{
  "packet_id": "noncoordinate-polynomial",
  "scope_summary": "The main conclusion is that the explicit polynomial F in C[h,u,v,w] has R/(F) isomorphic to C[X,Y,T] but has a critical point on F=-1, so it is not a coordinate; adjoining variables preserves both properties in every ambient dimension n>=4. The audit covers all three numbered results, every numbered equation, the internal presentation changes, the appendix's inverse maps, and concrete nilpotent and h=0 boundary cases. No demonstrated error was found in the self-contained argument. External literature statements remain unverified and are not prerequisites for the construction.",
  "audited_steps": [
    {
      "anchor": "Section 1, definition of coordinate and discussion of its fibres",
      "result": "checked",
      "evidence": "An automorphism with first component f identifies C[z_1,...,z_n]/(f-c) with a polynomial ring in n-1 variables for every c. The distinction between a polynomial zero fibre and a globally coordinate polynomial is therefore correctly stated."
    },
    {
      "anchor": "Equation (1.1), eq:xyz",
      "result": "checked",
      "evidence": "Expansion gives x^2=u^6+2hu^3v+h^2v^2 and y^3=-u^6+3hu^4w-3h^2u^2w^2+h^3w^3. Their sum is exactly h times the displayed S."
    },
    {
      "anchor": "Equation (1.2), eq:F",
      "result": "checked",
      "evidence": "Substitution of the polynomials x,y,s from (1.1) makes p and F genuine elements of C[h,u,v,w]. The constant term -1 is used consistently in the auxiliary algebra and ambient identity."
    },
    {
      "anchor": "Theorem 1.1, thm:main, assertion R/(F) isomorphic to C^[3]",
      "result": "checked",
      "evidence": "The presentation elimination in the proof identifies R/(F) with the algebra C of Lemma 2.1. Its base satisfies both lemma hypotheses and is C[X,Y]; hence C is C[X,Y,T]. The appendix provides mutually inverse C-algebra maps."
    },
    {
      "anchor": "Theorem 1.1, thm:main, assertion grad F(2,0,-1/2,1/2)=0",
      "result": "checked",
      "evidence": "At P, x=-1, y=s=1 and p=2. In variable order (h,u,v,w), dx=(-1/2,0,2,0), dy=(1/2,0,0,2), ds=(3/4,0,-2,3). Since (p_x,p_y,p_s)=(-2,3,-2), dp=(1,0,0,0)=dh, giving dF=0 independently of the ambient-identity argument."
    },
    {
      "anchor": "Theorem 1.1, thm:main, conclusion that F is not a coordinate; Section 3 final paragraph",
      "result": "checked",
      "evidence": "If an automorphism had first component F, its Jacobian at P would have a zero first row. Differentiating its composition with its polynomial inverse gives an identity matrix, forcing that Jacobian to be invertible. This is a direct contradiction and requires no external obstruction theorem."
    },
    {
      "anchor": "Section 1, paragraph following Theorem 1.1",
      "result": "checked",
      "evidence": "The computed values give F(P)=2-2-1=-1. The zero fibre is smooth because its C-algebra is a polynomial ring in three variables."
    },
    {
      "anchor": "Corollary 1.2, cor:dimension, and its Section 3 proof",
      "result": "checked",
      "evidence": "For n>4 the quotient is (R/(F))[z_1,...,z_{n-4}], a polynomial ring in n-1 variables. The extended gradient vanishes at (P,0,...,0), including every added-variable derivative. The same inverse-Jacobian argument excludes a coordinate. The case n=4 is Theorem 1.1."
    },
    {
      "anchor": "Lemma 2.1, lem:lift",
      "result": "checked",
      "evidence": "The proof establishes C isomorphic to A by eliminating W through a unit-ideal identity, then A isomorphic to B[T] through explicit inverse maps. No cancellation of h or y, division, domain assumption, or characteristic assumption is used."
    },
    {
      "anchor": "Equation (2.1), eq:base",
      "result": "conditional",
      "evidence": "The cusp relation and (h,y)=B are hypotheses, not consequences for an arbitrary B. Both are verified separately for the particular base (3.4), so the main application does not leave either prerequisite unverified."
