{
  "packet_id": "foulkes-howe-stabilization",
  "scope_summary": "The main conclusion is surjectivity of the explicitly defined canonical multiplication map Sym^b(Sym^a V) \u2192 Sym^a(Sym^b V) for every finite-dimensional complex V, a \u2265 2, and b \u2265 a(a\u22121). The review covers all five numbered theorem/lemma/proposition/corollary/remark statements, all ten numbered equations, and the supporting polarization, annihilator, differentiation, and induction arguments. No demonstrated error was found in the self-contained proof of the main theorem. The representation-comparison corollary is checked conditionally on the cited complete-reducibility theorem. External documents and their historical assertions were not inspected and remain explicit verification obligations.",
  "audited_steps": [
    {
      "anchor": "Equation (1), eq:average",
      "result": "checked",
      "evidence": "Permutation averaging is unchanged by permuting the input factors, so it descends to Sym^a W. The quotient projection sends every summand to w_1\u22efw_a, hence projection composed with averaging is the identity. Conversely, averaging fixes invariant tensors, giving the asserted inverse."
    },
    {
      "anchor": "Equation (2), eq:map, and the definition of A",
      "result": "checked",
      "evidence": "Multiplication in P_a preserves invariance and adds the common tensor-factor degrees. Its restriction to R_1 is commutative, so it factors through Sym^b R_1. With R_1 \u2245 Sym^a V and R_b \u2245 Sym^a(Sym^b V), this gives the displayed map. Since A is generated in degree one, its degree-b component is precisely the multiplication image."
    },
    {
      "anchor": "Theorem 1, thm:main, and its concluding proof",
      "result": "checked",
      "evidence": "Proposition 3 applies for exactly the theorem's a, b, and V. The independently checked annihilator criterion then gives A_b = R_b. The proof does not require the cited stabilization or geometric results."
    },
    {
      "anchor": "Introduction, identification with the RSW divided-power formulation",
      "result": "unverified",
      "evidence": "The manuscript's own map is well defined and its tensor-averaging construction is checked above. Whether the cited RSW formulation uses exactly these conventions was not checked against the external source; this does not affect the theorem about the explicitly defined map."
    },
    {
      "anchor": "Introduction, History and significance paragraph",
      "result": "unverified",
      "evidence": "The cited Chow-coordinate-ring and normalization identifications, Brion results, RSW regularity scope, and Landsberg problem statement were not externally inspected. The internal arithmetic translating d=b and n=a to d \u2265 n(n\u22121) is correct. These contextual assertions are not premises of the vanishing proof."
    },
    {
      "anchor": "Introduction, diagonal-kernel examples and Ikenmeyer/McKay propagation discussion",
      "result": "unverified",
      "evidence": "The externally attributed kernel examples and propagation theorem were not inspected. The examples' claimed pairs lie outside the present bound: 5 < 5\u00b74 and 6 < 6\u00b75. Neither external result is used in the main proof."
    },
    {
      "anchor": "Section 2, polarization identity",
      "result": "checked",
      "evidence": "For a monomial supported on a proper subset of {1,\u2026,k}, the alternating subset sum cancels. The only degree-k monomial containing every formal generator is u_1\u22efu_k, with coefficient k!. Consequently products, and then pure powers, span Sym^k V over C."
    },
    {
      "anchor": "Section 2, exact image description A_b = span{P^{\u2297a}}",
      "result": "checked",
      "evidence": "Polarization makes z(u)=u^{\u2297a} span R_1. Products of b spanning elements span the degree-b generated algebra, and componentwise multiplication gives z(u_1)\u22efz(u_b)=(u_1\u22efu_b)^{\u2297a}. This establishes both inclusions in the image description."
    },
    {
      "anchor": "Section 2, correspondence T_lambda and italicized annihilator criterion",
      "result": "checked",
      "evidence": "A symmetric a-linear form on E linearizes to a symmetric functional on E^{\u2297a}; restricting to invariant tensors gives lambda, and averaging preserves its values. Conversely, invariant tensors are spanned by averages of simple tensors. Therefore lambda annihilates A_b exactly when T_lambda(P,\u2026,P)=0 for all products P. Finite dimensionality implies that a proper A_b would have a nonzero annihilator."
    },
    {
      "anchor": "Lemma 2, lem:shift",
      "result": "checked",
      "evidence": "For the stated Fischer norm, differentiation is adjoint to coordinate multiplication, so D*=E. Direct expansion gives ED\u2212DE=N_U\u2212N_Y. Equation (3) therefore proves injectivity when p>q. Additional variables are unaffected by the commutator and remain part of the positive-definite norm."
