{Vhead}{V}$-cocycles lift through $\notn{Mmid}{M}$
$\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}}$	the scalar class $\chi_*e+\delta_\kappa+\Theta_q^0(a_{\chi,\kappa})$ obtained after completing the square

For the candidate source we write

\begin{equation*} \iota_A:H^2_{A,\notnfar{rhoEpi}{\rho}}(\F_2)\longrightarrow\F_2 \end{equation*}

for the functional induced by the degree-two trace

\begin{equation*} \notn{Dfox}{D}^2(u_{\mathrm t},u_{\mathrm w})=u_{\mathrm t}+u_{\mathrm w} \end{equation*}

of Proposition 5.10. For the local source we write

\begin{equation*} \iota_F=\operatorname{inv}_{\Qtwo}:H^2(\notnfar{GQ}{\GQ},\F_2)\longrightarrow\F_2. \end{equation*}

All quadratic obstruction functions below use the source-specific functional $\notn{iotaGamma}{\iota_\Gamma}\text{.}$

Subsection 6.1 The normalized equivariant base class

A $\notn{Cfin}{C}$-invariant quadratic form on $\notnfar{Vhead}{V}$ need not admit a $\notn{Cfin}{C}$-invariant factor set for a chosen section $\notnfar{Vhead}{V}\to E\text{.}$ The correction terms in the lifted $\notn{Cfin}{C}$-action are therefore included in the equivariant central extension class.

Lemma 6.1. Equivariant extraspecial connecting cocycle.

Let $\notnfar{Vhead}{V}$ be an elementary $\F_2[\notn{Cfin}{C}]$-module, let $q:\notnfar{Vhead}{V}\to\F_2$ be a $\notn{Cfin}{C}$-invariant quadratic form with polar form $\notn{bq}{b_q}\text{,}$ and let

\begin{equation*} f:\notnfar{Vhead}{V}\times \notnfar{Vhead}{V}\to\F_2, \qquad f(v,v)=q(v),\qquad f(v,w)+f(w,v)=\notn{bq}{b_q}(v,w), \end{equation*}

be a normalized factor set for the central extension

\begin{equation*} E_f=\notnfar{Vhead}{V}\times\F_2, \qquad (v,z)(w,t)=(v+w,z+t+f(v,w)). \end{equation*}

When an equivariant lift is chosen, its action in the normalized section is written by functions $m_c:\notnfar{Vhead}{V}\to\F_2$ satisfying

\begin{equation} m_c(v+w)+m_c(v)+m_c(w)=f(cv,cw)+f(v,w)\tag{6.1} \end{equation}

and

\begin{equation} m_{cd}(v)=m_c(dv)+m_d(v),\qquad m_1=0.\tag{6.2} \end{equation}

Equivalently,

\begin{equation*} c\cdot(v,z)=(cv,z+m_c(v)) \end{equation*}

defines an action by automorphisms of $E_f\text{.}$ The associated normalized base central cocycle on $\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}$ is

\begin{equation} \notnfar{kappaq0}{\kappa_q^0}((v,c),(w,d))=f(v,cw)+m_c(w).\tag{6.3} \end{equation}

Consequently, for a source $\notnfar{GammaSrc}{\Gamma}\text{,}$ a lower map $\notnfar{rhoEpi}{\rho}:\notnfar{GammaSrc}{\Gamma}\to \notn{Cfin}{C}\text{,}$ and a normalized $\notnfar{Vhead}{V}$-valued $1$-cocycle $b\text{,}$ the base connecting cocycle is

\begin{equation} (b,\notnfar{rhoEpi}{\rho})^*\notnfar{kappaq0}{\kappa_q^0}(g,h) =f(b(g),\notnfar{rhoEpi}{\rho}(g)b(h))+m_{\rho(g)}(b(h)).\tag{6.4} \end{equation}

If $\kappa$ is another normalized central cocycle on $\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}$ whose restriction to $\notnfar{Vhead}{V}$ is the same quadratic form $q\text{,}$ then, after replacing $\kappa$ by a cohomologous normalized cocycle and changing the normalized section, the following is an equality of normalized $2$-cochains:

\begin{equation} \kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_\gamma+ \operatorname{inf}_C^{\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}}\delta,\tag{6.5} \end{equation}

where

\begin{equation} \Gamma_\gamma((v,c),(w,d))=\gamma(c)(cw), \qquad \gamma\in Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee),\quad \delta\in Z^2(\notn{Cfin}{C},\F_2).\tag{6.6} \end{equation}

After graph pullback and application of the source functional $\notn{iotaGamma}{\iota_\Gamma}\text{,}$ the corresponding scalar quadratic obstruction is

\begin{equation} Q_{\kappa,\notnfar{GammaSrc}{\Gamma},\notnfar{rhoEpi}{\rho}}(b) =Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}(b) +\langle b,\notnfar{rhoEpi}{\rho}^*\gamma\rangle_\Gamma +\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\delta),\tag{6.7} \end{equation}

where

\begin{equation*} Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}(b)= \notn{iotaGamma}{\iota_\Gamma}\bigl((b,\notnfar{rhoEpi}{\rho})^*\notnfar{kappaq0}{\kappa_q^0}\bigr). \end{equation*}

Its polarization is the source pairing transported by $\notn{bq}{b_q}\text{.}$

Lean

Proof.

We verify the equivariant central extension, identify the filtration-one and inflated difference terms, and then compute the polarization.

Equation (6.1) is exactly the condition that $(v,z)\mapsto(cv,z+m_c(v))$ preserve the multiplication in $E_f\text{;}$ equation (6.2) says that these automorphisms form a genuine $\notn{Cfin}{C}$-action. Multiplication in the semidirect product attached to this lifted action is

\begin{equation} (v,c,z)(w,d,t)= (v+cw,cd,z+t+f(v,cw)+m_c(w)),\tag{6.8} \end{equation}

which gives (6.3). Pulling back this cocycle by the graph of a $1$-cocycle $b$ gives (6.4). This is the term which would be missing if one incorrectly assumed that the section $v\mapsto(v,0)$ were $\notn{Cfin}{C}$-equivariant.

For the difference statement, the quotient $\kappa-\notnfar{kappaq0}{\kappa_q^0}$ restricts trivially to $\notnfar{Vhead}{V}\times \notnfar{Vhead}{V}\text{.}$ In the normalized Lyndon–Hochschild–Serre filtration for the split extension $1\to \notnfar{Vhead}{V}\to \notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}\to \notn{Cfin}{C}\to1\text{,}$ such a class has only a filtration-one part in $Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee)$ and an inflated scalar part in $Z^2(\notn{Cfin}{C},\F_2)\text{.}$ The normalized representative is exactly (6.6); the cocycle condition for it is the condition that $\gamma$ be a $1$-cocycle. Pullback along $(b,\notnfar{rhoEpi}{\rho})$ gives (6.7). Polarization of the base term is obtained by adding the formulas for $b+b'\text{,}$ $b\text{,}$ and $b'\text{;}$ the $m_c$-contribution cancels because each $m_c$ has polar form $f(c-,c-)+f(-,-)\text{,}$ and the remaining term is $\notn{bq}{b_q}(b\smile b')\text{.}$ The difference terms in (6.7) are linear or inflated scalar terms and therefore do not change the polar form.

Lemma 6.2. Explicit involution-orbit cocycle.

Let $H$ be a finite group, let $g\in H$ have order $2\text{,}$ and let $M=\F_2[H]$ with left-regular action

\begin{equation*} (c x)_h=x_{c^{-1}h}. \end{equation*}

Choose a set $\mathcal R\subset H$ of representatives for the right cosets $u\langle g\rangle\text{.}$ For $c\in H$ and $u\in\mathcal R\text{,}$ write uniquely

\begin{equation} c^{-1}u=\pi_c(u)g^{\epsilon_c(u)}, \qquad \pi_c(u)\in\mathcal R, \quad \epsilon_c(u)\in\F_2.\tag{6.9} \end{equation}

Define

\begin{align} q_g(x)&=\sum_{u\in\mathcal R}x_u x_{ug},\tag{6.10}\\ f_g(x,y)&=\sum_{u\in\mathcal R}x_u y_{ug},\tag{6.11}\\ m_c^g(x)&=\sum_{u\in\mathcal R} \epsilon_c(u)x_{\pi_c(u)}x_{\pi_c(u)g}.\tag{6.12} \end{align}

Then

\begin{align} m_c^g(x+y)+m_c^g(x)+m_c^g(y) &=f_g(cx,cy)+f_g(x,y),\tag{6.13}\\ m_{cd}^g(x)&=m_c^g(dx)+m_d^g(x), \qquad m_1^g=0.\tag{6.14} \end{align}

Consequently

\begin{equation} \kappa_g^{\mathcal R}((x,c),(y,d)) =f_g(x,cy)+m_c^g(y)\tag{6.15} \end{equation}

is a normalized $2$-cocycle on $M\rtimes H\text{.}$ It vanishes on the zero section $H\hookrightarrow M\rtimes H\text{,}$ and its restriction to $M$ has square map $q_g\text{.}$

Proof.

We first compute the transition functions for the right-coset action, then verify the quadratic correction, and finally apply the general equivariant extension formula.

From (6.9) one obtains

\begin{equation} \pi_{cd}=\pi_d\circ\pi_c, \qquad \epsilon_{cd}(u)=\epsilon_c(u)+\epsilon_d(\pi_c(u)).\tag{6.16} \end{equation}

Moreover,

\begin{equation*} f_g(cx,cy) =\sum_{u\in\mathcal R} x_{\pi_c(u)g^{\epsilon_c(u)}} y_{\pi_c(u)g^{\epsilon_c(u)+1}}. \end{equation*}

For $\epsilon_c(u)=0$ this is the summand of $f_g(x,y)$ indexed by $\pi_c(u)\text{;}$ for $\epsilon_c(u)=1$ its orientation is reversed. Therefore

\begin{equation*} f_g(cx,cy)+f_g(x,y) =\sum_{u\in\mathcal R}\epsilon_c(u) \bigl(x_{\pi_c(u)}y_{\pi_c(u)g} +x_{\pi_c(u)g}y_{\pi_c(u)}\bigr), \end{equation*}

which is exactly the polarization of (6.12). This proves (6.13).

The product $x_vx_{vg}$ is unchanged when $v$ is replaced by $vg\text{.}$ Hence

\begin{equation*} m_c^g(dx) =\sum_{u\in\mathcal R}\epsilon_c(u) x_{\pi_d(\pi_c(u))}x_{\pi_d(\pi_c(u))g}. \end{equation*}

Adding $m_d^g(x)$ and using (6.16) gives (6.14). The cocycle assertion now follows directly from Lemma 6.1. Finally, $f_g(x,x)=q_g(x)$ and $m_1^g=0\text{,}$ proving the restriction and normalization claims.

Lemma 6.3. Base equivariant determinant class.

Let $\notnfar{Vhead}{V}$ be a nontrivial simple self-dual tame $\F_2[\notn{Cfin}{C}]$-module, let $H=H_V$ be its faithful tame image, and let $q:\notnfar{Vhead}{V}\to\F_2$ be a nonzero $H$-invariant nonsingular quadratic form. There is a normalized base class

\begin{equation*} \notnfar{kappaq0}{\kappa_q^0}\in Z^2(\notnfar{Vhead}{V}\rtimes H,\F_2) \end{equation*}

whose restriction to $\notnfar{Vhead}{V}$ has square map $q\text{.}$

Choose an $H$-split pair

\begin{equation*} \notnfar{Vhead}{V}\xrightarrow{i}W=\F_2[H]^N\xrightarrow{p}\notnfar{Vhead}{V}, \qquad pi=1, \end{equation*}

and put $q_W=q\circ p\text{.}$ In the left-regular coordinates $X_{j,h}$ of $W\text{,}$ the unique algebraic normal form of the $H$-invariant quadratic map $q_W$ is a sum of orbit polynomials of the following three types:

\begin{align*} S_j&=\sum_{h\in H}X_{j,h}^2,\\ C_{j,k,g}&=\sum_{h\in H}X_{j,h}X_{k,hg} \quad\text{when the corresponding unordered pair has trivial stabilizer},\\ E_{j,g}&=\sum_{u\in\mathcal R_g}X_{j,u}X_{j,ug}, \qquad g^2=1,\ g\ne1, \end{align*}

where $\mathcal R_g$ is any set of representatives for the right cosets of $\langle g\rangle$ in $H\text{.}$ For a square orbit use the $H$-invariant factor set

\begin{equation} f_{S_j}(x,y)=\sum_{h\in H}x_{j,h}y_{j,h}, \qquad m_c=0.\tag{6.17} \end{equation}

For a free orbit use

\begin{equation} f_{C_{j,k,g}}(x,y)=\sum_{h\in H}x_{j,h}y_{k,hg}, \qquad m_c=0.\tag{6.18} \end{equation}

For an involution-stabilized orbit use the cocycle (6.15) of Lemma 6.2 on the $j$th regular summand, with $\mathcal R=\mathcal R_g\text{.}$ Let $\kappa_{q_W}^0$ be the sum of the cocycles corresponding to the orbit polynomials occurring in $q_W\text{,}$ and put

\begin{equation} \notnfar{kappaq0}{\kappa_q^0}=(i\rtimes1)^*\kappa_{q_W}^0.\tag{6.19} \end{equation}

Then $\notnfar{kappaq0}{\kappa_q^0}$ vanishes on the zero section and $\operatorname{res}_V(\notnfar{kappaq0}{\kappa_q^0})$ has square map $q\text{.}$ If a lower target has quotient $\notn{Cfin}{C}$ acting through $H\text{,}$ we inflate this class along $\notn{Cfin}{C}\to H$ and keep the same notation.

Lean

Proof.

We construct the class on a split regular module by the three possible orbit types of quadratic monomials and then pull it back to $\notnfar{Vhead}{V}\text{.}$

If $\notnfar{Vhead}{V}$ is ramified, $\notnfar{Vhead}{V}$ is projective over $\F_2[H]$ by Lemma 6.11; if $\notnfar{Vhead}{V}$ is unramified, the $2$-primary subgroup of the cyclic unramified image acts trivially on every simple characteristic-$2$ module, so the faithful image $H$ has odd order and Maschke’s theorem applies. Thus the split pair exists.

The algebraic normal form of a quadratic map over $\F_2$ is determined by its values on the coordinate vectors and by its polar form. $H$-invariance makes its coefficients constant on the orbits of unordered degree-two coordinate monomials. Since the action on each regular coordinate set is free, the stabilizer of an unordered pair is either trivial or has order two and exchanges the two entries. The latter possibility is exactly $j=k$ with a nontrivial involution $g\text{,}$ and gives $E_{j,g}\text{.}$

The factor sets (6.17) and (6.18) are $H$-invariant, vanish on the zero section, and have square maps $S_j$ and $C_{j,k,g}\text{,}$ respectively. By Lemma 6.2, the explicit cocycle for $E_{j,g}$ has square map $E_{j,g}$ and also vanishes on the zero section. Their sum is therefore an actual normalized equivariant central cocycle on $W\rtimes H$ with fibre square map $q_W\text{.}$ Pullback along $i\rtimes1$ gives fibre square map $q_W(iv)=q(v)$ and preserves zero-section normalization.

Lemma 6.4. Uniqueness and isometry invariance of the normalized base class.

Let $\notnfar{Vhead}{V}\text{,}$ $H=H_V\text{,}$ and $q$ be as in Lemma 6.3. Then

\begin{equation} H^1(H,\notnfar{Vhead}{V}^\vee)=0.\tag{6.20} \end{equation}

Any two classes in $H^2(\notnfar{Vhead}{V}\rtimes H,\F_2)$ which restrict to the class with square map $q$ on $\notnfar{Vhead}{V}$ and restrict trivially along the zero section $H\hookrightarrow \notnfar{Vhead}{V}\rtimes H$ are equal. Consequently, changing the normalized factor set, the correction functions $m_c\text{,}$ the half-orbit transversals, or the $H$-split regular embedding and retraction changes $\notnfar{kappaq0}{\kappa_q^0}$ only by a normalized coboundary. In particular the base quadratic obstruction $Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}$ is well-defined.

Moreover, every $H$-equivariant isometry $u$ of $(\notnfar{Vhead}{V},q)$ fixes the normalized base class:

\begin{equation} (u\rtimes1)^*[\notnfar{kappaq0}{\kappa_q^0}]=[\notnfar{kappaq0}{\kappa_q^0}].\tag{6.21} \end{equation}

Proof.

We first prove the required cohomology vanishing, then use the low-degree Lyndon–Hochschild–Serre sequence for uniqueness, and finally deduce choice-independence and isometry invariance.

If $\notnfar{Vhead}{V}$ is unramified, its faithful image $H$ has odd order: the $2$-primary subgroup of the cyclic unramified image acts trivially on every simple characteristic-$2$ module. Hence positive-degree $\F_2[H]$-cohomology vanishes by averaging.

Assume that $\notnfar{Vhead}{V}$ is ramified, and let $I\triangleleft H$ be the odd tame-inertia subgroup. The fixed space $(\notnfar{Vhead}{V}^\vee)^I$ is an $H$-submodule. It is zero, because $\notnfar{Vhead}{V}$ is simple and inertia acts nontrivially. Since $|I|$ is odd, $H^j(I,\notnfar{Vhead}{V}^\vee)=0$ for $j>0\text{.}$ The Hochschild–Serre spectral sequence for $1\to I\to H\to H/I\to1$ therefore gives [Bro, Ch. VII, §6]

\begin{equation*} H^n(H,\notnfar{Vhead}{V}^\vee)=0\qquad(n\ge0). \end{equation*}

In particular (6.20) holds in both cases.

Put $G=\notnfar{Vhead}{V}\rtimes H\text{.}$ For an elementary abelian $2$-group $\notnfar{Vhead}{V}\text{,}$ the square-map correspondence

\begin{equation*} H^2(\notnfar{Vhead}{V},\F_2)\xrightarrow{\sim} \{\text{quadratic maps }\notnfar{Vhead}{V}\to\F_2\} \end{equation*}

is an isomorphism: the polarization recovers the commutator form, and the square and commutator data determine the central extension class. Thus two classes with square map $q$ have the same fibre restriction.

The low-degree Lyndon–Hochschild–Serre sequence for $1\to \notnfar{Vhead}{V}\to G\to H\to1$ contains the exact segment [Bro, Cor. VII.6.4]

\begin{equation} H^2(H,\F_2)\xrightarrow{\operatorname{inf}} \ker\!\left(H^2(G,\F_2)\longrightarrow H^2(\notnfar{Vhead}{V},\F_2)\right) \xrightarrow{\operatorname{edge}}H^1(H,\notnfar{Vhead}{V}^\vee).\tag{6.22} \end{equation}

Let $\xi$ be the difference of two classes with the stated normalizations. Its fibre restriction is zero. By (6.20) and exactness of (6.22), $\xi=\operatorname{inf}(\delta)$ for some $\delta\in H^2(H,\F_2)\text{.}$ Pullback along the zero section is a left inverse to inflation, while both classes vanish there. Hence $\delta=0$ and $\xi=0\text{.}$

At the filtration-one level one can see the same cancellation directly. If $\gamma=dh$ for $h\in \notnfar{Vhead}{V}^\vee\text{,}$ then the normalized $1$-cochain

\begin{equation} \ell_h(v,c)=h(v)\tag{6.23} \end{equation}

on $\notnfar{Vhead}{V}\rtimes H$ satisfies $d\ell_h=\Gamma_{dh}\text{,}$ with $\notnfar{GammaSrc}{\Gamma}$ as in (6.6).

If normalized cocycle representatives are used, their difference is $da$ for a normalized $1$-cochain $a\text{.}$ Since the difference vanishes on $H\times H\text{,}$ $a|_H$ is a $1$-cocycle. Subtracting its inflation makes $a$ vanish on the zero section without changing $da\text{.}$ Thus all choices in Lemma 6.3 give isomorphic normalized central extensions, and transport of lifts through that isomorphism leaves both the candidate relator obstruction and the local pullback class unchanged. This proves the choice-independence of $Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}\text{.}$

Finally, $(u\rtimes1)^*\notnfar{kappaq0}{\kappa_q^0}$ has the same fibre square map and the same zero-section restriction as $\notnfar{kappaq0}{\kappa_q^0}\text{,}$ so uniqueness proves (6.21).

The uniqueness assertion is deliberately made over the faithful image $H_V\text{.}$ After inflation to a larger quotient $\notn{Cfin}{C}$ acting through $H_V\text{,}$ the group $H^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee)$ need not vanish; its contribution is precisely the filtration-one term $\Gamma_\gamma$ in (6.5), and it remains part of the target-side affine data.

Subsection 6.2 Candidate-side calculation

Let $\notnfar{Vhead}{V}$ be a nontrivial self-dual simple tame module, let $q:\notnfar{Vhead}{V}\to\F_2$ be invariant and nonsingular, let $\notnfar{bq}{b_q}$ be its polar form, and let $\notnfar{kappaq0}{\kappa_q^0}$ be the base equivariant determinant class of Lemma 6.3. The candidate determinant calculation below is made in the lifted semidirect product (6.8); thus all $m_c$-terms are included.

Proposition 6.5. Complete base second-order word expansion.

With $\notn{opS}{\mathsf S},\notn{opT}{\mathsf T},\notn{Pnorm}{P},\notnfar{opU}{\mathsf U}=\notn{opS}{\mathsf S}^\notn{w}{\w}$ as above, evaluation of the actual relator (1.6) for the base class $\notnfar{kappaq0}{\kappa_q^0}$ gives

\begin{equation} Q_A^0(c)=q(c)+\notnfar{bq}{b_q}((\notn{Pnorm}{P}+1)c,\notnfar{opU}{\mathsf U}^{-1}c).\tag{6.24} \end{equation}

Equivalently,

\begin{equation} Q_A^0(c)= \begin{cases} q(c),&\notn{opT}{\mathsf T}=1,\\ q(c)+\notnfar{bq}{b_q}(c,\notnfar{opU}{\mathsf U}^{-1}c),&\notnfar{Vhead}{V}^{\notn{opT}{\mathsf T}}=0. \end{cases}\tag{6.25} \end{equation}

Its ramified polar form is

\begin{equation} b_A(c,c')=\notnfar{bq}{b_q}(c,(1+\notnfar{opU}{\mathsf U}+\notnfar{opU}{\mathsf U}^{-1})c').\tag{6.26} \end{equation}

For a general determinant class $\kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_{\gamma_\kappa}+\operatorname{inf}\delta_\kappa\text{,}$ one has

\begin{equation} Q_{A,\kappa,\notnfar{rhoEpi}{\rho}}(c)=Q_A^0(c) +\langle c,\notnfar{rhoEpi}{\rho}^*\gamma_\kappa\rangle_A +\iota_A(\notnfar{rhoEpi}{\rho}^*\delta_\kappa).\tag{6.27} \end{equation}

Lean

Proof.

We evaluate the square contribution of $h_0\text{,}$ the commutator contribution of $d_0$ and $z_0\text{,}$ and then polarize the resulting quadratic expression.

Normalize a cohomology representative so that only $\notn{x0}{x_0}$ varies, by $c\text{.}$ In terms of (6.8), the complete base contribution ledger is

\begin{equation*} \begin{array}{c|c|c} \text{factor}&\text{$\notnfar{Vhead}{V}$-coordinate}&\text{central coordinate}\\ \hline h_0&0&q(c)\\ u_1^{-1}&0&0\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0\\ d_0=(\notn{x0}{x_0}\notn{tau}{\tau})^\notn{w}{\w} \notn{x0}{x_0}^{-1}&(\notn{Pnorm}{P}+1)c&\ast\\ z_0=\notn{x0}{x_0}^{\notn{sigma}{\sigma}^\notn{w}{\w}}&\notnfar{opU}{\mathsf U}^{-1}c&\ast\\ {[d_0,z_0]}&0&\notnfar{bq}{b_q}((\notn{Pnorm}{P}+1)c,\notnfar{opU}{\mathsf U}^{-1}c)\\ \end{array} \end{equation*}

The starred entries contain ordinary $f$-terms and the $m$-terms of the lifted $\notn{Cfin}{C}$-action. They occur only as central coordinates of $d_0$ and $z_0\text{,}$ and central coordinates do not affect the commutator. Since the lower values of $d_0$ and $z_0$ at the tame map are both the identity, the commutator contribution is exactly the polar form of their $\notnfar{Vhead}{V}$-coordinates, $\notnfar{bq}{b_q}((\notn{Pnorm}{P}+1)c,\notnfar{opU}{\mathsf U}^{-1}c)\text{.}$

The first line is the extraspecial case of Lemma 5.3, but we recall why it remains valid with the $m$-terms present. The class-two identity Lemma 5.2 uses only that the lift of the relevant tame operator is an automorphism of the central extension and fixes the centre; it does not assume that the chosen section is equivariant. If $\notn{Pnorm}{P}=0\text{,}$ then $d_0$ and $\notn{x0}{x_0}$ have the same image in $\notnfar{Vhead}{V}\text{,}$ so the identity gives the square of $\notn{x0}{x_0}\text{,}$ namely $q(c)\text{.}$ If $\notn{Pnorm}{P}=1\text{,}$ then $\notn{opT}{\mathsf T}=1$ and the lift of $\notnfar{opU}{\mathsf U}$ is a central automorphism of order at most $2\text{,}$ so $g_0=\sigma_2^2$ acts trivially on the class-two coefficient group and the same identity applies.

The factors $u_1^{-1}$ and $\notnfar{x1}{x_1}^\notn{sigma}{\sigma}$ have zero $\notnfar{Vhead}{V}$-coordinate on the normalized representative, and hence no quadratic central contribution. Multiplying the table gives (6.24). Polarizing and using the $\notnfar{opU}{\mathsf U}$-invariance of $\notnfar{bq}{b_q}$ gives (6.26); the operator is invertible by the same polynomial argument used in Proposition 5.15. Finally, (6.27) is exactly (6.7) on the candidate source.

Subsubsection 6.2.1 The Arf invariant of the candidate form

Lemma 6.6. Wall doubling.

Let $(\notnfar{Vhead}{V},q)$ be nonsingular over $\F_2\text{,}$ with polar form $\notnfar{bq}{b_q}\text{,}$ and let $\notnfar{opU}{\mathsf U}\in O(\notnfar{Vhead}{V},q)$ have $2$-power order. Put

\begin{equation*} q_{\notnfar{opU}{\mathsf U}}(x)=q(x)+\notnfar{bq}{b_q}(x,\notnfar{opU}{\mathsf U}^{-1}x). \end{equation*}

Then $q_{\notnfar{opU}{\mathsf U}}$ is nonsingular and

\begin{equation} \notn{Arf}{\Arf}(q_{\notnfar{opU}{\mathsf U}})=\notn{Arf}{\Arf}(q)+\operatorname{rank}(1+\notnfar{opU}{\mathsf U})\pmod2.\tag{6.28} \end{equation}

Lean (nonsingularity)Lean (main)

Proof.

Let $N=1+\notnfar{opU}{\mathsf U}\text{,}$ $R=NV\text{,}$ and $K=\ker N\text{.}$ Since $q(\mathsf Ux)=q(x)\text{,}$

\begin{equation*} \notnfar{bq}{b_q}(x,\notnfar{opU}{\mathsf U}^{-1}x)=\notnfar{bq}{b_q}(x,\mathsf Ux)=q(Nx), \end{equation*}

so $q_{\notnfar{opU}{\mathsf U}}(x)=q(x)+q(Nx)\text{.}$ Define the nondegenerate Wall form on $R$ by

\begin{equation*} \notn{omgu}{\omega}(Nx,u)=\notnfar{bq}{b_q}(x,u). \end{equation*}

In $(\notnfar{Vhead}{V},q_{\notnfar{opU}{\mathsf U}})\perp(\notnfar{Vhead}{V},q)\text{,}$ the diagonal $\{(k,k):k\in K\}$ is totally singular. Its orthogonal quotient is $R\oplus R$ with quadratic form

\begin{equation*} \Phi_\omega(v,u)=\notn{omgu}{\omega}(v,v)+\notn{omgu}{\omega}(u,u)+\notn{omgu}{\omega}(v,u). \end{equation*}

If $v\mapsto\notn{omgu}{\omega}(v,v)$ is nonzero, split an anisotropic plane from $\Phi_\omega\text{;}$ if $\omega$ is alternating, split a symplectic plane from $R\text{,}$ which contributes a hyperbolic four-space. Induction gives $\notn{Arf}{\Arf}(\Phi_\omega)=\dim R\pmod2\text{,}$ proving (6.28).

Lemma 6.7. Invariant quadratic forms on a hermitian line.

Let $F/F_0$ be a quadratic extension of finite fields of characteristic $2\text{,}$ write $x\mapsto x^*$ for its involution, and let $\mathcal U_1=\{u\in F^\times:uu^*=1\}\text{.}$ Let $W$ be a one-dimensional $F$-space. Every nonsingular $\mathcal U_1$-invariant quadratic form $Q:W\to\F_2$ is uniquely of the form

\begin{equation*} Q(x)=\Tr_{F_0/\F_2}(a xx^*) \qquad (a\in F_0^\times). \end{equation*}

Lean

Proof.

Choose a generator $u_0$ of the cyclic group $\mathcal U_1\text{.}$ Since $2^m\equiv-1\pmod{2^m+1}$ and no smaller positive power of $2$ is $1$ modulo $2^m+1\text{,}$ the element $u_0$ has degree $2m$ over $\F_2\text{;}$ hence $\F_2[u_0]=F\text{.}$ Invariance gives

\begin{equation*} b_Q(dx,y)=b_Q(x,d^*y)\qquad(d\in F). \end{equation*}

Putting $\ell(y)=b_Q(1,y)\text{,}$ we obtain $b_Q(x,y)=\ell(x^*y)\text{.}$ Write $\ell(z)=\Tr_{F/\F_2}(cz)$ for the unique $c\in F\text{.}$ Symmetry of $b_Q$ forces $c=c^*\text{,}$ hence $c\in F_0\text{,}$ and therefore

\begin{equation*} b_Q(x,y)=\Tr_{F_0/\F_2} \bigl(c(xy^*+x^*y)\bigr). \end{equation*}

Conversely these forms are alternating, and they are nonsingular exactly when $c\ne0\text{.}$ The quadratic form $q_c(x)=\Tr_{F_0/\F_2}(cxx^*)$ has this polarization. Hence $Q-q_c$ is a $\mathcal U_1$-invariant linear form. Writing such a form as $x\mapsto\Tr_{F/\F_2}(bx)\text{,}$ invariance under $u_0$ gives $b(u_0-1)=0\text{,}$ so $b=0\text{.}$ Thus $Q=q_c\text{,}$ uniquely, and nonsingularity gives $c\in F_0^\times\text{.}$

Lemma 6.8. Ramified Hermitian model and Frobenius fixed space.

Let $I=\langle \notn{opT}{\mathsf T}\rangle$ be tame inertia, let $\notnfar{Vhead}{V}$ be a simple ramified self-dual $\F_2[H_V]$-module, and write

\begin{equation*} \notnfar{Vhead}{V}|_I\cong W^{\oplus s}, \qquad f=\dim_{\F_2}W=2^a r, \qquad r\text{ odd},\quad a\ge1. \end{equation*}

Let $q$ be an $H_V$-invariant nonsingular quadratic form with polar form $\notnfar{bq}{b_q}\text{.}$ Then

\begin{equation} \notn{Arf}{\Arf}(q)\equiv s\pmod2.\tag{6.29} \end{equation}

For $\notnfar{opU}{\mathsf U}=\notn{opS}{\mathsf S}^{\omega_2}$ one also has

\begin{equation} \dim_{\F_2}\notnfar{Vhead}{V}^{\notnfar{opU}{\mathsf U}}=rs, \qquad \operatorname{rank}(1+\notnfar{opU}{\mathsf U})=rs(2^a-1)\equiv s\pmod2.\tag{6.30} \end{equation}

Consequently the ramified candidate base form of (6.25) has $\notn{Arf}{\Arf}(Q_A^0)=0\text{.}$

Lean (Arf vanishing)Lean (Arf = s)Lean (main)

Proof.

We identify the invariant quadratic form with a Hermitian form, compute its Arf invariant one Hermitian line at a time, and then determine the fixed space of the $2$-primary Frobenius operator.

Put

\begin{equation*} F_V=\End_I(W)\cong\F_{2^f}. \end{equation*}

Choose an inertia character $\zeta$ over $\overline{\F}_2\text{;}$ then $W$ is the additive group of $F_V$ and $\notn{opT}{\mathsf T}$ acts by multiplication by $\zeta\text{.}$ Since $\notnfar{Vhead}{V}|_I$ has only the irreducible isomorphism type $W$ and $\notnfar{Vhead}{V}\cong \notnfar{Vhead}{V}^\vee\text{,}$ the inverse character $\zeta^{-1}$ lies in the Frobenius orbit of $\zeta\text{.}$ The character is nontrivial of odd order, so this orbit has even length and

\begin{equation*} \zeta^{-1}=\zeta^{2^{f/2}}. \end{equation*}

Set

\begin{equation*} F_{V,0}=\F_{2^{f/2}}, \qquad x^*=x^{2^{f/2}}. \end{equation*}

After an $I$-module identification $\notnfar{Vhead}{V}\cong F_V^s\text{,}$ the nondegenerate $I$-invariant form $\notnfar{bq}{b_q}$ is uniquely expressible as

\begin{equation} \notnfar{bq}{b_q}(x,y)=\Tr_{F_V/\F_2}(x^{*t}\mathsf H y), \qquad \mathsf H\in\GL_s(F_V),\quad \mathsf H^{*t}=\mathsf H.\tag{6.31} \end{equation}

Indeed, invariance under multiplication by $\zeta\text{,}$ nondegeneracy of the trace pairing, and symmetry identify the adjoint involution with $*$ and force the matrix to be Hermitian.

Define

\begin{equation} q_{\mathsf H}(x)=\Tr_{F_{V,0}/\F_2}(x^{*t}\mathsf H x).\tag{6.32} \end{equation}

The scalar $x^{*t}\mathsf Hx$ lies in $F_{V,0}\text{,}$ and

\begin{align*} q_{\mathsf H}(x+y)+q_{\mathsf H}(x)+q_{\mathsf H}(y) &=\Tr_{F_{V,0}/\F_2}(x^{*t}\mathsf Hy+y^{*t}\mathsf Hx)\\ &=\Tr_{F_V/\F_2}(x^{*t}\mathsf Hy)=\notnfar{bq}{b_q}(x,y). \end{align*}

Thus $q-q_{\mathsf H}$ is linear. It is $I$-invariant, while $(\notnfar{Vhead}{V}^\vee)^I=0$ because the inertia action is nontrivial and $\notnfar{Vhead}{V}$ is simple. Hence $q=q_{\mathsf H}\text{.}$

Hermitian diagonalization over the finite field extension $F_V/F_{V,0}$ gives

\begin{equation*} \mathsf H\sim\operatorname{diag}(a_1,\dots,a_s), \qquad a_i\in F_{V,0}^\times. \end{equation*}

Therefore $q$ is an orthogonal sum of the $s$ Hermitian-line forms

\begin{equation*} q_{a_i}(x)=\Tr_{F_{V,0}/\F_2}(a_i xx^*). \end{equation*}

Write $m=f/2\text{.}$ The norm $F_V^\times\to F_{V,0}^\times$ is surjective with fibres of size $2^m+1\text{,}$ and the trace-zero subspace of $F_{V,0}$ contains $2^{m-1}-1$ nonzero elements. Hence

\begin{equation*} \#q_{a_i}^{-1}(0) =1+(2^m+1)(2^{m-1}-1) =2^{2m-1}-2^{m-1}. \end{equation*}

Each Hermitian line has Arf invariant $1\text{.}$ Additivity of the Arf invariant proves (6.29).

It remains to compute the fixed space of $\notnfar{opU}{\mathsf U}\text{.}$ Over $\overline{\F}_2\text{,}$ the $f$ inertia weights form one Frobenius cycle, each weight space having multiplicity $s\text{.}$ Let $e_\omega$ be the residue modulo $f=2^ar$ of an integer representing $\omega_2\text{.}$ By the defining congruences of $\omega_2\text{,}$

\begin{equation*} e_\omega\equiv1\pmod{2^a}, \qquad e_\omega\equiv0\pmod r. \end{equation*}

Thus translation by $e_\omega$ on $\mathbb{Z}/f\mathbb{Z}$ has $r$ cycles, each of length $2^a\text{.}$ On one cycle write the successive maps as

\begin{equation*} E_0\xrightarrow{A_0}E_1\xrightarrow{A_1}\cdots \xrightarrow{A_{2^a-1}}E_0, \qquad \dim E_i=s. \end{equation*}

The product $A_{2^a-1}\cdots A_0$ is the action of $\notnfar{opU}{\mathsf U}^{2^a}$ on $E_0\text{.}$ The shift by $2^ae_\omega$ is trivial modulo $f\text{,}$ so $\notnfar{opU}{\mathsf U}^{2^a}$ centralizes inertia; it also commutes with $\notn{opS}{\mathsf S}\text{,}$ being a power of $\notn{opS}{\mathsf S}\text{.}$ Hence it is a central $2$-element of the faithful tame image. A cyclic $2$-group acting in characteristic $2$ has nonzero fixed vectors, and centrality makes its fixed space an $H_V$-submodule. Simplicity therefore forces $\notnfar{opU}{\mathsf U}^{2^a}=1$ on $\notnfar{Vhead}{V}\text{.}$ Consequently a $\notnfar{opU}{\mathsf U}$-fixed vector on a cycle is uniquely determined by its component in $E_0\text{,}$ and that cycle contributes dimension $s\text{.}$ There are $r$ cycles, proving (6.30).

Finally, Lemma 6.6 gives

\begin{equation*} \notn{Arf}{\Arf}(Q_A^0)=\notn{Arf}{\Arf}(q)+\operatorname{rank}(1+\notnfar{opU}{\mathsf U})=s+s=0 \quad\text{in }\F_2. \end{equation*}

These determinant computations combine into the candidate-side zero count.

Proposition 6.9. Candidate base determinant zero count.

If $d=\dim \notnfar{Vhead}{V}\text{,}$ then

\begin{equation} \#(Q_A^0)^{-1}(0)= \begin{cases} 2^{d-1}-2^{d/2-1},&\notnfar{Vhead}{V}\text{ unramified},\\ 2^{d-1}+2^{d/2-1},&\notnfar{Vhead}{V}\text{ ramified}. \end{cases}\tag{6.33} \end{equation}

Lean (unramified, free)Lean (unramified, abelian)Lean (unramified, cyclic)Lean (unramified)Lean (ramified)Lean

Proof.

We determine the unramified and ramified signs separately, using the Hermitian-line count in the first case and Lemma 6.8 in the second.

If $\notnfar{Vhead}{V}$ is unramified, $\notnfar{opU}{\mathsf U}=1$ and $Q_A^0=q\text{.}$ Put $F_V=\End_{H_V}(\notnfar{Vhead}{V})\cong\F_{2^{2m}}$ and let $F_{V,0}\cong\F_{2^m}$ be the fixed field of the adjoint involution $d\mapsto d^*$ defined by the polar form $\notnfar{bq}{b_q}\text{.}$ The simple cyclic module $\notnfar{Vhead}{V}$ is a one-dimensional $F_V$-space. The norm-one group

\begin{equation*} \mathcal U=\{u\in F_V^\times:uu^*=1\} \end{equation*}

preserves $\notnfar{bq}{b_q}\text{.}$ For $u\in\mathcal U\text{,}$ the function $x\mapsto q(ux)+q(x)$ is therefore linear. It is $H_V$-invariant because $u$ commutes with $H_V$ and $q$ is $H_V$-invariant. Since $\notnfar{Vhead}{V}$ is nontrivial simple, $(\notnfar{Vhead}{V}^\vee)^{H_V}=0\text{,}$ so the function is zero. Thus $\mathcal U\le O(\notnfar{Vhead}{V},q)\text{.}$

By Lemma 6.7,

\begin{equation*} q(x)=\Tr_{F_{V,0}/\F_2}(a xx^*),\qquad a\in F_{V,0}^\times. \end{equation*}

The norm fibres have size $2^m+1\text{,}$ so the number of zeros is $2^{2m-1}-2^{m-1}$ and the Arf invariant is $1\text{.}$

Assume $\notnfar{Vhead}{V}$ is ramified. The complete Hermitian and fixed-space calculation is Lemma 6.8; in particular $\notn{Arf}{\Arf}(Q_A^0)=0\text{.}$ For a nonsingular quadratic form of dimension $d\text{,}$ the identity

\begin{equation*} \#Q^{-1}(0)=2^{d-1}+(-1)^{\notn{Arf}{\Arf}(Q)}2^{d/2-1} \end{equation*}

now gives (6.33).

Subsection 6.3 Local calculation

This subsection uses the unit-filtration structure of dyadic local fields [Ser79, Chapter IV §2], the norm criteria for quadratic extensions [Ser79, Chapter V §2 and Chapter XIV §2], and local Tate duality [NSW, Theorem (7.2.6)].

Let $q:\notnfar{Vhead}{V}\to\F_2$ be the invariant nonsingular quadratic form under consideration, and let $\notnfar{kappaq0}{\kappa_q^0}$ be the base equivariant determinant class of Lemma 6.3. For a local cocycle $b:\notnfar{GQ}{G_{\Qtwo}}\to \notnfar{Vhead}{V}$ with lower map $\notnfar{rhoEpi}{\rho}\text{,}$ put

\begin{equation} Q^0_{\mathrm{loc}}([b])= \inv_{\Qtwo}\bigl[(b,\notnfar{rhoEpi}{\rho})^*\notnfar{kappaq0}{\kappa_q^0}\bigr].\tag{6.34} \end{equation}

This is the base quadratic connecting map. For a general determinant pushout class $\kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_{\gamma_\kappa}+\operatorname{inf}\delta_\kappa\text{,}$ the actual local obstruction is obtained from (6.34) by the difference formula (6.7); the affine terms are not folded into the base Gauss sum.

Let $H_V$ be the faithful tame image on $\notnfar{Vhead}{V}$ and let $K/\notn{Qtwo}{\Qtwo}$ be its tame splitting field. Put $M_K=K^\times/K^{\times2}$ and let $U_i$ be the image of $1+\mathfrak p_K^i\text{.}$ If $e=v_K(2)\text{,}$ then

\begin{equation} \gr_iM_K\cong \begin{cases} k_K,&1\le i<2e\text{ odd},\\ 0,&1\le i<2e\text{ even}, \end{cases}\tag{6.35} \end{equation}

and

\begin{equation} U_i^\perp=U_{2e-i+1}, \qquad -1\in U_e.\tag{6.36} \end{equation}

These are the standard dyadic square-class filtration and Hilbert-symbol orthogonality relations; see [Ser79, Chapter XIV, Sections 2--3].

