# Internal Multi-Agent Adversarial Review of AGNT Labs Technical Report IV

**Subject:** *Cyclic Rigidity in Dimension Two: A Newton-Edge Obstruction for Equivariant Keller Maps*
**Review date:** 2026-07-22
**Subject SHA-256:** `179f954ca9cd3d06e150842b682fe839a94e27552d9f42b2ba1cc1c26ccd8fed`
**ProofKit verification:** `pv_mrvbqobt_283e13ab61`

## Executive verdict

### Algebraic theorem: VALID

The internal adversarial panel found no invalid implication in the theorem

\[
muH_u g_y-H_y(g+mug_u)=c\in k^\times\Longrightarrow g\in k^\times
\]

for a characteristic-zero field \(k\), \(m\ge1\), and \(H,g\in k[u,y]\). Two agents reconstructed the proof independently; two hostile agents separately attacked proof obligations O1--O3 and O4--O6. The classification and explicit inverse also check.

### Published manuscript: MATERIAL CORRECTION REQUIRED

The live manuscript is not correct as written because it says

\[
F(x,y)=(H(x^m,y),xg(x^m,y))
\]

commutes with \(\tau_\zeta(x,y)=(\zeta x,y)\). Direct composition gives

\[
F\circ\tau_\zeta=(H,\zeta xg),\qquad
\tau_\zeta\circ F=(\zeta H,xg),
\]

so these are generally unequal.

The exact repair is either:

1. define different source and target actions
   \[
   \rho_s(\zeta)=\operatorname{diag}(\zeta,1),\qquad
   \rho_t(\zeta)=\operatorname{diag}(1,\zeta),
   \]
   for which \(F\circ\rho_s=\rho_t\circ F\); or
2. swap target coordinates and use
   \[
   \widetilde F(x,y)=(xg(x^m,y),H(x^m,y)),
   \]
   which genuinely commutes with \(\operatorname{diag}(\zeta,1)\).

The coordinate swap changes only the sign of the Jacobian and does not affect rigidity or automorphy.

A second correction is required in the collision discussion: the \(m\) points \((\zeta^j x_0,y_0)\) form a genuine collision only for \(m\ge2\). For \(m=1\), there is one point.

The Miyanishi comparison must also be narrowed. His main theorem assumes a **small** finite subgroup of \(\mathrm{GL}(2,\mathbb C)\). His reduction by the normal subgroup generated by pseudoreflections does not directly settle the pure cyclic pseudoreflection case: there \(G=N\), so the residual group \(G/N\) is trivial and the quotient-map automorphy question is not resolved merely by parity of the original \(m\).

## Six proof obligations

| ID | Obligation | Verdict | Reason |
|---|---|---|---|
| O1 | Maximal slope gives \(g_{[0]}=A(u^qy^p)\), \(A(0)\ne0\), \(\deg A\ge1\) | PASS | Weight zero means \(qd=pj\); coprimality gives \((j,d)=(qn,pn)\). The case \(p=0\) reduces to \(q=1,t=u\). |
| O2 | The maximal component of \(H\) exists, has \(\sigma\ge1\), and equals \(u^ry^s\Phi(t)\) | PASS | Polynomial support is finite; the boundary equation supplies a nonzero weight-one \(y\)-term; a fixed edge has one residue class modulo \((q,p)\). |
| O3 | No lower-weight pieces contribute at weight \(\sigma-1\) | PASS | Every bilinear term from weights \((\alpha,\beta)\) has weight \(\alpha+\beta-1\); \(\alpha\le\sigma,\beta\le0\), with equality only at the two selected edges. |
| O4 | Edge polynomial formula | PASS | Independent differentiation gives \(B=-ptA\Phi'-sA\Phi-mq\sigma tA'\Phi\); the mixed \(t^2A'\Phi'\) terms cancel and \(pr-qs=-q\sigma\). |
| O5 | Terminal coefficient cannot vanish | PASS | \([t^{D+N}]B=-a_D\phi_N(pN+s+mq\sigma D)\), where the parenthesis is a strictly positive integer and hence nonzero in characteristic zero. |
| O6 | Boundary case \(\sigma=1\) | PASS | Equality with the scalar \(c\) would require a constant top component, but the nonzero term of degree \(D+N\ge1\) forbids this. |

## Independent reproduction

A newly written script, not copied from the report artifacts, tested exact coefficient systems for

\[
g=1+au+by+cuy,\qquad a,b,c\in\{-1,0,1\},
\]

excluding the constant case, for \(m=1,\ldots,5\), with

\[
H=\sum_{0\le i\le4,\ 0\le j\le5}h_{ij}u^iy^j.
\]

It solved 130 exact rational rank-consistency systems and found zero nonconstant solutions. The same independent script checked 500 exact rational instances of the weighted-edge identity. These finite checks corroborate but do not prove the unbounded theorem.

The supplied archive was also replayed: all four reproduction scripts passed, and the 16-entry SHA-256 ledger matched before and after the review. The review did not modify the production report or artifacts.

