{
  "packet_id": "noncoordinate-polynomial",
  "scope_summary": "Main claim (Theorem 1.1): F = h - 1 - p(x,y,s) in R = C[h,u,v,w], with x,y,s given in (1.1) and p in (1.2), satisfies R/(F) \u2245 C^[3] and \u2207F(2,0,-1/2,1/2) = 0. Hence F is not a coordinate, which would make it a counterexample to the Abhyankar\u2013Sathaye conjecture for n = 4. Corollary 1.2 extends this to every n \u2265 4. I audited every proof-bearing item in full: the identity (1.1), Lemma 2.1 (every step), the shift identity (3.3), the presentation of B (3.4)\u2013(3.6), the unit-ideal claim, the identification C \u2245 R/(F), the critical-point computation (re-derived independently from partial derivatives), the Jacobian obstruction, Corollary 1.2, and the Appendix certificate (A.2) and inverse (A.4). Cited literature is used only for context. The one cited obstruction (Shpilrain\u2013Yu) is reproved inline by the chain-rule argument.",
  "audited_steps": [
    {
      "anchor": "eq (1.1), x^2+y^3=hs",
      "result": "checked",
      "evidence": "(u^3+hv)^2+(-u^2+hw)^3 = h[2u^3v+3u^4w+h(v^2-3u^2w^2)+h^2w^3]. The u^6 terms cancel and every remaining term divided by h matches S exactly."
    },
    {
      "anchor": "Lemma 2.1, substitution G=V+UW",
      "result": "checked",
      "evidence": "Expanding with y=-U^2+hW gives hG^2-2UyG+y^2W = hV^2+2U^3V+3U^4W-3hU^2W^2+h^2W^3 = S(h,U,V,W); the hU^2W^2 coefficient is 1-2-2=-3. Also U^3+h(G-UW)-x = hG-yU-x, using hUW = U(y+U^2). V\u21a6G is a triangular automorphism, so (2.3) holds."
    },
    {
      "anchor": "Lemma 2.1, eq (2.4) and the Bezout pair r,q",
      "result": "checked",
      "evidence": "y^2(y+U^2)-h(s-hG^2+2UyG) = (hG-yU)^2+y^3-hs = x^2+y^3-hs = 0 in A. rh+qy^2 = (1-\u03b2y)(1+\u03b2y)+\u03b2^2y^2 = 1."
    },
    {
      "anchor": "Lemma 2.1, W_0 and the ideal equality",
      "result": "checked",
      "evidence": "hW_0 = rha+qy^2a = a, using hb = y^2a. y^2W_0 = rhb+qy^2b = b. W-W_0 = r(hW-a)+q(y^2W-b), and hW-a = h(W-W_0), y^2W-b = y^2(W-W_0). So the ideal is principal (W-W_0) and A[W]/(\u00b7) \u2245 A."
    },
    {
      "anchor": "Lemma 2.1, eq (2.6) A \u2245 B[T]",
      "result": "checked",
      "evidence": "hG-yU = x(\u03b1h+\u03b2y) = x. On B[T]: \u03b1(hT-\u03b2x)+\u03b2(yT+\u03b1x) = T. In A: h(\u03b1U+\u03b2G) = \u03b1hU+\u03b2(x+yU) = U+\u03b2x and y(\u03b1U+\u03b2G) = \u03b1(hG-x)+\u03b2yG = G-\u03b1x, so both composites are the identity. No integrality or regularity assumption is used, so the lemma holds over any commutative ring."
    },
    {
      "anchor": "eq (3.3) shift identity",
      "result": "checked",
      "evidence": "(x+s^3)^2+(y-s^2)^3 = x^2+y^3+2s^3x-3s^2y^2+3s^4y = x^2+y^3-sp with p = -2s^2x+3sy^2-3s^3y."
    },
    {
      "anchor": "eqs (3.4)-(3.6), B \u2245 C[X,Y]",
      "result": "checked",
      "evidence": "Eliminating h=1+p turns the second relation into x^2+y^3-s-sp = X^2+Y^3-s. (x,y,s)\u21a6(X,Y,s) is a triangular automorphism of C[x,y,s], so B \u2245 C[X,Y,s]/(X^2+Y^3-s) \u2245 C[X,Y]."