Lemma 6.10. The middle unit layer is unramified.

Assume $K/\notn{Qtwo}{\Qtwo}$ is tame and put $e=v_K(2)\text{.}$ Then $e$ is odd, and tame inertia acts trivially on the middle graded piece

\begin{equation*} U_e/U_{e+1}. \end{equation*}

Consequently a ramified simple tame module does not occur in that graded piece.

Proof.

The ramification index of a tame extension of $\notn{Qtwo}{\Qtwo}$ is odd and equals the order of tame inertia. On the odd-depth piece $\gr_jM_K\text{,}$ an inertia element represented by $\zeta$ acts through the character $\zeta\mapsto\zeta^j\text{.}$ At $j=e$ this character is trivial. A ramified simple module has nontrivial inertia, so its isotypic multiplicity at the middle depth is zero.

Lemma 6.11. Faithful-image projectivity.

If $\notnfar{Vhead}{V}$ is ramified, then $\notnfar{Vhead}{V}$ and $\notnfar{Vhead}{V}^\vee$ are projective $\F_2[H_V]$-modules.

Lean (tame pair case)Lean (main)

Proof.

We use Clifford theory over an algebraic closure to describe the inertia weights, prove regularity on Sylow-$2$ weight blocks, and then descend projectivity to $\F_2\text{.}$

Let $\overline{\F}_2$ be an algebraic closure, let $I\triangleleft H_V$ be tame inertia, and choose an absolutely irreducible constituent $W_0$ of $\notnfar{Vhead}{V}\otimes_{\F_2}\overline{\F}_2\text{.}$ Since $|I|$ is odd, $W_0|_I$ is semisimple. Choose a nontrivial inertia character $\chi$ occurring in it, and let $H_\chi$ be its stabilizer. Clifford theory expresses $W_0$ as the induction of a simple $H_\chi$-module on which $I$ acts through $\chi\text{.}$ The quotient $H_\chi/I$ is cyclic. Since $\chi$ is $H_\chi$-invariant and $\overline{\F}_2$ contains all roots needed to satisfy the single cyclic relation, $\chi$ extends to a one-dimensional character of $H_\chi\text{.}$ After twisting by that extension, the inducing module factors through the cyclic quotient. Over $\overline{\F}_2$ every simple module for a cyclic group is one-dimensional, and its $2$-primary subgroup acts trivially. Thus every $\chi$-weight space in $W_0$ is one-dimensional.

Let the Frobenius orbit of $\chi$ have length $f=2^a r$ with $r$ odd, and let $\notn{Pnorm}{P}$ be a Sylow $2$-subgroup of $H_V\text{.}$ The image of $\notn{Pnorm}{P}$ in the cyclic quotient $H_V/I$ is cyclic. Its action on the orbit of $\chi$ has $2$-power orbit length $2^a\text{.}$ If an element of $\notn{Pnorm}{P}$ stabilizes $\chi\text{,}$ it acts on the one-dimensional $\chi$-weight space by a scalar of $2$-power order, hence trivially because $\overline{\F}_2^\times$ has no nontrivial $2$-power torsion. By conjugacy it then acts trivially on all inertia-weight spaces of $W_0\text{,}$ and on every Galois conjugate constituent as well. Faithfulness of the image $H_V$ therefore shows that this stabilizer is trivial. Consequently $|\notn{Pnorm}{P}|=2^a\text{,}$ and $\notn{Pnorm}{P}$ acts freely on the weights in blocks of size $2^a\text{.}$

On each block, transport a nonzero vector around the $\notn{Pnorm}{P}$-orbit. Because the generator of $\notn{Pnorm}{P}$ has order $2^a\text{,}$ the transported vectors form a regular $\overline{\F}_2[\notn{Pnorm}{P}]$-basis. Hence the restriction of every absolutely irreducible constituent, and therefore of $\notnfar{Vhead}{V}\otimes_{\F_2}\overline{\F}_2\text{,}$ is a direct sum of regular $\overline{\F}_2[\notn{Pnorm}{P}]$-modules. It is projective over $\overline{\F}_2[\notn{Pnorm}{P}]\text{.}$

Projectivity descends along the faithfully flat extension $\F_2\subset\overline{\F}_2\text{,}$ so $\notnfar{Vhead}{V}|_P$ is projective over $\F_2[\notn{Pnorm}{P}]\text{.}$ The Sylow criterion for modular projectivity now gives projectivity over $\F_2[H_V]\text{.}$ Replacing every inertia character by its inverse gives the identical Sylow restriction calculation for $\notnfar{Vhead}{V}^\vee\text{,}$ and hence proves projectivity of the dual module.

Remark 6.12. Why the faithful image is required.

Projectivity is asserted only for the faithful image $H_V\text{.}$ A larger tame quotient may contain an additional unramified $2$-power subgroup acting trivially on $\notnfar{Vhead}{V}\text{,}$ and projectivity over that larger group can fail. The base determinant connecting map depends only on $H_V\text{;}$ information involving the extra kernel belongs to the inflated and filtration-one components separated from the base determinant calculation.

Normalized-cochain conventions. From this point through Lemma 6.15, all coefficient modules are left modules and all cochains are normalized inhomogeneous cochains. The differential, cup product, semidirect-product convention, Shapiro map, corestriction, and all changes of sections and transversals are fixed in Appendix A. In particular, “equal as normalized cochains” means literal equality in the indicated cochain group; when a choice changes a formula, the corresponding normalized coboundary is the one displayed in Subsection A.7.

Lemma 6.13. Universal two-point normalization and the index-two Evens norm.

Let

\begin{equation*} J=\langle s\rangle\cong C_2, \qquad E_2=\F_2e_1\oplus\F_2e_s, \qquad s(\notnfar{x1}{x_1},x_s)=(x_s,\notnfar{x1}{x_1}), \end{equation*}

and put

\begin{equation*} q_{\mathrm{hyp}}(\notnfar{x1}{x_1},x_s)=x_1x_s, \qquad f_J(x,y)=x_1y_s, \qquad m_1=0, \qquad m_s(y)=y_1y_s. \end{equation*}

Then

\begin{equation} \kappa_J((x,c),(y,d))=f_J(x,cy)+m_c(y)\tag{6.37} \end{equation}

is a normalized $2$-cocycle on $E_2\rtimes J\text{.}$ It restricts to the quadratic map $q_{\mathrm{hyp}}$ on $E_2$ and to zero on the zero section $J\text{.}$ Its fibre extension is the dihedral group $D_8\text{:}$ the two coordinate lifts are involutions, their commutator is the central involution, and their product has square equal to that involution. Moreover

\begin{equation} [\kappa_J]=N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee) \quad\text{in }H^2(E_2\rtimes J,\F_2).\tag{6.38} \end{equation}

Now let $N\triangleleft G$ have index $2\text{,}$ identify $G/N$ with $J\text{,}$ and let $\alpha\in Z^1(N,\F_2)\text{.}$ Choose lifts $\widetilde1=1$ and $\widetilde s\in G\text{,}$ and define the normalized Shapiro cocycle

\begin{equation} b(\gamma)_u= \alpha\!\left(\widetilde u^{-1}\gamma \widetilde{\bar\gamma^{-1}u}\right), \qquad u\in J.\tag{6.39} \end{equation}

This is the degree-one normalized Shapiro map (A.12). The graph pullback of (6.37) is

\begin{equation} \nu_\alpha(\gamma,\eta) =b(\gamma)_1b(\eta)_{\bar\gamma^{-1}s} +\varepsilon(\bar\gamma)b(\eta)_1b(\eta)_s,\tag{6.40} \end{equation}

where $\varepsilon(1)=0$ and $\varepsilon(s)=1\text{.}$ Its class is the index-two Evens norm:

\begin{equation} [\nu_\alpha]=N_N^{G,\mathrm{Ev}}([\alpha])\in H^2(G,\F_2).\tag{6.41} \end{equation}

Equivalently, the normalization is

\begin{equation*} N_N^G(1+[\alpha]) =1+\operatorname{cor}_N^G[\alpha] +N_N^{G,\mathrm{Ev}}([\alpha]). \end{equation*}

Finally, if $g\in H$ is an involution, $M=\F_2[H]\text{,}$ $J=\langle g\rangle\text{,}$ and $\pi:M\to\F_2[J]$ selects the coordinates indexed by $1$ and $g\text{,}$ then for every right-coset transversal $\mathcal R\text{,}$

\begin{equation} [\kappa_g^{\mathcal R}] =\operatorname{cor}_{M\rtimes J}^{M\rtimes H} (\pi\rtimes1)^*[\kappa_J].\tag{6.42} \end{equation}

Lean (dihedral)Lean (Evens)

Proof.

We verify the universal cocycle and its fibre extension, identify its class with the Evens norm, pull it back to a general index-two inclusion, and then compare the regular-module half-orbit transfer.

Expanding $m_s$ on the two coordinates gives the two identities required in Lemma 6.1: for $v,w\in E_2\text{,}$

\begin{equation*} m_s(v+w)+m_s(v)+m_s(w)=f_J(sv,sw)+f_J(v,w), \qquad m_s(sv)+m_s(v)=0. \end{equation*}

Thus (6.37) is a normalized cocycle. Its square map is $x_1x_s\text{,}$ and it vanishes when both fibre coordinates are zero. If $\widetilde e_1,\widetilde e_s$ are the coordinate lifts, then $\widetilde e_1^2=\widetilde e_s^2=1\text{,}$ their commutator is the central generator, and $(\widetilde e_1\widetilde e_s)^2$ is that generator; hence the fibre extension is $D_8\text{.}$

For the swap module,

\begin{equation} H^1(J,E_2^\vee)=\frac{\ker(1+s)}{\operatorname{im}(1+s)}=0.\tag{6.43} \end{equation}

The low-degree exact sequence (6.22) therefore says that a class on $E_2\rtimes J$ is determined by its restriction to $E_2$ and its restriction to the zero section. Mackey’s formula for the standard Evens norm gives

\begin{equation*} \res_{E_2} N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee)=e_1^\vee e_s^\vee, \qquad \res_J N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee)=0. \end{equation*}

These are exactly the two restrictions of $[\kappa_J]\text{,}$ proving (6.38). Equivalently, this class is the degree-two component of the relative Stiefel–Whitney class $w(\operatorname{Ind}\ell)/w(\operatorname{Ind}1)$ under the Evens–Kahn normalization; see [Eve, Thm. 1], [Kah, Théorèmes 1--2], [Koz, Thm. 1.1].

For the general index-two inclusion, the graph

\begin{equation*} \Phi_\alpha:G\longrightarrow E_2\rtimes J, \qquad \gamma\longmapsto(b(\gamma),\bar\gamma) \end{equation*}

has inverse image of $E_2$ equal to $N\text{,}$ and the pullback of $e_1^\vee$ to $N$ is $\alpha\text{.}$ Naturality of the Evens norm in this pullback square and (6.38) give (6.41). Direct substitution of (6.39) into (6.37) gives (6.40) term by term. This is also the degree-two term in Evens’s normalized bar construction [Eve, §§4--5]. If $\widetilde s$ is replaced by $\widetilde s a$ with $a\in N\text{,}$ then (A.25) gives the exact change of the Shapiro cocycle, and (A.36) gives the resulting normalized $1$-cochain whose boundary is the change in (6.40). Thus the lift dependence is controlled at cochain level, not only in cohomology.

For (6.42), Mackey restriction of the right side to $\notnfar{Mmid}{M}$ is

\begin{equation*} \sum_{u\in H/J}x_u x_{ug}, \end{equation*}

and its zero-section restriction is zero. The explicit class $[\kappa_g^{\mathcal R}]$ has the same two restrictions. Moreover $H^1(H,\notnfar{Mmid}{M}^\vee)=0$ by Shapiro’s lemma, since $\notnfar{Mmid}{M}^\vee$ is coinduced. The same low-degree uniqueness argument proves equality. More precisely, (A.30) identifies the two representatives as normalized cochains for the chosen transversal, and (A.32) displays the normalized $1$-cochain whose boundary is the change under an arbitrary new transversal.

Lemma 6.14. Regular-module realization of the base local connecting map.

With notation as in Lemma 6.3, the base local determinant form satisfies, for every $x\in H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V})\text{,}$

\begin{equation} Q^0_{\mathrm{loc},q}(x)=Q^0_{\mathrm{loc},q_W}(i_*x).\tag{6.44} \end{equation}

Under Kummer theory and the normalized Shapiro cochain map, the right side is obtained by applying the orbit operations of Lemma 6.15 to the scalar Shapiro coordinates. Thus the regular-module computation evaluates the actual base equivariant central class $\notnfar{kappaq0}{\kappa_q^0}\text{,}$ including the $m_c$-terms; it is not merely a polynomial with the same polarization.

Lean

Proof.

The equality of classes is the pullback identity $\notnfar{kappaq0}{\kappa_q^0}=(i\rtimes1)^*\kappa_{q_W}^0$ from (6.19). Pulling that identity back along a Galois graph cocycle gives equality of normalized central $2$-cocycles over $\notnfar{GQ}{G_{\Qtwo}}\text{,}$ hence (6.44) after applying the local invariant map.

For the second assertion, use the explicit representatives in (6.17), (6.18), and (6.15). Direct substitution of the normalized Shapiro cocycle gives the square and free-orbit corestriction formulas. For a half-orbit, the correction term $m_c^g$ records exactly whether the action on a right coset of $\langle g\rangle$ preserves or reverses the chosen orientation of its two coordinates. Thus the graph pullback is the normalized corestriction of the two-point graph cocycle (6.40); by Lemma 6.13 that two-point class is the index-two Evens norm. The full cochain calculation is written out in Lemma 6.15.

Lemma 6.15. Quadratic orbit–stabilizer Shapiro.

Let $W=\F_2[H]^N$ have coordinates $X_{r,h}$ with $H$ acting by left translation on $h\text{.}$ If $K/F$ is the $H$-extension and $\alpha_r\in H^1(K,\F_2)$ is the scalar Shapiro coordinate, then the orbit classes of Lemma 6.3 evaluate as follows. For $g\in H\text{,}$ choose any lift $\widetilde g\in G_F$ and use the convention

\begin{equation*} (g\alpha_s)(n)=\alpha_s(\widetilde g^{-1}n\widetilde g) \qquad(n\in G_K). \end{equation*}

This is independent of the chosen lift and is the convention fixed in (A.13):

\begin{align} S_r&\longmapsto \cor_{K/F}(\alpha_r^2),\tag{6.45}\\ C_{r,s,g}&\longmapsto \cor_{K/F}(\alpha_r\smile g\alpha_s),\tag{6.46}\\ E_{r,g}&\longmapsto \cor_{K_0/F}\Evens_{K/K_0}(\alpha_r), \qquad K_0=K^{\langle g\rangle}.\tag{6.47} \end{align}

Lean (square)Lean (free)Lean (involution)Lean (free, aux)Lean (involution, aux)

Proof.

We treat square orbits, free orbits, and involution-stabilized orbits in that order, recording at each stage whether the identity is literal at cochain level or only after passage to cohomology.

Choose lifts $\widetilde h\in G_F$ of $h\in H\text{,}$ and write $\bar\gamma$ for the image of $\gamma\in G_F$ in $H\text{.}$ With the left-regular convention used here, the normalized Shapiro map is

\begin{equation*} b_r(\gamma)_h =\operatorname{Sh}(\alpha_r)(\gamma)_h =\alpha_r\!\left(\widetilde h^{-1}\gamma \widetilde{\bar\gamma^{-1}h}\right). \end{equation*}

For scalar cocycles $\alpha,\beta$ on $G_K\text{,}$ direct substitution into the normalized bar corestriction gives

\begin{equation} \sum_{h\in H} \operatorname{Sh}(\alpha)(\gamma)_h\, \operatorname{Sh}(\beta)(\eta)_{\bar\gamma^{-1}h} =\operatorname{cor}_{K/F}(\alpha\smile\beta)(\gamma,\eta).\tag{6.48} \end{equation}

The factor set (6.17) therefore gives (6.45). For the free orbit, the coordinate appearing in the monomial is the right shift $\mathsf R_g\operatorname{Sh}(\alpha_s)\text{.}$ By (A.15) this differs from $\operatorname{Sh}(g\alpha_s)$ by the explicit coboundary $dq_{\alpha_s,g}\text{,}$ and (A.16) displays the resulting normalized boundary after taking the cup product and trace. Therefore (6.46) holds in cohomology; it is a literal cochain identity whenever the chosen section is right-$g$-equivariant.

Now let $J=\langle g\rangle\text{,}$ let $\mathcal R$ be right-coset representatives for $H/J\text{,}$ and put $K_0=K^J\text{.}$ For $c\in H$ and $u\in\mathcal R\text{,}$ use the notation

\begin{equation*} c^{-1}u=\pi_c(u)g^{\epsilon_c(u)} \end{equation*}

of Lemma 6.2. Let $\rho_0:G_{K_0}\twoheadrightarrow J$ be the quotient map. The graph pullback of the explicit half-orbit cocycle (6.15) is

\begin{align} &\sum_{u\in\mathcal R} b_r(\gamma)_u\, b_r(\eta)_{\bar\gamma^{-1}ug}\notag\\ &\qquad+ \sum_{u\in\mathcal R}\epsilon_{\bar\gamma}(u) b_r(\eta)_{\pi_{\bar\gamma}(u)} b_r(\eta)_{\pi_{\bar\gamma}(u)g}.\tag{6.49} \end{align}

For completeness, choose the lifts compatibly so that $\widetilde{ug}=\widetilde u\,\widetilde g$ for $u\in\mathcal R\text{,}$ and put

\begin{equation*} \ell_u(\gamma)= \widetilde u^{-1}\gamma\, \widetilde{\pi_{\bar\gamma}(u)} \in G_{K_0}. \end{equation*}

Its image in $J$ is $g^{\epsilon_{\bar\gamma}(u)}\text{.}$ In these conventions the normalized corestriction of a $2$-cochain $\nu$ on $G_{K_0}$ is

\begin{equation} (\operatorname{cor}_{K_0/F}\nu)(\gamma,\eta) = \sum_{u\in\mathcal R} \nu\!\left( \ell_u(\gamma), \ell_{\pi_{\bar\gamma}(u)}(\eta) \right).\tag{6.50} \end{equation}

Let $b_J=\operatorname{Sh}_{K/K_0}(\alpha_r)\text{.}$ Compatibility of the transversals gives

\begin{equation*} b_J(\ell_u(\gamma))_1=b_r(\gamma)_u \end{equation*}

and, for $j\in\{1,g\}\text{,}$

\begin{equation*} b_J\!\left(\ell_{\pi_{\bar\gamma}(u)}(\eta)\right)_j = b_r(\eta)_{\pi_{\bar\gamma}(u)j}. \end{equation*}

Substituting the explicit two-point cocycle (6.40) into (6.50) gives exactly (6.49): its first summand is the oriented factor-set term, and its second summand is the orientation-reversal correction. Consequently

\begin{equation} (b_r,\notnfar{rhoEpi}{\rho})^*\kappa_g^{\mathcal R} =\operatorname{cor}_{K_0/F} \left((\operatorname{Sh}_{K/K_0}\alpha_r,\rho_0)^*\kappa_J\right)\tag{6.51} \end{equation}

as normalized cochains. By Lemma 6.13, the inner graph class is $N_{G_K}^{G_{K_0},\mathrm{Ev}}(\alpha_r)\text{,}$ which is denoted $\Evens_{K/K_0}(\alpha_r)$ in field notation. Equation (6.47) follows. With the compatible nested transversals just chosen, (6.51) is literal equality of normalized cochains. For arbitrary choices, (A.32) and (A.34) give the two explicit normalized $1$-cochains whose boundaries account for the change. Hence the formulas evaluate the actual orbit classes used in $\kappa_{q_W}^0\text{,}$ including all $m_c$-terms, without an unspecified convention-dependent ambiguity.

Lemma 6.16. Deep-unit Evens norm.

Let $L/k$ be an unramified quadratic extension of finite dyadic local fields and put $e=v_k(2)\text{.}$ Then

\begin{equation} \Evens_{L/k}(\alpha)=0 \qquad(\alpha\in U_{e+1}(L)\subset L^\times/L^{\times2}).\tag{6.52} \end{equation}

Lean

Proof.

We translate the Evens norm into the degree-two Evens–Kahn identity, reduce the trace forms to Hilbert symbols, and then cancel the remaining terms by norm and Steinberg relations.

By Lemma 6.13, the Evens norm is represented by the explicit two-point graph cocycle (6.40). The index-two Evens–Kahn formula identifies this normalization with the degree-two term in the relative Stiefel–Whitney identity [Kah, Théorèmes 1--2], [Koz, Thm. 1.1]

\begin{equation} w(\Tr_{L/k}\langle a\rangle)= w(\Tr_{L/k}\langle1\rangle) \bigl(1+\cor_{L/k}[a]+\Evens_{L/k}([a])\bigr).\tag{6.53} \end{equation}

Take a class in $U_{e+1}(L)$ and choose from the outset a representative

\begin{equation*} a=1+2b,\qquad b\in\mathfrak p_L. \end{equation*}

Write $L=k(\sqrt d)\text{,}$ where $d$ is the unramified quadratic square class, and write

\begin{equation*} a=u+v\sqrt d,\qquad n=N_{L/k}(a)=u^2-dv^2. \end{equation*}

\begin{equation*} t=\Tr_{L/k}(b),\qquad m=N_{L/k}(b), \end{equation*}

then

\begin{equation} u=1+t\in\mathcal O_k^\times, \qquad n=1+2t+4m.\tag{6.54} \end{equation}

In particular the divisions by $u$ below are legitimate. The trace forms diagonalize as

\begin{equation*} \Tr_{L/k}\langle a\rangle\simeq \left\langle2u,\frac{2dn}{u}\right\rangle, \qquad \Tr_{L/k}\langle1\rangle\simeq\langle2,2d\rangle. \end{equation*}

Taking degree two in (6.53) first gives

\begin{equation*} \Evens_{L/k}([a])=(2u,2dn/u)+(2,2d)+(d,n). \end{equation*}

We now keep every Hilbert-symbol cancellation visible. Bilinearity and $(x,x)=(x,-1)$ give

\begin{align*} (2u,2dn/u) &=(2u,2)+(2u,d)+(2u,n)+(2u,u),\\ (2u,2)+(2u,u)&=(2u,2u)=(2u,-1),\\ (2,2d)&=(2,-1)+(2,d). \end{align*}

Consequently

\begin{align*} \Evens_{L/k}([a]) &= (u,-1)+(u,d)+(2u,n)+(d,n)\\ &= (u,-d)+(2du,n). \end{align*}

Thus

\begin{equation} \Evens_{L/k}([a])=(u,-d)+(2du,n).\tag{6.55} \end{equation}

In more detail: the Kummer coordinates are part of the proof, not additional arithmetic hypotheses. Since the residue characteristic is not the field characteristic, an unramified quadratic extension has a presentation $L=k(\delta)$ with $\delta^2=d\in k^\times\text{,}$ and every deep unit can be written uniquely as $a=u+v\delta\text{.}$ The index-two condition ensures $\delta\notin k\text{;}$ together with $a\ne0\text{,}$ this shows that $n=u^2-dv^2$ is nonzero and that the divisions in the displayed diagonalization are legitimate.

Because $L/k$ is unramified and $b\in\mathfrak p_L\text{,}$ one has $v_k(m)\ge2\text{.}$ From (6.54),

\begin{equation} \frac{n}{2u-1} =1+\frac{4m}{2u-1}\in U_{2e+2}(k).\tag{6.56} \end{equation}

Here $2u-1$ is a unit and $v_k(4m)\ge2e+2>2v_k(2)\text{.}$ Hensel’s lemma, applied to $X^2-n/(2u-1)$ at $X=1\text{,}$ therefore gives $U_{2e+2}(k)\subset k^{\times2}\text{.}$ Thus $n$ and $2u-1$ have the same square class.

Unramifiedness is used here in two distinct ways. Equality of value groups gives the depth estimate that makes $n/(2u-1)$ a square by Hensel’s lemma. In the remaining line, surjectivity of the norm on units kills the symbols involving $d\text{;}$ the Steinberg relation then reduces the answer to two copies of $(u,-1)\text{,}$ which cancel in characteristic two.

Every unit of $k$ is a norm from its unramified quadratic extension, and $n=N_{L/k}(a)\text{.}$ Hence

\begin{equation*} (u,d)=0,\qquad(d,n)=0. \end{equation*}

Formula (6.55) becomes

\begin{equation*} (u,-1)+(2u,2u-1). \end{equation*}

The Steinberg relation gives $(2u,2u-1)=(2u,-1)\text{.}$ Finally $(2,-1)=0\text{:}$ if $i\in k\text{,}$ then $-1$ is a square; otherwise $2=N_{k(i)/k}(1+i)\text{.}$ Consequently $(2u,-1)=(u,-1)\text{,}$ and the two terms cancel.

Lemma 6.17. The deep half is totally singular.

Assume $\notnfar{Vhead}{V}$ is ramified and put

\begin{equation*} X_+=\Hom_{H_V}(\notnfar{Vhead}{V}^\vee,U_{e+1})\subset H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V}). \end{equation*}

Then

\begin{equation*} \dim X_+=\frac12\dim H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V}), \qquad Q^0_{\mathrm{loc}}|_{X_+}=0. \end{equation*}

Lean (dim, ext+duality)Lean (dim, duality)Lean (dim, residue lift)Lean (dimension)Lean (vanishing)

Proof.

We first count the contributing unit-filtration layers and then prove vanishing on the deep half by checking square, free, and involution orbits.

By Hochschild–Serre and Kummer theory,

\begin{equation*} H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V})\cong\Hom_{H_V}(\notnfar{Vhead}{V}^\vee,M_K), \end{equation*}

and $\Hom_{H_V}(\notnfar{Vhead}{V}^\vee,-)$ is exact on the unit filtration. Hilbert duality pairs the depth-$j$ and depth-$(2e-j)$ graded pieces, and self-duality of $\notnfar{Vhead}{V}$ gives equal $\notnfar{Vhead}{V}$-isotypic multiplicities in each pair. The only unpaired depth is $j=e\text{;}$ by Lemma 6.10 it has trivial inertia and contains no ramified copy of $\notnfar{Vhead}{V}\text{.}$ Hence exactly one member of every contributing pair lies above $e\text{,}$ proving the dimension statement.

In more detail: the dimension assertion is proved in the formalization as the cardinal identity $|X_+|^2=|H^1(\Qtwo,V)|\text{.}$ The regular-module embedding is constructed for both $V$ and $V^\vee\text{;}$ inverse Shapiro maps extend families from the regular module, while local duality identifies the deep part for one module with the quotient part for the other. A residue-trivial lift of tame inertia is what makes this comparison respect the middle unit layer. Since that layer is inertia-trivial, it contains no copy of the ramified simple module.

By Lemma 6.14, compute the base connecting map after an $H_V$-split embedding into a regular module, and apply Lemma 6.15 to the complete invariant monomial expansion of $q\circ p\text{.}$ For a class in $X_+$ every scalar coordinate $\alpha_r$ lies in $U_{e+1}(K)\text{.}$ Free orbits vanish by (6.36). Square orbits vanish because $\alpha_r^2=\alpha_r\smile(-1)$ and $-1\in U_e\text{.}$ For an involution orbit, tame inertia has odd order, so $K/K_0$ in (6.47) is unramified quadratic. The depth parameter is unchanged in an unramified subextension, and Lemma 6.16 makes the Evens norm vanish before corestriction. Every orbit contribution is therefore zero.

For the involution orbits, one needs slightly more than the abstract fact that the relevant subgroup has index two. The tame reciprocity orientation identifies the corresponding quadratic extension as unramified; only then does the deep-unit Evens norm vanish by Lemma 6.16. Thus the square, free, and involution orbit calculations use respectively the depth pairing, ordinary cup-product vanishing, and the oriented unramified norm calculation.

Subsection 6.4 Comparison of the Gauss signs

Proposition 6.18. Dyadic base determinant theorem.

If $d=\dim \notnfar{Vhead}{V}\text{,}$ then

\begin{equation} \#(Q^0_{\mathrm{loc}})^{-1}(0)= \begin{cases} 2^{d-1}-2^{d/2-1},&\notnfar{Vhead}{V}\text{ unramified},\\ 2^{d-1}+2^{d/2-1},&\notnfar{Vhead}{V}\text{ ramified}. \end{cases}\tag{6.57} \end{equation}

Equivalently, the base local Gauss sum is negative in the unramified case and positive in the ramified case. The candidate base form $Q_A^0$ has the same Gauss sum by Proposition 6.9.

Lean (vanishing core)Lean (ramified)Lean (unramified)Lean

Proof.

The ramified and unramified cases are treated separately: the first follows from the half-dimensional totally singular subspace, while the second is determined by the invariant Hermitian-line classification.

In the ramified case, local Euler–Poincaré and Tate duality give $\dim H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V})=d\text{.}$ By Lemma 6.17, the nonsingular base local quadratic space contains a totally singular subspace of dimension $d/2\text{,}$ so it is hyperbolic.

In more detail: the two conclusions of Lemma 6.17 are used as a Lagrangian criterion. The local determinant form is first shown to be nonsingular, and simplicity together with $|V|=2^{2m}$ supplies the exponent-two structure. A totally singular subgroup whose square has the cardinality of the whole quadratic space is maximal totally singular; hence the Arf invariant is zero, and the plus-type zero count follows without choosing a symplectic basis.

In the unramified case, the faithful image is cyclic of odd order. Put $F_V=\End_{H_V}(\notnfar{Vhead}{V})\text{.}$ Self-duality equips $F_V\cong\F_{2^{2m}}$ with the involution $x\mapsto x^*=x^{2^m}$ and fixed field $F_{V,0}\cong\F_{2^m}\text{.}$ Inflation–restriction and the local Euler characteristic identify $H^1(\notn{Qtwo}{\Qtwo},\notnfar{Vhead}{V})$ with a one-dimensional $F_V$-space. The norm-one group

\begin{equation*} \mathcal U=\{u\in F_V^\times:uu^*=1\} \end{equation*}

preserves the polar form of $q\text{.}$ For $u\in\mathcal U\text{,}$ the function $x\mapsto q(ux)+q(x)$ is linear and $H_V$-invariant. Since $(\notnfar{Vhead}{V}^\vee)^{H_V}=0\text{,}$ it vanishes; hence every $u\in\mathcal U$ is an $H_V$-equivariant $q$-isometry. By (6.21), $(u\rtimes1)^*[\notnfar{kappaq0}{\kappa_q^0}]=[\notnfar{kappaq0}{\kappa_q^0}]\text{.}$ Functoriality of graph pullback therefore gives

\begin{equation*} Q^0_{\mathrm{loc}}(u_*x)=Q^0_{\mathrm{loc}}(x) \qquad(u\in\mathcal U). \end{equation*}

Thus Lemma 6.7 applies to the base local form, and it is

\begin{equation*} Q(x)=\Tr_{F_{V,0}/\F_2}(a xx^*), \qquad a\in F_{V,0}^\times. \end{equation*}

The norm map $F_V^\times\to F_{V,0}^\times$ has fibres of size $2^m+1\text{.}$ Among the nonzero elements of $F_{V,0}\text{,}$ exactly $2^{m-1}-1$ have trace $0\text{.}$ Hence

\begin{equation*} \#Q^{-1}(0)=1+(2^m+1)(2^{m-1}-1) =2^{2m-1}-2^{m-1}, \end{equation*}

which is the minus-type number in (6.57).

Here one must also justify that the norm-one scalars act by isometries of the local form. The formal proof does not treat this as a literal equality of the chosen factor sets: naturality changes the graph cocycle by an explicit normalized coboundary, which is enough to preserve its cohomology class and therefore $Q^0_{\mathrm{loc}}\text{.}$ After this descent, $H^1(\Qtwo,V)$ is a one-dimensional space over the Schur field $F_V\text{,}$ and the Hermitian trace model applies.

Thus the candidate and local base determinant problems have the same Gauss sum. For an arbitrary determinant pushout class, the edge term $\gamma_\kappa$ and the inflated scalar term $\delta_\kappa$ remain in the obstruction formula; their behavior under transgression and shear is described in Subsection 6.5.

Corollary 6.19. Complete source interface.

Call a source a topologically finitely generated profinite group $\notnfar{GammaSrc}{\Gamma}\text{,}$ marked by four generators $\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1}\text{,}$ whose compatible tame and maximal pro-$2$ quotient maps assemble as in (4.2) into an epimorphism $\notnfar{bGamma}{b_\Gamma}:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow\notnfar{bdry}{\bdry}\text{.}$ Say that a source $\notnfar{GammaSrc}{\Gamma}$ meets the source interface if, for every finite lower quotient, every lower exact-image map, and every elementary characteristic-$2$ coefficient module, $\notnfar{GammaSrc}{\Gamma}$ and $\notnfar{GQ}{\GQ}$ have:

the same obstruction-space dimension and the same number of lifts when the obstruction vanishes;
natural perfect pairings for all short exact coefficient sequences, with adjoint connecting homomorphisms;
the same scalar cup–Bockstein form for the trivial module;
for every nonzero invariant nonsingular quadratic form on a simple self-dual head, the same base determinant Gauss sum for the class $\notnfar{kappaq0}{\kappa_q^0}\text{.}$

Then $\notnfar{GA}{\GA}$ meets the source interface.

Together with the marked maximal pro-$2$ identification in Proposition 3.10, these are the source-side inputs supplied to the target induction, and they are the only ones: after Section 5 the argument never differentiates the wild word again, so it applies verbatim to any source that meets the interface and carries the tame and pro-$2$ data of Section 3. That is exactly what a further presentation has to supply, and Appendix C supplies it for a second one, the square-commutator presentation; the statement above is used there in this abstract form, at a source other than $\notnfar{GA}{\GA}\text{.}$ This corollary is not by itself a determinacy statement: equality of base Gauss sums is used only after Fourier expansion. For a general pushout class

\begin{equation*} \kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_{\gamma_\kappa}+ \operatorname{inf}\delta_\kappa, \end{equation*}

the terms $\gamma_\kappa$ and $\delta_\kappa$ enter only through the scalar classes obtained by completing the square in Proposition 8.8. The corresponding signed sums are expressed as exact-image counts in central double covers of the lower target.

Lean (the interface)Lean (Γ_A meets it)Lean (induction over a source)Lean

Proof.

Items (i)–(iii) are Proposition 5.15, Proposition 5.16, Corollary 5.17 and (3.18); item (iv) is Proposition 6.9, Proposition 6.18. The later operations are restrictions, pushouts, exact-image partitions, finite Fourier transforms, and central-cover constructions internal to the finite target. The determinacy of those operations is isolated in Theorem 8.17 and then executed in Section 9; the central-cover identity (8.22) supplies the remaining affine information.

Subsection 6.5 Affine terms: transgression and shear

Lemma 6.20. No square-only invariant.

Let $\notnfar{Vhead}{V}$ be a nontrivial simple $\F_2[\notn{Cfin}{C}]$-module. Every nonzero element of $H^0(\notn{Cfin}{C},\operatorname{Sym}^2\notnfar{Vhead}{V}^\vee)$ has nonzero, hence nonsingular, polar form.

Lean

Proof.

The zero-polar submodule is the Frobenius-square copy of $\notnfar{Vhead}{V}^\vee\text{,}$ whose fixed subspace is zero. The radical of a nonzero invariant polar form is a submodule of the simple module $\notnfar{Vhead}{V}\text{.}$

Lemma 6.21. Determinant transgression relative to the fixed equivariant class.

Let $q$ be a nonsingular $\notn{Cfin}{C}$-invariant quadratic form on $\notnfar{Vhead}{V}\text{,}$ and assume that a zero-section-normalized equivariant class

\begin{equation*} \notnfar{kappaq0}{\kappa_q^0}\in H^2(\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C},\F_2) \end{equation*}

restricting to $q$ on $\notnfar{Vhead}{V}$ has been fixed. Let

\begin{equation*} 1\to \notnfar{Vhead}{V}\to G_\eta\to \notn{Cfin}{C}\to1 \end{equation*}

have extension class $\eta\in H^2(\notn{Cfin}{C},\notnfar{Vhead}{V})\text{.}$ Relative to the fixed equivariant lift $\notnfar{kappaq0}{\kappa_q^0}\text{,}$ the obstruction to extending the fibre class over $G_\eta$ is

\begin{equation} d_2(q)=\notnfar{bq}{b_q}^\flat{}_*\eta\in H^2(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee).\tag{6.58} \end{equation}

Consequently, if $q$ is the restriction of an actual class in $H^2(G_\eta,\F_2)\text{,}$ then $\eta=0$ and $G_\eta\cong \notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}$ over $\notn{Cfin}{C}\text{.}$

Lean Lean

Proof.

We choose sections for the extension, compare the induced automorphisms of the fibre extension, and identify their discrepancy with the transgressed class $\notnfar{bq}{b_q}^\flat{}_*(\eta)\text{.}$

Let

\begin{equation*} 1\longrightarrow\F_2\longrightarrow E_q\longrightarrow \notnfar{Vhead}{V}\longrightarrow1 \end{equation*}

be the central extension represented by $\notnfar{kappaq0}{\kappa_q^0}|_V\text{.}$ Central automorphisms of $E_q$ inducing the identity on $\notnfar{Vhead}{V}$ are canonically $\notnfar{Vhead}{V}^\vee\text{.}$ If $\widehat v$ is a lift of $v\in \notnfar{Vhead}{V}\text{,}$ inner conjugation by $\widehat v$ is the central automorphism

\begin{equation} w\longmapsto \notnfar{bq}{b_q}(v,w)=\notnfar{bq}{b_q}^\flat(v)(w).\tag{6.59} \end{equation}

Choose the zero-section-normalized representative of $\notnfar{kappaq0}{\kappa_q^0}\text{.}$ Its preimage of $\notn{Cfin}{C}$ is split, so conjugation by the chosen lifts of $c\in \notn{Cfin}{C}$ supplies coherent automorphisms

\begin{equation*} \alpha_c\in\Aut(E_q), \qquad \alpha_c\alpha_d=\alpha_{cd}, \end{equation*}

lifting the prescribed $\notn{Cfin}{C}$-action on $\notnfar{Vhead}{V}\text{.}$

Choose a normalized section $s:\notn{Cfin}{C}\to G_\eta$ and write

\begin{equation} s(c)s(d)=\eta(c,d)s(cd).\tag{6.60} \end{equation}

In a putative central extension of $G_\eta$ restricting to $E_q\text{,}$ lifts of the section elements would satisfy

\begin{equation*} \widehat s(c)\widehat s(d) =z(c,d)\widehat{\eta(c,d)}\widehat s(cd) \end{equation*}

for a scalar factor $z(c,d)\in\F_2\text{.}$ Compare the induced automorphisms of $E_q\text{.}$ The left side induces $\alpha_c\alpha_d=\alpha_{cd}\text{,}$ whereas the right side induces

\begin{equation*} \operatorname{Int}(\widehat{\eta(c,d)})\alpha_{cd}. \end{equation*}

By (6.59), their discrepancy is the $\notnfar{Vhead}{V}^\vee$-valued normalized $2$-cocycle

\begin{equation} (c,d)\longmapsto \notnfar{bq}{b_q}^\flat(\eta(c,d)).\tag{6.61} \end{equation}

Changing the section $s$ changes $\eta$ by a coboundary and changes (6.61) by the corresponding $\notnfar{Vhead}{V}^\vee$-coboundary. Changing the coherent lifts $\alpha_c$ by central automorphisms $\gamma(c)\in \notnfar{Vhead}{V}^\vee$ changes the discrepancy by $d\gamma\text{.}$ Hence the cohomology class of the obstruction is exactly $\notnfar{bq}{b_q}^\flat{}_*[\eta]\text{.}$ This is the $d_2^{0,2}$ differential after the intrinsic equivariance obstruction has been trivialized by the fixed split class $\notnfar{kappaq0}{\kappa_q^0}\text{,}$ proving (6.58).

If an actual class on $G_\eta$ restricts to $q\text{,}$ the obstruction vanishes. Since $q$ is nonsingular, $\notnfar{bq}{b_q}^\flat:\notnfar{Vhead}{V}\xrightarrow{\sim}\notnfar{Vhead}{V}^\vee$ is an isomorphism of $\notn{Cfin}{C}$-modules. Therefore $[\eta]=0\text{,}$ so the extension splits over $\notn{Cfin}{C}\text{.}$

For a split group $\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}\text{,}$ fix the base determinant class $\notnfar{kappaq0}{\kappa_q^0}$ of Lemma 6.3. After replacing a representative by a cohomologous normalized cocycle and changing the normalized section, every central class with restriction $q$ admits the following equality of normalized $2$-cochains:

\begin{equation} \kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_\gamma+ \operatorname{inf}_C^{\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}}\delta,\tag{6.62} \end{equation}

where $\gamma\in Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee)\text{,}$ $\delta\in Z^2(\notn{Cfin}{C},\F_2)\text{,}$ and $\Gamma_\gamma$ is given by (6.6).

Lemma 6.22. Marking-preserving shear of the difference data.

Let $a\in Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V})$ and define

\begin{equation*} s_a(v,c)=(v+a(c),c). \end{equation*}

If $\kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_\gamma+\operatorname{inf}\delta\text{,}$ then, in cohomology,

\begin{equation} s_a^*\kappa= \notnfar{kappaq0}{\kappa_q^0}+ \Gamma_{\gamma+\notnfar{bq}{b_q}^\flat a}+ \operatorname{inf}\bigl(\delta+ \Theta_q^0(a)+\gamma\smile a\bigr),\tag{6.63} \end{equation}

where

\begin{equation} \Theta_q^0(a)=(a,\id_C)^*\notnfar{kappaq0}{\kappa_q^0}\in Z^2(\notn{Cfin}{C},\F_2)\tag{6.64} \end{equation}

and

\begin{equation} (\gamma\smile a)(c,d)=\gamma(c)(c a(d)).\tag{6.65} \end{equation}

In particular, if $q\ne0\text{,}$ a unique cohomology class $a$ kills the edge $\gamma\text{,}$ and the scalar phase then becomes $\delta+\Theta_q^0(a)+\gamma\smile a\text{.}$

Lean

Proof.

The cocycle identity for $a$ is exactly the condition that $s_a$ be an automorphism over $\notn{Cfin}{C}\text{.}$ Substitute $v+a(c)$ and $w+a(d)$ into the normalized representative (6.62). The terms linear in $v$ or $w$ change the edge by the polar map $\notnfar{bq}{b_q}^\flat a\text{.}$ The term containing only the base class and the graph $a$ is $\Theta_q^0(a)\text{.}$ The old edge also contributes the scalar term $\gamma(c)(c a(d))\text{,}$ namely (6.65). All remaining changes are normalized coboundaries. Since $\notnfar{bq}{b_q}^\flat$ is an isomorphism, the edge can be killed uniquely in cohomology.