    },
    {
      "anchor": "Equation (2.2), eq:bezout",
      "result": "checked",
      "evidence": "The hypothesis (h,y)=B means precisely that coefficients alpha,beta in B exist with alpha h+beta y=1. The appendix supplies such coefficients for the main application."
    },
    {
      "anchor": "Equation (2.3), eq:Clinear",
      "result": "checked",
      "evidence": "Let Q=hW-y-U^2 and substitute V=G-UW. The first original relation equals hG-yU-x-UQ. Also S(h,U,G-UW,W)-(hG^2-2UyG+y^2W)=Q(-2UG+2yW+WQ). Thus the displayed presentation has exactly the same defining ideal, without dividing by h."
    },
    {
      "anchor": "Equation (2.4), eq:compat",
      "result": "checked",
      "evidence": "For a=y+U^2 and b=s-hG^2+2UyG, direct expansion gives y^2a-hb=(hG-yU)^2+y^3-hs. In A the first square is x^2, so this vanishes by (2.1)."
    },
    {
      "anchor": "Equation (2.5), eq:Wlinear, and the following ideal equality",
      "result": "checked",
      "evidence": "With r=alpha(1+beta y) and q=beta^2, rh+qy^2=1. Compatibility hb=y^2a implies h(ra+qb)=a and y^2(ra+qb)=b. Moreover W-W_0=r(hW-a)+q(y^2W-b), while the two original generators equal h(W-W_0) and y^2(W-W_0). Hence their ideal is exactly (W-W_0), even over a ring with zero divisors."
    },
    {
      "anchor": "Equation (2.6), eq:UG, and the final paragraph of Lemma 2.1",
      "result": "checked",
      "evidence": "U=hT-beta x and G=yT+alpha x satisfy hG-yU=x. Conversely, in A, h(alpha U+beta G)=U+beta x and y(alpha U+beta G)=G-alpha x. These identities recover U,G, and alpha(hT-beta x)+beta(yT+alpha x)=T."
    },
    {
      "anchor": "Lemma 2.1, concrete zero-divisor and h=0 boundary test",
      "result": "checked",
      "evidence": "Take B=C[e]/(e^2), h=e, x=1, y=-1, s=0, alpha=0, beta=-1. The inverse formulas give U=1+eT, G=-T, W=2T+eT^2, V=-3T-3eT^2. Then U^3+hV=1, -U^2+hW=-1, and S=0: the three contributions to S are -6T-24eT^2, 6T+27eT^2, and -3eT^2. Specializing e=0 also checks the h=0 boundary."
    },
    {
      "anchor": "Equation (3.1), eq:cusp",
      "result": "checked",
      "evidence": "This repeats the exact expansion checked at (1.1): x^2+y^3=hS=hs."
    },
    {
      "anchor": "Equation (3.2), eq:shift",
      "result": "checked",
      "evidence": "The substitution is an automorphism over C[s], with inverse x=X-s^3 and y=Y+s^2. Its invertibility does not require specializing or inverting s."
    },
    {
      "anchor": "Equation (3.3), eq:shiftidentity",
      "result": "checked",
      "evidence": "Expansion gives X^2+Y^3=x^2+y^3+2s^3x-3s^2y^2+3s^4y. The last three terms equal -s p(x,y,s), and the two s^6 terms cancel."
    },
    {
      "anchor": "Equation (3.4), eq:B",
      "result": "checked",
      "evidence": "The manuscript explicitly introduces independent generators before quotienting. Its relations impose h=1+p and x^2+y^3=sh, precisely the required base relation and zero-fibre condition."
    },
    {
      "anchor": "Equation (3.5), eq:Bplane",
      "result": "checked",
      "evidence": "Eliminating h changes the cusp relation into x^2+y^3-s-sp=0. By (3.3), this is X^2+Y^3-s=0. Eliminating the variable s then gives C[X,Y]."
    },
    {
      "anchor": "Equation (3.6), eq:Bparam",
      "result": "checked",
      "evidence": "Setting s=X^2+Y^3, x=X-s^3, y=Y+s^2, and h=1+p satisfies both relations of B. The reverse substitution recovers X,Y,s,h, so these are inverse presentations."