    },
    {
      "anchor": "Sec 3, (h,y)=B",
      "result": "checked",
      "evidence": "Modulo (h,y): p = -2s^2x, so 1-2s^2x = 0, and x^2 = sh = 0. Then 1 = 4s^4x^2 = 0. The explicit certificate (A.2) independently confirms \u03b1h+\u03b2y = 1 in B; I expanded both sides and they agree."
    },
    {
      "anchor": "Sec 3, C \u2245 R/(F)",
      "result": "checked",
      "evidence": "C = C[h,x,y,s,u,v,w]/(h-1-p, x^2+y^3-sh, x-(u^3+hv), y-(-u^2+hw), s-S). Eliminating x,y,s yields C[h,u,v,w]/(h-1-p(x,y,s), x^2+y^3-hS). The second generator is identically 0 by (1.1) and the first is F. Lemma hypotheses (3.4)-(2.1) hold in B, so R/(F) \u2245 B[T] \u2245 C[X,Y,T] as C-algebras."
    },
    {
      "anchor": "eq (3.8) and critical point P",
      "result": "checked",
      "evidence": "X^2+Y^3 = hs-sp = s(h-p) = s(1+F) in R. At P=(2,0,-1/2,1/2): x=-1, y=1, s=1/2+1/2=1, p=2+3-3=2, F=-1, X=Y=0. The differential identity gives dF(P)=0. Independent check: dx=-1/2dh+2dv, dy=1/2dh+2dw, ds=3/4dh-2dv+3dw, du-terms vanish, dp=-2ds-2dx+3dy=dh, so dF=dh-dp=0."
    },
    {
      "anchor": "Sec 3, non-coordinate argument",
      "result": "checked",
      "evidence": "If (F,G2,G3,G4) is an automorphism with inverse \u03c8, then J(\u03c8)(\u03c6(P))\u00b7J(\u03c6)(P) = I. So J(\u03c6)(P) is invertible, contradicting a zero first row at P."
    },
    {
      "anchor": "Corollary 1.2",
      "result": "checked",
      "evidence": "R[z]/(F) = (R/(F))[z] \u2245 C^[n-1]. \u2202F/\u2202z_i = 0 everywhere and the original partials vanish at P, so (P,0,...,0) is a critical point and the same Jacobian argument applies."
    },
    {
      "anchor": "Appendix eqs (A.3)-(A.4)",
      "result": "checked",
      "evidence": "w = \u03b1(1+\u03b2y)(y+u^2)+\u03b2^2(s-hg^2+2uyg) equals ra+qb with r=\u03b1(1+\u03b2y), q=\u03b2^2. u, g and v=g-uw match (2.6) and V=G-UW. All maps are polynomial."
    },
    {
      "anchor": "Sec 1, derivation D and Maubach kernel remark",
      "result": "unverified",
      "evidence": "D(x)=3u^2h-3u^2h=0 and D(y)=-2uh+2uh=0 were checked. The attribution to Maubach's paper was not checked. The proof does not use it."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "Sec 1, literature attributions (KVZ Thm 4.2(a), Maubach Cor 4.2-4.3, GGP Thm 4.11, Ghosh Thm 1.4, MaubachMonomial Thm 3.2 case (3), SY \u00a71)",
      "obligation": "Confirm the cited external statements match the descriptions. They are contextual only; no proof step depends on them."
    },
    {
      "anchor": "Abstract/Sec 1, novelty framing",
      "obligation": "Optionally confirm by computer algebra that (A.3) and (A.4) compose to the identity on R/(F) and on C[X,Y,T]. The symbolic lemma proof already establishes this, so this would be a redundant sanity check."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
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    "model_key": "claude-opus-5.5",
    "provider": "anthropic",
    "effort": "high",
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    "started_at": "2026-10-13T02:24:58.003218+00:00",
    "finished_at": "2026-10-13T02:26:12.745013+00:00",
    "elapsed_seconds": 74.742,
    "selected_attempt": "panel-high",
    "source_only_independent_first_pass": true
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