The shear fixes the quotient $\notn{Cfin}{C}\text{.}$ A marked subgroup containing $\notnfar{Vhead}{V}$ is the full inverse image of its image in $\notn{Cfin}{C}\text{,}$ so it is preserved. Scalar decorations kill the nontrivial simple module $\notnfar{Vhead}{V}\text{,}$ and are therefore preserved as well.

Section 7 A minimal non-scalar module layer in the wild kernel

Given a finite target whose marked $2$-kernel has a nontrivial composition factor, this section isolates the first such factor and the minimal normal subgroup that realizes it. The resulting quotient tower separates the elementary lifting stage from the final Frattini obstruction and shows that every nonzero scalar pushout induces a nonsingular quadratic form on the simple head.

Let $\notnfar{calY}{\mathcal Y}=(\notnfar{Ytarget}{Y},\notnfar{LY}{L_Y},\notnfar{piY}{\pi_Y},\notnfar{thetaY}{\theta_Y})$ be a boundary-framed target, so that $\notnfar{LY}{L_Y}=\ker(\notnfar{piY}{\pi_Y})$ is a finite $2$-group and $\notnfar{Htarget}{H}=\notnfar{Ytarget}{Y}/\notnfar{LY}{L_Y}\text{.}$ For a finite $2$-group $\notn{Kker}{K}\text{,}$ write

\begin{equation*} \notn{Frattini}{\Phi}(\notn{Kker}{K})=\notn{Kker}{K}^2[\notn{Kker}{K},\notn{Kker}{K}] \end{equation*}

for its Frattini subgroup. A chief factor is scalar when the tame quotient acts on it trivially. Thus, formally, a chief factor of $\notnfar{LY}{L_Y}$ is called scalar when it is the trivial $\F_2[\notnfar{Htarget}{H}]$-module.

Assume that $\notnfar{LY}{L_Y}$ has a non-scalar chief factor. Choose $\notnfar{Ytarget}{Y}$-normal subgroups

\begin{equation*} S\triangleleft P\le \notnfar{LY}{L_Y}, \qquad P/S\cong \notnfar{Vhead}{V}, \end{equation*}

where all chief factors below $S$ are scalar and $\notnfar{Vhead}{V}$ is the first non-scalar simple factor. Choose $\notn{Kker}{K}\triangleleft \notnfar{Ytarget}{Y}\text{,}$ $\notn{Kker}{K}\le P\text{,}$ minimal subject to $KS=P\text{,}$ and set

\begin{equation*} \notn{Rrad}{R}=\notn{Frattini}{\Phi}(\notn{Kker}{K}),\qquad \notnfar{Mmid}{M}=\notn{Kker}{K}/\notn{Rrad}{R},\qquad \notn{Tsub}{T}=(\notn{Kker}{K}\cap S)/\notn{Rrad}{R}, \end{equation*}

\begin{equation*} \notn{Bmid}{B}=\notnfar{Ytarget}{Y}/\notn{Rrad}{R},\qquad \notn{Cfin}{C}=\notnfar{Ytarget}{Y}/\notn{Kker}{K},\qquad \notnfar{Htarget}{H}=\notnfar{Ytarget}{Y}/\notnfar{LY}{L_Y},\qquad H_V=\im(\notn{Cfin}{C}\to\GL(\notnfar{Vhead}{V})). \end{equation*}

The quotient tower and the elementary module layer are

\begin{equation} \notnfar{Ytarget}{Y}\twoheadrightarrow \notn{Bmid}{B}=\notnfar{Ytarget}{Y}/\notn{Rrad}{R}\twoheadrightarrow \notn{Cfin}{C}=\notnfar{Ytarget}{Y}/\notn{Kker}{K} \twoheadrightarrow \notnfar{Htarget}{H}=\notnfar{Ytarget}{Y}/\notnfar{LY}{L_Y}, \qquad 0\longrightarrow \notn{Tsub}{T}\longrightarrow \notnfar{Mmid}{M}\longrightarrow \notnfar{Vhead}{V}\longrightarrow0.\tag{7.1} \end{equation}

Write $L_B=\notnfar{LY}{L_Y}/\notn{Rrad}{R}$ and $L_C=\notnfar{LY}{L_Y}/\notn{Kker}{K}\text{.}$ The four quotient levels used below are

Level	Role
$\notnfar{Htarget}{H}$	fixed tame quotient and boundary frame
$\notn{Cfin}{C}$	lower target for elementary $\notnfar{Mmid}{M}$-lifting, connecting maps, and scalar phase classes
$\notn{Bmid}{B}$	quotient on which the final $\notn{Rrad}{R}$-valued obstruction and its scalar pushouts are evaluated
$H_V$	faithful image used only for the pure quadratic-form calculation on the simple head $\notnfar{Vhead}{V}$

Lemma 7.1. Simple head.

There is an exact sequence of $\F_2[\notn{Cfin}{C}]$-modules

\begin{equation*} 0\to \notn{Tsub}{T}\to \notnfar{Mmid}{M}\to \notnfar{Vhead}{V}\to0, \end{equation*}

and

\begin{equation*} \notn{Tsub}{T}=\rad_{\F_2[\notn{Cfin}{C}]}\notnfar{Mmid}{M}, \qquad \notnfar{Mmid}{M}/\notn{Tsub}{T}\cong \notnfar{Vhead}{V}, \qquad (\notnfar{Mmid}{M}^\vee)^\notn{Cfin}{C}=0. \end{equation*}

Lean (head)Lean (radical)Lean (dual)Lean

Proof.

Since $\notn{Kker}{K}/(\notn{Kker}{K}\cap S)\cong \notnfar{Vhead}{V}$ is elementary abelian, $\notn{Rrad}{R}\le \notn{Kker}{K}\cap S\text{.}$ The displayed sequence follows. Since the quotient is simple, $\rad \notnfar{Mmid}{M}\le \notn{Tsub}{T}\text{.}$ If a maximal submodule $X<\notnfar{Mmid}{M}$ failed to contain $\notn{Tsub}{T}\text{,}$ then $X+\notn{Tsub}{T}=\notnfar{Mmid}{M}\text{,}$ so $X$ would map onto $\notnfar{Vhead}{V}\text{.}$ Writing $K_X$ for the inverse image of $X$ in $\notn{Kker}{K}\text{,}$ the subgroup $K_X$ would be proper and $\notnfar{Ytarget}{Y}$-normal while still satisfying $K_XS=P\text{,}$ contradicting minimality. Thus every maximal submodule contains $\notn{Tsub}{T}\text{.}$ The assertion about dual invariants follows because a nonzero map $\notnfar{Mmid}{M}\to\F_2$ would give a trivial simple quotient, whereas the unique simple head is $\notnfar{Vhead}{V}\text{.}$

Lemma 7.2. Frattini–centralizer collapse.

The subgroup $\notn{Rrad}{R}$ is central elementary abelian in $\notn{Kker}{K}\text{,}$ and $\notn{Kker}{K}^4=1\text{.}$

Lean Lean

Proof.

We first prove that $\notn{Kker}{K}$ centralizes its Frattini subgroup, then show that the fourth-power homomorphism vanishes, and finally identify $\notn{Rrad}{R}$ as central and elementary abelian.

Let $H_2$ be the maximal $2$-quotient of the finite tame group $\notnfar{Htarget}{H}\text{,}$ let $N=\ker(\notnfar{Htarget}{H}\to H_2)\text{,}$ and choose an odd Hall lift $\widetilde N$ in the preimage of $N$ in $\notnfar{Ytarget}{Y}\text{.}$ Along a chief series for $S\text{,}$ the action of $\widetilde N$ on every factor is trivial; coprime action gives

\begin{equation*} [S,\widetilde N]=1. \end{equation*}

Since $\notn{Rrad}{R}\le \notn{Kker}{K}\cap S\text{,}$ $\widetilde N$ centralizes $\notn{Rrad}{R}\text{.}$

Put $\mathcal C_R=C_K(\notn{Rrad}{R})\text{.}$ It is characteristic in $\notn{Kker}{K}$ and hence normal in $\notnfar{Ytarget}{Y}\text{.}$ The three-subgroup lemma gives $[\notn{Kker}{K},\widetilde N]\le\mathcal C_R\text{.}$ The group $N$ acts nontrivially on $\notnfar{Vhead}{V}\text{:}$ otherwise the action would factor through the $2$-group $\notnfar{Htarget}{H}/N\text{,}$ whose only simple module in characteristic $2$ is trivial. Hence $[\notnfar{Vhead}{V},N]=\notnfar{Vhead}{V}\text{,}$ so

\begin{equation*} [\notn{Kker}{K},\widetilde N]S=P. \end{equation*}

Thus $\mathcal C_RS=P\text{,}$ and minimality of $\notn{Kker}{K}$ gives $\mathcal C_R=\notn{Kker}{K}\text{.}$ Therefore $\notn{Rrad}{R}\le Z(\notn{Kker}{K})\text{.}$

Since $\notn{Kker}{K}/\notn{Rrad}{R}$ is elementary abelian, squares lie in $\notn{Rrad}{R}$ and commutators have order at most $2\text{.}$ The fourth-power map

\begin{equation*} f:\notn{Kker}{K}\to \notn{Rrad}{R}, \qquad f(k)=k^4, \end{equation*}

is a $\notnfar{Ytarget}{Y}$-equivariant homomorphism. The subgroup $\widetilde N$ fixes $\notn{Rrad}{R}\text{,}$ so $f([k,n])=1$ for $k\in \notn{Kker}{K}\text{,}$ $n\in\widetilde N\text{.}$ The $\notnfar{Ytarget}{Y}$-normal closure of $[\notn{Kker}{K},\widetilde N]$ still maps onto $\notnfar{Vhead}{V}$ modulo $S\text{,}$ hence equals $\notn{Kker}{K}$ by minimality. Thus $f=1$ and $\notn{Kker}{K}^4=1\text{.}$ Finally

\begin{equation*} \notn{Rrad}{R}=\notn{Frattini}{\Phi}(\notn{Kker}{K})=\notn{Kker}{K}^2[\notn{Kker}{K},\notn{Kker}{K}] \end{equation*}

is central of exponent $2\text{.}$

Lemma 7.3. Decorations vanish on the block.

Every homomorphism from $\notnfar{Ytarget}{Y}$ to an elementary abelian $2$-group vanishes on $\notn{Kker}{K}\text{.}$

Lean Lean

Proof.

It kills $\notn{Rrad}{R}=\notn{Kker}{K}^2[\notn{Kker}{K},\notn{Kker}{K}]$ and therefore induces a $\notn{Cfin}{C}$-module map $\notnfar{Mmid}{M}\to \notnfar{Etarget}{E}\text{.}$ By Lemma 7.1, $\Hom_C(\notnfar{Mmid}{M},\F_2)=0\text{,}$ so this map is zero.

Subsection 7.1 The determinant pushout on the simple head

Put

\begin{equation*} \notn{calX}{\mathcal X_R}=(\notn{Rrad}{R}^\vee)^\notn{Bmid}{B}=(\notn{Rrad}{R}^\vee)^\notn{Cfin}{C}. \end{equation*}

For $0\ne\lambda\in \notn{calX}{\mathcal X_R}\text{,}$ the pushout $K_\lambda=\notn{Kker}{K}/\ker\lambda$ is a central extension

\begin{equation*} 1\to\F_2\to K_\lambda\to \notnfar{Mmid}{M}\to1. \end{equation*}

Let $\notn{qlambda}{q_\lambda}:\notnfar{Mmid}{M}\to\F_2$ be its square map and $\notn{blambda}{b_\lambda}$ its polar form.

Proposition 7.4. Simple-head determinant.

Keep the minimal-block data of (7.1): $\notnfar{Mmid}{M}=K/R$ with radical submodule $\notn{Tsub}{T}=(K\cap S)/R$ and simple head $\notnfar{Mmid}{M}/\notn{Tsub}{T}\cong \notnfar{Vhead}{V}\text{,}$ all acted on by the finite quotient $\notn{Cfin}{C}=Y/K$ arising from the chosen minimal block; by Lemma 5.12, the action on $\notnfar{Vhead}{V}$ factors through the finite tame quotient $H\text{.}$ For every $0\ne\lambda\in \notn{calX}{\mathcal X_R}\text{,}$

\begin{equation*} \notn{qlambda}{q_\lambda}|_{T}=0, \qquad \notn{blambda}{b_\lambda}(\notn{Tsub}{T},\notnfar{Mmid}{M})=0. \end{equation*}

Hence $\notn{qlambda}{q_\lambda}$ descends to a nonzero nonsingular $\notn{Cfin}{C}$-invariant quadratic form

\begin{equation*} \ol q_\lambda:\notnfar{Vhead}{V}\to\F_2. \end{equation*}

Lean (main)Lean (commutator vanishing)Lean

Proof.

We identify the polar radical, rule out a nonzero restriction to $\notn{Tsub}{T}\text{,}$ and then prove that the descended form on $\notnfar{Vhead}{V}$ is nonzero and nonsingular.

The adjoint $\notn{blambda}{b_\lambda}^\flat:\notnfar{Mmid}{M}\to \notnfar{Mmid}{M}^\vee$ is $\notn{Cfin}{C}$-linear. Since $\notn{Tsub}{T}$ has only trivial composition factors while $\soc(\notnfar{Mmid}{M}^\vee)\cong \notnfar{Vhead}{V}^\vee$ is nontrivial simple, $\Hom_C(\notn{Tsub}{T},\notnfar{Mmid}{M}^\vee)=0\text{.}$ Thus $\notn{blambda}{b_\lambda}(\notn{Tsub}{T},\notnfar{Mmid}{M})=0\text{.}$

The restriction $\notn{qlambda}{q_\lambda}|_{T}$ is therefore a $\notn{Cfin}{C}$-invariant linear functional. Suppose it is nonzero. Quotienting $\notn{Tsub}{T}$ by its kernel yields a $\notn{Cfin}{C}$-module extension

\begin{equation*} 0\to\F_2t\to\ol M\to \notnfar{Vhead}{V}\to0, \end{equation*}

with $t$ in the polar radical and $\notn{qlambda}{q_\lambda}(t)=1\text{.}$ The kernel of $\notn{Cfin}{C}\to H_V$ acts on $\ol M$ by possible shears

\begin{equation*} m\longmapsto m+f(\ol m)t. \end{equation*}

Quadratic invariance and $\notn{qlambda}{q_\lambda}(t)=1$ force every such $f$ to vanish, so the extension factors through $H_V\text{.}$ If $\notnfar{Vhead}{V}$ is ramified, its odd inertia subgroup has no fixed vectors and $H^*(H_V,\notnfar{Vhead}{V}^\vee)=0$ by Hochschild–Serre. If $\notnfar{Vhead}{V}$ is unramified, $H_V$ is cyclic of odd order and the same vanishing follows from Maschke’s theorem. Hence

\begin{equation*} H^1(H_V,\notnfar{Vhead}{V}^\vee)=0, \end{equation*}

so the extension splits. Projection to the scalar summand gives a nonzero $\notn{Cfin}{C}$-map $\notnfar{Mmid}{M}\to\F_2\text{,}$ contradicting $(\notnfar{Mmid}{M}^\vee)^\notn{Cfin}{C}=0\text{.}$ Therefore $\notn{qlambda}{q_\lambda}|_{T}=0\text{.}$

In more detail: the vanishing of $q_\lambda$ on $T$ is the one part of the argument that uses the boundary frame, rather than only the abstract minimal block. If $q_\lambda(t)=1\text{,}$ conjugation on the extension $0\to\F_2t\to\bar M\to V\to0$ could a priori contain shears by elements of $V^\vee\text{.}$ Quadratic invariance kills the shears coming from the kernel of the action, so the extension is controlled by the faithful acting image $H_V\text{.}$ It is important here to pass to $H_V\text{,}$ not to make a normal-odd-inertia assertion about the original group $C\text{.}$ The framed tame quotient makes $H_V$ either cyclic of odd order or an extension with odd inertia, and in both cases $H^1(H_V,V^\vee)=0\text{.}$ The resulting splitting yields an invariant functional on $M\text{,}$ contradicting the simple-head construction.

The descended form is nonzero. Otherwise $K_\lambda$ would be elementary abelian, whereas

\begin{equation*} \notn{Frattini}{\Phi}(K_\lambda)=\notn{Rrad}{R}/\ker\lambda\ne1. \end{equation*}

Its polar radical is a submodule of the simple module $\notnfar{Vhead}{V}\text{;}$ if the radical were all of $\notnfar{Vhead}{V}\text{,}$ the form would be invariant linear, impossible. Thus it is nonsingular.

Section 8 Central covers, affine fibres, and Fourier inversion

Using the target tower of Section 7 and the source comparison from Section 5, Section 6, this section converts the lifting count into scalar Fourier coefficients and exact-image counts for smaller targets. The result is the closed recursion of Theorem 8.17.

Retain the tower and module sequence (7.1).

The induction repeatedly separates maps according to the subgroup they generate. The notation below retains only maps whose image is exactly the lower target.

Definition 8.1. Exact-image lower maps.

For a boundary-framed lower target $\notn{Cfin}{C}\text{,}$ let $\notn{XGamma}{X_\Gamma}(\notn{Cfin}{C})$ denote the finite set of boundary-framed exact-image epimorphisms $\notnfar{rhoEpi}{\rho}:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow \notn{Cfin}{C}$ at the current lower stage. All sums over $\notn{XGamma}{X_\Gamma}(\notn{Cfin}{C})$ are over exact-image lower maps; proper-image contributions are subtracted separately by induction.

Lean

Before the transform can be compared across the two sources, the scalar characters entering it must agree; they do, with the same count on both sides.

Lemma 8.2. Common scalar character group.

For both sources $\notnfar{GammaSrc}{\Gamma}\in\{\notnfar{GA}{\GA},\notnfar{GQ}{\GQ}\}$ one has

\begin{equation*} |\Hom_{\mathrm{cont}}(\notnfar{GammaSrc}{\Gamma},\F_2)|=8. \end{equation*}

Moreover scalar twisting by any such character preserves the boundary-framed condition for the central double covers used below.

Lean (Γ_A)Lean (local)Lean

Proof.

For the candidate, every map to $\F_2$ kills $\notn{tau}{\tau}$ because the tame relation forces the tame inertia generator to have odd order in finite quotients; hence it factors through the maximal pro-$2$ quotient $\notnfar{Pi}{\Pi}\text{.}$ For the local source, the same assertion follows from the characteristic wild subgroup and the common marked maximal pro-$2$ quotient. The quotient $\notnfar{Pi}{\Pi}$ has three mod-$2$ generators, represented by $\notn{sigma}{\sigma},\notn{x0}{x_0},\notnfar{x1}{x_1}\text{,}$ and one Demushkin relator with no linear mod-$2$ part, so

\begin{equation*} H^1(\notnfar{Pi}{\Pi},\F_2)\cong\F_2^3. \end{equation*}

Thus the scalar character group has order $2^3=8$ for both sources. The central covers used in the induction have central kernel killed by the fixed boundary and decoration, so multiplying a lift by a scalar character changes only the central coordinate and leaves the boundary-framed data unchanged.

Lemma 8.3. Central-cover exact-image transform.

Let $\notnfar{calY}{\mathcal Y}=(\notnfar{Ytarget}{Y},\notnfar{LY}{L_Y},\notnfar{piY}{\pi_Y},\notnfar{thetaY}{\theta_Y})$ be boundary-framed and let

\begin{equation*} p:\widetilde Y\twoheadrightarrow \notnfar{Ytarget}{Y} \end{equation*}

be a central double cover. Give it the pulled-back structure

\begin{equation*} \widetilde L=p^{-1}(\notnfar{LY}{L_Y}), \qquad \widetilde\pi=\pi_Yp, \qquad \widetilde\theta=\theta_Yp, \end{equation*}

and assume the central kernel lies in $\ker(\widetilde\pi,\widetilde\theta)\text{.}$ Fix an exact-image subgroup $J\le \notnfar{Ytarget}{Y}$ projecting onto $\notnfar{Htarget}{H}\text{.}$ If $u_\Gamma^\beta(p,J)$ counts boundary-framed exact-image maps to $J$ whose pullback cover is split, then

\begin{equation} 8u_\Gamma^\beta(p,J) =\sum_{\substack{\widetilde J\le p^{-1}(J)\\p(\widetilde J)=J}} \notnfar{eGamma}{e_\Gamma}^\beta(\widetilde J,\widetilde J\cap\widetilde L, \widetilde\pi|_{\widetilde J}, \widetilde\theta|_{\widetilde J}).\tag{8.1} \end{equation}

Every term on the right is therefore an ordinary exact-image object in the same global boundary-framed category.

Lean Lean

Proof.

An unobstructed map $f:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow J$ has a lift, and the set of all lifts is a torsor under $\Hom(\notnfar{GammaSrc}{\Gamma},\F_2)\text{.}$ By Lemma 8.2, this group has order $8$ for both sources, and scalar twisting preserves the boundary-framed condition. Twisting a lift changes only the new central coordinate. Because that coordinate lies in the kernel of both the tame marking and the decoration, every twist still satisfies the identical boundary equation. The exact image of a lift is a subgroup $\widetilde J\le p^{-1}(J)$ mapping onto $J\text{,}$ and every such exact-image lift occurs in precisely one summand. This is a literal partition of the eight lifts, proving (8.1); it also proves exact-image and decoration compatibility without an equidistribution assumption.

The transform is inverted by a finite Fourier average.

Lemma 8.4. Fourier inversion.

Let $\Lambda$ be the character group of a finite $\F_2$-obstruction space and let $o:X\to \Lambda^\vee\text{.}$ If

\begin{equation*} m_\lambda=\#\{x\in X:\langle\lambda,o(x)\rangle=0\}, \end{equation*}

then

\begin{equation} \#\{x:o(x)=0\} =\frac1{|\Lambda|}\sum_{\lambda\in \Lambda}(2m_\lambda-|X|).\tag{8.2} \end{equation}

Lean Lean

Proof.

Use $1_{o=0}=|\Lambda|^{-1}\sum_{\lambda\in \Lambda}(-1)^{\langle\lambda,o\rangle}\text{.}$

Lemma 8.5. Constrained quadratic Gauss transform.

Let $W,E$ be finite $\F_2$-vector spaces, let $L:W\twoheadrightarrow E\text{,}$ and let $Q:W\to\F_2$ be nonsingular with polar form $b_Q\text{.}$ For $\kappa\in E$ and $\epsilon\in\F_2\text{,}$ put

\begin{equation*} N(\kappa,\epsilon)=\#\{x\in W:Lx=\kappa,\ Q(x)=\epsilon\}. \end{equation*}

For $\chi\in E^\vee\text{,}$ let $a_\chi$ be uniquely determined by

\begin{equation*} b_Q(a_\chi,x)=\chi(Lx). \end{equation*}

Then

\begin{equation} N(\kappa,\epsilon) =\frac1{2|E|}\left( |W|+G(Q)\sum_{\chi\in E^\vee} (-1)^{\chi(\kappa)+\epsilon+Q(a_\chi)} \right),\tag{8.3} \end{equation}

where $G(Q)=\sum_{x\in W}(-1)^{Q(x)}\text{.}$

Lean (aggregated)Lean (main)Lean

Proof.

Expand both conditions with additive characters. Surjectivity of $L$ kills the nontrivial characters in the nonquadratic sum. In the quadratic sum, complete the square:

\begin{equation*} Q(x)+b_Q(a_\chi,x)=Q(x+a_\chi)+Q(a_\chi). \end{equation*}

This gives (8.3).

We now record the two pieces that replace an unrestricted count on all of $H^1(\notnfar{Vhead}{V})\text{.}$

The preimage of $\notn{Tsub}{T}$ in a scalar double cover need not have a normal complement to the new central involution. The cohomology class measuring that failure, and equivalently the variation of the scalar obstruction under $\notn{Tsub}{T}$-twists, is called the radical edge.

Lemma 8.6. Radical edge, variation formula, and descent.

Let $0\to \notn{Tsub}{T}\to \notnfar{Mmid}{M}\to \notnfar{Vhead}{V}\to0$ be the simple-head sequence, and let

\begin{equation*} p:\widetilde B\twoheadrightarrow \notnfar{Bmid}{B} \end{equation*}

be a central double cover whose restriction to $\notnfar{Mmid}{M}$ has quadratic form with polar radical $\notn{Tsub}{T}$ and whose restriction to $\notn{Tsub}{T}$ is zero. The cover determines a canonical edge class

\begin{equation*} [\varepsilon]\in H^1(\notnfar{Bmid}{B}/\notn{Tsub}{T},\notn{Tsub}{T}^\vee). \end{equation*}

Its restriction to $\notnfar{Vhead}{V}$ is zero, so it is inflated from a unique class

\begin{equation*} [\bar\varepsilon]\in H^1(\notn{Cfin}{C},\notn{Tsub}{T}^\vee). \end{equation*}

For every lower exact-image epimorphism $\notnfar{rhoEpi}{\rho}:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow \notn{Cfin}{C}$ and every unrestricted $\notnfar{Mmid}{M}$-lift $f\text{,}$ twisting by $u\in \notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})$ changes the scalar obstruction by

\begin{equation} \operatorname{ob}(f_u)=\operatorname{ob}(f) +\bigl\langle [u],\notnfar{rhoEpi}{\rho}^*[\bar\varepsilon]\bigr\rangle_\Gamma.\tag{8.4} \end{equation}

If $[\bar\varepsilon]\ne0\text{,}$ the free $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})$-action partitions the unrestricted $\notnfar{Mmid}{M}$-lifts into orbits in each of which exactly one half satisfy the central relation. If $[\bar\varepsilon]=0\text{,}$ the cover descends to a central double cover of $\notnfar{Bmid}{B}/\notn{Tsub}{T}\text{;}$ conversely, descent forces the edge class to vanish.

Lean (torsor, Γ_A)Lean (torsor, local)Lean (Γ_A)Lean (local)Lean (data)Lean

Proof.

We construct the edge cocycle, descend it from $\notnfar{Bmid}{B}/\notn{Tsub}{T}$ to $\notn{Cfin}{C}\text{,}$ derive the exact variation formula, and then analyze the nonzero and zero edge cases.

Let $z$ generate $\ker p\text{.}$ Since both the square and commutator forms vanish on $\notn{Tsub}{T}\text{,}$ the preimage $\widetilde T=p^{-1}(\notn{Tsub}{T})$ is elementary abelian. Choose a linear complement $s(\notn{Tsub}{T})$ to $\langle z\rangle$ in $\widetilde T\text{.}$

For $\bar b\in \notnfar{Bmid}{B}/\notn{Tsub}{T}\text{,}$ choose a lift $\widehat b\in\widetilde B$ and define $\varepsilon(\bar b)\in \notn{Tsub}{T}^\vee$ by

\begin{equation} \widehat b\,s(t)\,\widehat b^{-1} =s({}^{\bar b}t)\, z^{\varepsilon(\bar b)({}^{\bar b}t)}.\tag{8.5} \end{equation}

The definition is independent of the chosen lift $\widehat b\text{.}$ Applying (8.5) successively to two elements shows

\begin{equation*} \varepsilon(\bar b_1\bar b_2) =\varepsilon(\bar b_1)+{}^{\bar b_1}\varepsilon(\bar b_2), \end{equation*}

so $\varepsilon$ is a $1$-cocycle. Replacing $s(t)$ by $s(t)z^{\ell(t)}$ changes it by the coboundary $d\ell\text{;}$ hence its class is canonical.

If $\bar b\in \notnfar{Vhead}{V}=\notnfar{Mmid}{M}/\notn{Tsub}{T}\text{,}$ the exponent in (8.5) is the commutator pairing between a lift of $\bar b$ and $t\text{.}$ This is the polar form of the restricted quadratic form, and it is zero because $\notn{Tsub}{T}$ is the polar radical. Hence $\res_V[\varepsilon]=0\text{.}$ The group $\notnfar{Vhead}{V}=\notnfar{Mmid}{M}/\notn{Tsub}{T}$ acts trivially on $\notn{Tsub}{T}\text{,}$ since $\notnfar{Mmid}{M}$ is elementary abelian. Inflation–restriction for $1\to \notnfar{Vhead}{V}\to \notnfar{Bmid}{B}/\notn{Tsub}{T}\to \notn{Cfin}{C}\to1$ therefore gives a unique $[\bar\varepsilon]\in H^1(\notn{Cfin}{C},\notn{Tsub}{T}^\vee)$ inflating to $[\varepsilon]\text{.}$ Choose a normalized cocycle representative $\bar\varepsilon$ of this class. The difference between $\varepsilon$ and its inflation is a coboundary $d\ell$ for some $\ell\in \notn{Tsub}{T}^\vee\text{;}$ changing the complement $s(\notn{Tsub}{T})$ by $\ell$ therefore lets us assume from now on that $\varepsilon=\operatorname{inf}(\bar\varepsilon)$ as cocycles.

Fix a lower exact-image epimorphism $\notnfar{rhoEpi}{\rho}:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow \notn{Cfin}{C}\text{,}$ an $\notnfar{Mmid}{M}$-lift $f:\notnfar{GammaSrc}{\Gamma}\to \notnfar{Bmid}{B}\text{,}$ and lifts $\widehat f(g)\in\widetilde B\text{.}$ For $u\in \notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})$ put

\begin{equation*} f_u(g)=u(g)f(g),\qquad \widehat f_u(g)=s(u(g))\widehat f(g). \end{equation*}

Let $\omega_f$ and $\omega_{f_u}$ be the scalar central defects. Using (8.5), the cocycle identity $u(gh)=u(g)+\notnfar{rhoEpi}{\rho}(g)u(h)\text{,}$ and the fact that $s(\notn{Tsub}{T})$ is elementary abelian gives the exact identity

\begin{equation} \omega_{f_u}(g,h)=\omega_f(g,h) +\bar\varepsilon(\notnfar{rhoEpi}{\rho}(g))\bigl(\notnfar{rhoEpi}{\rho}(g)u(h)\bigr).\tag{8.6} \end{equation}

For the local source, applying the invariant map identifies the second term with the ordinary cup pairing. For the candidate source, its evaluation on the two relation defects is the adjoint pairing supplied by the direct finite-word Fox–Heisenberg chain map of Proposition 5.10. Thus (8.6) yields (8.4) in either source theory.

If $[\bar\varepsilon]\ne0\text{,}$ then $\notnfar{rhoEpi}{\rho}^*[\bar\varepsilon]\ne0\text{.}$ Indeed, a pulled-back coboundary gives the same coboundary formula on every element of $\notn{Cfin}{C}$ because $\notnfar{rhoEpi}{\rho}$ is surjective. Perfect degree-one duality therefore makes

\begin{equation*} [u]\longmapsto \bigl\langle[u],\notnfar{rhoEpi}{\rho}^*[\bar\varepsilon]\bigr\rangle_\Gamma \end{equation*}

a nonzero linear functional on $\notn{HGr}{H^1_{\Gamma,\rho}}(\notn{Tsub}{T})\text{,}$ and hence on $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})\text{.}$ This group acts freely on each fibre of the map from $\notnfar{Mmid}{M}$-lifts to their quotient modulo $\notn{Tsub}{T}\text{.}$ Equation (8.4) is one affine equation with nonzero linear part on every such orbit, so exactly half of its elements satisfy the scalar central relation.

In more detail: the half-count is a statement about a finite affine torsor, not merely about cohomology classes. Coboundaries lie in the kernel of the variation functional, so nonvanishing is detected on $H^1\text{;}$ perfect degree-one duality makes the functional nonzero, and a nonzero functional on a finite $\F_2$-space has fibres of equal size. The formal proof checks this separately for the two sources: local Tate duality supplies the nonzero class for $\GQ\text{,}$ while the finite-word obstruction map supplies it for $\GA\text{.}$ In both cases the scalar obstruction group has two elements, so precisely one half of each twist orbit satisfies the central relation.

Finally, if $[\bar\varepsilon]=0\text{,}$ then $[\varepsilon]=0\text{.}$ Alter the complement $s(\notn{Tsub}{T})$ so that the cocycle in (8.5) is identically zero. Then $s(\notn{Tsub}{T})\triangleleft\widetilde B\text{,}$ and

\begin{equation*} \widetilde B/s(\notn{Tsub}{T})\twoheadrightarrow \notnfar{Bmid}{B}/\notn{Tsub}{T} \end{equation*}

is a central double cover. Conversely, a descended cover supplies a normal complement to the preimage of $\notn{Tsub}{T}\text{,}$ so the edge class vanishes.

Vanishing is first used to change the chosen linear complement of the central kernel so that the edge cocycle itself, not only its class, is zero. That complement is then normal, and quotienting by it produces the descended double cover of $B/T\text{.}$ This normalization is what makes the two alternatives in the later recursion genuinely exhaustive.

Assume henceforth that the edge vanishes. By Lemma 6.21, the quotient $\notnfar{Bmid}{B}/\notn{Tsub}{T}$ is identified with $\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}\text{.}$ For the scalar pushout class under consideration, choose the base determinant class $\notnfar{kappaq0}{\kappa_q^0}$ attached to the descended nonsingular form on $\notnfar{Vhead}{V}\text{.}$ After a normalized change of section, write the descended class by the following equality of normalized $2$-cocycles:

\begin{equation} \kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_{\gamma_\kappa}+ \operatorname{inf}_C^{\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}}\delta_\kappa, \qquad \gamma_\kappa\in Z^1(\notnfar{Cfin}{C},\notnfar{Vhead}{V}^\vee),\quad \delta_\kappa\in Z^2(\notnfar{Cfin}{C},\F_2).\tag{8.7} \end{equation}

The original group $\notnfar{Bmid}{B}$ remains a possibly nonsplit extension of $\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}$ by $\notn{Tsub}{T}\text{.}$

Lemma 8.7. Affine $\notn{Tsub}{T}$-lifting equation.

Choose the splitting $\notnfar{Bmid}{B}/\notn{Tsub}{T}\cong \notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}$ above and let

\begin{equation*} e\in H^2(\notnfar{Cfin}{C},\notn{Tsub}{T}) \end{equation*}

be the class obtained by pulling $\notnfar{Bmid}{B}\to \notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}$ back along the zero section $\notnfar{Cfin}{C}\to \notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}\text{.}$ For a lower exact-image map $\notnfar{rhoEpi}{\rho}:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow \notnfar{Cfin}{C}\text{,}$ a class

\begin{equation*} c\in \notn{HGr}{H^1_{\Gamma,\rho}}(\notnfar{Vhead}{V}) \end{equation*}

is the $\notnfar{Vhead}{V}$-coordinate of an actual $\notnfar{Mmid}{M}$-lift if and only if

\begin{equation} \partial_{\Gamma,\notnfar{rhoEpi}{\rho}}(c)=\notnfar{rhoEpi}{\rho}^*e \quad\text{in }\notn{HGr}{H^2_{\Gamma,\rho}}(\notn{Tsub}{T}).\tag{8.8} \end{equation}

For every such class, the number of raw lifts above it is

\begin{equation} \mu=|B^1_{\Gamma,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})|\,|\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})|,\tag{8.9} \end{equation}

which is independent of $\notnfar{rhoEpi}{\rho}$ and has the same value on the two sources.

Lean Lean

Proof.

Represent the $\notn{Tsub}{T}$-extension of $\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}$ by a normalized cocycle. Its pure $\notnfar{Vhead}{V}\times \notnfar{Vhead}{V}$ component is zero because $\notnfar{Mmid}{M}$ is elementary abelian; its mixed component is the module extension $0\to \notn{Tsub}{T}\to \notnfar{Mmid}{M}\to \notnfar{Vhead}{V}\to0\text{;}$ and its restriction to the zero section is $e\text{.}$ Pulling this cocycle back along the graph of a $\notnfar{Vhead}{V}$-cocycle gives exactly $\partial c+\notnfar{rhoEpi}{\rho}^*e\text{,}$ proving (8.8) in characteristic two.

Each cohomology class has $|B^1(\notnfar{Vhead}{V})|$ cocycle representatives, and each solvable $\notn{Tsub}{T}$-defect equation has $|Z^1(\notn{Tsub}{T})|$ solutions. For the nontrivial simple $\notnfar{Vhead}{V}\text{,}$ $|B^1(\notnfar{Vhead}{V})|=2^{\dim \notnfar{Vhead}{V}}\text{;}$ Proposition 5.15, Proposition 5.16 give the common value of $|Z^1(\notn{Tsub}{T})|\text{.}$

Put

\begin{equation*} \notnfar{calX}{\mathcal X_T}=(\notn{Tsub}{T}^\vee)^\notnfar{Cfin}{C}\cong \notn{HGr}{H^2_{\Gamma,\rho}}(\notn{Tsub}{T})^\vee. \end{equation*}

For $\chi\in \notnfar{calX}{\mathcal X_T}\text{,}$ let

\begin{equation*} \gamma_\chi=\partial_C^\vee(\chi)\in H^1(\notnfar{Cfin}{C},\notnfar{Vhead}{V}^\vee), \qquad a_{\chi,\kappa}=(\notnfar{bq}{b_q}^\flat)^{-1}(\gamma_\chi+\gamma_\kappa) \in H^1(\notnfar{Cfin}{C},\notnfar{Vhead}{V}). \end{equation*}

Let

\begin{equation} \Theta_q^0(a_{\chi,\kappa})=(a_{\chi,\kappa},\id_C)^*\notnfar{kappaq0}{\kappa_q^0} \in H^2(\notnfar{Cfin}{C},\F_2)\tag{8.10} \end{equation}

be the base graph class, and define the single total target-side phase

\begin{equation} \notn{DeltaChiKappa}{\Delta_{\chi,\kappa}} =\chi_*e+\delta_\kappa+\Theta_q^0(a_{\chi,\kappa}) \in H^2(\notnfar{Cfin}{C},\F_2).\tag{8.11} \end{equation}

When the shear of Lemma 6.22 is applied with $a=a_{\chi,\kappa}\text{,}$ its cochain-level formula (6.63) contains the additional term $(\gamma_\kappa+\gamma_\chi)\smile a_{\chi,\kappa}\text{,}$ which (8.11) omits. The omission is harmless in cohomology: for cocycle representatives related by $\notnfar{bq}{b_q}^\flat a_{\chi,\kappa} =\gamma_\kappa+\gamma_\chi\text{,}$ the convention (6.65), the cocycle identity for $a_{\chi,\kappa}\text{,}$ and the $\notnfar{Cfin}{C}$-invariance of $q$ give

\begin{equation*} \bigl((\gamma_\kappa+\gamma_\chi)\smile a_{\chi,\kappa}\bigr)(c,d) = \notnfar{bq}{b_q}\bigl(a_{\chi,\kappa}(c),\,c\,a_{\chi,\kappa}(d)\bigr) = d(q\circ a_{\chi,\kappa})(c,d), \end{equation*}

so the dropped term is the normalized coboundary of the $1$-cochain $q\circ a_{\chi,\kappa}$ and vanishes in $H^2(\notnfar{Cfin}{C},\F_2)\text{.}$ Thus the last two terms of (8.11) are the full scalar phase of the sheared class in cohomology, not a truncation of it.

Proposition 8.8. Completed-square phase identity.

For either source, every lower exact-image map $\notnfar{rhoEpi}{\rho}\text{,}$ every $b\in \notn{HGr}{H^1_{\Gamma,\rho}}(\notnfar{Vhead}{V})\text{,}$ and every $\chi\in \notnfar{calX}{\mathcal X_T}\text{,}$

\begin{equation} Q_{\kappa,\notnfar{GammaSrc}{\Gamma},\notnfar{rhoEpi}{\rho}}(b) +\langle\partial_{\Gamma,\notnfar{rhoEpi}{\rho}} b+\notnfar{rhoEpi}{\rho}^*e,\chi\rangle =Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}(b+\notnfar{rhoEpi}{\rho}^*a_{\chi,\kappa}) +\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}}).\tag{8.12} \end{equation}

Here $Q^0$ is the base determinant form and $Q_\kappa$ is the obstruction for the actual scalar pushout class (8.7).

Lean Lean

Proof.

By Corollary 5.17,

\begin{equation*} \langle\partial_{\Gamma,\notnfar{rhoEpi}{\rho}}b,\chi\rangle =\langle b,\notnfar{rhoEpi}{\rho}^*\gamma_\chi\rangle_\Gamma. \end{equation*}

Using (6.7), the left side of (8.12) is

\begin{equation*} Q^0_{\Gamma,\notnfar{rhoEpi}{\rho}}(b) +\langle b,\notnfar{rhoEpi}{\rho}^*(\gamma_\kappa+\gamma_\chi)\rangle_\Gamma +\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\delta_\kappa) +\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*(\chi_*e)). \end{equation*}

By the definition of $a_{\chi,\kappa}\text{,}$ the middle linear term is the polar pairing $\notnfar{bq}{b_q}(b,\notnfar{rhoEpi}{\rho}^*a_{\chi,\kappa})\text{.}$ Completing the square for the base quadratic form gives

\begin{equation*} Q^0(b)+\notnfar{bq}{b_q}(b,\notnfar{rhoEpi}{\rho}^*a_{\chi,\kappa}) =Q^0(b+\notnfar{rhoEpi}{\rho}^*a_{\chi,\kappa}) +\notn{iotaGamma}{\iota_\Gamma}\bigl(\notnfar{rhoEpi}{\rho}^*\Theta_q^0(a_{\chi,\kappa})\bigr). \end{equation*}

Adding the two inflated scalar terms proves (8.12). The class $\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}}$ is defined on the finite target $\notnfar{Cfin}{C}\text{;}$ no comparison map between the two source $H^1$-spaces is involved.

Subsection 8.1 The exact-image recursion

Fix a boundary-framed target $\notnfar{calY}{\mathcal Y}=(\notnfar{Ytarget}{Y},\notnfar{LY}{L_Y},\notnfar{piY}{\pi_Y},\notnfar{thetaY}{\theta_Y})$ whose marked kernel has a non-scalar simple head, and retain the notation (7.1). Abbreviate $\notnfar{eGamma}{e_\Gamma}(\notnfar{Ytarget}{Y})=\notnfar{eGamma}{e_\Gamma}^\beta(\notnfar{calY}{\mathcal Y})\text{,}$ with the same convention for inherited exact-image subgroups and central covers. The successive finite operations are

\begin{equation*} \begin{gathered} \notnfar{Cfin}{C}\ \rightsquigarrow\ \text{$\notnfar{Mmid}{M}$-lifts} \ \rightsquigarrow\ \text{proper-image subtraction}\\ \rightsquigarrow\ \text{scalar tests of the $\notnfar{Rcent}{R}$-obstruction} \ \rightsquigarrow\ \text{phase covers} \ \rightsquigarrow\ \notnfar{Ytarget}{Y}. \end{gathered} \end{equation*}

The elementary stage counts all $\notnfar{Mmid}{M}$-lifts over $\notnfar{Cfin}{C}$ and removes their proper images. The remaining $\notnfar{Rcent}{R}$-valued obstruction is separated by invariant scalar pushouts; in the zero-edge case, completing the resulting quadratic sums replaces their scalar phases by central covers before the final Fourier inversion.