    },
    {
      "anchor": "Equation (3), eq:norm",
      "result": "checked",
      "evidence": "Using D*=E gives ||Dh||\u00b2\u2212||Eh||\u00b2=\u27e8(ED\u2212DE)h,h\u27e9. Bihomogeneity makes (N_U\u2212N_Y)h=(p\u2212q)h. If Dh=0 and p>q, then \u2212||Eh||\u00b2=(p\u2212q)||h||\u00b2 forces h=0. The inequality is essential: for two coordinates, h=U_1Y_2\u2212U_2Y_1 has p=q=1 and Dh=0 despite h\u22600."
    },
    {
      "anchor": "Proposition 3, prop:vanishing",
      "result": "checked",
      "evidence": "The induction decreases the number of multilinear slots from a to a\u22121 and applies only at an admissible new degree m. Both differential compositions are injective on their entire intermediate homogeneous spaces. Their terminal identities are supplied respectively by binary factorization and the induction hypothesis. The detailed degree and identity checks are recorded separately below."
    },
    {
      "anchor": "Equation (4), eq:diagonal",
      "result": "conditional",
      "evidence": "This is the proposition's hypothesis, not a derived identity. Its use is justified at every product P supplied by the binary-factor construction. No stronger assumption, such as vanishing on arbitrary elements of Sym^b V, is introduced."
    },
    {
      "anchor": "Proposition 3, opening induction and base cases",
      "result": "checked",
      "evidence": "For V=0 and b\u22651, Sym^b V=0. For a=1, the hypothesis forces a linear functional to vanish on a spanning set of products. The induction is simultaneous over all admissible b and finite dimensions, so invoking it at the new degree m creates no circular dependency."
    },
    {
      "anchor": "Equation (5), eq:rm",
      "result": "checked",
      "evidence": "For a\u22652, r=a\u22121\u22651 and m=b\u2212r. Thus r+m=b, as needed for x^rQ and t^rQ to belong to Sym^b V."
    },
    {
      "anchor": "Equation (6), eq:range",
      "result": "checked",
      "evidence": "The hypothesis b\u2265a(a\u22121) becomes b\u2265r(r+1). Subtracting r gives m\u2265r\u00b2, and r\u22651 gives m\u22651. These inequalities guarantee both a valid positive product degree and the later induction and injectivity bounds."
    },
    {
      "anchor": "Equation (7), eq:binary, and Step 1 factorization",
      "result": "checked",
      "evidence": "For c\u22600, choosing alpha^r=\u2212c gives \u220f_{j=0}^{r\u22121}(X\u2212alpha zeta^jY)=X^r+cY^r; c=0 is immediate. Substitution into Sym V remains valid when x and t are dependent or zero. Multiplication by Q therefore produces b vector factors. The coefficient of c in the zero polynomial T(P,\u2026,P) is a T(x^rQ,\u2026,x^rQ,t^rQ), and a is nonzero over C."
    },
    {
      "anchor": "Equation (8), eq:first-composition",
      "result": "checked",
      "evidence": "For fixed x, only the final slot of H_x depends on t. Successive differentiation therefore inserts y_1,\u2026,y_k there with coefficient b(b\u22121)\u22ef(b\u2212k+1). The existing occurrences of the y_i in Q are not differentiated because every differentiation is with respect to t. At k=m, b\u2212m=r, so Equation (7) makes the result zero."
    },
    {
      "anchor": "Proposition 3, Step 2 degree inequalities and cancellation",
      "result": "checked",
      "evidence": "Before operator i, the t-degree is b\u2212i+1 and the y_i-degree remains r: preceding operators involve different destination blocks. Thus p\u2212q=b\u2212i+1\u2212r=m\u2212i+1\u22651 for 1\u2264i\u2264m. Lemma 2 applies to each intermediate space, and injectivity of the composition recovers H_x=0."
    },
    {
      "anchor": "Equation (9), eq:inductive-vanishing, and Step 3",
      "result": "checked",
      "evidence": "Multiplication by fixed x^r is linear from Sym^m V to Sym^b V, making S_{x,t} a symmetric r-linear form. Step 2 supplies its diagonal vanishing on all degree-m products. The induction hypothesis applies because m\u2265r\u00b2\u2265r(r\u22121) and m\u22651. The resulting identity holds for every fixed x and t, so subsequent substitutions and conversion to a polynomial identity are justified."
