The axioms

The absolute Galois group $G_{\mathbb{Q}_2}$ is topologically finitely generated: some finite subset generates a dense subgroup. (The cited theorem gives $[k:\mathbb{Q}_p]+2$ generators for any $p$-adic local field $k$.) This is the input consumed by the reconstruction lemma, which upgrades the surjection-counting comparison to an isomorphism.

Citation. Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, Ch. VII §7.4, Theorem 7.4.1. The weaker $[k:\mathbb{Q}_p]+3$ bound of Jannsen–Wingberg (Satz 3.2, Lemma 3.3) would also suffice.

The maximal pro-$2$ quotient $G_{\mathbb{Q}_2}(2)$ is the rank-$3$ Demushkin group with $q=2$ in Labute's classification, together with the identification of its dualizing character with the cyclotomic character through this quotient, realized by a normalized isomorphism onto the marked presentation the paper uses (Lemma 3.4 → Proposition 1.1). This is a composite interface: classification, dualizing-character comparison, and normalization are bundled deliberately.

Citation. Labute, Classification of Demushkin groups, Théorème 4 case (2) and Théorème 8; dualizing = cyclotomic: Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, Ch. VII §7.5, (7.5.11)–(7.5.12); Serre, Structure de certains pro-p-groupes.

Local class field theory for $\mathbb{Q}_2$: the reciprocity map with its unramified normalization and the compatibilities consumed by the paper's marked-generator calculations (Lemma 3.5, eq. (13)).

Citation. Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, (7.1.1) and (7.1.5); Serre, Local Fields, Ch. XI–XIII.

Local Tate duality at $G_{\mathbb{Q}_2}$ and at $G_K$ for every finite $K/\mathbb{Q}_2$: an invariant isomorphism $H^2(G,\mu_n)\cong\mathbb{Z}/n$ making the evaluation cup pairings $H^i(G,\operatorname{Hom}(M,\mu_n))\times H^{2-i}(G,M)\to\mathbb{Z}/n$ perfect for every finite discrete $n$-torsion module $M$ and $i=0,1,2$.

Citation. Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, Ch. VII §7.2, Theorem 7.2.6; Serre, Galois Cohomology, II §5.2, Theorem 2; Milne, Arithmetic Duality Theorems, I.2.3.

For every finite discrete $G_{\mathbb{Q}_2}$-module $M$, the cohomology groups $H^i(G_{\mathbb{Q}_2},M)$ are finite for $i=0,1,2$ and $\#H^1 = \#H^0\cdot \#H^2\cdot 2^{\,v_2(\#M)}$; this is Tate's local Euler characteristic formula, which drives the dimension counts of the inductive step (§9.2, eq. (145)).

Citation. Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, Ch. VII §7.3, Theorem 7.3.1 (Tate); Serre, Galois Cohomology, II §5.7, Theorem 5; Milne, Arithmetic Duality Theorems, I.2.8.

For every $u\in\mathbb{Z}_2^\times$ there is a continuous automorphism of $\Delta=\pi_1^{\mathrm{pro}\text{-}2} (\mathbb{P}^1\smallsetminus\{0,1,\infty\})$ sending each peripheral generator to a conjugate of its $u$-th power; this is the group-theoretic conclusion of Lemma 3.7. Composite: Stix supplies the cyclotomic form of the action, and cyclotomic surjectivity (via B5) realizes every unit.

Citation. Stix, On cuspidal sections of algebraic fundamental groups, §3.3 and Definition 37; classical origin Deligne (MSRI 16, 1989).

Kahn's relative Stiefel–Whitney formula of paper eq. (111) in degrees $1$ and $2$ for the twisted trace forms $\operatorname{Tr}_{L/k}\langle a\rangle$ of a quadratic extension $L = k(\delta)$, over an arbitrary finite dyadic base $k$, stated for the isometry-class invariants themselves: $w_1(\operatorname{Tr}\langle a\rangle) = w_1(\operatorname{Tr}\langle 1\rangle) + \operatorname{cor}[a]$ and $w_2(\operatorname{Tr}\langle a\rangle) = w_2(\operatorname{Tr}\langle 1\rangle) + w_1(\operatorname{Tr}\langle 1\rangle)\cup\operatorname{cor}[a] + N^{\mathrm{Ev}}[a]$, with $N^{\mathrm{Ev}}$ the Evens norm of Lemma 6.13. Delzant well-definedness of $w_1, w_2$ is proved in the repository, and the census's earlier composite form (eq. (111) at the fixed diagonalizations of Lemma 6.16) is now the theorem evensKahn_dyadic, derived from this axiom together with B11a.

Citation. Kahn, Invent. Math. 78 (1984), Théorème 2 (with Théorème 1); Evens, Trans. Amer. Math. Soc. 108 (1963), Theorem 1; Kozlowski, Proc. Amer. Math. Soc. 91 (1984), Theorem 1.1.

A closed normal pro-$2$ subgroup $W\le G_{\mathbb{Q}_2}$ (wild inertia) with $G_{\mathbb{Q}_2}/W \cong \langle\sigma,\tau\mid\tau^{\sigma}=\tau^{2}\rangle$ (Iwasawa's presentation of the tame quotient), whose unramified coordinate is oriented against B5's reciprocity normalization: units land in tame inertia and $2$ maps to Frobenius in the geometric convention.

Citation. Existence: Neukirch–Schmidt–Wingberg, Cohomology of Number Fields, Ch. VII §7.5, Theorem 7.5.3 (Iwasawa), with (7.5.2); orientation: Serre, Local Fields, Ch. XIII §4, Proposition 13 and corollary.

Over any finite $k/\mathbb{Q}_2$, in Kummer-cup form: for $a,b\in k^\times$, the class $[a]\cup[b]$ vanishes in $H^2(G_k,\mathbb{F}_2)$ if and only if $b = x^2-ay^2$ has a solution in $k$; this is the symbol–norm criterion behind the local square-class calculation of §6.3.

Citation. Serre, Local Fields, Ch. XIV §2, Proposition 4(iii) (the criterion), Proposition 5 (the symbol is the cup product), and Proposition 7(iii).