Lemma 8.9. Elementary lifting and exact-image subtraction.

For every $\notnfar{rhoEpi}{\rho}\in \notnfar{XGamma}{X_\Gamma}(\notnfar{Cfin}{C})\text{,}$

\begin{equation*} \notn{HGr}{H^2_{\Gamma,\rho}}(\notnfar{Mmid}{M})=0, \qquad |\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notnfar{Mmid}{M})|=2^{2\dim \notnfar{Mmid}{M}}. \end{equation*}

Consequently

\begin{equation} \notnfar{eGamma}{e_\Gamma}(\notnfar{Bmid}{B})=2^{2\dim \notnfar{Mmid}{M}}\notnfar{eGamma}{e_\Gamma}(\notnfar{Cfin}{C}) -\sum_{\substack{J<\notnfar{Bmid}{B}\\J\twoheadrightarrow \notnfar{Cfin}{C}}}\notnfar{eGamma}{e_\Gamma}(J),\tag{8.13} \end{equation}

where each $J$ has the boundary-framed structure induced from $\notnfar{Bmid}{B}\text{.}$ For every proper $J<\notnfar{Bmid}{B}$ occurring in this sum,

\begin{equation} |J\cap L_B|=|J\cap \notnfar{Mmid}{M}|\,|L_C| \le \frac{|\notnfar{Mmid}{M}|}{2}|L_C| =\frac{|L_B|}{2}<|\notnfar{LY}{L_Y}|.\tag{8.14} \end{equation}

Lean (main)Lean (conditional)Lean (auxiliary)Lean

Proof.

The unique simple head of $\notnfar{Mmid}{M}$ is nontrivial, so $(\notnfar{Mmid}{M}^\vee)^\notnfar{Cfin}{C}=0$ by Lemma 7.1. Candidate deformation duality and local Tate duality therefore give the displayed vanishing and lift multiplicity. Every relation-compatible $\notnfar{Mmid}{M}$-lift has an exact image $J\le \notnfar{Bmid}{B}$ mapping onto $\notnfar{Cfin}{C}\text{;}$ partitioning by that exact image gives (8.13). If $J<\notnfar{Bmid}{B}\text{,}$ then $J\cap \notnfar{Mmid}{M}$ is a proper subgroup of the elementary abelian group $\notnfar{Mmid}{M}\text{,}$ and the exact sequence

\begin{equation*} 1\longrightarrow J\cap \notnfar{Mmid}{M}\longrightarrow J\cap L_B \longrightarrow L_C\longrightarrow1 \end{equation*}

gives (8.14).

Lemma 8.10. Scalar pushouts separate the complete obstruction.

Put

\begin{equation*} \notnfar{calX}{\mathcal X_R}=(\notn{Rrad}{R}^\vee)^\notnfar{Bmid}{B}=(\notn{Rrad}{R}^\vee)^\notnfar{Cfin}{C}. \end{equation*}

For either source, $\notnfar{calX}{\mathcal X_R}$ is canonically the dual of the $\notn{Rrad}{R}$-valued obstruction space. If $o$ is an $\notn{Rrad}{R}$-valued lifting obstruction, then

\begin{equation*} o=0\quad\Longleftrightarrow\quad \lambda_*o=0\text{ for every }\lambda\in\notnfar{calX}{\mathcal X_R}. \end{equation*}

For $0\ne\lambda\in\notnfar{calX}{\mathcal X_R}\text{,}$ the corresponding scalar pushout is the central double cover

\begin{equation*} p_\lambda:B_\lambda=\notnfar{Ytarget}{Y}/\ker\lambda\twoheadrightarrow \notnfar{Bmid}{B}. \end{equation*}

Lean

Proof.

By Lemma 7.2, $\notnfar{Rcent}{R}$ is elementary abelian and $\notn{Kker}{K}$ centralizes $\notnfar{Rcent}{R}\text{,}$ so the $\notnfar{Ytarget}{Y}$-action on $\notnfar{Rcent}{R}$ factors through $\notnfar{Cfin}{C}\text{.}$ The two duality theorems give

\begin{equation*} H^2_{A,\notnfar{rhoEpi}{\rho}}(\notnfar{Rcent}{R})^\vee\cong(\notnfar{Rcent}{R}^\vee)^\notnfar{Cfin}{C}, \qquad H^2_{F,\notnfar{rhoEpi}{\rho}}(\notnfar{Rcent}{R})^\vee\cong(\notnfar{Rcent}{R}^\vee)^\notnfar{Cfin}{C}. \end{equation*}

For $\lambda\in(\notnfar{Rcent}{R}^\vee)^\notnfar{Cfin}{C}\text{,}$ the subgroup $\ker\lambda$ is normal in $\notnfar{Ytarget}{Y}$ and $\notnfar{Cfin}{C}$ acts trivially on $\notnfar{Rcent}{R}/\ker\lambda\text{.}$ Thus $\notnfar{Ytarget}{Y}/\ker\lambda\to \notnfar{Ytarget}{Y}/\notnfar{Rcent}{R}$ is a central double cover, and its obstruction is the pairing with $\lambda\text{.}$ Perfectness proves the equivalence.

Separating the obstruction makes the final Frattini layer accessible to the same inversion.

Proposition 8.11. Fourier inversion for the final Frattini obstruction.

Assume $\notnfar{Rcent}{R}\ne1$ and put

\begin{equation*} z_R=|\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notnfar{Rcent}{R})| =2^{2\dim \notnfar{Rcent}{R}+\dim\notnfar{calX}{\mathcal X_R}}. \end{equation*}

For $\lambda\in\notnfar{calX}{\mathcal X_R}\text{,}$ let $m_{\Gamma,\lambda}(\notnfar{Bmid}{B})$ be the number of exact-image maps to $\notnfar{Bmid}{B}$ for which the scalar pushout $\lambda_*o_R$ vanishes. Then $m_{\Gamma,0}(\notnfar{Bmid}{B})=\notnfar{eGamma}{e_\Gamma}(\notnfar{Bmid}{B})$ and

\begin{equation} \boxed{ \notnfar{eGamma}{e_\Gamma}(\notnfar{Ytarget}{Y})=\frac{z_R}{|\notnfar{calX}{\mathcal X_R}|} \sum_{\lambda\in\notnfar{calX}{\mathcal X_R}} \bigl(2m_{\Gamma,\lambda}(\notnfar{Bmid}{B})-\notnfar{eGamma}{e_\Gamma}(\notnfar{Bmid}{B})\bigr).}\tag{8.15} \end{equation}

Lean

Proof.

For a fixed exact-image map to $\notnfar{Bmid}{B}\text{,}$ the set of $\notnfar{Rcent}{R}$-lifts, when nonempty, is a torsor of size $z_R\text{.}$ Every such lift is surjective onto $\notnfar{Ytarget}{Y}\text{:}$ if its image $J$ maps onto $\notnfar{Ytarget}{Y}/\notnfar{Rcent}{R}\text{,}$ then $(J\cap \notn{Kker}{K})\notnfar{Rcent}{R}=\notn{Kker}{K}\text{,}$ and $\notnfar{Rcent}{R}=\notn{Frattini}{\Phi}(\notn{Kker}{K})$ forces $J\cap \notn{Kker}{K}=\notn{Kker}{K}\text{,}$ hence $J=\notnfar{Ytarget}{Y}\text{.}$ Fourier inversion (8.2) on the full $\notnfar{Rcent}{R}$-valued obstruction, followed by the common lift multiplicity $z_R\text{,}$ gives (8.15).

Lemma 8.12. Proper-image subtraction for a scalar pushout.

Fix $0\ne\lambda\in\notnfar{calX}{\mathcal X_R}\text{.}$ Let $\notn{ZGl}{Z_{\Gamma,\lambda}}(\notnfar{Bmid}{B}/C)$ count all compatible lifts of exact-image maps to $C\text{,}$ without imposing generation in $\notnfar{Bmid}{B}\text{.}$ Then

\begin{equation} \boxed{ m_{\Gamma,\lambda}(\notnfar{Bmid}{B})=\notn{ZGl}{Z_{\Gamma,\lambda}}(\notnfar{Bmid}{B}/C) -\sum_{\substack{J<\notnfar{Bmid}{B}\\J\twoheadrightarrow C}} m_{\Gamma,\lambda}(J).}\tag{8.16} \end{equation}

For every proper $J$ in the sum,

\begin{equation} \boxed{ 8m_{\Gamma,\lambda}(J)= \sum_{\substack{\widetilde J\le p_\lambda^{-1}(J)\\ p_\lambda(\widetilde J)=J}} \notnfar{eGamma}{e_\Gamma}(\widetilde J).}\tag{8.17} \end{equation}

Lean

Proof.

Partition the relation-compatible maps to $\notnfar{Bmid}{B}$ by their exact image. This gives (8.16). For a fixed proper image $J\text{,}$ an unobstructed map has exactly eight scalar lifts; partitioning those lifts by their exact images in $p_\lambda^{-1}(J)$ is Lemma 8.3 and gives (8.17).

The recursion now splits according to the radical edge of the character.

Proposition 8.13. The nonzero radical-edge case.

If the radical edge of $p_\lambda$ is nonzero, then

\begin{equation} \boxed{ \notn{ZGl}{Z_{\Gamma,\lambda}}(\notnfar{Bmid}{B}/C) =2^{2\dim \notnfar{Mmid}{M}-1}\notnfar{eGamma}{e_\Gamma}(C).}\tag{8.18} \end{equation}

Lean

Proof.

By Lemma 8.6, the scalar obstruction varies by a nonzero affine functional on each free $\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})$-orbit of unrestricted $\notnfar{Mmid}{M}$-lifts. Exactly half of the $2^{2\dim \notnfar{Mmid}{M}}$ lifts over each $\notnfar{rhoEpi}{\rho}\in \notnfar{XGamma}{X_\Gamma}(\notnfar{Cfin}{C})$ satisfy the central relation, which is (8.18).

Proposition 8.14. The zero radical-edge case.

Assume that the radical edge vanishes. Write the descended scalar pushout class as

\begin{equation*} \kappa=\notnfar{kappaq0}{\kappa_q^0}+\Gamma_{\gamma_\kappa} +\operatorname{inf}\delta_\kappa \quad\text{on }\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}, \end{equation*}

and put

\begin{equation*} \notnfar{calX}{\mathcal X_T}=(\notn{Tsub}{T}^\vee)^\notnfar{Cfin}{C}, \qquad r=\dim\notnfar{calX}{\mathcal X_T}, \qquad d=\dim \notnfar{Vhead}{V}, \qquad \mu=|B^1_{\Gamma,\notnfar{rhoEpi}{\rho}}(\notnfar{Vhead}{V})|\,|\notn{Z1Gr}{Z^1_{\Gamma,\rho}}(\notn{Tsub}{T})|. \end{equation*}

For each lower map $\notnfar{rhoEpi}{\rho}\text{,}$

\begin{align} &\#\{c:\partial_{\Gamma,\notnfar{rhoEpi}{\rho}}c=\notnfar{rhoEpi}{\rho}^*e, \ Q_{\kappa,\notnfar{GammaSrc}{\Gamma},\notnfar{rhoEpi}{\rho}}(c)=0\}\notag\\ &\qquad=\frac1{2^{r+1}} \left(2^d+G(Q^0)\sum_{\chi\in\notnfar{calX}{\mathcal X_T}} (-1)^{\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}})}\right).\tag{8.19} \end{align}

Consequently

\begin{equation} \boxed{ \notn{ZGl}{Z_{\Gamma,\lambda}}(B/\notnfar{Cfin}{C}) =\frac{\mu}{2^{r+1}} \left(2^d \notnfar{eGamma}{e_\Gamma}(\notnfar{Cfin}{C}) +G(Q^0)\sum_{\chi\in\notnfar{calX}{\mathcal X_T}} s_\Gamma(\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}})\right),}\tag{8.20} \end{equation}

where

\begin{equation*} s_\Gamma(\zeta)= \sum_{\rho\in \notnfar{XGamma}{X_\Gamma}(\notnfar{Cfin}{C})} (-1)^{\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\zeta)}. \end{equation*}

Lean

Proof.

The vanishing edge and Lemma 6.21 identify $\notnfar{Bmid}{B}/\notn{Tsub}{T}$ with $\notnfar{Vhead}{V}\rtimes \notnfar{Cfin}{C}\text{.}$ By Lemma 8.7, the admissible $\notnfar{Vhead}{V}$-coordinates satisfy $\partial c=\notnfar{rhoEpi}{\rho}^*e$ and each occurs with multiplicity $\mu\text{.}$ The map $\partial_{\Gamma,\notnfar{rhoEpi}{\rho}}$ is surjective because $\notn{HGr}{H^2_{\Gamma,\rho}}(\notnfar{Mmid}{M})=0\text{.}$ Apply the constrained Gauss formula (8.3) and the completed-square identity (8.12); the common Gauss sum is supplied by Proposition 6.9, Proposition 6.18. This gives (8.19). Multiplication by $\mu$ and summation over $\notnfar{XGamma}{X_\Gamma}(\notnfar{Cfin}{C})$ give (8.20).

A scalar class $\zeta\in H^2(\notnfar{Cfin}{C},\F_2)$ is represented by a central double cover of $\notnfar{Cfin}{C}\text{.}$ We call it the phase cover because its splitting indicator realizes the sign $(-1)^{\notn{iotaGamma}{\iota_\Gamma}(\notnfar{rhoEpi}{\rho}^*\zeta)}$ appearing in the Fourier sum.

Lemma 8.15. Phase-cover transform.

For $\zeta\in H^2(\notnfar{Cfin}{C},\F_2)\text{,}$ let $p_\zeta:C_\zeta\twoheadrightarrow \notnfar{Cfin}{C}$ be the corresponding central double cover and let $n_{\Gamma,0}(\zeta)$ count the lower exact-image maps for which it is unobstructed. Then

\begin{equation} s_\Gamma(\zeta)=2n_{\Gamma,0}(\zeta)-\notnfar{eGamma}{e_\Gamma}(\notnfar{Cfin}{C})\tag{8.21} \end{equation}

and

\begin{equation} \boxed{ 8n_{\Gamma,0}(\zeta)= \sum_{\substack{\widetilde J\le C_\zeta\\ p_\zeta(\widetilde J)=\notnfar{Cfin}{C}}} \notnfar{eGamma}{e_\Gamma}(\widetilde J).}\tag{8.22} \end{equation}

Lean

Proof.

The first identity is the decomposition of the signed sum according to the two values of the scalar obstruction. The second is the exact eight-lift partition of Lemma 8.3, applied to $p_\zeta\text{.}$

Lemma 8.16. Strict decrease of recursive targets.

Every target occurring on the right sides of (8.13), (8.16), (8.17), and (8.22) has marked $2$-kernel strictly smaller than $|\notnfar{LY}{L_Y}|\text{.}$ More precisely,

\begin{equation*} |L_C|=\frac{|\notnfar{LY}{L_Y}|}{|\notn{Kker}{K}|}<|\notnfar{LY}{L_Y}|, \end{equation*}

\begin{equation} 2|J\cap L_B|\le |L_B|<|\notnfar{LY}{L_Y}|\tag{8.23} \end{equation}

for a proper scalar-cover stratum, and

\begin{equation} |L_{C_\zeta}|=2|L_C| =\frac{2|\notnfar{LY}{L_Y}|}{|\notn{Kker}{K}|}<|\notnfar{LY}{L_Y}|.\tag{8.24} \end{equation}

The proper-image bound in the elementary stage is (8.14).

Lean

Proof.

The first two inequalities follow from $\notnfar{Rcent}{R}<\notn{Kker}{K}$ and from the fact that a proper subgroup of the elementary abelian group $\notnfar{Mmid}{M}=\notn{Kker}{K}/\notnfar{Rcent}{R}$ has index at least two. A scalar double cover of a proper $J<\notnfar{Bmid}{B}$ has marked kernel $2|J\cap L_B|\text{,}$ giving (8.23). Since the non-scalar simple head has dimension at least two, $|\notn{Kker}{K}|\ge4\text{;}$ hence a phase cover has marked kernel as in (8.24).

Strict decrease of the marked $2$-kernel is what closes the recursion: every count reduces to boundary data and exact-image counts for smaller targets.

Theorem 8.17. Closed exact-image recursion.

Let $\notnfar{calY}{\mathcal Y}=(\notnfar{Ytarget}{Y},\notnfar{LY}{L_Y},\notnfar{piY}{\pi_Y},\notnfar{thetaY}{\theta_Y})$ be a boundary-framed marked target (Definition 4.1) and suppose $\notnfar{LY}{L_Y}$ has a non-scalar simple head, with the minimal-block tower (7.1) chosen as above. When $\notnfar{Rcent}{R}\ne1\text{,}$ the recursion uses, for each nonzero scalar functional $\lambda\in\notnfar{calX}{\mathcal X_R}\text{,}$ the descended nonsingular $C$-invariant form $\ol q_\lambda$ on $V$ of Proposition 7.4, its fixed equivariant base determinant class $\kappa^0_{\ol q_\lambda}$ of Lemma 6.3, and the radical data $T$ with its edge class $[\bar\varepsilon]$ of Lemma 8.6. The exact-image count $\notnfar{eGamma}{e_\Gamma}(\notnfar{Ytarget}{Y})$ is obtained from (8.13) and, when $\notnfar{Rcent}{R}\ne1\text{,}$ from (8.15)–(8.22). Every exact-image count on the right-hand sides belongs to a boundary-framed target with strictly smaller marked $2$-kernel. Hence the count is determined by the source interface of Corollary 6.19 and the exact-image counts for smaller targets.

Lean

Proof.

The elementary stage is Lemma 8.9. If $\notnfar{Rcent}{R}=1\text{,}$ then $\notnfar{Bmid}{B}=\notnfar{Ytarget}{Y}\text{.}$ Otherwise Lemma 8.10, Proposition 8.11 reduce the final $\notnfar{Rcent}{R}$-obstruction to its scalar pushouts. Proper images are removed by Lemma 8.12. The unrestricted scalar count is given by Proposition 8.13 when the radical edge is nonzero and by Proposition 8.14 when it vanishes. The signed scalar phases in the latter formula are exact-image counts by Lemma 8.15. Strict decrease is Lemma 8.16.

Section 9 Proof of the boundary-framed theorem

The terminal fibre-product description and the closed recursion of Theorem 8.17 cover all finite boundary-framed targets. We prove Theorem 4.2 by strong induction on $|\notnfar{LY}{L_Y}|\text{,}$ simultaneously for all such targets; every recursive target carries the same map to $\notnfar{Htarget}{H}\times \notnfar{Etarget}{E}\text{.}$

Lemma 9.1. Coprime-kernel subdirect products.

Let $A\twoheadrightarrow C$ and $B\twoheadrightarrow C$ be finite epimorphisms whose kernels have coprime orders. If a subgroup $J\le A\times_C B$ projects onto both $A$ and $B\text{,}$ then $J=A\times_C B\text{.}$

Lean

Proof.

The kernel of $J\to A$ is a subgroup of $\ker(\notnfar{Bmid}{B}\to \notnfar{Cfin}{C})\text{,}$ and the kernel of $J\to \notnfar{Bmid}{B}$ is a subgroup of $\ker(A\to \notnfar{Cfin}{C})\text{.}$ Comparing the two order formulas for $J$ shows that both kernels have their full possible orders, because the two ambient kernel orders are coprime. Thus $J$ contains both kernels and is the full fibre product.

Subsection 9.1 The terminal case: only trivial module factors

Lemma 9.2. Targets with only trivial module factors.

Suppose every chief factor of $\notnfar{LY}{L_Y}$ is a trivial $\notnfar{Htarget}{H}$-module. Let $H_2$ be the maximal $2$-quotient of $\notnfar{Htarget}{H}$ and let $N=\ker(\notnfar{Htarget}{H}\to H_2)\text{.}$ Then $N$ has odd order, has a unique normal lift $\widetilde N\triangleleft \notnfar{Ytarget}{Y}$ centralizing $\notnfar{LY}{L_Y}\text{,}$ and

\begin{equation} \notnfar{Ytarget}{Y}\cong \notnfar{Htarget}{H}\times_{H_2}Q, \qquad Q=\notnfar{Ytarget}{Y}/\widetilde N\text{ a finite $2$-group}.\tag{9.1} \end{equation}

Projection to $Q$ identifies exact-image subgroups of $\notnfar{Ytarget}{Y}$ projecting onto $\notnfar{Htarget}{H}$ with subgroups $Q'\le Q$ that map onto $H_2\text{.}$

Lean Lean

Proof.

We construct and characterize the odd-order complement, identify $\notnfar{Ytarget}{Y}$ as a fibre product, and then classify the exact-image subgroups.

The kernel $N$ has odd order by Lemma 3.1. Let $\notn{Pnorm}{P}$ be its inverse image in $\notnfar{Ytarget}{Y}\text{.}$ Schur–Zassenhaus gives a complement $\widetilde N\le \notn{Pnorm}{P}\text{.}$ Along a $\notnfar{Ytarget}{Y}$-chief series

\begin{equation*} 1=L_0\triangleleft L_1\triangleleft\cdots\triangleleft L_m=\notnfar{LY}{L_Y}, \end{equation*}

triviality of every factor gives $[L_i,\widetilde N]\le L_{i-1}\text{.}$ Hence $[\notnfar{LY}{L_Y},{}_m\widetilde N]=1\text{.}$ Coprime action gives

\begin{equation*} [\notnfar{LY}{L_Y},\widetilde N]=[\notnfar{LY}{L_Y},\widetilde N,\widetilde N] =\cdots=[\notnfar{LY}{L_Y},{}_m\widetilde N]=1. \end{equation*}

Thus $\notn{Pnorm}{P}=\notnfar{LY}{L_Y}\times\widetilde N\text{.}$ Schur–Zassenhaus conjugacy says that all complements are $\notnfar{LY}{L_Y}$-conjugate; since $\notnfar{LY}{L_Y}$ centralizes $\widetilde N\text{,}$ the complement is unique and therefore normal in $\notnfar{Ytarget}{Y}\text{.}$

In more detail: every element of the inverse image of $N$ has a commuting factorization $\tilde n\ell\text{,}$ with $\tilde n$ of odd order and $\ell$ of $2$-power order. Coprimality shows that such a product has odd order exactly when $\ell=1\text{.}$ Thus $\widetilde N$ is intrinsically the set of odd-order elements in this subgroup. This characterization is preserved by conjugation, giving normality and uniqueness at once.

The quotient $Q=\notnfar{Ytarget}{Y}/\widetilde N$ is a $2$-group. The two quotient maps define

\begin{equation*} \notnfar{Ytarget}{Y}\longrightarrow \notnfar{Htarget}{H}\times_{H_2}Q. \end{equation*}

Its kernel lies in both $\notnfar{LY}{L_Y}$ and $\widetilde N\text{,}$ hence is trivial, and the two groups have the same order. This proves (9.1).

The same parity separation explains the decoration: every homomorphism to the exponent-two group $E$ kills $\widetilde N\text{,}$ so the decoration descends to $Q\text{.}$ A subgroup of the resulting fibre product that surjects to both factors has kernels of coprime orders; the coprime subdirect-product lemma therefore forces it to be the whole fibre product.

If $J\le \notnfar{Ytarget}{Y}$ projects onto $\notnfar{Htarget}{H}$ and $Q'=\operatorname{pr}_Q(J)\text{,}$ then $Q'\twoheadrightarrow H_2\text{.}$ Inside $\notnfar{Htarget}{H}\times_{H_2}Q'\text{,}$ the subgroup $J$ projects onto both factors. The two kernels have odd and $2$-power orders, so Lemma 9.1 gives $J=\notnfar{Htarget}{H}\times_{H_2}Q'\text{.}$ Conversely, for every subgroup $Q'\le Q$ mapping onto $H_2\text{,}$ the fibre product $\notnfar{Htarget}{H}\times_{H_2}Q'$ is a subgroup of $\notnfar{Ytarget}{Y}$ projecting onto $\notnfar{Htarget}{H}\text{.}$

Let $q:Q\twoheadrightarrow H_2$ be the quotient map. The tame frame $\alpha:\notnfar{TA}{\TA}\twoheadrightarrow \notnfar{Htarget}{H}$ induces, through the full unramified marking of Proposition 3.14, a map

\begin{equation*} \alpha_2:\notnfar{Pi}{\Pi}\twoheadrightarrow H_2. \end{equation*}

Every boundary decoration kills the odd group $\widetilde N$ and factors through a map $\bar\theta:Q\to \notnfar{Etarget}{E}\text{.}$

For either source, the universal property of the pullback gives a bijection between maps $f:\notnfar{GammaSrc}{\Gamma}\to \notnfar{Ytarget}{Y}$ with fixed tame frame and maps $g:\notnfar{Pi}{\Pi}\to Q$ satisfying

\begin{equation} qg=\alpha_2, \qquad \bar\theta g=\bar\psi.\tag{9.2} \end{equation}

This uses the full $\mathbb{Z}_2$ marking: if $H_2=C_{2^m}\text{,}$ the first equation fixes the reduction of the unramified character modulo $2^m\text{,}$ not merely its parity.

The kernels of $\notnfar{Htarget}{H}\to H_2$ and $Q\to H_2$ have coprime orders, so Lemma 9.1 applies to the pullback $\notnfar{Htarget}{H}\times_{H_2}Q\text{:}$ the map $f$ is surjective exactly when $g$ is, and projection to $Q$ identifies every exact-image subgroup of $\notnfar{Ytarget}{Y}$ projecting onto $\notnfar{Htarget}{H}$ with a subgroup of $Q$ satisfying (9.2). Thus the terminal exact-image problems are genuinely identical for the two sources, by the fully marked isomorphism of their maximal pro-$2$ quotients.

Subsection 9.2 The inductive step

The cohomological count in this step rests on the local Euler–Poincaré characteristic [NSW, Theorem (7.3.1)]; see also [Mil, Theorem I.2.8].

Assume that $\notnfar{LY}{L_Y}$ has a non-scalar simple head. The target tower and module sequence are those of (7.1). By Lemma 8.9, Lemma 8.16, the elementary lifting stage expresses $\notnfar{eGamma}{e_\Gamma}(\notnfar{Bmid}{B})$ in terms of $\notnfar{eGamma}{e_\Gamma}(\notnfar{Cfin}{C})$ and exact-image counts with smaller marked kernels. If $\notnfar{Rcent}{R}=1\text{,}$ then $\notnfar{Bmid}{B}=\notnfar{Ytarget}{Y}$ and the induction closes at this stage.

Suppose $\notnfar{Rcent}{R}\ne1\text{.}$ The closed recursion of Theorem 8.17 expresses $\notnfar{eGamma}{e_\Gamma}(\notnfar{Ytarget}{Y})$ through the common lift multiplicities and Gauss signs of Corollary 6.19, together with exact-image counts for the smaller targets listed in Lemma 8.16. The induction hypothesis makes every such count equal for $\notnfar{GA}{\GA}$ and $\notnfar{GQ}{\GQ}\text{.}$ Thus

\begin{equation*} e_{\notnfar{GA}{\GA}}^\beta(\notnfar{Ytarget}{Y})=e_{\notnfar{GQ}{\GQ}}^\beta(\notnfar{Ytarget}{Y}), \end{equation*}

which completes the inductive step and proves Theorem 4.2.

Section 10 Passage to all finite quotients

The fixed-frame equality is now summed over the possible tame quotient maps of an arbitrary finite group. This gives equality of all finite epimorphism counts, to which the reconstruction lemma applies.

Lemma 10.1. Exhaustion by tame boundary frames.

Let $G$ be finite and put $L=O_2(G)\text{.}$ For either source $\notnfar{GammaSrc}{\Gamma}\text{,}$ every epimorphism $f:\notnfar{GammaSrc}{\Gamma}\twoheadrightarrow G$ determines a unique tame boundary frame

\begin{equation*} \notnfar{TA}{\TA}\twoheadrightarrow G/L. \end{equation*}

Conversely, with decoration $\notnfar{Etarget}{E}=0\text{,}$ a boundary-framed epimorphism to $G$ with a fixed tame frame is exactly an ordinary epimorphism to $G$ inducing that frame. Distinct tame frames give disjoint sets of epimorphisms.

Lean (main)Lean (count identity)Lean

Proof.

The characteristic marked wild subgroup maps into $O_2(G)=L\text{,}$ so every epimorphism has a well-defined quotient map to $G/L\text{.}$ The descended map $\notnfar{TA}{\TA}\to G/L$ is continuous: the projection $\Gamma\twoheadrightarrow\notnfar{TA}{\TA}$ is a continuous surjection from a compact profinite group onto a Hausdorff quotient, hence a topological quotient map, and the composite $\Gamma\to G/L$ is continuous. By Proposition 3.2, this quotient map is a tame frame. Conversely, when $\notnfar{Etarget}{E}=0$ there is no extra decoration, and the boundary-framed condition records only the induced tame quotient map and the marked $2$-kernel. Thus a boundary-framed epimorphism with target $G$ is precisely an ordinary epimorphism inducing the chosen frame. The induced quotient map $\notnfar{TA}{\TA}\to G/L$ is unique, so the fixed-frame sets are disjoint as the frame varies.

Let $G$ be finite and put $L=O_2(G)\text{.}$ By Lemma 10.1, the ordinary surjection set is the disjoint union of the fixed-frame surjection sets over all tame frames $\notnfar{TA}{\TA}\twoheadrightarrow G/L\text{.}$ Applying Theorem 4.2 to each frame with $\notnfar{Etarget}{E}=0\text{,}$ summing over the finite set of frames, and using Proposition 2.3, gives

\begin{equation} |\Sur(\notnfar{GA}{\GA},G)|=|\Sur(\notnfar{GQ}{\GQ},G)|\tag{10.1} \end{equation}

for every finite group $G\text{.}$

The candidate is topologically generated by four elements. The one-sided reconstruction lemma Lemma 2.5 applied to (10.1) gives $\notnfar{GA}{\GA}\cong\notnfar{GQ}{\GQ}\text{,}$ proving Theorem 1.2.

Section 11 Further questions

The presentation, the counting route behind it, and the formalization each suggest continuations. We record the questions we consider most tractable or most consequential.

A natural continuation of the formalization is to drive the census of external interfaces (nine at the time of writing; see Subsection 1.3) to zero and to upstream the reusable theory. The project has already discharged several earlier assumptions internally: the dyadic Hilbert-symbol formula, Kummer surjectivity, and the unit-filtration counts are now theorems of the repository. Each remaining interface (local Euler characteristic, local Tate duality, local reciprocity, the oriented Demushkin classification [Lab], and the rest of the list in Subsection 1.3) is a well-scoped target, so the paper induces a prioritized worklist for formalizing local class field theory at $p=2\text{.}$

Uniformity across dyadic fields. Extend Theorem 1.2 to every finite extension $K/\Qtwo$ without rebuilding the proof field by field. The boundary formalism and counting induction are already uniform, while the marked Demushkin group [Lab], its cyclotomic orientation, and the quadratic comparison vary through the standard local data $[K:\Qtwo]\text{,}$ the image of the cyclotomic character, and the filtered square-class group with its Hilbert pairing. The central question is whether the candidate relator and the quadratic Gauss-sign calculation can be formulated once over this package of invariants. A positive answer would give a uniform theorem for all dyadic fields; a negative answer should identify the additional invariant and reduce the problem to finitely many structural types, rather than one calculation per field. The seven quadratic extensions of $\Qtwo\text{,}$ which lie outside Diekert’s hypotheses [Die], are a useful stress test for such a uniform formulation; Schneider’s partial structural results [Sch] provide a second comparison point.
Constraints on Jannsen–Wingberg-type presentations. Classify the allowable wild relator in a dyadic analogue of the Jannsen–Wingberg presentation. Here Jannsen–Wingberg shape means the structural ansatz of [NSW, Theorem 7.5.14]: tame generators $\sigma,\tau$ with $\sigma\tau\sigma^{-1}=\tau^q\text{;}$ wild generators whose closed normal subgroup is pro-$p\text{;}$ and one further relation expressing $x_0^\sigma$ as a cyclotomic norm word in $x_0,\tau\text{,}$ a $p$-power, and commutators whose partners are constrained by the tame cyclotomic action. Determine, modulo boundary-preserving changes of wild generators, which $2$-adic analogues of this ansatz can present $G_{\Qtwo}\text{.}$ Low central quotients, the evaluated Fox Jacobian, and the quadratic Gauss sums should give necessary constraints on the relator. Constraints of this kind would also give criteria for evaluating the independently proposed candidate presentations mentioned in the introduction.
Deformation rings from the presentation. Compute universal deformation rings of residual representations $\bar\rho:G_{\Qtwo}\to\mathrm{GL}_d(k)$ directly from the presentation. Substitute universal framed matrices into the two relator words and complete at $\bar\rho\text{;}$ the evaluated Fox Jacobian is the linearization of these equations, matching Mazur’s description of the tangent space by $H^1(G_{\Qtwo},\operatorname{ad}\bar\rho)$ and of the relation ideal by an obstruction map into $H^2$ [Maz, Section 1.6, Proposition 2], while higher Fox terms retain the full power-series equations. Compare the resulting completed rings, and their quotients imposing Hodge conditions, with the formal completions of the Emerton–Gee stack and its crystalline or semistable substacks [EG, Section 1.1 and Theorem 1.2.4]. This would turn the local geometry of their moduli spaces at $G_{\Qtwo}$-representations into explicit word calculus: first recover known two-dimensional crystalline deformation rings as a consistency check, then compute new $p=2$ examples.
Explicit counting formulas. Extract from the Fourier-inversion stage an explicit algorithm, ideally a closed form, for $|\Sur(G_{\Qtwo},G)|$ for an arbitrary finite group $G\text{,}$ refined by inertia and higher ramification data. The quotient $|\Sur(G_{\Qtwo},G)|/|\operatorname{Aut}(G)|$ counts $G$-Galois extensions, whereas Monge’s formula [Mon, Theorem 1] counts isomorphism classes of extensions with prescribed ramification index $e$ and inertia degree $f\text{,}$ equivalently conjugacy classes of the corresponding open subgroups. Find an explicit passage between these counts using subgroup and normalizer data, and reconcile it with the Krasner–Serre mass-formula tradition [Ser78]; then refine the comparison by discriminant and upper ramification breaks. Tables for small $G\text{,}$ cross-linked to the LMFDB’s $2$-adic fields data and a machine-readable form of the presentation, would make both kinds of enumeration computable from the same source.
Ramification filtration and canonical wild generators. Give the wild generators $x_0,x_1$ a canonical Galois-theoretic characterization up to conjugacy, and locate the ramification filtration in upper numbering word-theoretically in the presentation. The presentation retains only the tame–wild dichotomy; recovering the filtration would connect it to the local anabelian results [Moc], in which the filtration is the essential extra datum.
The outer automorphism group. Compute $\operatorname{Out}(G_{\Qtwo})\text{,}$ or at least its action on the marked tame–pro-$2$ data, directly from the presentation. Automorphism groups of $2$-adic local Galois groups are studied in [Nis]; a finite presentation plus the Fox toolkit makes this a concrete word problem.
The counting route as a general technique. Establish the counting route as a general profinite technique. The pipeline (a candidate assembled from admissible finite quotients, the one-sided reconstruction lemma, a lifting-theory comparison, and Fourier induction) uses nothing specific to $\Qtwo$ before the quadratic stage. Test cases: a machine-checkable re-proof of the odd-$p$ theorem [JW], and maximal pro-$p$ Galois groups with restricted ramification, where Koch-type presentations supply candidates. Separately: is there an a priori bound on the induction, turning the count comparison into a finite decision procedure?

Appendix A Normalized cochains, Shapiro maps, corestriction, and the Evens norm

The formulas in Lemma 6.13, Lemma 6.15 depend on choices of left action, coset representatives, and normalized sections. This appendix fixes those conventions at cochain level and gives the explicit normalized boundaries produced by every change of section or transversal.

The general bar and transfer constructions are standard; compare [Bro, Chapter III and Chapter VII] and [Eve].

Subsection A.1 Normalized inhomogeneous cochains

All coefficient modules are left modules. If $G$ is profinite and $M$ is a finite discrete continuous left $G$-module, then $C^n(G,M)$ denotes the group of continuous normalized inhomogeneous $n$-cochains. Thus a cochain of positive degree vanishes whenever one of its arguments is $1\text{,}$ and $C^0(G,M)=M\text{.}$ In additive notation the differential is

\begin{align} (df)(g_1,\ldots,g_{n+1}) &=g_1f(g_2,\ldots,g_{n+1}) +\sum_{i=1}^{n}(-1)^i f(g_1,\ldots,g_ig_{i+1},\ldots,g_{n+1})\notag\\ &\qquad+(-1)^{n+1}f(g_1,\ldots,g_n),\tag{A.1} \end{align}

and, in degree zero,

\begin{equation} (dm)(g)=gm-m.\tag{A.2} \end{equation}

All coefficient groups in the normalization-sensitive calculations of Lemma 6.13, Lemma 6.15 are $\F_2$-vector spaces, so the minus signs in (A.1) disappear there.

Let $\mu:A\times B\to C$ be a $G$-equivariant bilinear pairing of left $G$-modules. Our cup-product convention is

\begin{equation} (f\smile g)(g_1,\ldots,g_{p+q}) =\mu\!\left( f(g_1,\ldots,g_p), (g_1\cdots g_p)g(g_{p+1},\ldots,g_{p+q}) \right).\tag{A.3} \end{equation}

In particular, for trivial scalar coefficients, $(\alpha\smile\beta)(n_1,n_2)=\alpha(n_1)\beta(n_2)\text{.}$

We use the following three levels of assertion.

“Equal as normalized cochains” means equality in $C^n(G,M)\text{.}$
“Equal up to the displayed normalized coboundary” means $f-g=dh$ for the particular normalized cochain $h$ that is written down.
“Equal in cohomology” means only $[f]=[g]$ in $H^n(G,M)\text{.}$

No cochain-level assertion in the main text is intended to pass silently from one of these levels to another.

Subsection A.2 Conjugation and semidirect products

The group-theoretic exponent convention remains

\begin{equation*} x^g=g^{-1}xg. \end{equation*}

It is distinct from the notation $gm$ for the given left action of $G$ on a module $M\text{.}$ We write the semidirect product as $M\rtimes G\text{,}$ with elements $(m,g)$ and multiplication

\begin{equation} (m,g)(n,h)=(m+gn,gh), \qquad (m,g)^{-1}=(-g^{-1}m,g^{-1}).\tag{A.4} \end{equation}

Consequently

\begin{equation*} (0,g)(m,1)(0,g)^{-1}=(gm,1), \qquad (m,1)^{(0,g)}=(g^{-1}m,1). \end{equation*}

Thus the manuscript’s right-conjugation notation acts on the normal module by $g^{-1}\text{,}$ whereas the semidirect-product law uses the specified left action by $g\text{.}$

Let $\rho:\Gamma\to G$ and $b:\Gamma\to M\text{.}$ The graph map

\begin{equation} \Phi_{b,\rho}:\Gamma\longrightarrow M\rtimes G, \qquad \gamma\longmapsto(b(\gamma),\rho(\gamma))\tag{A.5} \end{equation}

is a homomorphism if and only if

\begin{equation} b(\gamma\eta)=b(\gamma)+\rho(\gamma)b(\eta).\tag{A.6} \end{equation}

This is the graph-pullback convention used in (6.4), (6.40), and (6.51).

Subsection A.3 The left-regular coordinate convention

For a finite group $H$ we identify $\F_2[H]$ with the functions $x=(x_h)_{h\in H}$ and use the left-regular action

\begin{equation} (cx)_h=x_{c^{-1}h}.\tag{A.7} \end{equation}

Hence

\begin{equation*} (\bar\gamma x)_h=x_{\bar\gamma^{-1}h}. \end{equation*}

This is the source of the index $\bar\gamma^{-1}h$ in the Shapiro cocycle and in the cup/corestriction formula. Right multiplication of the coordinate index is a different operator:

\begin{equation} (\mathsf R_gx)_h=x_{hg}.\tag{A.8} \end{equation}

The operators $\mathsf R_g$ commute with the left-regular $H$-action. The monomial $x_ux_{ug}$ is therefore naturally indexed by cosets written $u\langle g\rangle\text{,}$ which explains the right multiplication by $g$ and the coset convention in Lemma 6.2.

Subsection A.4 The normalized Shapiro cochain map

Let

\begin{equation*} 1\longrightarrow N\longrightarrow G\xrightarrow{\;\pi\;}H\longrightarrow1 \end{equation*}

be exact with $H$ finite. Choose a normalized section $h\mapsto\widetilde h\text{,}$ with $\widetilde1=1\text{,}$ and write $\bar\gamma=\pi(\gamma)\text{.}$ Define the transition element

\begin{equation} \lambda_h(\gamma) =\widetilde h^{-1}\gamma \widetilde{\bar\gamma^{-1}h}\in N.\tag{A.9} \end{equation}

It is normalized and satisfies

\begin{equation} \lambda_h(\gamma\eta) =\lambda_h(\gamma)\lambda_{\bar\gamma^{-1}h}(\eta).\tag{A.10} \end{equation}

For a normalized scalar cochain $f\in C^n(N,\F_2)$ put

\begin{equation} \bigl(\operatorname{Sh}_{\widetilde H}^n f\bigr) (\gamma_1,\ldots,\gamma_n)_h =f\bigl( \lambda_{h_0}(\gamma_1),\ldots, \lambda_{h_{n-1}}(\gamma_n) \bigr), \quad h_i=\overline{\gamma_1\cdots\gamma_i}^{\,-1}h.\tag{A.11} \end{equation}

Equation (A.10) gives $d\operatorname{Sh}_{\widetilde H} =\operatorname{Sh}_{\widetilde H}d$ directly. Thus (A.11) fixes a full normalized cochain map, although the main text uses it only in degree one, together with the degree-two corestriction formula below.