    },
    {
      "anchor": "Equation (10), eq:second-composition",
      "result": "checked",
      "evidence": "The variable v_i occurs only in its own input slot. Applying D_{x,v_i} exactly r times sends v_i^b to (b!/m!)x^r v_i^m. Differentiation with respect to v_i cannot act on previously inserted x factors. Multiplying the r coefficients gives (b!/m!)^r, and Equation (9), with Q_i=v_i^m, makes the result zero."
    },
    {
      "anchor": "Proposition 3, Step 4 accumulated degrees",
      "result": "checked",
      "evidence": "More precisely, before the next derivative in slot i, after j prior derivatives in that slot, p=b\u2212j and q=(i\u22121)r+j, where 1\u2264i\u2264r and 0\u2264j\u2264r\u22121. Hence p\u2212q\u2265b\u2212r\u00b2\u2212r+2\u22652. Equivalently, the manuscript's bounds p\u2265m+1\u2265r\u00b2+1 and q\u2264r\u00b2\u22121 hold. Every individual operator is injective despite the shared destination x."
    },
    {
      "anchor": "Proposition 3, final pure-power spanning argument",
      "result": "checked",
      "evidence": "Injectivity of the second composition gives T(v_1^b,\u2026,v_r^b,t^b)=0 for all v_i and every fixed t. Since t was arbitrary, this covers all a-tuples of pure powers. Polarization spans each input space, and multilinearity then forces T=0."
    },
    {
      "anchor": "Corollary 4, cor:comparison",
      "result": "conditional",
      "evidence": "Averaging and multiplication commute with GL(V), so Theorem 1 supplies an equivariant surjection. Given complete reducibility of the finite-dimensional polynomial source representation over C, an invariant complement to its kernel exists and maps isomorphically onto the target; inversion gives the claimed embedding. The numerical specialization is correct: a=6 gives a(a\u22121)=30. The cited external proof of complete reducibility was not inspected."
    },
    {
      "anchor": "Remark 5, final scope remark",
      "result": "checked",
      "evidence": "The two injectivity tests involve only block degrees and not the number of coordinates. Finite dimensionality supplies the annihilator argument, and V=0 was separately handled. The stated uses of C are present: positive Fischer norm, splitting of X^r+cY^r, polynomial identity from pointwise vanishing, and nonzero integer/factorial coefficients. The remark does not assert extension to other fields."
    },
    {
      "anchor": "Boundary checks for Theorem 1 and Proposition 3",
      "result": "checked",
      "evidence": "For dim V=1, both spaces in the map are one-dimensional and multiplication sends their natural generators to one another; for V=0 they vanish in the relevant positive degrees. At a=2,b=2, r=m=1, the first shift has degrees (2,1) and the second (2,0). At a=3,b=6, r=2,m=4: the first shifts have source degrees 6,5,4,3 against destination degree 2, and the second shifts have degree pairs (6,0),(5,1),(6,2),(5,3). All required strict inequalities hold at these boundary parameters."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "Corollary 4; Macdonald, Chapter I, Appendix A, Sections 7\u20138",
      "obligation": "If verification of every external prerequisite is required, inspect or independently prove complete reducibility for finite-dimensional polynomial GL(V)-representations over C. The manuscript's splitting deduction is valid conditional on this theorem; the cited proof has not been audited here."
    },
    {
      "anchor": "Introduction; RSW Section 2 and Theorem 2.3",
      "obligation": "Check the cited divided-power conventions and the asserted identification of A with the Chow coordinate ring and R with its normalization. These external identifications are not needed for the proved statement about the explicitly defined multiplication map."
    },
    {
      "anchor": "Introduction, History and significance; Brion1993, Brion Theorem 3.3, RSW Theorem 3.9, Landsberg Section 7.6 and Problem 7.19",
      "obligation": "Inspect the precise hypotheses and conclusions of the cited stabilization, regularity, geometric, and polynomial-bound statements before certifying the manuscript's historical comparisons and claim that it answers the cited problem."
    },
    {
      "anchor": "Introduction; CIM Theorem 6(b),(c) and Remark 7; Ikenmeyer Theorem 1.3 and Sections 3,5",
      "obligation": "Check the externally attributed diagonal-kernel examples and propagation theorem, including map direction and dimension assumptions. They are contextual and do not enter the main induction."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "gpt-6.1-sol",
    "model_key": "sol",
    "provider": "openai-responses",
    "effort": "high",
    "packet_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21",
    "started_at": "2026-10-13T02:15:03.181379+00:00",
    "finished_at": "2026-10-13T02:18:51.148405+00:00",
    "elapsed_seconds": 227.967,
    "selected_attempt": "panel",
    "source_only_independent_first_pass": true
  }
}