There are many valid presentations for a profinite group; the paper proves two of the candidates below, and they are not the only ones that have been proposed for $\operatorname{Gal}(\overline{\mathbb{Q}}_2 / \mathbb{Q}_2)$. Each candidate in the proven and further-candidate sections passed the finite-quotient verifier, which compares counts of surjections onto a battery of finite test groups against the counts for the true Galois group; passing is strong evidence of correctness, but not a proof. Each card carries a status badge from a fixed five-level scale, defined on the comparison page: solid badges are theorem-level, outlined badges are experimental, and red badges mark refuted proposals.
In each presentation, $\omega_2 \in \widehat{\mathbb{Z}}$ projects to $1$ in the $\mathbb{Z}_2$ component and $0$ in the $\mathbb{Z}_p$ component for odd $p$. We also write $\omega_{\mathrm{odd}}$ for the complementary idempotent, projecting to $0$ in the $\mathbb{Z}_2$ component and to $1$ in every odd-primary component.
Three of the presentations below (the proven collector presentation, the square-commutator candidate, and the new twisted-square candidate) are scored head to head in a five-part complexity model: see the comparison for the methodology, the scores, and which presentation wins for which purpose.
Proven
The absolute Galois group of $\mathbb{Q}_2$ is topologically generated by $\sigma,\ \tau,\ x_0,\ x_1$, where the closed normal subgroup generated by $x_0, x_1$ is pro-$2$, subject to the two relations $$\tau^{\sigma}=\tau^{2} \qquad\text{and}\qquad h_0\,u_1^{-1}\,x_1^{\sigma}\,c_0=1,$$ where the auxiliary words are $$\sigma_2=\sigma^{\omega_2},\qquad u_i=(x_i\tau)^{\omega_2}\ \ (i=0,1),\qquad d_0=u_0x_0^{-1},$$ $$z_0=x_0^{\sigma_2},\qquad c_0=[d_0,z_0],\qquad g_0=\sigma_2^{2},\qquad d_g=d_0^{g_0},$$ $$h_c=[d_g,d_0],\qquad h_0=x_0^{g_0}x_0\,d_g d_0 d_0^{2}h_c.$$
This is the presentation proved in the paper, named here for the class-two collector word $h_0$ from which its wild relator is assembled.
Read the statement in the paper · explore the formalizations · see how it compares.
Further candidates
Each of the presentations below is topologically generated by $\sigma, \tau, x_0, x_1$ as above with the tame relation $\tau^\sigma = \tau^2$, and $\omega_2$ is as above. The first of them, the square-commutator presentation, has since been proved as well; the paper carries that proof as an appendix.
Put
$$\begin{aligned} \sigma_2&=\sigma^{\omega_2}, & a&=(x_0^{-3}\tau)^{\omega_2},\\ y_1&=x_1^{\sigma_2}, & c&=x_1^{-1}y_1^{-1}x_1y_1. \end{aligned}$$Second relator:
$$(x_0^{\sigma})^{-1}\,a\,x_1^{2}\,c=1.$$Named for the literal square-and-commutator block $x_1^2c$ that closes its relator.
Proved and formalized on July 26, 2026. The theorem is unconditional, and it rests on the same nine named external interfaces as the collector presentation and on nothing else. The proof is not a change of generators away from the proven presentation (no triple of words identifies the two maximal pro-$2$ quotients) but a re-run of the paper’s argument over a second source.
Read the proof in the paper · the Lean theorem · see how it compares.
Put
$$\sigma_2=\sigma^{\omega_2},\qquad u_i=(x_i\tau)^{\omega_2}\ \ (i=0,1),\qquad d=u_0x_0^{-1}.$$Second relator (with $x^g=g^{-1}xg$ and $[x,y]=x^{-1}y^{-1}xy$):
$$(x_0d^{-1})^{\sigma_2^{2}}\,x_0d\;u_1^{-1}\,x_1^{\sigma}\,[d,x_0^{\sigma_2}]=1.$$Proposed by GPT while constructing the complexity comparison, and named for its principal block $(x_0d^{-1})^{\sigma_2^2}x_0d$, the twisted square $y^{\sigma_2^2}y\,d^2$ for $y=x_0d^{-1}$. It keeps the collector presentation’s transparent marked pro-$2$ boundary while replacing the long collector word by that twisted square, in the spirit of the square-commutator presentation’s directness; in the comparison’s model it strictly improves the collector presentation. It comes with a proof sketch along the paper’s architecture, and it passed the verifier on July 26, 2026, predicting the correct count for every one of the 5,402 test groups.