For $\alpha\in Z^1(N,\F_2)$ this specializes to

\begin{equation} \operatorname{Sh}_{\widetilde H}(\alpha)(\gamma)_h =\alpha\!\left( \widetilde h^{-1}\gamma \widetilde{\bar\gamma^{-1}h} \right).\tag{A.12} \end{equation}

Indeed, by (A.10),

\begin{align*} \operatorname{Sh}(\alpha)(\gamma\eta)_h &=\operatorname{Sh}(\alpha)(\gamma)_h +\operatorname{Sh}(\alpha)(\eta)_{\bar\gamma^{-1}h}\\ &=\operatorname{Sh}(\alpha)(\gamma)_h +(\bar\gamma\operatorname{Sh}(\alpha)(\eta))_h, \end{align*}

which is exactly the left-module cocycle identity. Formula (6.39) is (A.12) for $H=C_2\text{.}$

Subsection A.5 The convention for $g\alpha$

Let $g\in H\text{,}$ choose its section lift $\widetilde g\text{,}$ and let $\alpha\in Z^1(N,\F_2)\text{.}$ We define

\begin{equation} (g\alpha)(n)=\alpha(\widetilde g^{-1}n\widetilde g) \qquad(n\in N).\tag{A.13} \end{equation}

Because $\alpha$ has trivial scalar coefficients, this does not depend on replacing $\widetilde g$ by another lift. It is the usual left $H$-action on $H^1(N,\F_2)$ expressed in the manuscript’s conjugation convention.

For $h\in H$ put

\begin{equation} a_h^{(g)}=\widetilde{hg}^{-1}\widetilde h\widetilde g\in N, \qquad q_{\alpha,g}(h)=\alpha(a_h^{(g)}).\tag{A.14} \end{equation}

If $k=\bar\gamma^{-1}h\text{,}$ then

\begin{equation*} \lambda_{hg}(\gamma) =a_h^{(g)}\, \widetilde g^{-1}\lambda_h(\gamma)\widetilde g\, (a_k^{(g)})^{-1}. \end{equation*}

Applying $\alpha$ gives the exact normalized-cochain identity

\begin{equation} \operatorname{Sh}(g\alpha) =\mathsf R_g\operatorname{Sh}(\alpha)+dq_{\alpha,g},\tag{A.15} \end{equation}

where $(dq_{\alpha,g})(\gamma)_h =q_{\alpha,g}(h)+q_{\alpha,g}(\bar\gamma^{-1}h)\text{.}$ If the section is right-$g$-equivariant, namely $\widetilde{hg}=\widetilde h\widetilde g$ for all relevant $h\text{,}$ then $q_{\alpha,g}=0$ and (A.15) is strict:

\begin{equation*} \operatorname{Sh}(g\alpha)(\gamma)_h =\operatorname{Sh}(\alpha)(\gamma)_{hg}. \end{equation*}

No such equivariance is assumed silently when it is unavailable.

For later use, let $A=\operatorname{Sh}(\alpha)$ and $B=\operatorname{Sh}(\beta)\text{.}$ With the $H$-invariant trace pairing $\langle x,y\rangle=\sum_hx_hy_h\text{,}$ the free-orbit cochain formed with the right-shifted coordinate $\mathsf R_gB$ differs from the cochain formed with $\operatorname{Sh}(g\beta)$ by the displayed boundary

\begin{equation} \operatorname{tr}\bigl(A\smile R_gB\bigr) -\operatorname{tr}\bigl(A\smile\operatorname{Sh}(g\beta)\bigr) =d\mathfrak b_{\alpha,\beta,g},\tag{A.16} \end{equation}

where

\begin{equation} \mathfrak b_{\alpha,\beta,g}(\gamma) =\sum_{h\in H} A(\gamma)_h\, q_{\beta,g}(\bar\gamma^{-1}h).\tag{A.17} \end{equation}

This follows from (A.3), (A.15), and the Leibniz rule. It is the precise cochain correction behind (6.46).

Subsection A.6 Corestriction in degrees one and two

Let $K\leq G$ have finite index and let $\Omega=G/K\text{,}$ with cosets written $t_\omega K\text{.}$ Choose a normalized transversal $\{t_\omega\}_{\omega\in\Omega}$ and define

\begin{equation} \ell_\omega(\gamma) =t_\omega^{-1}\gamma t_{\gamma^{-1}\omega}\in K.\tag{A.18} \end{equation}

Then

\begin{equation} \ell_\omega(\gamma\eta) =\ell_\omega(\gamma)\ell_{\gamma^{-1}\omega}(\eta).\tag{A.19} \end{equation}

For scalar normalized cochains, the bar-corestriction map is

\begin{equation} (\operatorname{cor}_{\mathcal T}^n f)(\gamma_1,\ldots,\gamma_n) =\sum_{\omega\in\Omega} f\bigl( \ell_{\omega_0}(\gamma_1),\ldots, \ell_{\omega_{n-1}}(\gamma_n) \bigr), \quad \omega_i=(\gamma_1\cdots\gamma_i)^{-1}\omega.\tag{A.20} \end{equation}

In degrees one and two this reads

\begin{align} (\operatorname{cor}_{\mathcal T}\alpha)(\gamma) &=\sum_{\omega\in\Omega}\alpha(\ell_\omega(\gamma)),\tag{A.21}\\ (\operatorname{cor}_{\mathcal T}\nu)(\gamma,\eta) &=\sum_{\omega\in\Omega} \nu\bigl(\ell_\omega(\gamma), \ell_{\gamma^{-1}\omega}(\eta)\bigr).\tag{A.22} \end{align}

Equation (A.19) gives

\begin{equation} d\operatorname{cor}_{\mathcal T} =\operatorname{cor}_{\mathcal T}d.\tag{A.23} \end{equation}

Equivalently, when $K=N\triangleleft G$ and $H=G/N\text{,}$ corestriction is the coordinate-sum map after the Shapiro map.

In particular, for $\alpha,\beta\in Z^1(N,\F_2)\text{,}$

\begin{equation} \operatorname{cor}_N^G(\alpha\smile\beta)(\gamma,\eta) =\sum_{h\in H} \operatorname{Sh}(\alpha)(\gamma)_h\, \operatorname{Sh}(\beta)(\eta)_{\bar\gamma^{-1}h}.\tag{A.24} \end{equation}

This is the exact normalized-cochain identity used in (6.48).

For (6.50), take $G=G_F\text{,}$ $K=G_{K_0}\text{,}$ $\Omega=H/J\text{,}$ and representatives written $uJ\text{.}$ The relation $\bar\gamma^{-1}u=\pi_{\bar\gamma}(u)g^{\epsilon_{\bar\gamma}(u)}$ means that $\gamma^{-1}(uJ)=\pi_{\bar\gamma}(u)J\text{,}$ so (A.22) becomes exactly (6.50).

Subsection A.7 Changing sections and transversals

We first record the degree-one Shapiro correction. Let $\widetilde h'=\widetilde h a_h$ with $a_h\in N$ and $a_1=1\text{.}$ Then

\begin{equation*} \lambda_h'(\gamma)=a_h^{-1}\lambda_h(\gamma) a_{\bar\gamma^{-1}h}. \end{equation*}

For $\alpha\in Z^1(N,\F_2)$ define the normalized $0$-cochain $q_\alpha\in\F_2[H]$ by $q_\alpha(h)=\alpha(a_h)\text{.}$ Then

\begin{equation} \operatorname{Sh}_{\widetilde H'}(\alpha) -\operatorname{Sh}_{\widetilde H}(\alpha) =dq_\alpha \quad\text{in }C^1(G,\F_2[H]).\tag{A.25} \end{equation}

Thus the change of Shapiro lift is not left as an unspecified cohomology ambiguity.

For all degrees there is an explicit prism homotopy. Let $K\leq G$ be as in Subsection A.6, and let $t_\omega'=t_\omega a_\omega\text{,}$ $a_\omega\in K\text{,}$ be a second normalized transversal indexed by the same coset set $\Omega\text{.}$ For a normalized $n$-cochain $f$ on $K\text{,}$ write

\begin{equation*} f^{\mathrm h}(y_0,\ldots,y_n) =f(y_0^{-1}y_1,\ldots,y_{n-1}^{-1}y_n) \end{equation*}

whenever all consecutive ratios lie in $K\text{.}$ Given $\gamma_1,\ldots,\gamma_{n-1}\in G\text{,}$ put

\begin{equation*} p_i=\gamma_1\cdots\gamma_i, \qquad \omega_i=p_i^{-1}\omega, \qquad y_i=p_i t_{\omega_i}, \qquad y_i'=p_i t_{\omega_i}'=y_i a_{\omega_i}. \end{equation*}

Define

\begin{equation} \bigl(\mathfrak c_{\mathcal T,\mathcal T'}^n f\bigr) (\gamma_1,\ldots,\gamma_{n-1}) =\sum_{\omega\in\Omega}\sum_{i=0}^{n-1} f^{\mathrm h}(y_0,\ldots,y_i,y_i',\ldots,y_{n-1}').\tag{A.26} \end{equation}

Over general coefficients the inner sum has the usual signs $(-1)^i\text{;}$ over $\F_2$ they are all $+1\text{.}$ The simplicial prism identity gives

\begin{equation} \operatorname{cor}_{\mathcal T'}f -\operatorname{cor}_{\mathcal T}f =d\mathfrak c_{\mathcal T,\mathcal T'}f +\mathfrak c_{\mathcal T,\mathcal T'}(df).\tag{A.27} \end{equation}

This is the promised cochain-level change-of-transversal formula.

For a normalized $2$-cochain $\nu\text{,}$ the degree-one transition cochain in (A.26) is especially concrete. Write $\ell_\omega,\ell_\omega'$ for the transition elements associated to the two transversals. Then

\begin{equation} \begin{split} \bigl(\mathfrak c_{\mathcal T,\mathcal T'}\nu\bigr)(\gamma) =\sum_{\omega\in\Omega}\bigl(& \nu(a_\omega,\ell_\omega'(\gamma))\\ &+\nu(\ell_\omega(\gamma),a_{\gamma^{-1}\omega}) \bigr). \end{split}\tag{A.28} \end{equation}

It is normalized. If $d\nu=0\text{,}$ then

\begin{equation} \operatorname{cor}_{\mathcal T'}\nu -\operatorname{cor}_{\mathcal T}\nu =d\bigl(\mathfrak c_{\mathcal T,\mathcal T'}\nu\bigr).\tag{A.29} \end{equation}

The same prism without the outer sum over $\Omega$ gives the corresponding homotopy between the two Shapiro cochain maps; in degree one it is precisely (A.25).

We now specialize these formulas to the two cocycles used in the manuscript. Let $J=\langle g\rangle\text{,}$ $M=\F_2[H]\text{,}$ and write $p:M\to\F_2[J]$ for the projection to the coordinates $1,g\text{.}$ Put

\begin{equation*} \theta_g=(p\rtimes1)^*\kappa_J \in Z^2(M\rtimes J,\F_2). \end{equation*}

For a transversal $\mathcal R$ of the cosets $uJ\text{,}$ represented in $M\rtimes H$ by $\widehat u=(0,u)\text{,}$ direct substitution into (A.22) gives the literal cochain equality

\begin{equation} \kappa_g^{\mathcal R} =\operatorname{cor}_{\mathcal R}\theta_g.\tag{A.30} \end{equation}

Indeed, the transition element for $(x,c)\in M\rtimes H$ is

\begin{equation*} \widehat u^{-1}(x,c)\widehat{\pi_c(u)} =\bigl(u^{-1}x,g^{\epsilon_c(u)}\bigr), \end{equation*}

and its two selected coordinates are $x_u,x_{ug}\text{;}$ the two terms of $\kappa_J$ reproduce (6.11) and (6.12).

Let $\mathcal R'$ be a second transversal, indexed so that $u'=uj_u$ with $j_u\in J\text{,}$ and put $A_u=(0,j_u)\in M\rtimes J\text{.}$ Formula (A.28) therefore gives the explicit normalized $1$-cochain

\begin{equation} \begin{split} \Lambda^g_{\mathcal R,\mathcal R'}(x,c) =\sum_{u\in\mathcal R}\bigl(& \theta_g(A_u,\ell_u'(x,c))\\ &+\theta_g(\ell_u(x,c),A_{\pi_c(u)}) \bigr) \end{split}\tag{A.31} \end{equation}

for which

\begin{equation} \kappa_g^{\mathcal R'}-\kappa_g^{\mathcal R} =d\Lambda^g_{\mathcal R,\mathcal R'}.\tag{A.32} \end{equation}

Thus the half-orbit change of transversal has a specified boundary, not only an abstract uniqueness argument.

Likewise, in the field-theoretic situation of Lemma 6.15, put

\begin{equation*} \theta_\alpha =\bigl(\operatorname{Sh}_{K/K_0}(\alpha),\rho_0\bigr)^*\kappa_J \in Z^2(G_{K_0},\F_2) \end{equation*}

and define $\nu_{\alpha,\mathcal R}=\operatorname{cor}_{\mathcal R}\theta_\alpha\text{.}$ If the lifts of the representatives change from $\widetilde u$ to $\widetilde u'=\widetilde u a_u$ with $a_u\in G_{K_0}\text{,}$ then

\begin{equation} \begin{split} \Lambda^\alpha_{\mathcal R,\mathcal R'}(\gamma) =\sum_{u\in\mathcal R}\bigl(& \theta_\alpha(a_u,\ell_u'(\gamma))\\ &+\theta_\alpha(\ell_u(\gamma),a_{\pi_{\bar\gamma}(u)}) \bigr) \end{split}\tag{A.33} \end{equation}

satisfies

\begin{equation} \nu_{\alpha,\mathcal R'}-\nu_{\alpha,\mathcal R} =d\Lambda^\alpha_{\mathcal R,\mathcal R'}.\tag{A.34} \end{equation}

With the compatible nested transversals chosen in the proof of Lemma 6.15, equation (6.51) is a literal equality of normalized cochains. Equations (A.32) and (A.34) give the correction for arbitrary choices.

Finally, consider the index-two formula itself. If the lift $\widetilde s$ is replaced by $\widetilde s'=\widetilde s a\text{,}$ $a\in N\text{,}$ then (A.25) gives $b'=b+dq$ with $q_1=0$ and $q_s=\alpha(a)\text{.}$ Put $A=(q,1)\in E\rtimes J$ and

\begin{equation*} \Phi_b(\gamma)=(b(\gamma),\bar\gamma), \qquad \Phi_{b'}(\gamma)=(b'(\gamma),\bar\gamma). \end{equation*}

Then

\begin{equation*} \Phi_{b'}(\gamma)=A^{-1}\Phi_b(\gamma)A. \end{equation*}

For $p\in E\rtimes J$ define

\begin{equation} r_A(p)= \kappa_J(A^{-1},A) +\kappa_J(A^{-1},p) +\kappa_J(A^{-1}p,A).\tag{A.35} \end{equation}

This is the central coordinate of conjugation by the canonical lift of $A$ in the extension represented by $\kappa_J\text{.}$ Therefore

\begin{equation} \nu_{\alpha,\widetilde s'}-\nu_{\alpha,\widetilde s} =d\bigl(r_A\circ\Phi_b\bigr).\tag{A.36} \end{equation}

This displays the normalized boundary asserted after (6.40).

Subsection A.8 The index-two specialization and the Evens normalization

Let $J=\langle s\rangle\cong C_2$ act on $E_2=\F_2e_1\oplus\F_2e_s$ by swapping the two coordinates. Put

\begin{equation*} q_{\mathrm{hyp}}(\notnfar{x1}{x_1},x_s)=x_1x_s, \qquad f_J(x,y)=x_1y_s, \qquad m_1=0, \qquad m_s(y)=y_1y_s, \end{equation*}

and

\begin{equation} \kappa_J((x,c),(y,d))=f_J(x,cy)+m_c(y).\tag{A.37} \end{equation}

The identities

\begin{align*} m_s(v+w)+m_s(v)+m_s(w)&=f_J(sv,sw)+f_J(v,w),\\ m_s(sv)+m_s(v)&=0 \end{align*}

show directly, after expanding $d\kappa_J((x,c),(y,d),(z,e))\text{,}$ that $d\kappa_J=0\text{.}$ The cochain is normalized. On the zero section it is zero, and on the fibre

\begin{equation*} \kappa_J((x,1),(x,1))=f_J(x,x)=x_1x_s, \end{equation*}

so its square map is $q_{\mathrm{hyp}}\text{.}$ The lifts of $e_1,e_s$ are involutions, their commutator is the central involution, and their product has nontrivial square; the fibre extension is therefore $D_8\text{.}$

We normalize the homogeneous index-two Evens norm by requiring

\begin{equation} N_N^G(1+[\alpha]) =1+\operatorname{cor}_N^G[\alpha] +N_N^{G,\mathrm{Ev}}([\alpha])\tag{A.38} \end{equation}

for $[\alpha]\in H^1(N,\F_2)\text{.}$ For the universal inclusion $E_2\leq E_2\rtimes J\text{,}$ Mackey restriction gives

\begin{equation*} \operatorname{res}_{E_2} N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee) =e_1^\vee e_s^\vee, \qquad \operatorname{res}_J N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee)=0. \end{equation*}

Since

\begin{equation*} H^1(J,E_2^\vee)=\ker(1+s)/\operatorname{im}(1+s)=0, \end{equation*}

the low-degree sequence for $1\to E_2\to E_2\rtimes J\to J\to1$ shows that these two restrictions determine the class. They are precisely the fibre and zero-section restrictions of (A.37); hence

\begin{equation} [\kappa_J]=N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee).\tag{A.39} \end{equation}

Now let $N\triangleleft G$ have index two and let $b=\operatorname{Sh}(\alpha)$ be the cocycle (A.12). By Subsection A.2, the graph $\gamma\mapsto(b(\gamma),\bar\gamma)$ is a homomorphism. Pulling back (A.37) gives, literally,

\begin{equation} \nu_\alpha(\gamma,\eta) =b(\gamma)_1b(\eta)_{\bar\gamma^{-1}s} +\varepsilon(\bar\gamma)b(\eta)_1b(\eta)_s,\tag{A.40} \end{equation}

where $\varepsilon(1)=0$ and $\varepsilon(s)=1\text{.}$ Naturality of the Evens norm in this pullback square and (A.39) give

\begin{equation*} [\nu_\alpha]=N_N^{G,\mathrm{Ev}}([\alpha]). \end{equation*}

For profinite Galois groups, every cochain in this construction factors through a finite quotient, and the displayed finite-group normalization is applied there.

The convention (A.38) is also the one used in the Evens–Kahn formula: for a quadratic extension $L/k\text{,}$ the degree-two term in the relative total Stiefel–Whitney identity is

\begin{equation*} w(\operatorname{Tr}_{L/k}\langle a\rangle) =w(\operatorname{Tr}_{L/k}\langle1\rangle) \bigl(1+\operatorname{cor}_{L/k}[a] +N_{L/k}^{\mathrm{Ev}}([a])\bigr). \end{equation*}

Thus the normalization in this appendix is exactly the normalization used in (6.53); no additional sign or inverse convention enters Lemma 6.16.

Subsection A.9 Convention audit for the orbit formulas

The following table records where each convention-sensitive formula in Lemma 6.13, Lemma 6.15 is fixed.

Manuscript formula	Convention-dependent input	Appendix result
(6.39)	normalized Shapiro section and the index $\bar\gamma^{-1}h$	Subsection A.3, Subsection A.4
(6.40)	semidirect graph pullback and the orientation of the two coordinates	Subsection A.2, Subsection A.8
(6.46)	the definition of $g\alpha$ and the right-coordinate shift	Subsection A.5
(6.48)	cup product and normalized bar corestriction	Subsection A.1, Subsection A.6
(6.50)	the coset action and degree-two transition elements	Subsection A.6, Subsection A.7
(6.51)	compatibility of the two Shapiro stages and change of transversal	Subsection A.5, Subsection A.7, Subsection A.8

Appendix B Finite Fox–Heisenberg rules and word ledger

This appendix starts from the finite semidirect, Heisenberg, and extraspecial groups used in the first- and second-order calculations. It records the literal word-evaluation identities that supply the Jacobian, mixed pairing, and candidate quadratic form in Section 5, Section 6.

The rules below are those used in Lemma 5.4, Lemma 5.5, Lemma 5.14, Proposition 6.5. They are identities of finite word evaluations before passage to cohomology; no formal Fox calculus over an infinite completed group is invoked. For each finite coefficient calculation one may replace $\omega_2$ by any ordinary integer representative modulo the exponent of the finite semidirect, Heisenberg, or extraspecial group being evaluated; Lemma 5.1 then shows that the resulting first and mixed second derivatives are independent of the representative.

Let $G$ act on $\notn{Amod}{A}$ and $\notn{Amod}{A}^\vee\text{,}$ and evaluate a word in

\begin{equation*} G\ltimes(\notn{Amod}{A}\times \notn{Amod}{A}^\vee\times\F_2). \end{equation*}

If $\bar u$ is its base value, $D_u,D_u^\vee$ are its two first derivatives, and $\beta_u$ is its mixed central coordinate, then direct multiplication in the finite Heisenberg group gives

\begin{align} D_{uv}&=D_u+\bar uD_v, &D_{u^{-1}}&=-\bar u^{-1}D_u,\tag{B.1}\\ D_{u^g}&=\bar g^{-1}D_u+\bar g^{-1}(\bar u-1)D_g, &D_{u^n}&=\sum_{j=0}^{n-1}\bar u^jD_u,\notag\\ \beta_{uv}&=\beta_u+\beta_v+(D_u^\vee)(\bar uD_v), &\beta_{u^{-1}}&=\beta_u+(\bar u^{-1}D_u^\vee)(\bar u^{-1}D_u).\tag{B.2} \end{align}

The formulas for conjugates, commutators, and powers follow recursively. When $\bar u$ has odd order $e$ and the exponent is the $2$-primary idempotent, the power sum is the norm projector

\begin{equation*} \notn{Pnorm}{P}=1+\bar u+\cdots+\bar u^{e-1}. \end{equation*}

We now spell out the first-order ledger at a tame lower map. Write $\notn{opS}{\mathsf S},\notn{opT}{\mathsf T}$ for the lower actions of $\notn{sigma}{\sigma},\notn{tau}{\tau}\text{,}$ put $\notnfar{opU}{\mathsf U}=\notn{opS}{\mathsf S}^{\omega_2}\text{,}$ and let $(a,b,c,d)$ be the lift variables on $(\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1})\text{.}$ The $\omega_2$-power rule gives

\begin{equation} \notn{Dfox}{D}(u_i)=\notn{Pnorm}{P}(Dx_i+\notn{Dfox}{D}\notn{tau}{\tau}),\tag{B.3} \end{equation}

and hence

\begin{equation} \notn{Dfox}{D}(d_0)=\notn{Pnorm}{P} b+(\notn{Pnorm}{P}+1)c, \qquad \notn{Dfox}{D}(z_0)=\notnfar{opU}{\mathsf U}^{-1}c.\tag{B.4} \end{equation}

The tame relator gives the unnormalized row

\begin{equation} \notn{Dfox}{D}(\notn{tau}{\tau}^\notn{sigma}{\sigma}\notn{tau}{\tau}^{-2}) =\notn{opS}{\mathsf S}^{-1}(1+\notn{opT}{\mathsf T})a+(\notn{opS}{\mathsf S}^{-1}+1+\notn{opT}{\mathsf T})b.\tag{B.5} \end{equation}

The wild relator has the following first-order table:

\begin{equation*} \begin{array}{c|cccc} \toprule \text{factor} & a&b&c&d\\ \midrule h_0&0&0&0&0\\ u_1^{-1}&0&\notn{Pnorm}{P}&0&\notn{Pnorm}{P}\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0&0&\notn{opS}{\mathsf S}^{-1}\\ {[d_0,z_0]}&0&0&0&0\\ \bottomrule \end{array} \end{equation*}

Thus

\begin{equation} L_w=\notn{Pnorm}{P} b+(\notn{Pnorm}{P}+\notn{opS}{\mathsf S}^{-1})d.\tag{B.6} \end{equation}

Together, (B.5) and (B.6) are the Jacobian displayed in Lemma 5.5.

The only non-obvious entry in this ledger is the first line. The variation of $g_0=\sigma_2^2$ contributes only in order at least two, because the lower values of $\notn{x0}{x_0}$ and $d_0$ at the tame map are both $1\text{.}$ If $\notn{Pnorm}{P}=0\text{,}$ then $Dd_0=\notn{Dfox}{D}\notn{x0}{x_0}=c\text{,}$ and the exact class-two identity of Lemma 5.2 shows that $h_0$ has the same elementary and central shadow as $\notn{x0}{x_0}^2\text{.}$ If $\notn{Pnorm}{P}=1\text{,}$ then $\notn{opT}{\mathsf T}=1$ and the $2$-primary operator $\notnfar{opU}{\mathsf U}$ acts trivially on the simple characteristic-$2$ quotient; its lift to the Heisenberg or extraspecial coefficient group is a central automorphism of order at most $2\text{,}$ so $g_0=\sigma_2^2$ acts trivially on the full coefficient group. Again Lemma 5.2 applies. This is the precise reason that $h_0$ may be replaced by $\notn{x0}{x_0}^2$ in the linear, mixed cup, and determinant calculations.

Write $b_{\mathrm{cl}}$ for the bilinear commutator pairing of the chosen class-two coefficient group. For the mixed Heisenberg and extraspecial calculations on normalized representatives supported on $\notn{x0}{x_0}\text{,}$ the ledger (stated inline in Section 5 and repeated here as part of the appendix’s complete record) is

\begin{equation*} \begin{array}{c|c|c} \toprule \text{factor} & \notnfar{Vhead}{V}\text{-coordinate} & \text{central coordinate}\\ \midrule h_0&0&\lambda(c)\text{ in }H(\notn{Amod}{A})\text{, or }q(c)\text{ for }\notnfar{kappaq0}{\kappa_q^0}\\ u_1^{-1}&0&0\\ \notnfar{x1}{x_1}^\notn{sigma}{\sigma}&0&0\\ d_0=(\notn{x0}{x_0}\notn{tau}{\tau})^{\omega_2}\notn{x0}{x_0}^{-1}&(\notn{Pnorm}{P}+1)c&*\\ z_0=\notn{x0}{x_0}^{\sigma_2}&\notnfar{opU}{\mathsf U}^{-1}c&*\\ {[d_0,z_0]}&0&b_{\mathrm{cl}}((\notn{Pnorm}{P}+1)c,\notnfar{opU}{\mathsf U}^{-1}c)\\ \bottomrule \end{array} \end{equation*}

The starred central coordinates of $d_0$ and $z_0$ are immaterial because these words occur only inside the commutator. The table gives both the mixed Hessian in Lemma 5.14 and the exact base determinant word expansion

\begin{equation*} Q_A^0(c)=q(c)+\notnfar{bq}{b_q}((\notn{Pnorm}{P}+1)c,\notnfar{opU}{\mathsf U}^{-1}c) \end{equation*}

of Proposition 6.5. The starred entries may contain the non-diagonal action corrections $m_c$ from Lemma 6.1; they are central coordinates of $d_0$ and $z_0$ and hence do not affect the commutator. No relator-by-relator Stokes identity is used: the correction terms in the finite-word Stokes formula cancel only after summing the tame and wild relators, exactly as in Proposition 5.8.

Appendix C The square-commutator presentation

A second four-generator, two-relator presentation of $\notnfar{GQ}{\GQ}$ was proposed independently of the one proved in Theorem 1.2. This appendix proves that it, too, presents $\notnfar{GQ}{\GQ}\text{.}$

The proof is not an identification of the two presentations. No Nielsen or Tietze transformation carrying one to the other is known, and Proposition C.6 shows that none of the expected kind exists: already their maximal pro-$2$ quotients cannot be matched by any triple of words. What is reused instead is the argument, through the interface isolated in Corollary 6.19. Three of the interface’s four inputs are computations of the kind carried out in Section 5 and Section 6, with the new word in place of (1.6). The fourth input, the fully marked maximal pro-$2$ quotient, is genuinely different: the new pro-$2$ quotient is not the boundary group $\notnfar{Pi}{\Pi}$ of Proposition 3.10 but a differently presented Demushkin group, and identifying it needs the classification of Demushkin groups at one instance rather than the explicit Nielsen change of (3.16). That instance is proved in Subsection C.5 rather than quoted.

Subsection C.1 The presentation and the theorem

The conventions of Section 1 remain in force throughout: $x^g=g^{-1}xg\text{,}$ $[x,y]=x^{-1}y^{-1}xy\text{,}$ and $\notn{w}{\w}\in\notn{Zhat}{\Zhat}$ is the idempotent that extracts the $2$-primary component. Put

\begin{equation} \sigma_2=\notn{sigma}{\sigma}^{\notn{w}{\w}}, \qquad a=(\notn{x0}{x_0}^{-3}\notn{tau}{\tau})^{\notn{w}{\w}}, \qquad y_1=\notnfar{x1}{x_1}^{\sigma_2}, \qquad c=[\notnfar{x1}{x_1},y_1].\tag{C.1} \end{equation}

Definition C.1. The square-commutator candidate.

Let $F$ be the free profinite group on $\notn{sigma}{\sigma},\notn{tau}{\tau},\notn{x0}{x_0},\notnfar{x1}{x_1}\text{.}$ Call an epimorphism $\varphi:F\twoheadrightarrow G$ to a finite group square-commutator admissible if the distinguished tuple generates $G\text{,}$ satisfies (1.5) and

\begin{equation} (\notn{x0}{x_0}^{\notn{sigma}{\sigma}})^{-1}\,a\,\notnfar{x1}{x_1}^2\,c=1,\tag{C.2} \end{equation}

and the normal closure of $\varphi(\notn{x0}{x_0}),\varphi(\notnfar{x1}{x_1})$ is a $2$-group. Let $\mathcal Q_R$ be the class of all such quotients and set

\begin{equation} N_R=\bigcap_{\varphi\in\mathcal Q_R}\ker\varphi, \qquad \Gamma_R=F/N_R.\tag{C.3} \end{equation}

For a finite group $G$ write $\mathrm{ac}_R(G)$ for the number of square-commutator admissible markings of $G\text{,}$ that is, of generating quadruples in $G^4$ satisfying (1.5) and (C.2) whose last two entries lie in the $2$-core, as in Proposition 2.3.

Lean (relator word)Lean (relation)Lean (the group)Lean (marking count)

This is the construction of Section 2 with (C.2) in place of (1.6). Nothing in Lemma 2.1, Lemma 2.2 or Proposition 2.3 uses any property of the wild relator beyond its being a word in the four generators, so all three hold verbatim for $\Gamma_R\text{:}$ finitely many admissible quotients have admissible subdirect product, every finite quotient of $\Gamma_R$ is again admissible, and epimorphisms $\Gamma_R\twoheadrightarrow G$ correspond to square-commutator admissible markings of $G\text{.}$ In particular $\Gamma_R$ is topologically generated by four elements, so Lemma 2.5 applies to it.

Remark C.2.

The name records the shape of (C.2): the relator closes with the literal block $\notnfar{x1}{x_1}^2c\text{,}$ a square times a commutator, where the relator (1.6) of Theorem 1.2 closes with the class-two collector word $h_0$ and the commutator $c_0\text{.}$ The two presentations share their generators, their tame relation (1.5) and their pro-$2$ condition; only the wild relator differs.

Theorem C.3. Replacement theorem.

$\Gamma_R\cong\notnfar{GQ}{\GQ}\text{.}$ Equivalently,

\begin{equation} |\Sur(\Gamma_R,G)|=|\Sur(\notnfar{GQ}{\GQ},G)|\tag{C.4} \end{equation}

for every finite group $G\text{;}$ equivalently again, $\mathrm{ac}_R(G)=|\Sur(\notnfar{GQ}{\GQ},G)|$ for every finite group $G\text{.}$ Consequently the two candidates present the same group, and their marking counts agree: $\mathrm{ac}_R(G)=\mathrm{ac}(G)$ for every finite $G\text{.}$

Lean (literal relators)Lean (counting form)Lean (fixed-frame equality)Lean (counts agree)

Proof.

By Proposition C.5 the source interface of Corollary 6.19 holds for $\Gamma_R\text{,}$ so Theorem 4.2 and its proof apply with $\Gamma_R$ in place of $\notnfar{GA}{\GA}\text{.}$ Summing over tame boundary frames as in Lemma 10.1 gives (C.4). Since $\Gamma_R$ is topologically generated by four elements, Lemma 2.5 converts (C.4) into an isomorphism. The last assertion combines (C.4) with (10.1) and Proposition 2.3.

Remark C.4.

The counts $\mathrm{ac}_R(G)$ were computed and compared with $|\Sur(\notnfar{GQ}{\GQ},G)|$ for small $G$ long before any proof: for $C_2,C_4,V_4,D_4,Q_8$ both sides are $7,24,42,144,144\text{.}$ Such agreement is evidence and not proof; Theorem C.3 upgrades it to an identity valid for every finite group.

Subsection C.2 What is reused, and what has to be redone

The induction of Sections 7–10 takes place entirely inside the finite target: it manipulates lower quotients, exact-image partitions, pushouts, finite Fourier transforms and central double covers, and after Section 5 it never differentiates the wild word again. Its dependence on the source is confined to the items listed in Corollary 6.19 together with the tame and pro-$2$ data of Section 3. Replacing the source therefore amounts to re-verifying a fixed, finite list.

Proposition C.5. The interface at the square-commutator relator.

The source $\Gamma_R$ satisfies every item of Corollary 6.19, and in addition:

its tame quotient is $\notnfar{TA}{\TA}\text{,}$ with the unramified marking of Proposition 3.2 and Proposition 3.14;
its fully marked maximal pro-$2$ quotient is $(G_{\Qtwo}(2),\notn{nuur}{\nu_{\mathrm{ur}}})\text{.}$

Lean (the interface)Lean (induction over a source)Lean (deformation duality)Lean (base quadratic form)

Proof.

Item (i) is the argument of Proposition 3.2 verbatim: in the quotient by the wild subgroup the elements $\notn{x0}{x_0},\notnfar{x1}{x_1}$ vanish, and Lemma 3.1 gives $\notn{tau}{\tau}^{\notn{w}{\w}}=1\text{,}$ so $a=1$ and (C.2) disappears exactly as (1.6) does. Item (ii) is Proposition C.12, proved in Subsection C.4.

For the remaining items the calculations of Section 5 and Section 6 are repeated with (C.2) in place of (1.6), using the word ledgers of Appendix B, which are statements about evaluation in the finite semidirect, Heisenberg and extraspecial groups and are available for any word. The first-order calculation is the shortest of these: the evaluated wild row of the Fox Jacobian of Lemma 5.5 for (C.2) is the row of (1.6) with its two wild columns interchanged. Consequently the lifting complex has the same dimensions, and the argument of Proposition 5.15 gives items (i) and (ii) of Corollary 6.19 unchanged; the trivial-module collapse and the Stokes endpoint of Proposition 5.8 give item (iii); and the second-order expansion of Proposition 6.5, applied to (C.2), produces a base quadratic form with the zero counts of Proposition 6.9, hence item (iv).

The square-commutator relator is the shorter word at every seam, and the calculations are correspondingly shorter: it has no analogue of the auxiliary words $h_0,d_0,z_0,g_0$ of (1.1)–(1.3), its square block is the bare $\notnfar{x1}{x_1}^2\text{,}$ and its ramified base evaluation lands on the Wall form of Lemma 6.6 without the telescoping corrections that (1.6) requires; several auxiliary hypotheses used in the main text are correspondingly unnecessary here. In the formalization the independence of the induction from the source is made literal: the interface is a structure, the induction is proved once over an abstract instance of it, and the two sources are two instantiations.

Subsection C.3 No identification by words

For the source of Theorem 1.2 the maximal pro-$2$ quotient is identified with the standard dyadic Demushkin group by the explicit Nielsen change (3.16). The corresponding step for $\Gamma_R$ cannot be taken that way. Write

\begin{equation} \notn{D0}{D_0}=\angles{\notn{Agen}{A},S,Y\;\middle|\;\notn{Agen}{A}^2S^4[S,Y]}_{\mathrm{pro}-2}\tag{C.5} \end{equation}

for the group of (1.4), and let $D_R$ be the maximal pro-$2$ quotient of $\Gamma_R\text{,}$ computed in Lemma C.8 below.

Proposition C.6. Obstruction to a word identification.

Let $G,H\in\{\notn{D0}{D_0},D_R\}\text{.}$

Every continuous epimorphism $G\twoheadrightarrow H$ is an isomorphism.
Every continuous isomorphism intertwines the canonical orientations of the two groups.
There is no triple of words in the generators of $\notn{D0}{D_0}$ whose images generate $D_R$ and satisfy its relation, and none in the generators of $D_R$ for $\notn{D0}{D_0}\text{.}$ The same holds for words in the four generators of $\notnfar{GA}{\GA}$ or of $\Gamma_R\text{.}$

Proof.

Both groups have $\dim_{\F_2}H^1(-,\F_2)=3$ and $\dim_{\F_2}H^2(-,\F_2)=1\text{.}$ Let $f:G\twoheadrightarrow H$ be continuous with $K=\ker f\neq1\text{.}$ Inflation is injective on $H^1\text{,}$ hence bijective by equality of dimensions, so the restriction $H^1(G)\to H^1(K)^G$ is zero. The five-term exact sequence then makes transgression injective on $H^1(K)^G\text{,}$ which is nonzero because $K$ is a nontrivial pro-$2$ group; so transgression is onto $H^2(H)\cong\F_2\text{,}$ and exactness makes inflation $H^2(H)\to H^2(G)$ vanish. But the cup product on $H^1(G)$ is the inflation of the cup product on $H^1(H)\text{,}$ and would therefore be zero, contradicting the nondegeneracy recorded in Lemma C.9 and in Proposition 1.1. Hence $K=1\text{,}$ which is (i). Assertion (ii) holds because the dualizing module of a Demushkin group is unique and $\Aut(\mathbb{Z}_2)$ is abelian, so the orientation character is canonical.

For (iii), a word triple as described would define an epimorphism, hence by (i) an isomorphism, hence by (ii) one intertwining the orientations. By Proposition C.10 the orientation of $D_R$ takes the values $S,X,Y$ on its generators; these are units of the cubic field $K=\mathbb{Q}(X)$ of discriminant $-59\text{,}$ with

\begin{equation*} N_{K/\mathbb{Q}}(X)=-1, \qquad N_{K/\mathbb{Q}}(S)=1/4, \qquad N_{K/\mathbb{Q}}(Y)=-1, \end{equation*}

so the orientation value of any word in the generators of $D_R$ has norm in $\pm4^{\mathbb{Z}}\text{.}$ An isomorphism $\notn{D0}{D_0}\to D_R$ would have to carry $Y$ to an element of orientation value $\notn{eta}{\eta}=(-3)^{-1}\text{,}$ of norm $-1/27\text{;}$ impossible. In the other direction the orientation of $\notn{D0}{D_0}$ takes the rational values $-1,1,\notn{eta}{\eta}$ on its generators, so every word value is rational, whereas $X$ is a cubic irrational. Finally, in any pro-$2$ quotient the tame generator dies and exponentiation by $\notn{w}{\w}$ is the identity, so words in the four generators factor through words in the three pro-$2$ generators and the same obstruction applies.

Remark C.7.

Proposition C.6 is an obstruction and not the report of an unsuccessful search, and it is worth recording why a search would not have settled the question. The orientation congruences that a word triple must satisfy are solvable modulo $2^k$ for every $k\text{,}$ because $X$ topologically generates $1+4\mathbb{Z}_2\text{,}$ and the exponent sums forced at depth $k$ grow like $2^{k-3}\text{.}$ A lifting search therefore produces approximate solutions of unbounded length indefinitely, and no finite computation distinguishes that from a solution just out of reach. Unlike the rest of this appendix, the proposition is not part of the formalized chain: it motivates the route taken below but is not used in the proof of Theorem C.3.

Subsection C.4 The maximal pro-$2$ quotient and its marking

Lemma C.8. Pro-$2$ quotient of the square-commutator candidate.

The maximal pro-$2$ quotient of $\Gamma_R$ is

\begin{equation} D_R=\angles{s,x,y\;\middle|\;(x^s)^{-1}x^{-3}y^2[y,y^s]}_{\mathrm{pro}-2},\tag{C.6} \end{equation}

where $s,x,y$ are the images of $\notn{sigma}{\sigma},\notn{x0}{x_0},\notnfar{x1}{x_1}\text{.}$

Lean

Proof.

In a finite $2$-group quotient Lemma 3.1 forces $\notn{tau}{\tau}=1\text{,}$ and $\notn{w}{\w}$ acts as the identity. Hence $\sigma_2=\notn{sigma}{\sigma}\text{,}$ $a=\notn{x0}{x_0}^{-3}\text{,}$ $y_1=\notnfar{x1}{x_1}^{\notn{sigma}{\sigma}}$ and $c=[\notnfar{x1}{x_1},\notnfar{x1}{x_1}^{\notn{sigma}{\sigma}}]\text{,}$ so (C.2) becomes exactly the relator of (C.6). Conversely every finite quotient of (C.6) is square-commutator admissible once $\notn{tau}{\tau}$ is set to $1\text{.}$

Lemma C.9. Demushkin invariants of $D_R$.

$D_R$ is a Demushkin pro-$2$ group of rank $3$ with $q(D_R)=2\text{.}$ In the basis of $H^1(D_R,\F_2)$ dual to $(s,x,y)\text{,}$ the Gram matrix of the cup–Bockstein form is

\begin{equation} \begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix},\tag{C.7} \end{equation}

which is nonsingular.

Lean (Demushkin)Lean (rank)Lean (the invariant q)

Proof.

The presentation has three generators and one relator, so $\dim H^1=3$ and $\dim H^2\le1\text{.}$ From $x^s=x[x,s]$ and $y^s=y[y,s]$ one gets $[y,y^s]=[y,[y,s]]$ and hence

\begin{equation*} (x^s)^{-1}x^{-3}y^2[y,y^s]=[x,s]^{-1}\,x^{-4}\,y^2\,[y,[y,s]]. \end{equation*}

Both $x^{-4}$ and $[y,[y,s]]$ lie in the third step of the filtration (C.9), so the initial form of the relator is $y^2[x,s]\text{.}$ This reads off (C.7): the squared generator contributes the diagonal Bockstein entry at $y\text{,}$ and the commutator contributes the off-diagonal pair at $(s,x)\text{.}$ The matrix is nonsingular, so $\dim H^2=1$ and the cup product $H^1\times H^1\to H^2$ is a perfect pairing, which is the Demushkin condition. Finally the exponent sums of the relator on $(s,x,y)$ are $(0,-4,2)\text{,}$ so

\begin{equation*} D_R^{\mathrm{ab}}\cong\mathbb{Z}_2^2\times\mathbb{Z}/2, \end{equation*}

and $q(D_R)\text{,}$ the order of the torsion subgroup of the abelianization, is $2\text{.}$

At $p=2$ the cup square $u\smile u$ equals the Bockstein $\beta(u)$ and is additive in $u\text{,}$ so (C.7) is the Gram matrix of a symmetric bilinear form carrying the Bockstein on its diagonal, not the polar form of a quadratic form; invariants of quadratic type must not be read off from it.