    },
    {
      "anchor": "Section 3, paragraph beginning 'The elements h,y generate the unit ideal in B'",
      "result": "checked",
      "evidence": "In B/(h,y), the cusp equation gives x^2=0 and h=1+p gives 1=2s^2x. Squaring yields 1=4s^4x^2=0. Thus the quotient is zero and (h,y)=B. Appendix identity (A.2) independently provides an explicit certificate."
    },
    {
      "anchor": "Section 3, presentation of C and elimination paragraph preceding (3.7)",
      "result": "checked",
      "evidence": "In the seven-generator presentation, the three monic relations solve for x=u^3+hv, y=-u^2+hw, and s=S. After substitution, x^2+y^3-sh is identically zero and h-1-p becomes exactly F. Therefore the resulting quotient is R/(F), not merely a localization or a surjective image."
    },
    {
      "anchor": "Equation (3.7), eq:fiber",
      "result": "checked",
      "evidence": "The preceding presentation gives R/(F) isomorphic to C; the verified base hypotheses permit Lemma 2.1, giving C isomorphic to B[T]; and (3.5) gives B[T] isomorphic to C[X,Y,T]. All maps fix C."
    },
    {
      "anchor": "Equation (3.8), eq:ambient",
      "result": "checked",
      "evidence": "Combining (3.1) and (3.3) gives X^2+Y^3=hs-sp=s(h-p)=s(1+F). This identity holds in the original polynomial ring, not just on F=0."
    },
    {
      "anchor": "Section 3, evaluation at P and differentiation of (3.8)",
      "result": "checked",
      "evidence": "At P, S=2(1/4)+4(1/8)=1, x=-1, y=1, p=2, X=Y=0, and F=-1. In 2X dX+3Y^2 dY=(1+F)ds+s dF, every term except dF vanishes, so dF(P)=0. The separate component calculation above confirms this conclusion."
    },
    {
      "anchor": "Equation (A.1), eq:alphabeta",
      "result": "checked",
      "evidence": "Both coefficients are polynomials. Their claimed Bezout property follows from the free-polynomial identity (A.2), checked separately below; they require no inverses of h or y."
    },
    {
      "anchor": "Equation (A.2), eq:certificate",
      "result": "checked",
      "evidence": "Put A=1+2s^2x. Expanding the right-hand side gives Ah+4s^5h+A(3s^3y-3sy^2)-4s^4y^3-1, because A(-1+2s^2x)-4s^4x^2=-1. This is exactly alpha h+beta y-1. Both defining relations vanish in B."
    },
    {
      "anchor": "Equation (A.3), eq:fibercoords",
      "result": "checked",
      "evidence": "X,Y are the verified base coordinates, and T=alpha u+beta(v+uw) is precisely the inverse parameter in Lemma 2.1 with G=v+uw. Their expressions are polynomials on the entire zero fibre."
    },
    {
      "anchor": "Equation (A.4), eq:explicitinverse",
      "result": "checked",
      "evidence": "The formulas reproduce (2.6) and W_0 from (2.5). Starting on the fibre, hT-beta x=u and yT+alpha x=g follow from alpha h+beta y=1 and hg-yu=x; uniqueness of W then recovers w and v. Starting from independent X,Y,T, the reverse composite gives alpha u+beta g=T and restores X,Y by (3.6). Thus both composites are identities."
    },
    {
      "anchor": "Appendix, claim that no open subset is removed; concrete h=0 point of F=0",
      "result": "checked",
      "evidence": "Every inverse expression is polynomial. For an explicit boundary check, choose a with a^5=-1/4 and take (h,u,v,w)=(0,a,1/(2a^2),0). Then x=a^3, y=-a^2, s=a, p=-1, and F=0. Here alpha=-1/2, beta=4a^3, beta y=1, and T=3a/2. Formula (A.4) returns u=a, g=-2a^3=v, w=0, confirming that h=0 is not excluded."
    },
    {
      "anchor": "Section 1, invariant-theoretic paragraph defining D",
      "result": "checked",
      "evidence": "Directly, D(h)=0, D(x)=3hu^2-3hu^2=0, and D(y)=-2hu+2hu=0. Applying D to hs=x^2+y^3 gives hD(s)=0. Since h is not a zero divisor in R, D(s)=0."
    },
    {
      "anchor": "Section 1, Abhyankar--Moh--Suzuki assertion for n=2",
      "result": "unverified",
      "evidence": "The cited external proofs are not supplied. This contextual result is not used in any step of the construction."