## Novelty audit

| Source | Relation | Calibrated finding |
|---|---|---|
| Miyanishi, *Transformation Groups* 28 (2023), 951--971 | Closest equivariant precedent | Main theorem treats small finite groups of even order over \(\mathbb C\); it does not directly subsume a pure cyclic pseudoreflection group. |
| Razar, *Israel J. Math.* 32 (1979), 97--106 | Geometric special-case precedent | Fiber/place-at-infinity criteria, not the cyclic PDE or classification. |
| Abhyankar--Moh, *J. Reine Angew. Math.* 276 (1975), 148--166 | One-place/embedding background | Structural tool, not an anticipation of the theorem. |
| Moh, *J. Reine Angew. Math.* 340 (1983), 140--212 | Newton/root-configuration precedent | Bounded-degree and infinity analysis, not this all-degree cyclic normal form. |
| Bass--Connell--Wright, *Bull. AMS* 7 (1982), 287--330 | General reduction precedent | Does not preserve or prove the displayed cyclic normal form. |
| Glidewell--Hurst--Lee--Li, arXiv:2205.12792 | Newton-polygon/Magnus-formula method | Methodological overlap only; no exact theorem located. |
| Nguyen, *Quaest. Math.* 48(2) (2025) | Recent Newton-polygon classes/degree 104 | Different hypotheses; no exact cyclic PDE overlap located. |

**Defensible wording:** Report IV appears to provide a new direct rigidity theorem and classification for this particular cyclic pseudoreflection normal form, with a distinct terminal-coefficient Newton-edge proof. No exact antecedent was located in the audited sources.

**Not defensible:** an unconditional world-first claim, a claim to resolve unrestricted \(JC_2\), or a claim that Miyanishi already covers the Report IV class whenever \(m\) is even.

Primary links:

- Miyanishi: https://doi.org/10.1007/s00031-022-09727-7 and https://arxiv.org/abs/2110.06709
- Bass--Connell--Wright: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-7/issue-2/The-Jacobian-conjecture--Reduction-of-degree-and-formal-expansion/bams/1183549636.full
- Moh: https://eudml.org/doc/152524
- Glidewell et al.: https://arxiv.org/abs/2205.12792
- Nguyen: https://arxiv.org/abs/1902.05923

## Infinity bridge

The panel separated three candidate bridge statements.

1. **PROVED locally:** In residue characteristic zero, inertia in a finite Galois extension of a DVR is tame and cyclic, acting on the cotangent parameter by roots of unity. Source: Stacks Project, Tag 09E3, Lemma 15.114.5.
2. **OPEN:** A branch at infinity of a minimal nonproper Keller map can be simultaneously linearized in compatible completed source and target coordinates into the Report IV two-coordinate cyclic normal form.
3. **DISPROVED without extra hypotheses:** The polynomial terminal-coefficient obstruction extends unchanged to arbitrary completed power series. Set
   \[
   m=p=q=\sigma=s=1,\quad r=0,\quad A(t)=1+t,\quad \Phi(t)=\frac1{1+t}.
   \]
   Then
   \[
   B(t)=-tA\Phi'-A\Phi-tA'\Phi=-1,
   \]
   even though \(A\) is nonconstant. Infinite series have no terminal coefficient.

The first real missing implication is therefore

\[
\text{cyclic inertia on one completed parameter}
\not\Rightarrow
\text{compatible global polynomial pseudoreflection normal form}.
\]

Even a completed normal form would not be enough by itself; one must also recover bounded support, finite pole order, or another extremal invariant replacing the missing terminal coefficient.

## Mandatory correction set before external circulation

1. Correct the equivariance statement or swap the target coordinates everywhere.
2. Restrict the collision paragraph to \(m\ge2\).
3. Replace “Miyanishi controls broad even-order equivariant classes” with the exact small-group statement and explain why the pure pseudoreflection case is not directly subsumed.
4. Regenerate the HTML, artifact manifest, and SHA-256 ledger after the textual correction.
5. Circulate the corrected LaTeX/PDF, not the current live v1.0, to human referees.
6. Keep the novelty claim calibrated until a specialist literature review is obtained.

## Internal review traces

- O1--O3 hostile review: `bcfbd98e-7a93-4abe-8ea9-77231dd7e1be`
- Independent theorem reproduction: `c923400e-9f43-4706-9d9e-ef9fb1d9df1d`
- Geometric/invariant-theoretic referee: `273626f0-88e9-4df6-9cf5-a9dc904243c1`
- O4--O6 hostile review: `ce0b4c26-248c-4a2b-8924-59f4b4499283`
- Fast manuscript verifier: `d3e1f6e0-34d4-4688-8274-fda331170050`
- Novelty audit: `79fbf738-2c96-4926-945d-bed0aa37d6fd`
- Infinity bridge attack: `8829af4a-c473-4d53-8daf-975ea5925b27`

## Corrected manuscript outcome

A corrected conventional LaTeX manuscript was produced at `../report-iv-arxiv-v1/report-iv.tex` and compiled to an eight-page PDF. It incorporates the mandatory equivariance, collision, Miyanishi, and edge-factorization corrections. Two independent agents rechecked the corrected file after the final edits; both returned **PASS**. The live HTML v1.0 was deliberately left untouched pending Nathan's approval.

## Final panel decision

**Accept the algebraic theorem. Reject the current live HTML wording until corrected. Accept the new LaTeX/PDF as the internally reviewed circulation draft. Do not claim that the infinity bridge has been established.**