Proposition C.10. The orientation of $D_R$.

$D_R$ carries a unique continuous character $\chi_R:D_R\to\mathbb{Z}_2^\times$ describing its action on the dualizing module. Its values on the generators are

\begin{equation} \chi_R(s)=S,\qquad \chi_R(x)=X,\qquad \chi_R(y)=Y=-X^2,\tag{C.8} \end{equation}

where $X\in\mathbb{Z}_2$ is the unique root of $Z^3+2Z^2+1$ with $X\equiv1\pmod2\text{,}$ and $S=-X^3/(X^2+X+1)\text{;}$ numerically $X\equiv5$ and $S\equiv13\pmod{16}\text{.}$ Moreover $\chi_R$ is surjective, with image $\{\pm1\}\times(1+4\mathbb{Z}_2)=\mathbb{Z}_2^\times\text{.}$

Lean (the character)Lean (orientation)Lean (image)

Proof.

The orientation of a Demushkin group is characterized by the property that every $\chi$-crossed derivation of the free pro-$2$ group kills the relator; see [Ser63, no. 9.3, pp. 153--154]. Evaluating the crossed derivations valued in $\mathbb{Z}_2(\chi)\rtimes\mathbb{Z}_2^\times$ on the relator of (C.6) gives three coordinate equations, together with the relation $Y^2=X^4$ forced by the exponent sums. The branch $Y=+X^2$ is excluded, because it forces $(X-1)(X^2-X-1)=0$ with $X^2-X-1$ a unit, while $X=1$ is inconsistent with the remaining equation. On the branch $Y=-X^2$ the system reduces to $X^3+2X^2+1=0\text{,}$ which has a unique root congruent to $1$ modulo $2$ by Hensel’s lemma, and then $S=-X^3/(X^2+X+1)\text{.}$ Surjectivity follows from Burnside’s basis theorem applied to the image, together with the fact that $X$ is not a square modulo $8\text{.}$

Proposition C.11. Demushkin classification at rank $3$ and $q=2$.

Let $D$ be a Demushkin pro-$2$ group of rank $3$ with $q(D)=2\text{,}$ carrying a continuous surjective character $\chi:D\to\mathbb{Z}_2^\times$ that is an orientation of $D\text{.}$ Then $D\cong \notn{D0}{D_0}$ as topological groups.

Lean (statement)Lean (theorem)Lean (levelwise limit)Lean (Hopficity)

Proof.

This is the instance $n=3\text{,}$ $q=2$ of the classification of Demushkin groups, [Lab, Thms. 4(2) and 8]. The surjectivity hypothesis pins the remaining invariant: for a Demushkin group of this shape the image of the orientation is $\{\pm1\}\times(1+2^f\mathbb{Z}_2)\text{,}$ and surjectivity says $f=2\text{.}$ The instance is proved directly in Subsection C.5.

Proposition C.12. Marked identification of the pro-$2$ quotient.

Let $\nu_R:D_R\twoheadrightarrow\mathbb{Z}_2$ be defined by $\nu_R(s)=1$ and $\nu_R(x)=\nu_R(y)=0\text{.}$ Then the marked pairs $(G_{\Qtwo}(2),\notn{nuur}{\nu_{\mathrm{ur}}})$ and $(D_R,\nu_R)$ are isomorphic.

Lean

Proof.

By Lemma C.9 and Proposition C.10 the group $D_R$ satisfies the hypotheses of Proposition C.11, so $D_R\cong \notn{D0}{D_0}\text{;}$ and $\notn{D0}{D_0}\cong G_{\Qtwo}(2)$ is Proposition 1.1. It remains to match the markings. By Proposition C.6(ii) every continuous isomorphism intertwines the orientations, so the composite $G_{\Qtwo}(2)\to D_R$ is determined on the abelianization up to those automorphisms of $\notn{D0}{D_0}$ that preserve its orientation. Expressing the unramified character of the target in the coordinates of (C.6) leaves a pair of unknowns, and these are solved by a generation argument modulo $2$ together with a parity computation for the $s$-coordinate modulo $16\text{,}$ using the values (C.8). Composing with the resulting automorphism gives an isomorphism carrying $\notn{nuur}{\nu_{\mathrm{ur}}}$ to $\nu_R\text{.}$

Subsection C.5 The classification instance

This subsection proves Proposition C.11. The argument does not invoke the uniqueness statement of the classification. It re-runs the classification at the single instance needed, by constructing continuous epimorphisms in both directions level by level along a central tower and closing with Hopficity, exactly as in Lemma 2.5. Only the two relators

\begin{equation*} \notn{r0}{r_0}=\notn{Agen}{A}^2S^4[S,Y], \qquad r_2=(x^s)^{-1}x^{-3}y^2[y,y^s] \end{equation*}

occur, and the two directions are mirror images of one another.

For a pro-$2$ group $G$ put

\begin{equation} \lambda_1=G, \qquad \lambda_{k+1}=\lambda_k^2[\lambda_k,G], \qquad Q_k=G/\lambda_k, \qquad Z_k=\lambda_k/\lambda_{k+1}.\tag{C.9} \end{equation}

This is the filtration used in [Ser63, Section 6]. Its steps are open and characteristic, their intersection is trivial when $G$ is topologically finitely generated, each $Z_k$ is elementary abelian, and $Z_k$ is central in $Q_{k+1}\text{.}$ The following consequences of $[\lambda_a,\lambda_b]\subseteq\lambda_{a+b}$ hold for $k\ge3$ and are used throughout: $[\lambda_{k-1},\lambda_{k-1}]\subseteq\lambda_{k+1}\text{;}$ squaring induces an $\F_2$-linear map $Z_{k-1}\to Z_k\text{;}$ and, for $v\in\lambda_{k-1}\text{,}$ the element $[v,g]$ depends only on the classes of $v$ and $g$ and is central of order at most $2$ modulo $\lambda_{k+1}\text{.}$ At $k=2$ these fail, which is why every statement below carries the hypothesis $k\ge3$ and the level $k=3$ is treated as a base case rather than by the calculus.

Definition C.13. Defect and defect homomorphism.

Let $k\ge3\text{,}$ let $D$ be the group in which we work, and let $T$ be a triple of elements of $Q_k$ that generates $Q_k$ and at which the tested relator evaluates to $1\text{.}$ Lift $T$ to $Q_{k+1}$ and evaluate the relator there; the result lies in $Z_k$ and is independent of the lift, because all three exponent sums of each relator are even. Call it the defect $\delta_k(T)\in Z_k\text{.}$ Modifying $T$ slotwise by $w\in(\lambda_{k-1})^3$ preserves both clauses and shifts the defect by the defect homomorphism, given in additive notation on $Z_k$ by

\begin{equation} \bar d_k(w_1,w_2,w_3)=w_1^2+[w_1,\notn{Agen}{A}]+[w_2,\notn{Ygen}{Y}]+[w_3,\notnfar{Sgen}{S}]\tag{C.10} \end{equation}

for $\notn{r0}{r_0}\text{,}$ and by

\begin{equation} \bar d_k(u,v,w)=w^2+[w,y]+[u,x]+[v,s]\tag{C.11} \end{equation}

for $r_2\text{.}$

Lean (the filtration)Lean (defect)Lean (defect homomorphism)

In (C.10) the factor $S^4$ is inert, its whole contribution being a square of an element of $\lambda_k\text{;}$ the factor $\notn{Agen}{A}^2$ contributes the diagonal term $w_1^2+[w_1,\notn{Agen}{A}]\text{,}$ and $[S,Y]$ contributes the two cross terms. In (C.11) the block $(x^s)^{-1}x^{-3}$ contributes the two cross terms but nothing in the squaring direction, and $[y,y^s]$ is inert altogether, its first-order terms cancelling in pairs because both of its slots carry the same letter. In each shape exactly one generator is twisted: it occupies the coupled slot $v^2+[v,\cdot]\text{,}$ while the other two carry the plain bracket slots.

Remark C.14.

These are the transported analogues, at $q=2$ and in the conventions of this paper, of the defect homomorphism of [Ser63, Lemme 6.3], which is stated for the normal form available when $q\neq2$ and is surjective there. At $q=2$ it is not: the cokernel of $\bar d_k$ is two-dimensional at every level. That failure is the whole difficulty of this subsection, and it is repaired by Theorem C.15 together with the congruence clause of (C.14).

Theorem C.15. Span theorem.

Let $F$ be the free pro-$2$ group on three generators, filtered by (C.9), and let $\bar d_k$ be either (C.10) or (C.11) evaluated at the generators. For every $k\ge3\text{,}$

\begin{equation} Z_k(F)=\im\bar d_k +\angles{g_i^{2^{k-1}},\,g_j^{2^{k-1}}},\tag{C.12} \end{equation}

where $g_i,g_j$ are the two untwisted generators: $\notnfar{Sgen}{S},\notn{Ygen}{Y}$ for $\notn{r0}{r_0}$ and $s,x$ for $r_2\text{.}$ The corresponding power of the twisted generator already lies in $\im\bar d_k\text{.}$ The same identity holds in $Z_k(D)$ for any quotient $D$ of $F$ and any generating triple, because the filtration (C.9) is verbal.

Lean (free form, first shape)Lean (free form, second shape)Lean (descent)

Proof.

Because the filtration is verbal, $Z_k$ is generated by the classes of $v^2$ and $[v,g]$ with $v\in\lambda_{k-1}$ and $g\in F\text{.}$ We reduce both families into the right-hand side of (C.12).

Bracket atoms. For fixed $v$ the class of $[v,g]$ is multiplicative in $g$ and central, so $g$ may be taken to be a generator. Against either untwisted generator, $[v,g_i]$ is by definition a value of $\bar d_k\text{,}$ namely the one with $v$ in the corresponding slot and $1$ in the others. Against the twisted generator $t$ one has, also by definition, $[v,t]=(v^2)^{-1}\bar d_k(v,1,1)\text{.}$ So the bracket atoms lie in $\im\bar d_k$ together with the square atoms.

Square atoms, by induction on $k\text{.}$ Let $v\in\lambda_{k-1}\text{.}$ By the statement one level down, the image of $v$ in $Z_{k-1}$ is a product of values of $\bar d_{k-1}$ and of the two level-$(k-1)$ powers. Lift that factorization back into $\lambda_{k-1}\text{;}$ the ambiguity is an element of $Z_k\text{,}$ a central involution, which dies on squaring. Squaring is multiplicative on $\lambda_{k-1}$-classes because $[\lambda_{k-1},\lambda_{k-1}]\subseteq\lambda_{k+1}\text{,}$ so it suffices to square each factor. A lifted power $g_i^{2^{k-2}}$ squares to the level-$k$ power $g_i^{2^{k-1}}\text{,}$ and a lifted value of $\bar d_{k-1}$ squares to a value of $\bar d_k$ by the transport identity

\begin{equation} \bar d_k(w)^2=\bar d_k(w^2),\tag{C.13} \end{equation}

read slotwise. Its proof is the same layer calculus: the four factors of $\bar d_k(w)$ lie one step deeper, squares of products of such elements are products of squares, and the correction term in $[u,g]^2=[u^2,g]$ is $[[u,g],u]\in[\lambda_{k-1},\lambda_{k-2}]\subseteq\lambda_{2k-3}\text{,}$ which is trivial in $Q_{k+1}$ as soon as $k\ge4\text{.}$ This closes the induction, and no termination measure beyond $k$ is needed, because the recursion is exactly the statement one level down.

Base case. For $k=3$ the computation takes place in $Q_4\text{,}$ where the layer calculus is exact: $(uv)^4=u^4v^4[v,u]^2\text{,}$ and $[u,v]^2$ is bimultiplicative. Both atom families then close under the same reductions, using $[u^2,z]=[u,z][[u,z],u][u,z]$ for the squares of bracket atoms, and the diagonal identity $t^4=\bar d_3(t^2,1,1)\text{,}$ valid because $t$ commutes with $t^2\text{,}$ for the twisted generator’s fourth power.

Remark C.16.

The spanning statement is asserted for $q=2$ in [Ser63, Section 7], where the generator powers are printed with exponent $2^h$ at level $h\text{.}$ The level-$h$ classes are those of the $2^{h-1}$-st powers; the $2^h$-th powers lie one step deeper and are zero in $Z_h\text{.}$ With the exponents corrected, (C.12) is that statement, and Theorem C.15 supplies a proof of it, for the two relator shapes at issue, in the form needed here. Which pair of generators carries the powers is not a matter of choice: it is determined by the relator, and the wrong pair fails already at $k=3\text{.}$

Fix now one of the two directions: a target group $D$ with its orientation $\chi\text{,}$ and the relator of the other group as the tested word. For $k\ge2$ one has $\chi(\lambda_k)\subseteq1+2^{k+1}\mathbb{Z}_2\text{,}$ so $\chi$ has a well-defined level-$k$ shadow $\hat\chi_k$ on $Q_k\text{.}$ Let $t_1,t_2,t_3$ be the orientation values of the tested group’s own generators: $-1,1,\notn{eta}{\eta}$ with $\notn{eta}{\eta}=(-3)^{-1}$ for $\notn{r0}{r_0}\text{,}$ and $S,X,Y$ as in (C.8) for $r_2\text{.}$ Put

\begin{equation} S_k=\left\{T:\{1,2,3\}\to Q_k \;\middle|\; \begin{array}{l} \text{the tested relator vanishes at }T,\\ T\text{ generates }Q_k,\\ \hat\chi_k(T_i)\equiv t_i\ \bmod\ 2^k\ \ (i=1,2,3) \end{array}\right\}.\tag{C.14} \end{equation}

The third clause is what makes the induction work. It is a condition on the trajectory rather than on the level, and it is where the invariant that distinguishes the possible orientation images enters.

Lemma C.17. Stage lemma.

Let $k\ge3$ and $T\in S_k\text{.}$ Then

reachability: $\delta_k(T)\in\im\bar d_k\text{;}$
digit adjustment: the map from $\ker\bar d_k$ to $(\mathbb{Z}/2)^2$ recording the two fresh orientation digits of the corrected triple is surjective.

Consequently some modification of $T$ lifts into $S_{k+1}\text{,}$ and $S_k\neq\emptyset$ for every $k\ge3\text{.}$

Lean (reachability)Lean (digit adjustment)Lean (stage step)Lean (the levelwise sets)

Proof.

(i) By Theorem C.15, applied at the generating triple $T\text{,}$

\begin{equation*} \delta_k(T)=\bar d_k(w)\cdot t_p^{\alpha}t_q^{\beta}, \qquad \alpha,\beta\in\F_2, \end{equation*}

where $t_p,t_q$ are the two power classes. For each untwisted generator $z$ let $D_z$ be the $\chi$-crossed derivation of the free pro-$2$ group taking the value $1$ at $z$ and $0$ at the other two generators. Every $\chi$-crossed derivation kills the relator of $D$ (that is exactly the statement that $\chi$ is an orientation), so $D_z$ descends to $D\text{.}$ Reducing $D_z$ modulo a suitable power of $2\text{,}$ which may be done inside the finite semidirect product $\mathbb{Z}/2^N\rtimes(\mathbb{Z}/2^N)^\times\text{,}$ gives an additive functional $\varphi^z$ on $Z_k$ that kills $\im\bar d_k\text{;}$ and the $2\times2$ matrix formed by the two functionals on the two power classes is invertible over $\F_2\text{,}$ by the generation clause alone. Finally the congruence clause forces $\varphi^z(\delta_k(T))=0\text{:}$ written out, this vanishing is the orientation identity of the tested group at its own generator values, namely $3+\notn{eta}{\eta}^{-1}=0$ for $\notn{r0}{r_0}$ and the relation among $S,X,Y$ of (C.8) for $r_2\text{.}$ Hence $\alpha=\beta=0\text{.}$

(ii) The digit in the twisted slot is automatic. Applying $\chi$ to the relator clause one level up, and using $\chi(\lambda_j)\subseteq1+2^{j+1}\mathbb{Z}_2$ at $j=k+1\text{,}$ gives a relation modulo $2^{k+2}$ among the orientation values of the lifted triple; the corresponding identity holds exactly at the targets, since $(-1)^2\cdot1^4=1$ and $X^{-4}Y^2=1\text{.}$ Comparing the two forces the fresh digit of the twisted slot to vanish, with no adjustment and with no data from level $k+2\text{.}$ The remaining two digits are moved by explicit elements of $\ker\bar d_k\text{:}$ modifications whose slots are $2^{k-2}$-nd powers of the canonical lifts themselves, so that the brackets they would contribute vanish because an element commutes with its own powers. The digit each such modification carries is computed by lifting the exponent from the exact valuations

\begin{equation*} v_2(\notn{eta}{\eta}-1)=v_2(X-1)=v_2(S-1)=2, \end{equation*}

which place a nonzero digit at position exactly $k\text{.}$ Two such moves span $(\mathbb{Z}/2)^2\text{.}$

For the last assertion, combine (i) and (ii): correct $T$ by a modification realizing the inverse of $\delta_k(T)\text{,}$ then adjust inside $\ker\bar d_k\text{.}$ Generation survives because modifications lie in $\lambda_{k-1}\subseteq\notn{Frattini}{\Phi}(Q_k)\text{.}$ The base case $S_3\neq\emptyset$ is checked directly in $Q_3\text{,}$ a group of order $2^8\text{:}$ the triples $(y,sx,x)$ and $(SY,Y,\notn{Agen}{A})$ serve in the two directions.

Remark C.18.

The congruence clause is necessary and not merely convenient. Its two cokernel coordinates measure exactly the top orientation deviation digits of $T$ in the two untwisted slots, with the slots transposed, so the defect is reachable precisely when the clause holds. The sharpest illustration is the group $\angles{A,S,Y\;\middle|\;A^2S^8[S,Y]}\text{,}$ which has the same rank, the same $q$ and the same abelianization as $\notn{D0}{D_0}\text{,}$ and whose orientation differs only in the image. For it the analogue of $S_4$ is empty, because the relevant orientation value is $(1-8)^{-1}\equiv1$ rather than $\notn{eta}{\eta}\equiv5$ modulo $8\text{,}$ so no triple can achieve the required congruence depth. This is the point at which the surjectivity hypothesis of Proposition C.11 is consumed.

We can now prove Proposition C.11. Each $S_k$ is a finite set, the restriction maps $S_{k+1}\to S_k$ are defined because all three clauses weaken, and each $S_k$ is nonempty by Lemma C.17. Compactness of an inverse limit of nonempty finite sets, exactly as in the proof of Lemma 2.5, supplies a compatible family. Such a family is a continuous homomorphism from the free pro-$2$ group on three generators to $D$ that kills the tested relator and is surjective at every level, hence a continuous epimorphism from the tested group onto $D\text{.}$ Running the construction in both directions and composing gives a continuous surjective endomorphism of a topologically finitely generated profinite group, which is an isomorphism by Hopficity. Hence $D\cong \notn{D0}{D_0}\text{.}$

Applied to $D=D_R\text{,}$ which satisfies the hypotheses by Lemma C.9 and Proposition C.10, this yields $D_R\cong \notn{D0}{D_0}$ and completes the proof of Proposition C.12, hence of Proposition C.5 and of Theorem C.3.

Subsection C.6 Inputs

Theorem C.3 rests on the same classical inputs as Theorem 1.2 and on no others. The two places where a further input might have been expected are the classification of Demushkin groups and the identification of the pro-$2$ quotient. The first is used only through Proposition C.11, which is proved in Subsection C.5 from the filtration (C.9) and the orientation calculus rather than quoted; the second reduces to it. Everything else is either an item of Corollary 6.19, re-verified for (C.2) in Proposition C.5, or a part of the induction that never sees the wild word.

In the formalization this is checked mechanically rather than asserted: the axiom set reported for the formalized form of Theorem C.3 is identical to the one reported for Theorem 1.2, and the formalized form of Proposition C.11 reports no classical input at all. The whole chain is formalized without gaps.

Appendix D Worked examples: three small targets

The presentation turns extension-counting questions into finite computations: by Proposition 2.3, surjections $\GA\twoheadrightarrow G$ are exactly the generating quadruples $(\sigma,\tau,x_0,x_1)\in G^4$ satisfying the tame relation (1.5), the wild relation (1.6), and $x_0,x_1\in O_2(G)\text{,}$ while $|\Sur(\GQ,G)|=|\operatorname{Aut}(G)|\cdot\#\{G$-extensions of $\Qtwo\}\text{.}$ This appendix evaluates the auxiliary words (1.1)–(1.3) on three small targets, identifies the branch of the induction that each exercises, and compares the resulting counts with the known extension counts, as tabulated in the LMFDB’s database of $2$-adic fields [bib-LMFDB]; the same counts are the ground truth used by the finite-quotient verifier. Throughout, recall $x^g=g^{-1}xg\text{,}$ $[x,y]=x^{-1}y^{-1}xy\text{,}$ and that exponentiation by $\w$ extracts the $2$-primary part of an element of finite order: $g^{\w}=g$ when $g$ has $2$-power order, and $g^{\w}=1$ when $g$ has odd order.

One observation applies to every finite $2$-group target and explains why such targets probe exactly the maximal pro-$2$ quotient: the tame relation forces $\tau\mapsto1\text{.}$ Indeed, conjugate elements have equal orders, so $\tau^\sigma=\tau^2$ forces $|\tau|=|\tau^2|\text{,}$ which for an element of $2$-power order gives $\tau=1\text{.}$

Subsection D.1 A $2$-group target: the cyclic group of order four

Let $G=\mathbb Z/4\text{,}$ written multiplicatively as $\langle g\rangle\text{.}$ Here $O_2(G)=G\text{,}$ and $\tau\mapsto1$ by the observation above. Since every element has $2$-power order, $\sigma_2=\sigma^{\w}=\sigma$ and $u_i=(x_i\tau)^{\w}=x_i\text{.}$ The word ledger collapses:

\begin{equation*} u_0=x_0,\quad d_0=u_0x_0^{-1}=1,\quad z_0=x_0^{\sigma_2}=x_0,\quad c_0=[d_0,z_0]=1,\quad d_g=1,\quad h_c=1, \end{equation*}

and, because $G$ is abelian,

\begin{equation*} h_0=x_0^{g_0}\,x_0\,d_gd_0d_0^2h_c=x_0^2. \end{equation*}

The wild relation (1.6) therefore reads $x_0^2\,u_1^{-1}\,x_1^{\sigma}\,c_0=x_0^2\,x_1^{-1}x_1=x_0^2=1\text{:}$ on this target the relator sees only the elementary shadow $x_0^2$ of $h_0\text{,}$ exactly as in Lemma 5.3. An admissible quadruple is thus $(\sigma,\tau,x_0,x_1)=(g,1,x_0,x_1)$ with $x_0\in\{1,g^2\}$ and $x_1$ arbitrary, or any quadruple with $x_1$ a generator. Counting generating triples $(\sigma,x_0,x_1)$ with $x_0^2=1\text{:}$ the pair $(\sigma,x_1)$ must contain a generator of $G$ ($16-4=12$ ways) and $x_0$ is one of $2$ elements, so

\begin{equation*} |\Sur(\GA,\mathbb Z/4)|=2\cdot12=24. \end{equation*}

On the Galois side, local class field theory gives $\Qtwo^\times\cong 2^{\mathbb Z}\times\{\pm1\}\times(1+4\mathbb Z_2)\text{,}$ so $\Qtwo$ has exactly $12$ cyclic quartic extensions and $|\Sur(\GQ,\mathbb Z/4)|=12\cdot|\operatorname{Aut}(\mathbb Z/4)|=24\text{.}$ The counts agree. (For $G=\mathbb Z/2$ the same collapse makes the relator vacuous and yields $2^3-1=7$ surjections: the seven quadratic extensions of $\Qtwo\text{.}$)

Subsection D.2 A tame target: the symmetric group $S_3$

Let $G=S_3\text{,}$ so $O_2(G)=1$ and admissibility forces $x_0,x_1\mapsto1\text{.}$ Every auxiliary word then evaluates to the identity: $u_i=\tau^{\w}=1$ because $\tau$ has odd order (it is conjugate to its square), and consequently $d_0=z_0=c_0=d_g=h_c=h_0=1\text{,}$ so (1.6) holds automatically. The count is governed entirely by the $(\sigma,\tau)$-frame: we need generating pairs with $\tau^\sigma=\tau^2\text{.}$ Taking $\tau=(123)\text{,}$ the condition $\sigma^{-1}(123)\sigma=(132)$ holds exactly for the three transpositions $\sigma\in\{(12),(13),(23)\}\text{;}$ with the two choices of $3$-cycle $\tau$ this gives

\begin{equation*} |\Sur(\GA,S_3)|=6, \end{equation*}

and each such pair does generate $S_3\text{.}$ On the Galois side there is exactly one $S_3$-extension of $\Qtwo$ (the splitting field of $x^3-2\text{,}$ the Galois closure of the tamely ramified cubics), so $|\Sur(\GQ,S_3)|=1\cdot|\operatorname{Aut}(S_3)|=6\text{.}$ This target illustrates the tame frame of Section 3: $\sigma$ maps to geometric Frobenius in the residue extension and $\tau$ to a tame generator, with $\tau^\sigma=\tau^q\text{,}$ $q=2\text{.}$

Subsection D.3 A mixed target: the symmetric group $S_4$

Let $G=S_4\text{.}$ Now $L=O_2(G)=V_4=\{1,(12)(34),(13)(24),(14)(23)\}$ with $G/L\cong S_3\text{,}$ so a surjection carries a nontrivial elementary $2$-layer over a tame frame. Consider the quadruple

\begin{equation*} \sigma=(12),\qquad \tau=(123),\qquad x_0=(12)(34),\qquad x_1=1. \end{equation*}

Both tame data are as in the $S_3$ example (lifted along $S_4\to S_3$), and $x_0,x_1\in O_2(G)\text{.}$ The words evaluate as follows. Since $\sigma$ has order $2\text{,}$ $\sigma_2=\sigma$ and $g_0=\sigma_2^2=1\text{.}$ The product $x_0\tau=(12)(34)(123)=(134)$ has odd order, so $u_0=(x_0\tau)^{\w}=1$ and likewise $u_1=\tau^{\w}=1\text{.}$ Then

\begin{equation*} d_0=u_0x_0^{-1}=x_0,\qquad z_0=x_0^{\sigma_2}=((12)(34))^{(12)}=(12)(34)=x_0, \end{equation*}

so $c_0=[d_0,z_0]=[x_0,x_0]=1\text{,}$ $d_g=d_0^{g_0}=d_0\text{,}$ $h_c=[d_g,d_0]=1\text{,}$ and

\begin{equation*} h_0=x_0^{g_0}\,x_0\,d_gd_0d_0^2h_c =x_0^2\,d_0^2\,d_0^2=1. \end{equation*}

The wild relation $h_0u_1^{-1}x_1^\sigma c_0=1$ holds, the four elements generate $S_4$ (the frame generates a copy of $S_3$ and $(12)\cdot(12)(34)=(34)$ supplies the fourth point), and the quadruple is admissible.

This target exercises the first genuinely mixed branch of the induction. In the tower (7.1) the marked kernel is $L_Y=V_4\text{,}$ whose unique chief factor is the standard two-dimensional simple $\F_2[S_3]$-module, which is non-scalar, so the recursion of Theorem 8.17 applies with $K=V_4$ and $R=\Phi(V_4)=1\text{:}$ only the elementary stage (8.13) is active, with no Frattini or scalar-phase layer. The exact-image bookkeeping is visible concretely: maps whose image is a proper subgroup containing the frame (for example the four point-stabilizer copies of $S_3\text{,}$ with $x_0,x_1\mapsto1$) are subtracted by Lemma 8.12, leaving the count of surjections with image exactly $S_4\text{.}$ Summing the resulting counts over the six tame frames as in Lemma 10.1 gives

\begin{equation*} |\Sur(\GA,S_4)|=72=3\cdot|\operatorname{Aut}(S_4)|, \end{equation*}

matching the three $S_4$-extensions of $\Qtwo\text{,}$ the Galois closures of the three quartic fields with Galois group $S_4$.

Subsection D.4 The finite-quotient verifier’s input

The same comparisons, run mechanically over a battery of 5,402 potentially $2$-realizable finite groups, constitute the finite-quotient verification described at github.com/roed314/padicIGP. The verifier consumes the presentation as the following relation file, whose lines are exactly the words (1.1)–(1.3) and the relator (1.6); the integer $40491355905$ represents $\w$ modulo $85667662080=2^8\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\text{,}$ which suffices for every group in the battery:

sigma,tau,x0,x1

sigma2 = sigma^40491355905
g0 = sigma2^2
u0 = (x0*tau)^40491355905
u1 = (x1*tau)^40491355905
d0 = u0*x0^-1
z0 = x0^sigma2
c0 = d0^-1*z0^-1*d0*z0
dg = d0^g0
hc = dg^-1*d0^-1*dg*d0
h0 = x0^g0*x0*dg*d0*d0^2*hc

h0*u1^-1*x1^sigma*c0

For the four groups above, the verifier’s ground-truth extension counts are $7$ (for $\mathbb Z/2$), $12$ (for $\mathbb Z/4$), $1$ (for $S_3$), and $3$ (for $S_4$), in agreement with the quadruple counts computed here. Passing all 5,402 comparisons is strong evidence for the presentation but is not a proof; the proof is the body of this paper.

Appendix E Background

This appendix collects short background summaries for the external inputs of the paper. Each subsection states the facts in the form they are used, with citations to specific locations in the references, and points back to the statements that use them.

Subsection E.1 Profinite group preliminaries

A profinite group is an inverse limit of finite groups, and the single topological input we use is compactness, in the following combinatorial form: the inverse limit of an inverse system of nonempty finite sets over a directed poset is nonempty [RZ, Proposition 1.1.4]. The proof is exactly the finite intersection property of closed subsets of the compact product: partial compatible families form nonempty closed sets, any finitely many of which meet, so a fully compatible family exists. This is the argument invoked in Lemma 2.5 to assemble a single epimorphism out of epimorphisms onto the members of a directed system of finite quotients. The sets being intersected there are finite because a continuous homomorphism from a topologically finitely generated profinite group to a finite group is determined by its values on a fixed finite generating set; in particular a finitely generated profinite group has only finitely many open normal subgroups of each index [RZ, Proposition 2.5.1].

Finitely generated profinite groups are Hopfian: every continuous epimorphism $\varphi:G\to G$ of a topologically finitely generated profinite group $G$ onto itself is an isomorphism [RZ, Proposition 2.5.2]. The mechanism is again finiteness: $\varphi^{-1}$ induces an injection of the finite set of open normal subgroups of index at most $n$ into itself, hence a bijection, so $\ker\varphi$ lies in every open normal subgroup. Hopficity is used twice in this paper: in Lemma 2.5, where the composite of the two constructed epimorphisms $P\twoheadrightarrow Q\twoheadrightarrow P$ is a surjective endomorphism and therefore an isomorphism, and in Lemma 3.8, where endomorphisms already known to be surjective are upgraded to automorphisms. (Continuity of the inverse is automatic for compact groups.)

The Frattini subgroup $\PhiG(G)$ of a profinite group is the intersection of its maximal open subgroups. A closed subgroup $H$ with $H\PhiG(G)=G$ equals $G$ [RZ, Corollary 2.8.5], and for a pro-$p$ group $\PhiG(G)=\overline{G^p[G,G]}\text{,}$ so that the Frattini quotient $G/\PhiG(G)$ is a vector space over $\F_p$ [RZ, Lemma 2.8.7]; for a finite $2$-group $K$ this is the formula $\PhiG(K)=K^2[K,K]$ used at the head of Section 7. Together these give the pro-$p$ Burnside basis theorem: a subset of a pro-$p$ group generates it if and only if its image spans the Frattini quotient, and minimal generating sets correspond to bases of $G/\PhiG(G)\text{;}$ this is the “Frattini argument” of [NSW, Proposition (3.9.1)], where it is expressed as $d(G)=\dim H^1(G,\F_p)\text{.}$ In Lemma 3.8 this is how surjectivity is checked: the endomorphism $\theta_u$ sends a free basis to a set that is again a basis modulo the Frattini subgroup, and the images of the three generators under $\Psi_u$ form a triangular basis of the Frattini quotient; Hopficity then converts surjectivity into automorphy.

Fibre products of profinite groups exist and are computed naively: the pullback of $\beta_1:H_1\to H$ and $\beta_2:H_2\to H$ is the closed subgroup of $H_1\times H_2$ of pairs with equal image in $H$ [RZ, Section 2.10]. Two facts about them are used in Section 4, where the boundary (4.1) is such a fibre product. First, for closed normal subgroups $U,V$ of $\Gamma$ the natural square exhibits $\Gamma/(U\cap V)$ as the fibre product of $\Gamma/U$ and $\Gamma/V$ over $\Gamma/UV$ [RZ, Exercise 2.10.1]; consequently, the image of a pair of epimorphisms is the fibre product of the two targets over their largest common quotient compatible with the pair. Second, and equivalently, a pair of epimorphisms $\varphi_i:\Gamma\to H_i$ with $\beta_1\varphi_1=\beta_2\varphi_2$ maps onto the fibre product precisely when $\ker(\beta_1\varphi_1) =\ker(\varphi_1)\ker(\varphi_2)$ [RZ, Lemma 2.10.2]. This is the content of the phrase profinite Goursat in Section 4: surjectivity of $b_\Gamma$ reduces to the statement that the largest common quotient of $\TA$ and $\Pi$ glued by the pair is $\mathbb{Z}_2$ itself, which holds because every common quotient is pro-$2\text{,}$ hence a quotient of the maximal pro-$2$ quotient $\mathbb{Z}_2$ of $\TA\text{,}$ and the full $\mathbb{Z}_2$ occurs.

Sylow theory extends to profinite groups by the compactness argument of the first paragraph: $p$-Sylow subgroups exist and are conjugate [RZ, Corollary 2.3.6], and likewise $\pi$-Hall subgroups of prosolvable groups [RZ, Corollary 2.3.7]. The form we use is the Schur–Zassenhaus theorem: a closed normal Hall subgroup $K$ of a profinite group $G$ admits a complement, and any two complements are conjugate [RZ, Theorem 2.3.15]. In Lemma 9.2 and Lemma 7.2 it is applied to finite targets: the preimage of the odd-order kernel $N=\ker(H\to H_2)$ in $Y$ is an extension of $N$ by the finite $2$-group $L_Y\text{,}$ so Schur–Zassenhaus provides an odd-order complement $\widetilde N\text{,}$ the odd Hall lift, mapping isomorphically onto $N\text{.}$ Since the preimage is the product of $L_Y$ and $\widetilde N\text{,}$ the conjugating element between two complements may be taken in $L_Y\text{;}$ this is the $L_Y$-conjugacy used to pin $\widetilde N$ down as normal once it is known to centralize $L_Y\text{.}$

Free pro-$2$ groups enter through three standard facts. A free pro-$C$ group on a finite set is characterized by the usual universal property [RZ, Lemma 3.3.4], and any generating set of the same finite cardinality is again a basis [RZ, Lemma 3.3.5]; combined with the Frattini criterion above, this is the change-of-basis principle behind the substitution $P=S^3\text{,}$ $T=S^{-3}A^{-2}$ in Lemma 3.8 and the Nielsen change (3.16) of Proposition 3.11. Next, powers with profinite exponents: for $x$ in a profinite group and $\lambda\in\Zhat\text{,}$ the power $x^\lambda$ is defined through the unique continuous epimorphism from $\Zhat$ onto the closed procyclic subgroup generated by $x\text{,}$ and the exponent laws follow from the integer case by continuity [RZ, Section 4.1]. On a pro-$2$ group the exponent acts through $\mathbb{Z}_2\text{,}$ so for odd $m$ the power map $x\mapsto x^m$ is a bijection with inverse $x\mapsto x^{m^{-1}}\text{,}$ $m^{-1}\in\mathbb{Z}_2^\times\text{;}$ this is the unique-cube-root step of Lemma 3.8. Finally, HNN extensions exist in the pro-$2$ category: for a variety $\mathcal C$ of finite groups (such as finite $2$-groups) and a continuous isomorphism $f:A\to B$ of closed subgroups of a pro-$\mathcal C$ group $H\text{,}$ there is a unique pro-$\mathcal C$ HNN extension $\mathrm{HNN}(H,A,f)\text{,}$ characterized by the universal property that continuous homomorphisms out of it correspond to pairs consisting of a homomorphism on $H$ and an image of the stable letter intertwining $A$ and $B$ [RZ, Section 9.4, Proposition 9.4.1]. The relation (3.9) exhibits $D_0$ as such an extension of the free pro-$2$ group $F_{S,A}$ with stable letter $Y\text{,}$ and it is through this universal property that the automorphism $\theta_u$ of the base is extended over $Y$ in Lemma 3.8.

Subsection E.2 The Hochschild–Serre spectral sequence and low-degree exact sequence

To an extension of profinite groups $1\to N\to G\to Q\to 1\text{,}$ with $N$ a closed normal subgroup, and a discrete $G$-module $A\text{,}$ the Lyndon–Hochschild–Serre spectral sequence associates

\begin{equation*} E_2^{pq}=H^p(Q,H^q(N,A))\Longrightarrow H^{p+q}(G,A). \end{equation*}

For profinite $G$ and continuous cochain cohomology this is [NSW, Theorem 2.4.1]; for abstract groups it is constructed in [Bro, Chapter VII, §6]. All coefficient modules used in this paper ($\F_2\text{,}$ $V\text{,}$ $V^\vee$) are finite and discrete, so the continuous theory of [NSW] applies verbatim over $G_{\Qtwo}\text{,}$ and the two theories agree on the finite quotients where the normalized-cochain calculations of Lemma 6.1 and Lemma 6.4 take place.

The bottom of the spectral sequence is summarized by the five-term exact sequence of low degree

\begin{equation} 0\to H^1(Q,A^N)\xrightarrow{\operatorname{inf}} H^1(G,A)\xrightarrow{\operatorname{res}} H^1(N,A)^{Q}\xrightarrow{\operatorname{tg}} H^2(Q,A^N)\xrightarrow{\operatorname{inf}} H^2(G,A),\tag{E.1} \end{equation}

valid for closed normal $N$ [NSW, Proposition 1.6.7]; it is the edge-morphism sequence of any first-quadrant $E_2$-spectral sequence [NSW, Proposition 2.1.1]. The homological version for abstract groups is [Bro, Corollary VII.6.4]. The transgression $\operatorname{tg}$ has a direct cochain description (extend a $Q$-invariant cocycle on $N$ to a cochain on $G$ and take its coboundary [NSW, Proposition 1.6.6]) and coincides with the differential $d_2^{0,1}$ [NSW, Theorem 2.4.3]. When $H^i(N,A)=0$ for $0<i<n\text{,}$ the same sequence holds with $H^1$ and $H^2$ replaced by $H^n$ and $H^{n+1}$ (inflation–restriction) [NSW, Proposition 1.6.7]; degree-one instances underlie the descent of the radical edge class in Lemma 8.6 and the identification of $H^1(\Qtwo,V)$ in the unramified case of Proposition 6.18.

The spectral sequence refines (E.1) to a decreasing filtration $H^2(G,A)\supseteq F^1\supseteq F^2\supseteq 0$ on $H^2\text{:}$ the quotient $H^2/F^1$ embeds in $E_2^{0,2}=H^2(N,A)^Q\text{,}$ so $F^1$ consists exactly of the classes restricting to zero on the fibre $N\text{;}$ the bottom piece is $F^2=\operatorname{im}(\operatorname{inf})\text{;}$ and the intermediate graded piece $F^1/F^2$ embeds in $E_2^{1,1}=H^1(Q,H^1(N,A))$ [NSW, §§2.1-2.2]. Consequently there is an exact segment

\begin{equation} H^2(Q,A^N) \xrightarrow{\operatorname{inf}} \ker\bigl(H^2(G,A)\to H^2(N,A)\bigr) \longrightarrow H^1(Q,H^1(N,A)),\tag{E.2} \end{equation}

which for $1\to V\to G\to H\to 1$ with $A=\F_2$ is the sequence (6.22) used to prove uniqueness of the normalized base class in Lemma 6.4. A class in $F^1$ is said to have filtration one; its leading term is its image in $E_2^{1,1}\text{.}$ For the split extension $1\to V\to V\rtimes C\to C\to 1$ with central coefficients, a filtration-one class is represented at normalized-cochain level by a “linear-in-the-fibre” cocycle $\Gamma_\gamma$ together with an inflated scalar term, and this normalized representative is the difference decomposition (6.5) of Lemma 6.1. In the split case inflation is a split injection and all differentials into the horizontal edge vanish [NSW, Proposition 2.4.5].

Degree-two classes with central coefficients are the same thing as central extensions: choosing a set-theoretic section $s$ with $s(1)=1$ turns an extension of $G$ by $A$ into a normalized factor set $f(g,h)$ defined by $s(g)s(h)=f(g,h)\,s(gh)\text{,}$ and cohomologous normalized factor sets classify the extension up to equivalence [Bro, Chapter IV, §3]. Every class of a profinite group with finite discrete coefficients contains a normalized continuous $2$-cocycle [NSW, Chapter I, §2]. The equivariant factor sets, correction functions $m_c\text{,}$ and square maps attached to the quadratic modules of this paper are all built in this normalized framework; see Lemma 6.1 and the convention audit in Appendix A. The transgression obstruction computed relative to a fixed equivariant class in Lemma 6.21 is the analogous differential $d_2^{0,2}$ on the fibre $H^2\text{,}$ evaluated by the same section-comparison method; its role in the Gauss-sum comparison appears in Subsection 6.4.

The vanishing arguments in this paper follow a single coprimality pattern. If $U\le G$ has finite index then $\operatorname{cor}\circ\operatorname{res}$ is multiplication by $(G:U)\text{,}$ so positive-degree cohomology of a finite group is annihilated by the group order [Bro, III.10.1-10.2], [NSW, Propositions 1.6.1-1.6.2]. Hence an odd-order group has no higher cohomology with characteristic-$2$ coefficients. Feeding this into the Hochschild–Serre spectral sequence of $1\to I\to H\to H/I\to 1$ with $I$ of odd order and $(V^\vee)^I=0$ collapses every $E_2$-term and gives $H^*(H,V^\vee)=0\text{.}$ This is the argument used for the ramified case in Lemma 6.4 and Proposition 7.4, and, combined with Kummer theory and Shapiro’s lemma, for the unit-filtration computation of $H^1(\Qtwo,V)$ in Lemma 6.17.