    },
    {
      "anchor": "Section 1, Sathaye assertion concerning equations linear in one of three variables",
      "result": "unverified",
      "evidence": "The exact external theorem and its hypotheses were not inspected. It has no dependency role in the internal proof."
    },
    {
      "anchor": "Section 1, Russell [Theorem 2.3] assertion for arbitrary fields",
      "result": "unverified",
      "evidence": "The extension to arbitrary fields requires checking the cited theorem, including its quotient and coefficient hypotheses. The construction does not invoke it."
    },
    {
      "anchor": "Section 1, KVZ [Theorem 4.2(a)] and Maubach [Corollaries 4.2--4.3]",
      "result": "unverified",
      "evidence": "The claimed affirmative result for a(x)y+b(x,z,t), a nonzero, is external. Its precise scope and assumptions were not verified, and it is not applied to F."
    },
    {
      "anchor": "Section 1, GGP [Theorem 4.11]",
      "result": "unverified",
      "evidence": "The source only describes additional coefficient and divisor hypotheses without stating them. Their exact content remains an external contextual obligation, not a missing hypothesis in Lemma 2.1 or Theorem 1.1."
    },
    {
      "anchor": "Section 1, Ghosh [Theorem 1.4]",
      "result": "unverified",
      "evidence": "The cited characteristic-zero embedding result, including its independent-variable, b nonzero, and distinct-root conditions, was not inspected. No internal deduction depends on it."
    },
    {
      "anchor": "Section 1, MaubachMonomial [proof of Theorem 3.2, case (3)] attribution",
      "result": "unverified",
      "evidence": "The invariance of h,x,y,s was checked directly, but their attribution to the cited kernel calculation was not verified. The manuscript does not use that external calculation to prove the fibre isomorphism."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "Section 1, Abhyankar--Moh--Suzuki Theorem citations AM and Suzuki",
      "obligation": "Inspect the cited proofs or exact theorem statements supporting the affirmative n=2 coordinate assertion. This is contextual and not needed for the construction."
    },
    {
      "anchor": "Section 1, Sathaye [pp. 1--2] and Kraft [Section 4]",
      "obligation": "Verify the historical attribution of the coordinate-versus-all-fibres distinction and the stated survey coverage. Neither attribution is an internal proof prerequisite."
    },
    {
      "anchor": "Section 1, Sathaye and Russell [Theorem 2.3]",
      "obligation": "Check each external linear-plane rectifiability result, including the field, quotient-isomorphism, and coefficient assumptions, before accepting the literature summary."
    },
    {
      "anchor": "Section 1, KVZ [Theorem 4.2(a)] and Maubach [Corollaries 4.2--4.3]",
      "obligation": "Inspect both sources to confirm that the stated four-variable affine-three-space quotient condition yields ambient rectification under exactly the summarized hypotheses."
    },
    {
      "anchor": "Section 1, EEO and GGPthreefold citations",
      "obligation": "Verify the stated relationship of these works to the linear hypersurface problem. No particular theorem from them is supplied or used."
    },
    {
      "anchor": "Section 1, GGP [Theorem 4.11]",
      "obligation": "Retrieve the exact coefficient and divisor hypotheses and check that the contextual description accurately represents the external theorem."
    },
    {
      "anchor": "Section 1, Ghosh [Theorem 1.4]",
      "obligation": "Check the exact variable count, quotient hypothesis, coefficient conditions, field assumptions, and conclusion against the cited version."
    },
    {
      "anchor": "Section 1, Shpilrain--Yu [Section 1]",
      "obligation": "Verify the attributed historical question and use of critical loci to distinguish embeddings. The manuscript's own critical-point obstruction was proved directly and does not depend on this citation."
    },
    {
      "anchor": "Section 1, MaubachMonomial [proof of Theorem 3.2, case (3)]",
      "obligation": "Inspect the cited calculation to verify the attribution of h,x,y,s. Their annihilation by D and every algebraic step needed for the main theorem were checked independently."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "gpt-6.1-sol",
    "model_key": "sol",
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    "effort": "high",
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    "started_at": "2026-10-13T02:12:27.693222+00:00",
    "finished_at": "2026-10-13T02:19:21.452181+00:00",
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