Put
$$\begin{aligned} \sigma_2&=\sigma^{\omega_2}, & u&=x_1^{\sigma_2}x_1^{-1},\\ c&=u\sigma^{2}, & \theta&=(x_0\tau)^{\omega_2}x_0^{-1}. \end{aligned}$$Second relator:
$$x_1^{c}\,\sigma^{-2}x_1\sigma\,x_0^{-1}\sigma x_0\,\theta=1.$$Evidence: passed the verifier on all 5,402 test groups, and an AI-checked but unformalized proof has been proposed. There is no human-refereed or machine-checked proof.
Put
$$\begin{aligned} \sigma_2&=\sigma^{\omega_2}, & s&=\sigma_2^{-1}\sigma,\\ a&=x\sigma^{-2}, & a_y&=(y\tau)^{\omega_2},\\ f_y&=a_y^{-1}y, & e_1&=x^{-1}x^{\sigma},\\ e_2&=x^{-1}x^{\sigma^2}, & a_x&=(xs)^{\omega_2},\\ h_1&=a_x^{-1}x^{\sigma}, & h_2&=a_x^{-1}x^{\sigma^2}. \end{aligned}$$Second relator:
$$a^{2}\sigma^{3}\sigma^{y}e_1^{-4}e_2^{3}h_1^{2}h_2^{-2}f_y=1.$$Evidence: passed the verifier on all 5,402 test groups. No proof has been proposed.
Rejected candidates
The six proposals below failed some step in verification (“local test” indicates tests run by the model). Each used $\sigma,\tau,x_0,x_1$, with the closed normal subgroup generated by $x_0,x_1$ intended to be pro-$2$.
Six rejected candidate families and their failure evidence
Relations common to both retained variants
$$\tau^\sigma=\tau^2,$$ $$x_1^2[\sigma,x_0][\tau,x_0][\tau,x_1]=1.$$One retained variant additionally imposed the shadow relations
$$x_0^{\omega_{\mathrm{odd}}}=1,\qquad x_1^{\omega_{\mathrm{odd}}}=1.$$The other retained variant omitted both shadow relations.
Roe initially reported 931 failures among 2,952 processed groups; the retained full output contains 1,578 mismatches. Contemporary proof development used ChatGPT-5.2-Pro; the computation used Claude Code with Opus 4.6. The retained artifacts do not identify a sole generator.
Relations
$$\tau^\sigma=\tau^2,$$ $$x_1^2[\sigma,x_1][\tau,x_1] (x_0^\sigma)^{-1}(x_0\tau)^{40491355905}=1.$$ $$x_0^{\omega_{\mathrm{odd}}}=1,\qquad x_1^{\omega_{\mathrm{odd}}}=1.$$It passed all 1,403 reported local cases, then Roe’s verifier found 127 mismatches. The candidate came from a human-directed systematic search. The retained records do not identify the generating model.
Relations
$$\tau^\sigma=\tau^2,$$ $$x_1^2[\sigma_2,x_1](x_0^\sigma)^{-1} (x_0\tau)^{\omega_2}=1,$$ $$x_0^{\omega_{\mathrm{odd}}}=1,\qquad x_1^{\omega_{\mathrm{odd}}}=1.$$After the correction, Roe reported seven failures and 21 timeouts. Those results refuted the candidate.
Relations
$$\tau^\sigma=\tau^2,$$ $$x_0^2x_1^4\bigl((x_1\tau)^{\omega_2}\bigr)^{-1} x_1^\sigma=1.$$The local test found zero surjections onto $\operatorname{GL}_2(3)$ where 384 were required, so this splice was rejected immediately.
Relations
$$\tau^\sigma=\tau^2,$$ $$x_0^2x_1^4u_1^{-1}x_1^\sigma d_0d_1^{-1}=1.$$A cup-product rank calculation gave 17 instead of the required 23. The retained evidence identifies A1 as an internal candidate.
Relations
$$\tau^\sigma=\tau^2,$$ $$x_0^2x_1^4u_1^{-1}x_1^\sigma c_0=1.$$A2 passed all 1,639 runnable rows in the local cluster sweep; 64 custom groups were unavailable. A marked-abelianization contradiction later refuted the word itself.