{
  "packet_id": "noncoordinate-polynomial",
  "scope_summary": "The manuscript constructs an explicit counterexample to the Abhyankar-Sathaye conjecture in dimension 4 (and higher). The main claims are that the zero fibre of the polynomial F is isomorphic to affine three-space, and that F is not a coordinate polynomial because it has a critical point in the ambient space. All sections, lemmas, and the appendix have been fully audited.",
  "audited_steps": [
    {
      "anchor": "Equation (1.1)",
      "result": "checked",
      "evidence": "Direct expansion of x^2+y^3 = (u^3+hv)^2 + (-u^2+hw)^3 yields u^6 + 2u^3hv + h^2v^2 - u^6 + 3u^4hw - 3u^2h^2w^2 + h^3w^3 = h(2u^3v + 3u^4w + h(v^2-3u^2w^2) + h^2w^3) = hs. This matches the definition of s."
    },
    {
      "anchor": "Lemma 2.1, S(h,U,G-UW,W) expansion",
      "result": "checked",
      "evidence": "Substituting V = G-UW into S gives 2U^3G + U^4W + hG^2 - 2hUWG - 2hU^2W^2 + h^2W^3. Modulo hW = y+U^2, this becomes 2U^3G + U^4W + hG^2 - 2UG(y+U^2) - 2U^2W(y+U^2) + W(y+U^2)^2. Expanding and canceling terms exactly yields hG^2 - 2UyG + y^2W."
    },
    {
      "anchor": "Equation (2.4)",
      "result": "checked",
      "evidence": "y^2a - hb = y^2(y+U^2) - h(s-hG^2+2UyG) = y^3 - hs + y^2U^2 + h^2G^2 - 2hUyG. Using x^2+y^3=hs, this is -x^2 + (hG-yU)^2. In A, hG-yU=x, so this evaluates to -x^2+x^2 = 0."
    },
    {
      "anchor": "Equation (2.5)",
      "result": "checked",
      "evidence": "Since \\alpha h + \\beta y = 1, r = \\alpha(1+\\beta y) and q = \\beta^2 give rh+qy^2 = (1-\\beta y)(1+\\beta y) + \\beta^2 y^2 = 1. Then hW_0 = h(ra+qb) = rha + qhb = rha + qy^2a = (rh+qy^2)a = a. Similarly, y^2W_0 = b."
    },
    {
      "anchor": "Equation (2.6)",
      "result": "checked",
      "evidence": "The map U = hT - \\beta x, G = yT + \\alpha x satisfies hG - yU = h(yT+\\alpha x) - y(hT-\\beta x) = (\\alpha h + \\beta y)x = x. The inverse T = \\alpha U + \\beta G correctly recovers T, U, and G."
    },
    {
      "anchor": "Equation (3.3)",
      "result": "checked",
      "evidence": "X^2+Y^3 = (x+s^3)^2 + (y-s^2)^3 = x^2+y^3 + 2xs^3 - 3y^2s^2 + 3ys^4. Since p(x,y,s) = -2s^2x + 3sy^2 - 3s^3y, we have s p(x,y,s) = -2s^3x + 3s^2y^2 - 3s^4y. Thus X^2+Y^3 = x^2+y^3 - s p(x,y,s)."
    },
    {
      "anchor": "Equation (3.5)",
      "result": "checked",
      "evidence": "In B, h = 1+p(x,y,s). Thus X^2+Y^3 = x^2+y^3 - s(h-1) = x^2+y^3 - sh + s. Since x^2+y^3-sh = 0 in B, X^2+Y^3 = s. The translation is invertible, so B is isomorphic to C[X,Y,s]/(X^2+Y^3-s) which is isomorphic to C[X,Y]."
    },
    {
      "anchor": "Proof of Theorem 1.1, unit ideal generation",
      "result": "checked",
      "evidence": "Modulo (h,y), x^2+y^3-sh=0 implies x^2=0. h = 1+p(x,y,s) implies 0 = 1 - 2s^2x. Squaring gives 1 = 4s^4x^2 = 0. Thus B/(h,y) is the zero ring."
    },
    {
      "anchor": "Equation (3.7)",
      "result": "checked",
      "evidence": "In R, x^2+y^3 = hs. So X^2+Y^3 = x^2+y^3 - s p(x,y,s) = hs - s p(x,y,s) = s(h - p(x,y,s)). Since F = h - p(x,y,s) - 1, this is exactly s(1+F)."
    },
    {
      "anchor": "Proof of Theorem 1.1, critical point",
      "result": "checked",
      "evidence": "Differentiating X^2+Y^3 = s(1+F) gives 2X dX + 3Y^2 dY = (1+F) ds + s dF. At P=(2,0,-1/2,1/2), we have x=-1, y=1, s=1, p=2, F=-1, X=0, Y=0. Substituting these values yields 0 = 0 ds + 1 dF, so dF(P) = 0."
    },
    {
      "anchor": "Proof of Theorem 1.1, noncoordinate conclusion",
      "result": "checked",
      "evidence": "A coordinate polynomial must be the first component of an automorphism, meaning its gradient can never vanish. Since \\nabla F(P) = 0, F cannot be a coordinate."
    },
    {
      "anchor": "Equation (A.2)",
      "result": "checked",
      "evidence": "Expanding the RHS gives h(1+2s^2x+4s^5) - (1+2s^2x)(1+p) - 4s^4x^2 - 4s^4y^3. The h term is \\alpha h. The remainder simplifies to -1 - 2s^2x - p(1+2s^2x) - 4s^4x^2 - 4s^4y^3. Substituting p gives -1 + y((3s^3-3sy)(1+2s^2x) - 4s^4y^2) = \\beta y - 1. Thus \\alpha h + \\beta y = 1 in B."
    }
  ],
  "objections": [],
  "unverified_obligations": [],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "gemini-3.1-pro-preview",
    "model_key": "gemini",
    "provider": "google",
    "effort": "high",
    "packet_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c",
    "started_at": "2026-10-13T02:12:27.694326+00:00",
    "finished_at": "2026-10-13T02:16:00.805563+00:00",
    "elapsed_seconds": 213.112,
    "selected_attempt": "panel",
    "source_only_independent_first_pass": true
  }
}