Subsection E.3 Shapiro’s lemma, corestriction, and the normalized bar resolution

Shapiro’s lemma states that for a closed subgroup $H$ of a profinite group $G$ and a discrete $H$-module $A$ there is a canonical isomorphism

\begin{equation*} H^n(G,\operatorname{Ind}_H^G A)\;\cong\;H^n(H,A) \end{equation*}

for all $n\ge 0$ [NSW, Proposition 1.6.4], [Bro, Proposition III.6.2]. In this paper the standing instance is a Galois extension $K/F$ with finite group $H$ and the coinduced module $\F_2[H]\text{:}$ Shapiro identifies $H^1(G_F,\F_2[H])$ with $H^1(G_K,\F_2)\text{,}$ whose elements are the Kummer “scalar coordinates” $\alpha_r$ of Lemma 6.15. What the classical statement does not fix is a cochain-level realization; the choice-audited normalized Shapiro cochain map $\operatorname{Sh}$ used throughout Lemma 6.13, Lemma 6.14, and Lemma 6.15 is constructed in Appendix A (see (A.11) and its degree-one form (A.12)), together with the exact normalized coboundaries produced by any change of section.

For an open subgroup $H\le G$ (index $l$), restriction is supplemented by the wrong-way corestriction (transfer) $\operatorname{cor}_H^G:H^n(H,A)\to H^n(G,A)\text{,}$ with $\operatorname{cor}\circ\operatorname{res}=l\text{;}$ it can be defined by dimension shifting, by Shapiro’s lemma applied to the canonical surjection $\operatorname{Ind}_H^G A\to A\text{,}$ or directly on resolutions [Bro, Chapter III, §9]. On cochains it is a sum over a chosen coset transversal of translated arguments: for homogeneous continuous cochains this explicit formula is given in [NSW, Chapter I, §5.4], and [NSW, Proposition 1.6.5] identifies it as the coordinate-sum map under Shapiro. The inhomogeneous normalized version with transversal transition elements $\ell_\omega(\gamma)=t_\omega^{-1}\gamma t_{\gamma^{-1}\omega}\text{,}$ its compatibility $d\operatorname{cor}=\operatorname{cor}d\text{,}$ and the exact cup-product identity $\operatorname{cor}(\alpha\smile\beta)=\sum_h \operatorname{Sh}(\alpha)_h\,\operatorname{Sh}(\beta)_{\bar\gamma^{-1}h}$ are written out in Appendix A ((A.20), (A.24)); these are the “normalized bar corestriction” formulas invoked in Lemma 6.15 and Lemma 6.14, and, through Kummer theory, in Lemma 6.17.

Normalization refers to the normalized (reduced) standard resolution: the bar resolution of [Bro, Chapter I, §5] modulo its degenerate subcomplex, whose cochains are the inhomogeneous cochains vanishing whenever some argument equals $1\text{;}$ every $H^2$-class of a profinite group with finite coefficients is represented by a normalized continuous cocycle [NSW, Chapter I, §2]. Working systematically with normalized representatives is what makes the zero-section and fibre normalizations of Lemma 6.4 meaningful at cochain level rather than only in cohomology; the paper’s conventions are fixed once and for all in Appendix A.

The multiplicative companion of the additive transfer is the Evens norm $N_H^{G,\mathrm{Ev}}\text{,}$ defined via the monomial embedding of $G$ into the wreath product $S_l\wr H$ and a resolution of the form $W\otimes X^{\otimes l}$ [Eve, §§2, 4-5]. Its low-degree expansion

\begin{equation*} N_{H\to G}(1+\chi)=1+\operatorname{tr}_{H\to G}(\chi)+\cdots +N_{H\to G}(\chi) \end{equation*}

exhibits the ordinary transfer as the linear term [Eve, §7, Theorem 1], which at index two is the normalization identity (6.41) of Lemma 6.13. Instantiating Evens’s construction on normalized bar resolutions yields the explicit degree-two cochains recorded in Appendix A and evaluated on involution orbits in Lemma 6.15.

Subsection E.4 Local Tate duality and the Euler–Poincaré characteristic

For a $p$-adic local field $k\text{,}$ local class field theory provides the invariant map, a canonical isomorphism $\inv_k:H^2(G_k,\ol{k}^\times)\xrightarrow{\;\sim\;}\mathbb{Q}/\mathbb{Z}\text{;}$ see [NSW, Corollary (7.1.4)]. For every $n\ge1$ (the hypothesis $\operatorname{char}(k)\nmid n$ is vacuous in characteristic zero, so $n$ may share the residue characteristic), the Kummer sequence computes $H^1(k,\mu_n)\cong k^\times/k^{\times n}$ and $H^2(k,\mu_n)\cong\tfrac{1}{n}\mathbb{Z}/\mathbb{Z}\text{,}$ with $H^i(k,\mu_n)=0$ for $i\ge 3\text{;}$ see [NSW, Theorem (7.1.8)]. In particular $H^2(\GQ,\mu_2)\cong\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$ is the group of order two, which we identify with $\F_2\text{.}$ Since $\mu_2=\{\pm1\}\subset\Qtwo\text{,}$ the module $\mu_2$ carries the trivial Galois action, so $\mu_2\cong\F_2$ as $\GQ$-modules: the mod-$2$ cyclotomic twist is trivial, as used in Proposition 5.16. Composing a cup product with $\inv$ is what turns a degree-two class into a scalar; this is the step “applying the invariant map” in the variation formula of Lemma 8.6.

Local Tate duality is the statement that this scalar pairing is perfect. Let $A$ be a finite $G_k$-module and set $A'=\Hom(A,\mu)\text{,}$ where $\mu$ is the group of all roots of unity in $\ol{k}\text{,}$ the dualizing module of $G_k$ ([NSW, Theorem (7.2.4)]). Then the cup product

\begin{equation*} H^i(k,A')\times H^{2-i}(k,A)\longrightarrow H^2(k,\mu)\cong\mathbb{Q}/\mathbb{Z} \end{equation*}

induces, for $0\le i\le 2\text{,}$ isomorphisms of finite abelian groups $H^i(k,A')\cong H^{2-i}(k,A)^*\text{;}$ see [NSW, Theorem (7.2.6)] and, in the form $H^r(G,M^D)\cong H^{2-r}(G,M)^*$ with $M^D=\Hom(M,K_s^\times)\text{,}$ [Mil, Theorem I.2.1 and Corollary I.2.3]. When $A$ is killed by two, every homomorphism from $A$ to $\mu$ or to $K_s^\times$ takes values in $\mu_2\text{,}$ so $A'=\Hom(A,\mu_2)=A^\vee:=\Hom(A,\F_2)$ with the contragredient action, and duality becomes a perfect $\F_2$-valued pairing $H^i(\Qtwo,A)\times H^{2-i}(\Qtwo,A^\vee)\to H^2(\Qtwo,\mu_2)=\F_2\text{.}$ This is the pairing of Proposition 5.16; its case $i=1$ is the perfect degree-one duality invoked in Lemma 8.6, where nonvanishing of a class in $H^1(\Qtwo,T^\vee)$ is detected by pairing it against $H^1(\Qtwo,T)\text{.}$

The local Euler–Poincaré characteristic computes the alternating product of the (finite) orders $h^i(k,A)=\#H^i(k,A)\text{:}$

\begin{equation*} \chi(k,A)=\frac{h^0(k,A)\,h^2(k,A)}{h^1(k,A)}=\|\#A\|_k, \end{equation*}

where $\|x\|_k=q^{-v(x)}$ is the normalized absolute value of $k$ ($q$ the residue cardinality); see [NSW, Theorem (7.3.1)]. Equivalently $\chi(G,M)=(R:mR)^{-1}$ with $R$ the ring of integers and $m=\#M\text{;}$ see [Mil, Theorem I.2.8]. For $k=\Qtwo$ and an $\F_2[\GQ]$-module $V$ of dimension $d$ this reads $\|2^d\|_{\Qtwo}=(\mathbb{Z}_2:2^d\mathbb{Z}_2)^{-1}=2^{-d}\text{,}$ that is,

\begin{equation*} \dim H^1(\Qtwo,V)=\dim H^0(\Qtwo,V)+\dim H^2(\Qtwo,V)+d. \end{equation*}

This dimension count is the input cited in the inductive step of Subsection 2 and in the numerical formula of Proposition 5.16.

The paper applies these two facts together to self-dual modules. Suppose $V$ is a nontrivial simple $\F_2$-module of dimension $d$ on which $\GQ$ acts through a finite quotient, and $V\cong V^\vee\text{.}$ Simplicity gives $H^0(\Qtwo,V)=V^{\GQ}=0\text{;}$ duality in degree zero gives $H^2(\Qtwo,V)\cong H^0(\Qtwo,V^\vee)^*=0$ because $V^\vee\cong V$ is again nontrivial simple; and the Euler–Poincaré formula then yields $\dim H^1(\Qtwo,V)=d\text{.}$ This is the count used in both cases of Proposition 6.18: in the ramified case it produces the $d$-dimensional local quadratic space, and in the unramified case the same $d$-dimensional space is identified with a line over the Schur field $F_V\text{.}$ For self-dual $V$ the degree-one pairing is a perfect pairing of $H^1(\Qtwo,V)$ with itself; it is this self-duality that stands behind the cardinal identity $|X_+|^2=|H^1(\Qtwo,V)|$ of Lemma 6.17. The finer statement used there, that under the Kummer description of $H^1$ the pairing matches depth-$j$ and depth-$(2e-j)$ layers of the unit filtration (“Hilbert duality”), is a computation in local class field theory and is discussed with the dyadic square-class background.

Subsection E.5 Local reciprocity, Kummer theory, and the dyadic Hilbert symbol

Local class field theory furnishes the reciprocity map $\rec:\Qtwo^\times\to G_{\Qtwo}^{\mathrm{ab}}\text{,}$ an isomorphism onto a dense subgroup under which norm subgroups of finite abelian extensions correspond to the open subgroups of finite index; see [NSW, Corollary (7.1.5) and Theorem (7.2.3)] and [Ser79, Chapters XI--XIII]. Two normalizations are in circulation, differing by inversion. We use the classical (arithmetic) one: a uniformizer acts on the maximal unramified extension as the Frobenius substitution $x\mapsto x^q\text{,}$ and the units map onto inertia; this is [Ser79, Chapter XIII, §4, Proposition 13 and its Corollary], and likewise the convention of [Neu, Chapter V, §2]. Thus $\rec(2)$ is arithmetic Frobenius, exactly the normalization fixed in Lemma 3.5; because the unramified character $\nu_{\mathrm{ur}}$ of that lemma is normalized by geometric Frobenius (the inverse generator), one gets $\nu_{\mathrm{ur}}(\rec(2))=-1\text{.}$ On the multiplicative side, $\Qtwo^\times\cong 2^{\mathbb Z}\times\{\pm1\}\times(1+4\mathbb Z_2)$ with $1+4\mathbb Z_2\cong\mathbb Z_2\text{,}$ topologically generated by any unit $u$ with $v_2(u-1)=2\text{;}$ see [Ser73, Chapter II, §3.2--3.3, Proposition 8 and Theorem 2] and [Ser79, Chapter XIV, §4, Examples after Proposition 10]. Since $-3\equiv5\pmod 8\text{,}$ the unit $-3$ is such a generator, which is why the marked basis $\rec(-4),\rec(1/2),\rec(-3)$ of Lemma 3.5 and the abelianized relation in Subsection 3.4 present the pro-$2$ completion of $\Qtwo^\times$ as $\langle2\rangle_{\mathbb Z_2}\oplus\langle-1\rangle_{C_2}\oplus\langle-3\rangle_{\mathbb Z_2}\text{.}$

The $2$-adic cyclotomic character $\chi:G_{\Qtwo}\to\mathbb Z_2^\times$ is the action on $\mu_{2^\infty}\text{.}$ The maximal abelian extension of $\Qtwo$ is the compositum of the unramified tower with the totally ramified cyclotomic tower $\Qtwo(\mu_{2^\infty})\text{;}$ see [Ser79, Chapter XIV, §7, Theorem 2]. Under the arithmetic normalization the norm residue symbol acts on $2^{n}$-th roots of unity by $(u\,2^{v_2(a)},\Qtwo(\zeta)/\Qtwo)\,\zeta=\zeta^{u^{-1}}\text{:}$ units act by the inverse, and $\rec(2)$ fixes the cyclotomic tower; see [Neu, Chapter V, Theorem (2.4)] and [Ser79, Chapter XIV, §7, Remark and Exercise]. This is the computation $\chi(\rec(u))=u^{-1}\text{,}$ $\chi(\rec(2))=1$ behind the orientation row of Lemma 3.5, and it explains why the pair $(\nu_{\mathrm{ur}},\chi)$ sees the unramified and cyclotomic coordinates of $G_{\Qtwo}^{\mathrm{ab}}$ separately.

For any field $K$ of characteristic different from $2\text{,}$ Kummer theory identifies $H^1(G_K,\mu_2)\cong K^\times/K^{\times2}\text{,}$ the class of $a$ corresponding to the cocycle $s\mapsto s(\sqrt a)/\sqrt a\text{;}$ see [Ser79, Chapter X, §3(b)] and, for local fields together with $H^2(G_K,\mu_2)\cong\tfrac12\mathbb Z/\mathbb Z\text{,}$ [NSW, Theorem (7.1.8)(ii) and Corollary (7.1.4)]. Under this identification the cup product of two Kummer classes is the class of the symbol $(a,b)\text{,}$ and applying the invariant map gives the Hilbert symbol: $(a,b)=1$ if and only if $b$ is a norm from $K(\sqrt a)\text{;}$ see [Ser79, Chapter XIV, §2, Propositions 5 and 7] and [Neu, Chapter V, Proposition (3.1)]. For a local $K$ the pairing is a nondegenerate bilinear form on the $\F_2$-vector space $K^\times/K^{\times2}\text{;}$ see [Ser79, Chapter XIV, §2, Proposition 7(vi)] and, for $\mathbb R$ and $\mathbb Q_p\text{,}$ [Ser73, Chapter III, Theorem 2]; this is the mod-$2$ instance of local Tate duality [NSW, Theorem (7.2.6)]. Cohomological squares are Bockstein values, whence $(a,a)=(a,-1)\text{;}$ see [Ser79, Chapter XIV, §2, Exercise 1]. This is the precise sense of “under Kummer theory, cup product is the Hilbert symbol” in Lemma 3.5, and of the scalar cup/Bockstein ledger (3.18) in Subsection 3.4.

Over $\Qtwo$ everything is explicit. A unit is a square exactly when it is congruent to $1$ modulo $8\text{,}$ so $\Qtwo^\times/\Qtwo^{\times2}$ is of type $(2,2,2)$ with representatives $\{\pm1,\pm5,\pm2,\pm10\}\text{;}$ see [Ser73, Chapter II, §3.3, Theorem 4 and Corollary] and [Ser79, Chapter XIV, §4, Lemma 3]. Writing $\epsilon(u)=\tfrac{u-1}{2}$ and $\omega(u)=\tfrac{u^2-1}{8}$ modulo $2\text{,}$ the dyadic Hilbert symbol is

\begin{equation*} (2^\alpha u,2^\beta v)_2=(-1)^{\epsilon(u)\epsilon(v)+\alpha\omega(v)+\beta\omega(u)}, \end{equation*}

which is [Ser73, Chapter III, §1.2, Theorem 1]; the same values appear generator by generator in [Ser79, Chapter XIV, §4, Case (ii)]. Evaluated on the square-class basis $(-1,2,-3)$ of Lemma 3.5 (recall $-3\equiv5\pmod8$), the only nontrivial pairings are $(-1,-1)_2=-1$ and $(2,-3)_2=-1\text{,}$ which is the table producing the quadratic initial form $\alpha^2+\beta\gamma+\gamma\beta$ there and the pairing matrix of (3.18).

For a general dyadic field $k$ with $e=v_k(2)\text{,}$ write $U_i$ for the image of $1+\mathfrak p^i$ in $k^\times/k^{\times2}\text{.}$ The graded pieces of the unit filtration itself are $U^{(0)}/U^{(1)}\cong\bar k^\times$ and $U^{(i)}/U^{(i+1)}\cong\bar k\text{;}$ see [Ser79, Chapter IV, §2, Proposition 6]. Squares are governed by the quadratic defect: the defect of a unit is $0\text{,}$ or $4\mathfrak o\text{,}$ or $\mathfrak p^d$ with $d$ odd and $4\mathfrak o\subset\mathfrak p^d\subset\mathfrak o\text{,}$ every such value occurs, and the defect-$4\mathfrak o$ class is precisely the class generating the unramified quadratic extension; see [OMe, 63:2 through 63:6]. Deep units are squares: $1+4\mathfrak p\subseteq k^{\times2}\text{,}$ i.e. $U_{2e+1}=1\text{,}$ by the Local Square Theorem [OMe, 63:1], equivalently by the isomorphism $x\mapsto x^2:U^{(m)}\to U^{(m+e)}$ for $m>e$ of [Ser79, Chapter XIV, §4, Proposition 9]; the quantitative Hensel statement behind it (a root exists once $v(f(a_0))>2v(f'(a_0))$) is [FV, Chapter II, (1.3), Corollaries 3--4], and it is in exactly this form that Hensel’s lemma extracts the square root of $n/(2u-1)\in U_{2e+2}$ in the proof of Lemma 6.16. Consequently only the odd depths below $2e$ and the unramified layer at depth $2e$ carry square classes, the stratification recorded at (6.35); and since $v_k(-1-1)=e\text{,}$ one has $-1\in U_e\text{.}$ The complementary-depth orthogonality $U_i^\perp=U_{2e-i+1}$ of (6.36) is the standard duality bookkeeping for this stratification: it combines the norm criterion for the vanishing of a symbol with the conductor filtration of norm groups [Ser79, Chapter XV, §2, Corollary 2 to Theorem 1] and the square-class count against nondegeneracy [OMe, 63:9 and 63:13].

Two further classical inputs drive the cancellations in Lemma 6.16 and Lemma 6.17. First, the Steinberg relations $(a,-a)=1$ and $(a,1-a)=1$ hold for any Hilbert symbol; see [Ser79, Chapter XIV, §2, Proposition 4(iv) and Proposition 7(iv)], [Ser73, Chapter III, Proposition 2], or [Neu, Chapter V, Proposition (3.2)(v)]. They yield the moves used there: $(x,x)=(x,-1)\text{,}$ and $(2u,2u-1)=(2u,-1)$ from $1-2u=-(2u-1)\text{.}$ Second, in an unramified extension the norm is surjective on principal units, level by level: $N(U_L^{(n)})=U_K^{(n)}$ for $n\ge1\text{;}$ see [Ser79, Chapter V, §2, Propositions 1--3]. For the unramified quadratic extension $E=k(\sqrt d)$ of a dyadic local field, every unit of $k$ is a norm: $N(E^\times)=\mathcal O_k^\times\,k^{\times2}$ and $N(\mathcal O_E^\times)=\mathcal O_k^\times\text{,}$ hence $(u,d)=1$ for all units $u\text{;}$ see [OMe, 63:16 and 63:11a]. These are precisely the two uses of unramifiedness isolated in the proof of Lemma 6.16 (the depth estimate for Hensel’s lemma and the killing of the symbols involving the unramified class) and, through the Kummer-theoretic identification $H^1(\Qtwo,V)\cong\Hom_{H_V}(V^\vee,M_K)\text{,}$ they feed the unit-filtration computation of Lemma 6.17.

Subsection E.6 Demushkin groups and Labute’s classification

A pro-$p$ group $G$ is a Demushkin group if $H^1(G,\F_p)$ is finite dimensional, $H^2(G,\F_p)$ is one dimensional, and the cup product $H^1(G,\F_p)\times H^1(G,\F_p)\to H^2(G,\F_p)$ is a non-degenerate bilinear form; see [Lab, p. 106] and [NSW, Definition (3.9.9)]. The first two conditions say that $G$ is a one-relator pro-$p$ group: $G\cong F/(r)\text{,}$ where $F$ is a free pro-$p$ group of rank $n=\dim_{\F_p}H^1(G,\F_p)$ and $(r)$ is the closed normal subgroup generated by a single relator. Two invariants are read off from the abelianization $G^{\mathrm{ab}}\cong \mathbb{Z}_p^{\,n-1}\times \mathbb{Z}_p/q\mathbb{Z}_p\text{:}$ the rank $n$ and the power $q=q(G)$ of $p$ (with $q=0$ when $G^{\mathrm{ab}}$ is torsion free); see [NSW, p. 232]. The only finite Demushkin group is $\mathbb{Z}/2\mathbb{Z}\text{,}$ the absolute Galois group of $\mathbb{R}$ ([NSW, Proposition (3.9.10)]); all others are infinite of cohomological dimension $2\text{.}$ The example that drives Section 3 is the maximal pro-$2$ quotient $G_{\Qtwo}(2)\text{,}$ a Demushkin group with $n=3$ and $q=2$ ([NSW, Theorem (7.5.11)(ii) and Chapter VII §5, p. 417]).

The classification is by normal forms for the relator. Write $[x,y]=x^{-1}y^{-1}xy\text{.}$ For $q\neq2$ (the case treated by Demushkin) the rank $n$ is even and one can choose a basis $x_1,\dots,x_n$ of $F$ with

\begin{equation*} r=x_1^{q}[x_1,x_2][x_3,x_4]\cdots[x_{n-1},x_n], \end{equation*}

so $G$ is determined by $(n,q)\text{;}$ see [NSW, Theorem (3.9.11)]. For $q=2$ this fails, and Serre introduced a finer invariant, the image of a canonical character $\chi\colon G\to\mathbb{Z}_2^\times$ described below. When $q=2$ and $n$ is odd there is a basis with

\begin{equation*} r=x_1^{2}\,x_2^{2^f}\,[x_2,x_3][x_4,x_5]\cdots[x_{n-1},x_n], \qquad f\in\{2,3,\dots,\infty\}, \end{equation*}

and then $\im(\chi)=\{\pm1\}\times(1+2^f\mathbb{Z}_2)\text{;}$ this case is due to Serre, see [Ser63, Théorème 3.2, p. 147], [Lab, p. 107 and Theorem 3], and [NSW, Theorem (3.9.19)(i)]. (For $q=2$ and $n$ even there are two further families, Labute’s main theorem; see [Lab, Theorem 1].) The invariants are complete: two Demushkin groups with the same $n$ and $\im(\chi)$ are isomorphic, and indeed any two relators in $F$ yielding such groups differ by an automorphism of $F$ ([Lab, Theorem 2]). Basis changes of this kind are the Nielsen transformations used in Proposition 3.11. Taking $n=3\text{,}$ $q=2\text{,}$ $f=2$ gives the relator $A^2S^4[S,Y]\text{,}$ that is, the standard rank-three dyadic Demushkin group $D_0$ of (3.1) studied in Subsection 3.1.

Every Demushkin group carries a canonical character $\chi\colon G\to\mathbb{Z}_p^\times\text{,}$ called its orientation in this paper. Serre defined it as the unique continuous homomorphism such that, letting $I(\chi)$ denote $\mathbb{Z}_p$ with $G$ acting through $\chi\text{,}$ the maps $H^1(G,I/p^{j+1}I)\to H^1(G,I/p^{j}I)$ are surjective for all $j\ge1$ ([Lab, §3, Proposition 6 and Theorem 4]). Conceptually, an infinite Demushkin group has cohomological dimension $2$ and is precisely a Poincaré duality group of dimension $2$ ([NSW, p. 231 and Theorem (3.9.15)]): its dualizing module is isomorphic to $\mathbb{Q}_p/\mathbb{Z}_p$ as an abelian group, $G$ acts on it through $\Aut(\mathbb{Q}_p/\mathbb{Z}_p)=\mathbb{Z}_p^\times\text{,}$ and the resulting character is $\chi$ ([NSW, Definition (3.7.1)]; the identification of Serre’s invariant with the dualizing-module character is due to Tate, see [Lab, p. 107, Remark 1] and [Ser63, no. 9.3, pp. 153-154]). Since the dualizing module is constructed directly from the cohomology of the open subgroups of $G\text{,}$ namely as a direct limit of the duals $H^n(U,-)^\ast$ ([NSW, Theorem (3.4.1) and Chapter III §7]), every isomorphism of Demushkin groups intertwines the orientations and every automorphism preserves them. This functoriality is exactly what is invoked in Proposition 3.9 and in Subsection 3.3. On the odd-rank $q=2$ normal form above, the orientation takes the values $\chi(x_1)=-1\text{,}$ $\chi(x_2)=1\text{,}$ $\chi(x_3)=(1-2^f)^{-1}\text{,}$ and $\chi(x_i)=1$ for $i>3$ ([Lab, Theorem 4, proof, case (2)]); for $D_0$ (where $f=2$) this is the character $(-1,\,1,\,(-3)^{-1})$ on $(A,S,Y)$ appearing in Lemma 3.4. The invariant $q(G)$ is recovered from $\chi$ as the largest power of $p$ with $\im(\chi)\subseteq 1+q\mathbb{Z}_p$ ([Lab, Corollary to Theorem 4]).

The arithmetic source of these groups is local. Let $K$ be a $p$-adic field of degree $N$ over $\mathbb{Q}_p$ containing the $p$-th roots of unity, and let $G_K(p)$ be the Galois group of its maximal pro-$p$ extension $K(p)\text{.}$ Then $G_K(p)$ is a Demushkin group of rank $N+2$ whose dualizing module is the group $\mu_{p^\infty}$ of all $p$-power roots of unity in $K(p)$ ([NSW, Theorem (7.5.11)(ii)]); consequently its canonical orientation is the $p$-adic cyclotomic character (see also [Lab, §5, p. 131]). By local class field theory $G_K(p)^{\mathrm{ab}}$ is the pro-$p$ completion of $K^\times\text{,}$ so $q(G_K(p))=\#\mu(K)(p)$ ([Lab, §5, p. 130]), and under the Kummer identification $H^1(G_K(p),\F_p)\cong K^\times/K^{\times p}$ the cup product is the Hilbert symbol ([Lab, §5, p. 131]; [Ser79, Chapter XIV §§1-2]), which is how the dyadic Hilbert-symbol table enters Lemma 3.5. For $K=\Qtwo$ one has $N=1$ odd and $q=2\text{,}$ and the classification yields three generators $x,y,z$ with the single relation $x^2y^4[y,z]=1$ ([Lab, Theorem 8]; [NSW, Chapter VII §5, p. 417]), with $\im(\chi)=\{\pm1\}\times(1+4\mathbb{Z}_2)=\mathbb{Z}_2^\times$ ([Lab, §5, pp. 131-132]). This is the abstract identification $G_{\Qtwo}(2)\cong D_0$ of Lemma 3.4, with cyclotomic orientation.

It is worth isolating what the classification does not provide. It guarantees some isomorphism $G_{\Qtwo}(2)\to D_0\text{,}$ unique at best up to composition with automorphisms, and it retains no memory of the arithmetic marking: the full unramified character $\nu_{\mathrm{ur}}\colon G_{\Qtwo}(2)\twoheadrightarrow\mathbb{Z}_2$ is not among the classifying invariants. The boundary construction of this paper requires the pair $(G_{\Qtwo}(2),\nu_{\mathrm{ur}})$ up to isomorphism of pairs, a strictly finer datum: see Remark 3.15, and see Remark 3.13 for a familiar Labute-style presentation that admits no compatible marking even though the underlying abstract group is correct. Remark 3.6 shows that the abelianized relation together with the quadratic initial form still does not pin down the marked identification. The canonical orientation restores the missing rigidity: because every abstract isomorphism intertwines the canonical orientations, its effect on the abelianization automatically preserves the orientation character, and Proposition 3.9 shows that all such automorphisms of the abelianization lift to $\Aut(D_0)\text{,}$ so an arbitrary abstract isomorphism can be corrected to a marked one (Subsection 3.3).

Finally, the phrase quadratic initial form refers to the leading term of a relator along a central filtration of the free group. For a power $q$ of $p\text{,}$ the descending $q$-central series of a free pro-$p$ group $F$ is defined by $F^1=F$ and $F^{i+1}=(F^i)^q[F^i,F]$ ([NSW, Definition (3.8.1)]; [Lab, §2, p. 112]). Every relator $\rho\in F^2=F^q[F,F]$ has a representation

\begin{equation*} \rho\equiv \prod_{j=1}^{n} x_j^{\,q a_j}\; \prod_{1\le k<l\le n}[x_k,x_l]^{\,a_{kl}} \pmod{F^3}, \end{equation*}

with the exponents $a_{kl}$ (and, for $q\neq0\text{,}$ the $a_j$) uniquely determined in $\mathbb{Z}_p/q\mathbb{Z}_p$ ([NSW, Proposition (3.9.13)(i)]); this collection of exponents is the quadratic initial form of $\rho\text{.}$ It computes the degree-two cohomology of $G=F/(\rho)\text{:}$ in the dual basis $\chi_1,\dots,\chi_n$ of $H^1(G,\mathbb{Z}/q)$ and with the trace isomorphism $\mathrm{tr}_\rho$ on $H^2(G,\mathbb{Z}/q)\text{,}$ the cup products are $\mathrm{tr}_\rho(\chi_k\cup\chi_l)=-a_{kl}$ for $k<l$ and the Bockstein satisfies $\mathrm{tr}_\rho(B\chi_j)=-a_j$ ([NSW, Propositions (3.9.13)(ii) and (3.9.14)]; [Lab, §2, Proposition 3]); for $q=2$ the diagonal cup squares $\mathrm{tr}_\rho(\chi_j\cup\chi_j)$ read off the same exponents $a_j$, which is why Lemma 3.5 speaks of a single Bockstein/cup form such as $\alpha^2+\beta\gamma+\gamma\beta\text{.}$ For the standard relator $r_0=A^2S^4[S,Y]$ the factor $S^4=(S^2)^2$ lies in $F^3\text{,}$ so the initial form is $A^2+[S,Y]\text{,}$ as used in Remark 3.6. Section 3 states such reductions modulo the third term of the Zassenhaus filtration $F_{(n)}=\{g\mid g-1\in I^n\}\text{,}$ where $I$ is the augmentation ideal of $\F_p[[F]]$ ([NSW, p. 230]; proved in [Koc, §7.4, Theorems 7.11 and 7.12]). From the two containments $(F_{(i)})^p\subseteq F_{(ip)}$ and $[F_{(i)},F]\subseteq F_{(i+1)}$ recorded there, one checks by induction that $F^i\subseteq F_{(i)}$ (the containment is also recorded in the proof of [Koc, Theorem 7.13]), so a congruence modulo $F^3$ implies the corresponding congruence modulo the third Zassenhaus term.

Subsection E.7 Extraspecial groups, Heisenberg models, and the square-map correspondence

Commutators are normalized as $[x,y]=x^{-1}y^{-1}xy\text{,}$ so that $xy=yx[x,y]\text{;}$ this is the convention of [Asc, Section 8] and the one fixed in Lemma 5.2. The general expansion identities are $[ab,c]=[a,c]^b[b,c]$ and $[a,bc]=[a,c][a,b]^c$ [Asc, (8.5.4)]. In a group of class at most two the conjugations act trivially on commutators, so the commutator map becomes bimultiplicative; more precisely, if $z=[x,y]$ centralizes $x$ and $y\text{,}$ then $[x^n,y^m]=z^{nm}$ and

\begin{equation} (yx)^n=z^{n(n-1)/2}y^nx^n \qquad (n\ge 0)\tag{E.3} \end{equation}

[Asc, (8.6)]. For $n=2$ this is the collector identity $(xy)^2=x^2y^2[y,x]$ used throughout the class-two calculations; when commutators are central involutions the direction of the commutator is immaterial. A central extension of an elementary abelian $2$-group by an elementary abelian $2$-group is exactly this setting: squares and commutators land in the central subgroup [Asc, (23.2)], so the group has class at most two, commutators have order at most two, and fourth powers vanish; these are the hypotheses under which Lemma 5.2 rearranges (E.3) into its exact collector form. The same commutator toolkit supplies the three-subgroup lemma [Asc, (8.7)] invoked in Lemma 7.2.

A central extension $1\to\F_2\to\widetilde V\to V\to 1$ of an elementary abelian $2$-group $V$ is described by a normalized factor set $f:V\times V\to\F_2\text{,}$ and its equivalence class by the cohomology class of $f\text{:}$ this is the classical dictionary between extensions with abelian kernel and $H^2$ [Bro, Chapter IV, Theorem 3.12]. Two functions on $V$ are extracted from the group $\widetilde V\text{.}$ The squaring map $\widetilde v\mapsto\widetilde v^2$ takes central values and depends only on the image $v\in V\text{,}$ defining $q:V\to\F_2\text{;}$ the commutator induces a pairing $b(v,w)=[\widetilde v,\widetilde w]\text{,}$ bilinear by the class-two expansions above. The collector identity $(xy)^2=x^2y^2[x,y]$ says precisely that

\begin{equation*} q(v+w)=q(v)+q(w)+b(v,w), \end{equation*}

so $q$ is a quadratic map whose polar form is the commutator pairing, the computation carried out in [Asc, (23.10), proof]. Conversely, any factor set with $f(v,v)=q(v)$ realizes a prescribed quadratic map $q\text{,}$ as in the explicit models of Lemma 6.1. Cohomologous factor sets yield the same $q\text{,}$ a class with $q=0$ is split (all lifts are then commuting involutions, so a basis of $V$ lifts to a complement), and every quadratic map arises; the resulting bijection

\begin{equation*} H^2(V,\F_2)\;\xrightarrow{\;\sim\;}\; \{\text{quadratic maps } V\to\F_2\} \end{equation*}

is the square-map correspondence of Lemma 6.4: the polarization recovers the commutator form, and the square and commutator data determine the extension class.

A finite $p$-group is special if its Frattini subgroup, center, and derived subgroup coincide, and extraspecial if in addition the center is cyclic, hence of order $p$ [Asc, Section 23 and (23.7)]. Extraspecial $2$-groups are exactly the fibre extensions above whose commutator pairing is nondegenerate [Asc, (23.10)]: on $V=E/Z$ the commutator form is symplectic, the squaring map is a quadratic form $q$ polarizing to it, $\dim V=2n$ is even, and $|E|=2^{1+2n}\text{.}$ Under this dictionary a subgroup between $Z$ and $E$ is elementary abelian exactly when its image in $V$ is totally singular [Asc, (23.10.4)]. Over $\F_2$ a nondegenerate quadratic space of dimension $2n$ is isometric to exactly one of $D^n$ (hyperbolic) or $D^{n-1}Q$ (one definite plane) [Asc, (21.2)], and correspondingly there are exactly two extraspecial groups of order $2^{1+2n}\text{:}$ the central products $D_8^n$ and $D_8^{n-1}Q_8\text{,}$ distinguished by their $2$-rank [Asc, (23.14)], with central products taken with identified centers [Asc, (11.1), (11.2)]. The building blocks are the two-dimensional cases: the hyperbolic plane $q_{\mathrm{hyp}}(x_1,x_2)=x_1x_2$ corresponds to the dihedral group $D_8$ (the two coordinate lifts are involutions, their commutator is the central involution, and their product has order four), while the definite plane corresponds to $Q_8\text{,}$ all of whose noncentral elements have order four [Asc, (23.14), proof, and (21.1)]. The dihedral case is exactly the fibre extension computed in Lemma 6.13.

The coefficient group in which the paper computes cup products is the finite Heisenberg model of Subsection 5.2: for an elementary module $A\text{,}$

\begin{equation*} \mathcal H(A)=A\times A^\vee\times\F_2, \qquad (a,\lambda,z)(a',\lambda',z') =(a+a',\lambda+\lambda',z+z'+\lambda(a')), \end{equation*}

with diagonal action of the acting group. In the language above, $\mathcal H(A)$ is the central extension of $V=A\oplus A^\vee$ given by the factor set $((a,\lambda),(a',\lambda'))\mapsto\lambda(a')\text{;}$ its square map is the hyperbolic quadratic form $q(a,\lambda)=\lambda(a)\text{,}$ and its commutator pairing is the evaluation pairing $\lambda(a')+\lambda'(a)\text{.}$ Since that pairing is nondegenerate, $\mathcal H(A)$ is extraspecial of hyperbolic type, a central product of $\dim A$ copies of $D_8$ by the classification just recalled [Asc, (23.10), (23.14)]; the Heisenberg presentation is its standard polarized form, adapted to the splitting $V=A\oplus A^\vee\text{.}$ This finiteness is what the deformation section exploits: the Stokes identities of Lemma 5.7 and Proposition 5.8 are proved by conjugating finite word evaluations in $\mathcal H(A)\rtimes C\text{,}$ with no cellular diagonal approximation, and coefficient naturality is checked in the mixed model $A\times A'^\vee\times\F_2$ of Lemma 5.6.

The splitting theory used for the odd part of finite targets is classical. The Schur–Zassenhaus theorem states that a normal Hall subgroup $H$ of a finite group $G$ has a complement, and that all complements are conjugate; the solvability hypothesis in the standard statement can be removed by the odd order theorem [Asc, (18.1) and Remarks]. For solvable groups, Hall’s theorem provides Hall $\pi$-subgroups for every set of primes $\pi\text{,}$ again conjugate and containing every $\pi$-subgroup [Asc, (18.2), (18.5)]. In Lemma 7.2 and Lemma 9.2 these are applied to the preimage of the odd-order kernel $N=\ker(H\to H_2)\text{,}$ an extension of $N$ by a finite $2$-group: Schur–Zassenhaus furnishes the odd Hall lift $\widetilde N\text{,}$ and conjugacy of complements combines with the centralizing argument to make the lift unique and normal. The accompanying coprime-action identity $[G,A]=[G,A,A]$ for a $2'$-group of automorphisms of a $2$-group [Asc, (24.5)] is what collapses the chief-series commutator chains in both lemmas. The profinite form of Schur–Zassenhaus, used when the same splitting is needed before passage to a finite quotient, is recalled in Subsection E.1 with the Ribes–Zalesskii reference.

Subsection E.8 Quadratic forms over $\F_2$ and the Arf invariant

Over a field of characteristic $2\text{,}$ a quadratic form on a vector space $V$ is a function $q:V\to F$ satisfying $q(ax+by)=a^2q(x)+b^2q(y)+ab\,b_q(x,y)$ for a (then unique) bilinear form $b_q\text{;}$ equivalently $b_q(x,y)=q(x+y)+q(x)+q(y)\text{,}$ the polar form of $q\text{,}$ which is symmetric and alternating [Gro, Chapter 12, pp. 113--114] [Wal, §1.5, (1.5.1)]. In contrast with odd characteristic, $q$ is not determined by $b_q\text{.}$ The radical $\operatorname{rad}V$ is the kernel of $b_q\text{;}$ following Grove, $q$ is nondefective when the radical is zero, which forces $\dim V$ to be even [Gro, Chapter 12, p. 114]. Nondefectiveness is what this paper calls nonsingularity of $q\text{.}$ A vector $v\ne0$ with $q(v)=0$ is singular, and a subspace on which $q$ vanishes identically is totally singular; a totally singular subspace is in particular totally isotropic for $b_q$ [Gro, Chapter 12, p. 114]. In the equivariant setting of Section 6 the radical is the pivot: for a form invariant under a group acting irreducibly, the radical of a nonzero invariant polar form is a proper submodule, hence zero. This step closes both Lemma 6.20 and the nonsingularity argument of Proposition 7.4.

A hyperbolic pair is a pair $(u,v)$ of singular vectors with $b_q(u,v)=1\text{,}$ spanning a hyperbolic plane; this is stronger than a hyperbolic pair for the alternating form alone, for which we reserve the term symplectic plane [Gro, Chapter 12, p. 115]. Every singular vector outside the radical lies in a hyperbolic pair [Gro, Proposition 12.1], and a nondefective plane with no singular vector (an anisotropic plane) has, in a suitable basis, the form $q(a_1v_1+a_2v_2)=a_1^2+a_1a_2+ba_2^2$ with $x^2+x+b$ irreducible [Gro, Proposition 12.7]. Witt theory holds in characteristic $2\text{:}$ isometries between subspaces of a nondefective space extend to the whole space, all maximal totally singular subspaces have a common dimension, the Witt index, and over a perfect field every nondefective form of even dimension is an orthogonal sum of hyperbolic planes with at most one anisotropic plane [Gro, Theorems 12.9 and 12.10, Corollaries 12.11 and 12.12]. Moreover a totally singular subspace of dimension $k$ always sits inside an orthogonal sum of $k$ hyperbolic planes, so a nondefective space of dimension $2m$ containing a totally singular subspace of dimension $m$ is hyperbolic [Gro, Proposition 14.6 and Corollary 14.8]. This Lagrangian criterion is how the ramified case of Proposition 6.18 pins down the local determinant form: a totally singular subgroup of the maximal possible size forces the hyperbolic (plus) type without any choice of symplectic basis.

For a nondefective quadratic space with symplectic basis $\{u_i,v_i\}_{i\le m}\text{,}$ the class of $\sum_{i}q(u_i)q(v_i)$ in $F/\{a^2+a:a\in F\}$ is independent of the chosen symplectic basis; it is the Arf invariant of $q$ [Gro, Theorem 13.13 and p. 124]. Since concatenating symplectic bases of orthogonal summands gives a symplectic basis of the sum, the defining expression is additive over orthogonal sums, the additivity invoked in Lemma 6.8. Over a finite field the subgroup $\{a^2+a\}$ has index $2\text{,}$ so the Arf invariant is a bit, and for $F=\F_2$ it is literally an element of $\F_2$ [Gro, p. 124]. It is the complete invariant in even dimension over a perfect field: the hyperbolic normal form has Arf invariant $0\text{,}$ the form with an anisotropic plane $x^2+x+b$ has Arf invariant $b\text{,}$ and two such forms are equivalent exactly when the invariants agree [Gro, Proposition 13.14]. Wall calls this invariant the pseudo-discriminant, computes it as $\sum_i b_{2i-1}b_{2i}$ in normalized coordinates, and records the same classification, attributing it to Arf [Wal, §3.6, Lemma 3.6.2]. The two even-dimensional types over $\F_q$ are the plus type (hyperbolic, Witt index $n/2$) and the minus type (Witt index $n/2-1$), with orthogonal groups $O_+$ and $O_-$ [Gro, Chapter 14, p. 150] [Wal, §3.7, (3.7.1)].

The zero counts behind the paper’s Gauss signs are classical. Writing $s_n$ for the number of vectors with $q(v)=0$ (including $0$) in a nondefective space of even dimension $n$ over $\F_q\text{,}$ one has $s_n=q^{n-1}+q^{m}-q^{m-1}$ for the hyperbolic type with $n=2m\text{,}$ and $s_n=q^{n-1}-q^{m+1}+q^{m}$ for the minus type with $n=2m+2$ [Gro, Proposition 14.47(b),(c)]. For $q=2$ and $d=\dim V$ this reads

\begin{equation*} \#q^{-1}(0)=2^{d-1}+(-1)^{\Arf(q)}2^{d/2-1}, \end{equation*}

since the hyperbolic type is the type of Arf invariant $0\text{;}$ equivalently, the quadratic Gauss sum $\sum_{x\in V}(-1)^{q(x)}=2\,\#q^{-1}(0)-2^{d}$ equals $(-1)^{\Arf(q)}2^{d/2}\text{.}$ This is the identity used in Proposition 6.9 and Proposition 6.18, where the paper also re-derives the minus-type count directly on a Hermitian line by fibring over the norm map, giving $1+(2^m+1)(2^{m-1}-1)=2^{2m-1}-2^{m-1}$ zeros.

An isometry of $(V,q)$ preserves $b_q\text{,}$ so the orthogonal group $O(V,q)$ is a subgroup of the symplectic group of $b_q\text{,}$ and in particular every isometry has determinant $1$ [Gro, Chapter 14, p. 127]; the determinant is replaced by Dickson’s pseudodeterminant, a surjective homomorphism $\delta:O(V,q)\to\F_2\text{,}$ independent of all choices, which takes the value $1$ on every orthogonal transvection and whose kernel is by definition $SO(V,q)$ [Gro, Corollary 14.4, Proposition 14.5 and pp. 130--131]. For an isometry $\mathsf U$ write $N=1+\mathsf U\text{.}$ Wall attaches to $\mathsf U$ the subspace $R=NV\text{,}$ its space, and the form $\omega$ on $R$ defined by $\omega(Nx,Ny)=b_q(x,Ny)\text{;}$ this form is well defined and nondegenerate, and the orthogonal complement of $R$ is the fixed space $\ker N$ [Wal, §1.1, (1.1.2), Lemma 1.1.1 and Corollary]. In the characteristic-$2$ orthogonal case the invariance of $q$ strengthens this to $b_q(x,Nx)=q(Nx)\text{,}$ so $\omega(u,u)=q(u)$ for $u\in R\text{:}$ the Wall form remembers the quadratic form on $R$ [Wal, §1.5, (1.5.5)--(1.5.6) and Lemma 1.5.1], and equivalence of Wall forms detects conjugacy in the orthogonal group [Wal, §1.5, Theorem 1.5.2]. This is exactly the structure exploited by Lemma 6.6: the doubled form $q_{\mathsf U}(x)=q(x)+b_q(x,\mathsf U^{-1}x)=q(x)+q(Nx)$ is compared with $q$ through the Wall form of $\mathsf U\text{,}$ and an induction on $\omega\text{,}$ splitting an anisotropic plane or a symplectic plane at each step, shows that the Arf invariant shifts by $\dim R=\operatorname{rank}(1+\mathsf U)$ modulo $2\text{.}$ The proof is self-contained, so the classical identification of this rank parity with $\delta(\mathsf U)$ is never needed.

Hermitian forms enter through the unramified and ramified models of the invariant form. For a quadratic extension $F/F_0$ with involution $x\mapsto x^*\text{,}$ every Hermitian space has an orthogonal basis with diagonal entries in $F_0\text{,}$ over a finite field even an orthonormal one, so all Hermitian spaces of a given dimension over a finite field are equivalent [Gro, Theorems 10.2 and 10.3, Corollary 10.4]; moreover the trace $F\to F_0$ is onto [Gro, Proposition 10.6] and the norm $F^\times\to F_0^\times$ is onto with all fibres of size $q+1$ [Gro, Chapter 11, p. 109]. Composing a Hermitian form with the trace gives the $\F_2$-valued quadratic trace forms $q_c(x)=\Tr_{F_0/\F_2}(c\,xx^*)\text{;}$ Lemma 6.7 shows these are the only nonsingular quadratic forms on a line invariant under the norm-one circle group, and Lemma 6.8 diagonalizes the invariant Hermitian matrix to split the ramified form into such lines, each of Arf invariant $1$ by the zero count above. Finally, the algebraic normal form of a quadratic map over $\F_2\text{,}$ the unique reduced polynomial representative determined by the values on coordinate vectors together with the polar form, is established where it is used, in Lemma 6.3, whose orbit polynomials are its invariant incarnation.

Subsection E.9 Modular representations, Clifford theory, and Schur fields

Representation theory over $\F_2$ is governed by the interplay between the odd part and the $2$-part of the acting group. If a finite group $H$ has odd order, then $|H|$ is invertible in $\F_2$ and Maschke’s theorem applies: every invariant subspace of an $\F_2[H]$-module is a direct summand [Web, Theorem 1.2.1], so every finite-dimensional $\F_2[H]$-module is semisimple [Web, Corollary 1.2.5]; the same averaging over $H$ shows that $|H|$ annihilates positive-degree cohomology, so $H^n(H,M)=0$ for $n>0$ when $|H|$ is odd and $M$ is an $\F_2[H]$-module [Bro, Corollary III.10.2]. At the opposite pole, a $2$-group has only the trivial simple module in characteristic $2$ [Web, Proposition 6.2.1], and consequently the largest normal $2$-subgroup $O_2(H)$ is the common kernel of the action of $H$ on all simple $\F_2[H]$-modules [Web, Corollary 6.2.2]. The paper’s recurring unramified reduction combines the two poles: the $2$-primary subgroup of a cyclic unramified image acts trivially on every simple characteristic-$2$ module, so the faithful image $H_V$ of a simple unramified module has odd order, and semisimplicity together with the cohomological vanishing is available. This is the form in which Maschke’s theorem enters Lemma 6.3, Lemma 6.4, Proposition 7.4, and Proposition 6.18.

For a finite-dimensional module $U$ over $\F_2[H]\text{,}$ with $H$ now arbitrary, the radical $\rad U$ is the intersection of the maximal submodules, equivalently the smallest submodule with semisimple quotient, and the socle $\soc U$ is the sum of the simple submodules, the largest semisimple submodule [Web, Lemma 6.3.1 and Proposition 6.3.4]. The largest semisimple quotient $U/\rad U$ is called the head (or top) of $U\text{;}$ Webb says “radical quotient”. We say $U$ has simple head $V$ when $U/\rad U\cong V$ is simple, that is, when $\rad U$ is the unique maximal submodule. These notions organize Section 7: a chief series of the finite $2$-kernel $L_Y\text{,}$ i.e. a maximal chain of subgroups normal under the full ambient action, has elementary abelian factors admitting no proper nonzero invariant subgroups; each chief factor is thus a simple $\F_2[H]$-module for the tame quotient $H\text{,}$ called scalar exactly when it is the trivial module. Isolating the first non-scalar chief factor produces the simple-head sequence $0\to T\to M\to V\to 0$ of Lemma 7.1, with $T=\rad M$ and $\Hom_C(M,\F_2)=0\text{;}$ the trivial action of odd Hall lifts along chief series with scalar factors below drives Lemma 7.2, and the same sequence carries the radical-edge descent of Lemma 8.6 and the inductive step in Subsection 9.2.

Clifford’s theorem controls restriction to a normal subgroup $N\triangleleft H\text{:}$ over any field, the restriction to $N$ of a simple module $U$ is semisimple [Web, Theorem 5.3.1], and it decomposes into isotypic homogeneous components $S_1^{a_1}\oplus\cdots\oplus S_r^{a_r}$ which $H$ permutes transitively, with equal multiplicities and dimensions; moreover $U$ is induced from the stabilizer of a component [Web, Theorem 5.3.2]. When $N$ is abelian and the field algebraically closed, every simple $N$-module is a one-dimensional character [Web, Theorem 5.3.3], so the restriction splits into $\chi$-weight spaces indexed by characters $\chi$ of $N\text{,}$ permuted by conjugation; the stabilizer of a weight is its inertia group in the classical terminology adopted here (Webb writes $\operatorname{Stab}_G$). Lemma 6.11 runs this analysis over $\overline{\F}_2$ for the odd cyclic tame-inertia subgroup $I\triangleleft H_V\text{:}$ the inertia characters occurring in an absolutely irreducible constituent have one-dimensional weight spaces, they are permuted in orbits by $H_V$ and by Frobenius, and the Sylow-$2$ subgroup moves the weights freely in blocks of $2$-power size. The same weight picture, with self-duality exchanging $\chi$ and $\chi^{-1}\text{,}$ underlies the Hermitian model of Lemma 6.8.

A module is projective if and only if it is a direct summand of a free module, if and only if every epimorphism onto it splits [Web, Proposition 7.1.3]. Over $\F_2[P]$ with $P$ a $2$-group, finitely generated projective modules are free [Web, Theorem 8.1.1]: this is the regularity established block-by-block on Sylow-$2$ weight orbits in Lemma 6.11. Restriction and induction preserve projectivity [Web, Lemma 8.1.2], and when the index of a subgroup $P\le H$ is odd (for instance $P$ a Sylow $2$-subgroup, or $P=1$ in an odd-order group), every module is relatively $P$-projective, so that $U$ is projective over $\F_2[H]$ if and only if the restriction $U|_P$ is projective [Web, Proposition 11.3.5 and Corollary 11.3.6]. Projectivity is likewise insensitive to the ground field: for a field extension $E/F\text{,}$ a module $U$ is projective if and only if $E\otimes_F U$ is projective [Web, Chapter 9, Exercise 17], and by the Noether–Deuring theorem an isomorphism of modules established after an arbitrary extension of scalars already holds over the ground field [Web, Theorem 9.2.8]. Together these descend the free Sylow-block structure from $\overline{\F}_2$ back to $\F_2\text{,}$ completing Lemma 6.11. For a simple module $V$ that is either projective (ramified case) or semisimple with all its $\F_2[H]$-companions (unramified case, odd $|H|$), every embedding into and surjection from a free module splits; this produces the $H$-split pairs $V\to\F_2[H]^N\to V$ with $pi=1$ of Lemma 6.3, whose left-regular coordinates (the basis of the regular representation indexed by the elements of $H$ [Web, Section 1.1]) carry the orbit polynomials of the base determinant class.

By Schur’s lemma the endomorphism ring of a simple module is a division ring [Web, Theorem 2.1.1]; simplicity survives every extension of scalars precisely when the endomorphisms reduce to the scalars [Web, Proposition 9.2.5], and any field containing enough roots of unity is a splitting field [Web, Theorem 9.2.7]. All the endomorphism rings used in this paper are taken over cyclic groups, where the division ring is automatically commutative: for $C$ cyclic and $S$ a simple $F[C]$-module, $\End_{F[C]}(S)$ is a field, and extending scalars to it splits $S$ into absolutely simple summands [Web, Chapter 9, Exercise 3]. Over $\F_2$ this yields the finite Schur fields $F_V$ of the paper: a simple module $W$ of an odd-order cyclic group with $\dim_{\F_2}W=f$ is a one-dimensional vector space over $F_V=\End(W)\cong\F_{2^f}\text{,}$ a generator acting as multiplication by a primitive character value $\zeta\text{,}$ and the $f$ weights of $W\otimes_{\F_2}\overline{\F}_2$ form a single Frobenius orbit $\{\zeta^{2^i}\}\text{.}$ Self-duality $V\cong V^\vee$ ([Web, p. 28] for the dual module) places $\zeta^{-1}$ in this orbit, forcing the orbit length $f=2m$ to be even and making $x\mapsto x^*=x^{2^{m}}$ the involutive automorphism of $F_V$ with fixed field $F_{V,0}\cong\F_{2^{m}}\text{;}$ on endomorphisms this involution is the adjoint with respect to the invariant polar form. The norm and trace of $F_V/F_{V,0}\text{,}$ and the norm-one unitary scalars $\mathcal U\le F_V^\times$ acting by isometries, carry the Hermitian-line computations of Lemma 6.8, Proposition 6.9, and Proposition 6.18, and feed the isometry invariance of the normalized base class in Lemma 6.4.

Subsection E.10 Free differential calculus and the Fox Jacobian

Free differential calculus is Fox’s calculus of derivations in the group ring of a free group. A derivation of a group ring is an additive map $D$ satisfying $D(u\cdot v)=Du\cdot v^{o}+u\cdot Dv\text{,}$ where $v^{o}$ denotes the coefficient sum; on group elements this reads $D(gh)=Dg+g\,Dh$ [Fox, §1, (1.1)-(1.2)]. For a free group with basis $x_1,\dots,x_d\text{,}$ each generator carries a partial derivative $\partial/\partial x_j\text{,}$ characterized by $\partial x_k/\partial x_j=\delta_{jk}\text{,}$ and every derivation of the free group ring is uniquely of the form $f\mapsto\sum_j(\partial f/\partial x_j)\,h_j$ with prescribed values $h_j$ on the generators [Fox, §2, Theorem, (2.1)-(2.2)]. The fundamental formula

\begin{equation*} f\;=\;f(1)+\sum_{j}\frac{\partial f}{\partial x_j}\,(x_j-1) \end{equation*}

recovers an element of the free group ring from its augmentation and its derivatives [Fox, (2.3)]; the augmentation-zero elements form what Fox calls the fundamental ideal, the modern augmentation ideal [Fox, §1].

On words the product rule takes the recursive form $\partial(uv)/\partial x_j =\partial u/\partial x_j+u\,\partial v/\partial x_j\text{,}$ with $\partial(u^{-1})/\partial x_j=-u^{-1}\,\partial u/\partial x_j$ [Fox, (1.2)', (1.5), (1.6)], and the derivative of a power is a power sum:

\begin{equation*} \frac{\partial x^{p}}{\partial x}=1+x+\cdots+x^{p-1}, \qquad p\ge 1, \end{equation*}

the norm element of the cyclic group generated by the base [Fox, (2.4)], with a practical rule and a chain rule for substituted words [Fox, (2.5)-(2.6)]. Evaluated through a homomorphism to a finite semidirect product $C\ltimes A\text{,}$ so that the group-ring coefficients act on the kernel $A\text{,}$ these identities become exactly the finite Fox rules $D_{uv}=D_u+\bar uD_v\text{,}$ $D_{u^{-1}}=-\bar u^{-1}D_u\text{,}$ $D_{u^{n}}=\sum_{j=0}^{n-1}\bar u^{\,j}D_u$ of Appendix B (see (B.1)) and the raw finite Fox rules of Lemma 5.4: identities of literal finite word evaluations, for which no calculus over a completed group ring need be invoked. In particular, for an $\w$-power the evaluated power sum collapses: an integer representative of $\w$ modulo the relevant finite exponent is odd and divisible by the odd order $e$ of the lower operator $\mathsf T\text{,}$ so over $\F_2$ the power rule yields $P=1+\mathsf T+\cdots+\mathsf T^{e-1}\text{;}$ as $e$ is odd, $P$ is an idempotent projecting onto the $\mathsf T$-invariants, the norm projector of Lemma 5.5.

For a group presented by generators $x_1,\dots,x_d$ and relators $r_1,\dots,r_m\text{,}$ the array $(\partial r_i/\partial x_j)$ is the Jacobian matrix of the presentation. Fox already observed that the derivatives are the degree-one cocycles of the free group and that the computation of the cohomology of a presented group in dimensions $1,2,3,\dots$ rests on its Jacobian matrices, as carried out by Lyndon for one-relator groups and by Artin and Tate for certain finite groups [Fox, pp. 547-548]. This is the linearization used in Section 5: varying the generators of a lift to $C\ltimes A$ by $(a_1,\dots,a_d)\in A^d$ changes each relator defect to first order by the evaluated row $\sum_j(\partial r_i/\partial x_j)^{\rho}\,a_j\text{,}$ and the fundamental formula applied to $r_i$ gives $\sum_j(\partial r_i/\partial x_j)^{\rho}(\mathsf X_j-1) =\rho(r_i)-1=0\text{,}$ so the relator rows annihilate the simultaneous conjugations and the three terms of (5.1) form a complex whose degree-one cohomology classifies first-order lifts; the rows displayed in Lemma 5.5 are these evaluated derivatives for the tame and wild relators. In degree two the relator values retain full information by the classical pro-$p$ facts: a minimal system of relations has cardinality $\dim_{\F_p}H^2(G,\F_p)$ [Koc, Theorem 6.13], and for a minimal presentation $1\to R\to F\to G\to 1$ the transgression $H^1(R)^{G}\to H^2(G,\mathbb{Z}/q)$ is an isomorphism, so a degree-two class is determined by the values at the relators of the corresponding homomorphism on $R$ [Koc, §7.8]. In the completed setting the same derivative map is the boundary of the relation-module sequence

\begin{equation*} 0\longrightarrow R^{\mathrm{ab}}(p) \longrightarrow \mathbb{Z}_p[[G]]^{d} \longrightarrow \mathbb{Z}_p[[G]] \longrightarrow \mathbb{Z}_p\longrightarrow 0, \end{equation*}

the profinite analogue of Lyndon’s theorem: a relation maps to its differential, whose coordinates in the free module on $dx_1,\dots,dx_d$ are its partial derivatives, and $dx_j\mapsto x_j-1$ realizes the fundamental formula [NSW, Proposition (5.6.4) and Theorem (5.6.6)].

The pro-$p$ version of the calculus, which the finite ledgers of Appendix B shadow, lives in the complete group algebra $\F_p[[G]]=\varprojlim \F_p[G/N]$ [Koc, Definition 7.1]. For a free pro-$p$ group $F$ on $s_1,\dots,s_m\text{,}$ the Magnus embedding identifies $\F_p[[F]]$ with the algebra of formal noncommutative power series in variables $x_i$ via $s_i\mapsto 1+x_i$ [Koc, Theorem 7.16, with Definition 4.3], matching Fox’s identification of the formal Taylor expansion by higher derivatives with the Magnus representation on $x_j=s_j-1$ [Fox, §3, (3.4)]; a system of relations $\{r_i\}$ generates the kernel of $\F_p[[F]]\to\F_p[[G]]$ as the closed ideal on the $r_i-1$ [Koc, Theorem 7.17]. Second-order word coefficients compute cup products: writing a relator in collected normal form $r_i\equiv\prod_\nu s_\nu^{a_\nu q}\prod_{\nu<\mu} [s_\nu,s_\mu]^{a_{\nu\mu}}$ modulo the third step of the $(q)$-filtration, evaluation at $r_i$ through the inverse transgression sends $\chi_\nu\cup\chi_\mu$ to $-a_{\nu\mu}$ for $\nu<\mu$ [Koc, Theorems 7.22-7.23]; this is the classical prototype for extracting cup products from central word coordinates, the mechanism behind the finite Fox recursion formulas and Heisenberg central coordinate of Subsection 5.2. A systematic topological version of Fox’s calculus for completed group rings (continuous derivations of $\mathbb{Z}_p[[G]]$ into compact modules, the differential module, its freeness of rank $d$ over a free pro-$p$ group) is developed in [NSW, Chapter V, §6, (5.6.1)-(5.6.4)]. This body of results is what makes free differential calculus the established pro-$p$ tool invoked in Subsection 1.2; the deformation section’s device is to evaluate the resulting Jacobian on each finite tame module and read off lifting counts.

Subsection E.11 Peripheral inertia and the arithmetic action on fundamental groups

For a geometrically connected curve $U$ over a field $k$ of characteristic zero, the étale fundamental group sits in an extension

\begin{equation*} 1\longrightarrow\pi_1(\overline U)\longrightarrow\pi_1(U) \longrightarrow\operatorname{Gal}(\bar k/k)\longrightarrow 1, \end{equation*}

where $\overline U$ is the base change to an algebraic closure, and conjugation in this extension induces the arithmetic outer action $\operatorname{Gal}(\bar k/k)\to\operatorname{Out}(\pi_1(\overline U))$ of the absolute Galois group on the geometric fundamental group [Sti, Section 1.1, (1.1)]. For $X=\mathbb{P}^1_{\mathbb{Q}}\setminus\{0,1,\infty\}$ and a prime $\ell$ the same applies to the maximal pro-$\ell$ quotient of the geometric fundamental group, a free pro-$\ell$ group of rank $2\text{:}$ Ihara realizes it as $\pi_1=\operatorname{Gal}(M/\overline{\mathbb{Q}}(t))$ with $M$ the maximal pro-$\ell$ extension of $\overline{\mathbb{Q}}(t)$ unramified outside $\{0,1,\infty\}\text{,}$ and obtains the outer representation $\varphi_{\mathbb{Q}}\colon G_{\mathbb{Q}}\to\operatorname{Out}\pi_1$ from the extension $1\to\pi_1\to\operatorname{Gal}(M/\mathbb{Q}(t))\to G_{\mathbb{Q}}\to1$ [Iha, Section 1(A), (1.1) and (1.2)].

The cusps carry the peripheral structure of this action. A cusp $y$ of an affine hyperbolic curve, that is, a point of the boundary divisor $Y=X\setminus U\text{,}$ gives rise to inertia subgroups $I_{\tilde y}\subset\pi_1(\overline U)\text{,}$ one for each prolongation $\tilde y$ of $y$ to the universal pro-étale cover. They satisfy $I_{\gamma\tilde y}=\gamma I_{\tilde y}\gamma^{-1}\text{,}$ and the prolongations of a fixed cusp form a single orbit, so $y$ determines a well-defined conjugacy class of peripheral subgroups, and this class is preserved by the arithmetic action whenever the cusp is $k$-rational [Sti, Definition 10 and Section 3.3]. Each inertia group is free procyclic, its normalizer in $\pi_1(U)$ is the decomposition group $D_{\tilde y}\text{,}$ an extension of $\operatorname{Gal}(\bar k/\kappa(y))$ by $I_{\tilde y}\text{,}$ and the conjugation action of $D_{\tilde y}$ on $I_{\tilde y}$ is through the cyclotomic character: the inertia group acts on the tower obtained by extracting compatible roots of a local parameter at the cusp, and the resulting identification $I_{\tilde y}\cong\Zhat(1)$ is equivariant for conjugation and for the cyclotomic character [Sti, Section 3.3, Proposition 11, Corollary 15, Proposition 16, and Definitions 34 and 37].

For $\mathbb{P}^1_{\mathbb{Q}}\setminus\{0,1,\infty\}$ all three cusps are rational, and one may choose inertia generators $x,y,z$ above $0,1,\infty$ with $xyz=1$ such that $x,y$ freely generate the pro-$\ell$ fundamental group; the arithmetic action then satisfies Ihara’s formula

\begin{equation*} \varphi_{\mathbb{Q}}(\sigma)x\sim x^{\chi(\sigma)},\qquad \varphi_{\mathbb{Q}}(\sigma)y\sim y^{\chi(\sigma)},\qquad \varphi_{\mathbb{Q}}(\sigma)z\sim z^{\chi(\sigma)}, \end{equation*}

with $\sim$ denoting conjugacy in $\pi_1$ and $\chi$ the $\ell$-adic cyclotomic character [Iha, Section 1(A), (1.4)]. Taking $\ell=2\text{,}$ this is exactly the input to Lemma 3.7, whose generators $P,T,C$ with $C=(PT)^{-1}$ play the roles of $x,y,z\text{:}$ surjectivity of the $2$-adic cyclotomic character onto $\mathbb{Z}_2^\times$ supplies, for every unit $u\text{,}$ an outer automorphism of the geometric pro-$2$ fundamental group sending each of the three peripheral generators to a conjugate of its $u$-th power.

Subsection E.12 Complexes, cones, and cup products at the cochain level

All homological algebra in this section is finite-dimensional linear algebra over $\F_2\text{,}$ carried by three-term complexes such as the word complex (5.1). Two standard devices organize it. First, a chain map $f:X^\bullet\to Y^\bullet$ of complexes (a degreewise linear map commuting with the differentials) has a mapping cone $\Cone(f)\text{,}$ formed degree by degree: its degree-$n$ term is $X^{n+1}\oplus Y^n\text{,}$ and over $\F_2$ its differential is simply $(x,y)\mapsto(d_Xx,\,f(x)+d_Yy)\text{,}$ the sign in the general formula being invisible because $2=0\text{.}$ Second, a degreewise short exact sequence of complexes yields a long exact cohomology sequence whose connecting homomorphism is natural; applied to the degreewise-split sequence $0\to Y^\bullet\to\Cone(f)\to X^\bullet[1]\to0\text{,}$ it shows that $f$ is a quasi-isomorphism (it induces isomorphisms on all cohomology; Brown says weak equivalence) exactly when $\Cone(f)$ is acyclic, meaning its cohomology vanishes in every degree. See [Bro, Chapter I, Section 0].

The long exact sequence gives a two-out-of-three principle: in a degreewise short exact sequence of complexes, if two of the three complexes are acyclic, then so is the third. Since cones are formed degree by degree, a map of degreewise short exact sequences of complexes produces a degreewise short exact sequence of cones. These two remarks are the entire homological content of the exact cone dévissage of Lemma 5.11: a short exact sequence of coefficient modules induces compatible short exact sequences of word complexes and of their duals, hence of the cones $K(-)\text{,}$ and acyclicity of $K(V)$ for the simple modules $V$ propagates along a composition series to every elementary module, with no semisimplicity hypothesis.

Duality enters through the exact functor $\Hom_{\F_2}(-,\F_2)\text{.}$ For a three-term complex $X^0\to X^1\to X^2$ of finite-dimensional $\F_2$-vector spaces, the shifted dual $\Hom_{\F_2}(X^\bullet,\F_2)[-2]$ is again a three-term complex in degrees $0$ through $2\text{,}$ with the transposed differentials in reversed order, and its degree-$i$ cohomology is $H^{2-i}(X^\bullet)^\vee\text{:}$ over a field the $\operatorname{Ext}$-term of the universal coefficient sequence vanishes [Bro, Chapter I, Section 0]. Consequently a chain map from $X^\bullet$ to a shifted dual, such as (5.17) of Proposition 5.10, is a quasi-isomorphism if and only if the induced pairings $H^i\times H^{2-i}\to\F_2$ are perfect; equivalently, by the cone criterion above, if and only if the cone (5.20) is acyclic. This is the form in which Proposition 5.15 is stated and proved.

Finally, the cup products. At cochain level the cup product of inhomogeneous cochains is given by the explicit formula

\begin{equation*} (f\smile g)(g_1,\ldots,g_{p+q}) =f(g_1,\ldots,g_p)\cdot(g_1\cdots g_p)\,g(g_{p+1},\ldots,g_{p+q}), \end{equation*}

as in [NSW, Chapter I, Section 4, Proposition (1.4.8)] and, for continuous cochains on profinite groups, [NSW, Chapter II, Section 7]. Classically this formula is not an ansatz but the shadow of a diagonal approximation: an augmentation-preserving equivariant chain map $\Delta:F\to F\otimes F$ on a projective resolution $F\text{,}$ unique up to equivariant homotopy, through which the cup product is defined as the cross product followed by $\Delta\text{.}$ On the bar resolution the standard choice is the Alexander–Whitney map $\Delta(g_0,\ldots,g_n)=\sum_{p}(g_0,\ldots,g_p)\otimes(g_p,\ldots,g_n)\text{,}$ the algebraic form of a cellular approximation to the diagonal of the classifying space; see [Bro, Chapter V, Sections 1 and 3]. The remark opening Subsection 5.2 is to be read against this background: the Stokes identities feeding Proposition 5.10 are identities of finite word evaluations, proved by conjugation in the finite Heisenberg group before any passage to cohomology, so no resolution, and in particular no choice of cellular diagonal approximation, enters the candidate side. The displayed cochain formula is needed only on the Galois side, where the classical cup product supplies the pairing of Proposition 5.16.

Subsection E.13 The Evens norm and the Evens–Kahn formula

Let $H\le G$ be a subgroup of finite index $l\text{.}$ A transversal of $H$ in $G$ determines Frobenius’s monomial embedding $\Phi\colon G\to \mathcal S_l\wr H$ into the wreath product with the symmetric group, and Evens’s multiplicative transfer, or norm map, sends a class $\alpha\in H^n(H,\F_2)$ to the pullback along $\Phi$ of its Nakaoka wreath power on $\mathcal S_l\wr H\text{;}$ the resulting class $N_H^G(\alpha)\in H^{ln}(G,\F_2)$ does not depend on the transversal [Eve, §§2, 4--5]. The norm is multiplicative, $N_H^G(\alpha\smile\beta)=N_H^G(\alpha)\smile N_H^G(\beta)$ [Eve, Proposition 4], but it is neither additive nor degree-preserving. With integral coefficients sign problems confine the construction to even degrees; with $\F_2$-coefficients, the only case used in this paper, it is available in every degree [Kah, I.2].

For an index-two subgroup $N\triangleleft G$ and a class $[\alpha]\in H^1(N,\F_2)\text{,}$ the norm of the unit $1+[\alpha]$ expands into homogeneous components as

\begin{equation*} N_N^G(1+[\alpha]) =1+\cor_N^G[\alpha]+N_N^{G,\mathrm{Ev}}([\alpha]), \end{equation*}

where the degree-one term is the ordinary additive transfer and the degree-two term is the index-two Evens norm appearing throughout Section 6. This is the two-sheeted case of the general expansion of $N(1+x)$ into partial norms, whose first term is the transfer and whose top term is the norm [Eve, Theorem 1], [Kah, Proposition I.2.4]. Lemma 6.13 makes this class explicit: on the universal two-point model $E_2\rtimes J$ (the swap module over $J\cong C_2$) the norm of the coordinate character is represented by a normalized $2$-cocycle whose fibre extension is the dihedral group $D_8\text{,}$ and the general index-two case is recovered by pulling this model back along a graph homomorphism, using naturality of the norm in index-preserving pullback squares [Eve, Proposition 3].

Restrictions of the norm are computed by Evens’s double-coset rule, the multiplicative analogue of Mackey’s formula: for $K\le G\text{,}$ with $\F_2$-coefficients,

\begin{equation*} \res_K\,N_H^G(\alpha) =\prod_{KgH\in K\backslash G/H} N_{K\cap {}^{g}H}^{K}\bigl(\res_{K\cap {}^{g}H}({}^{g}\alpha)\bigr) \end{equation*}

[Eve, Proposition 2]. In the two-point model the two instances used in Lemma 6.13 read: restriction to the index-two subgroup itself is the product of the two conjugates, $\res_{E_2}N_{E_2}^{E_2\rtimes J,\mathrm{Ev}}(e_1^\vee) =e_1^\vee e_s^\vee\text{,}$ while restriction to a complement passes through the trivial intersection subgroup and vanishes. Since $H^1(J,E_2^\vee)=0\text{,}$ the low-degree exact sequence shows that these two restrictions already determine a class on $E_2\rtimes J\text{;}$ this uniqueness device is how Lemma 6.13 identifies the explicit cocycle with the Evens norm.

A real representation $\rho$ of a finite group carries Stiefel–Whitney classes $w_i(\rho)\text{,}$ those of the associated vector bundle on the classifying space; for a real representation of a Galois group with open kernel one inflates from a finite quotient [Kah, II.1.2], and for a quadratic form $q\simeq\langle a_1,\dots,a_r\rangle$ one has Delzant’s classes $w(q)=\prod_i(1+(a_i))\text{,}$ with $w_1$ the discriminant and $w_2$ the Hasse–Witt invariant [Kah, II.1.3]. The Evens–Kahn formula, named after the total Chern class formula of Evens and D. S. Kahn for induction from prime-index subgroups, computes the total Stiefel–Whitney class of an induced representation along an index-two subgroup through the norm: Kozlowski constructs a multiplicative transfer for double coverings on the units of mod-$2$ cohomology and proves that the total Stiefel–Whitney class intertwines it with the $KO$-transfer [Koz, Theorem 1.1], while B. Kahn, working algebraically in Evens’s framework, obtains

\begin{equation*} w(\operatorname{Ind}\rho) =N(w(\rho)) +\sum_{i\ge0}N(w_i(\rho))\bigl((1+\mu)^{r-i}+1\bigr), \end{equation*}

where $r$ is the rank and $\mu\in H^1(G,\F_2)$ classifies the double covering [Kah, Théorème I.4.2], [Gui, Appendix A.1]. Over the absolute Galois group of a field of characteristic not two the correction terms vanish and the formula becomes exact: $w(\operatorname{Ind}\rho)=N(w(\rho))\cdot w(\operatorname{Ind}1)^{r}\text{,}$ and likewise for trace forms $w(\Tr_{E/F}q)=N(w(q))\cdot w(\Tr_{E/F}\langle1\rangle)^{r}$ [Kah, Théorèmes 1--2]. For a rank-one character $\ell$ the relative class $w(\operatorname{Ind}\ell)/ w(\operatorname{Ind}1)$ is thereby normalized to $N(1+\ell)=1+\cor\ell+\Evens(\ell)\text{;}$ its degree-two component is exactly the index-two Evens norm. This is the Evens–Kahn normalization in which Lemma 6.13 identifies the universal two-point class.

For a quadratic extension of fields $L/k$ of characteristic different from $2\text{,}$ writing $\Evens_{L/k}$ for the index-two Evens norm of the corresponding Galois groups, the rank-one case of the trace-form comparison reads

\begin{equation*} w(\Tr_{L/k}\langle a\rangle) =w(\Tr_{L/k}\langle1\rangle) \bigl(1+\cor_{L/k}[a]+\Evens_{L/k}([a])\bigr), \end{equation*}

the degree-two Evens–Kahn identity invoked in Lemma 6.16. Both trace forms are binary and explicitly diagonalizable, so isolating the degree-two component expresses $\Evens_{L/k}([a])$ as a sum of Hilbert symbols in the diagonal entries; compare Kahn’s closed formula $N((a))=(\Tr_{L/k}a,\,-d\,N_{L/k}a)+(2,d)\text{,}$ valid when $\Tr_{L/k}a\ne0\text{,}$ with $d$ the discriminant of $L/k$ [Kah, Lemme II.2.1]. Section Section 6 runs this reduction inside the orbit decomposition of the regular module, where involution orbits contribute $\cor_{K_0/F}\Evens_{K/K_0}(\alpha_r)$ (Lemma 6.15, via Lemma 6.14): Lemma 6.16 shows that over dyadic local fields the norm $\Evens_{L/k}$ annihilates the deep units $U_{e+1}(L)$ of an unramified quadratic extension, which is what makes the involution-orbit contributions vanish on the deep half in Lemma 6.17.

Subsection E.14 Characters of finite abelian groups and Fourier inversion

A character of a finite abelian group $G$ is a homomorphism from $G$ to the multiplicative group $\mathbb{C}^*\text{;}$ the characters form the dual group $\widehat G=\Hom(G,\mathbb{C}^*)$ [Ser73, Ch. VI, §1.1, Definition 1]. The dual is again a finite abelian group, of the same order as $G\text{,}$ and the evaluation map $x\mapsto(\chi\mapsto\chi(x))$ is an isomorphism of $G$ onto its bidual [Ser73, Ch. VI, §1.1, Propositions 2 and 3]. When $G$ is a finite $\F_2$-vector space every character takes values in $\{\pm1\}\text{,}$ so $\widehat G$ is identified with the $\F_2$-linear dual $G^\vee=\Hom(G,\F_2)\text{,}$ the character attached to $\lambda\in G^\vee$ being $x\mapsto(-1)^{\langle\lambda,x\rangle}\text{.}$

The orthogonality relations state that for a fixed character $\chi\in\widehat G$ the sum $\sum_{x\in G}\chi(x)$ equals $|G|$ if $\chi=1$ and $0$ otherwise, and dually that for a fixed element $x\in G$ the sum $\sum_{\chi\in\widehat G}\chi(x)$ equals $|G|$ if $x=1$ and $0$ otherwise [Ser73, Ch. VI, §1.2, Proposition 4 and its Corollary]. The second relation is a finite Fourier inversion: averaging all characters at $x$ yields the indicator of $x=1\text{.}$ In the $\pm1$-valued form used in this paper (Lemma 8.4), if $\Lambda$ is the character group of a finite $\F_2$-obstruction space and $o:X\to\Lambda^\vee$ is defined on a finite set $X\text{,}$ then

\begin{equation*} \#\{x\in X: o(x)=0\} =\frac1{|\Lambda|}\sum_{\lambda\in\Lambda}\bigl(2m_\lambda-|X|\bigr), \end{equation*}

where $m_\lambda=\#\{x\in X:\langle\lambda,o(x)\rangle=0\}\text{;}$ each summand $2m_\lambda-|X|$ is the signed character sum $\sum_{x\in X}(-1)^{\langle\lambda,o(x)\rangle}\text{.}$

In the exact-image recursion of Subsection 8.1 the set $X$ consists of boundary-framed exact-image maps, and $o$ is the obstruction to lifting them one Frattini layer up the target tower. The recursion separates $o$ into scalar tests, one invariant scalar character at a time, each nonzero character being realized by a central double cover of the target. In the same way a scalar class $\zeta\in H^2(C,\F_2)$ is represented by a central double cover of the lower target $C\text{,}$ the phase cover, whose splitting indicator realizes the sign $(-1)^{\iota_\Gamma(\rho^*\zeta)}$ appearing in the Fourier sum. Averaging these signed splitting indicators over the scalar phases isolates exactly the maps with vanishing obstruction, and the phase-cover transform (Lemma 8.15) converts each signed sum back into ordinary exact-image counts: writing $n_{\Gamma,0}(\zeta)$ for the number of lower maps for which the phase cover is unobstructed, so that its pullback cover splits, the signed sum equals $2n_{\Gamma,0}(\zeta)-e_\Gamma(C)\text{,}$ and the eight scalar twists of each unobstructed lift (Lemma 8.2) are partitioned by their exact images inside the cover. In this way every Fourier coefficient of the lifting count becomes an exact-image count for a strictly smaller target, closing the recursion.

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Symbol	Meaning
\(q\)	a \(\notn{Cfin}{C}\)-invariant nonsingular quadratic form on the simple self-dual module \(\notnfar{Vhead}{V}\)
\(\notnfar{kappaq0}{\kappa_q^0}\)	the zero-section-normalized equivariant central \(2\)-cocycle on \(\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}\) whose fibre square map is \(q\)
\(\gamma\)	the filtration-one class in \(Z^1(\notn{Cfin}{C},\notnfar{Vhead}{V}^\vee)\text{;}\) its contribution on \(\notnfar{Vhead}{V}\rtimes \notn{Cfin}{C}\) is the cocycle \(\Gamma_\gamma\)
\(\delta\)	the scalar class in \(Z^2(\notn{Cfin}{C},\F_2)\) inflated from the lower quotient \(\notn{Cfin}{C}\)
\(e\)	the class in \(H^2(\notn{Cfin}{C},\notn{Tsub}{T})\) of the affine \(\notn{Tsub}{T}\)-extension governing which \(\notnfar{Vhead}{V}\)-cocycles lift through \(\notn{Mmid}{M}\)
\(\notn{DeltaChiKappa}{\Delta_{\chi,\kappa}}\)	the scalar class \(\chi_*e+\delta_\kappa+\Theta_q^0(a_{\chi,\kappa})\) obtained after completing the square

Level	Role
\(\notnfar{Htarget}{H}\)	fixed tame quotient and boundary frame
\(\notn{Cfin}{C}\)	lower target for elementary \(\notnfar{Mmid}{M}\)-lifting, connecting maps, and scalar phase classes
\(\notn{Bmid}{B}\)	quotient on which the final \(\notn{Rrad}{R}\)-valued obstruction and its scalar pushouts are evaluated
\(H_V\)	faithful image used only for the pure quadratic-form calculation on the simple head \(\notnfar{Vhead}{V}\)

A Presentation of the Absolute Galois Group of \(\mathbb{Q}_2\)

Section 1 Introduction and main theorem

Proposition 1.1. Marked dyadic Demushkin normalization.

Theorem 1.2. Presentation theorem.

Subsection 1.1 Strategy of the proof

Acknowledgements.

Subsection 1.2 What is new

Subsection 1.3 The formalizations and their trust base

Section 2 The candidate as a profinite group

Subsection 2.1 The marked quotient construction

Lemma 2.1. Finite subdirect closure.

Proof.

Lemma 2.2. Cofinality.

Proof.

Proposition 2.3. Epimorphic finite-quotient semantics.

Proof.

Remark 2.4.

Subsection 2.2 A reconstruction lemma

Lemma 2.5. One-sided profinite reconstruction.

Proof.

Section 3 The tame and maximal pro-\(2\) quotients

Lemma 3.1. Finite tame quotients.

Proof.

Proposition 3.2. Common tame quotient.

Proof.

Lemma 3.3. The wild subgroup is the characteristic \(2\)-core.

Proof.

Subsection 3.1 The standard rank-three dyadic Demushkin group

Lemma 3.4. Standard dyadic group and its orientation.

Proof.

Lemma 3.5. Marked abelianization, orientation, and initial form.

Proof.

Remark 3.6. Why the orientation cannot be omitted.

Subsection 3.2 Automorphisms preserving the orientation

Lemma 3.7. Cyclotomic powering of the three peripheral classes.

Proof.

Lemma 3.8. Square-root and HNN lifting.

Proof.

Proposition 3.9. Lifting the orientation-preserving abelian automorphisms.

Proof.

Subsection 3.3 Matching the unramified marking

Proof of Proposition 1.1.

Subsection 3.4 The maximal pro-\(2\) quotient of the candidate

Proposition 3.10. Maximal pro-\(2\) quotient.

Proof.

Proposition 3.11. Nielsen transform to the pro-\(2\) boundary.

Proof.

Corollary 3.12. Fully marked form.

Proof.

Remark 3.13. The integral marking is not detected by the mod-\(2\) cup form.

Proposition 3.14. Fully marked tame and pro-\(2\) quotients.

Proof.

Remark 3.15. Why the full marking is load-bearing.

Section 4 The common tame–pro-\(2\) boundary and the global induction

Definition 4.1. Boundary-framed marked target.

Theorem 4.2. Boundary-framed exact-image theorem.

Section 5 Linear lifting theory and duality

Subsection 5.1 The evaluated Fox Jacobian

Lemma 5.1. Finite-exponent independence.

Proof.

Lemma 5.2. Exact class-two identity.

Proof.

Lemma 5.3. The elementary and class-two shadow of \(h_0\).

Proof.

Lemma 5.4. Full finite-word ledger.

Lemma 5.5. Evaluated Jacobian.

Proof.

Subsection 5.2 The Fox–Heisenberg cup calculation

Lemma 5.6. Strict coefficient naturality.

Proof.

Lemma 5.7. Finite-word Stokes formula.

Proof.

Proposition 5.8. Traced finite-word Stokes identity.

Proof.

Remark 5.9. The trace is essential.

Proposition 5.10. The Fox–Heisenberg chain map.

Proof.

Lemma 5.11. Exact cone dévissage.

Proof.

Lemma 5.12. Simple characteristic-two modules are tame.