{
  "packet_id": "foulkes-howe-stabilization",
  "scope_summary": "Main claim (Theorem 1): for a>=2, b>=a(a-1) and every finite-dimensional complex V, the multiplication map mu_{a,b,V}: Sym^b(Sym^a V) -> Sym^a(Sym^b V) is surjective. I audited the full mathematical content: the image description and the unnumbered annihilator criterion (Sec. 2), the polarization identity, Lemma 2 (differential injectivity), Proposition 3 (Steps 1-4 and the induction), the proof of Theorem 1, Corollary 4, and Remark 5. Literature and history claims in the Introduction are not proof-bearing and were not audited against their sources. I also checked boundary cases: a=2; (a,b)=(5,5), where the map is cited as failing; and the base cases m=1 and V=0.",
  "audited_steps": [
    {
      "anchor": "Sec. 2, polarization identity",
      "result": "checked",
      "evidence": "This is the standard inclusion-exclusion identity. Expanding sum_J (-1)^{k-|J|}(sum_{j in J} u_j)^k kills every monomial that omits an index. The multilinear monomial u_1...u_k has coefficient k!. Since k! is nonzero over C, pure powers span Sym^k V."
    },
    {
      "anchor": "Sec. 2, A_b = span{P^{\u2297a}}",
      "result": "checked",
      "evidence": "The z(u)=u^{\u2297a} span R_1 = (V^{\u2297a})^{S_a} by polarization with k=a. Componentwise multiplication in (Sym V)^{\u2297a} gives z(u_1)...z(u_b)=(u_1...u_b)^{\u2297a}. Products of b elements from a spanning set span A_b."
    },
    {
      "anchor": "Sec. 2, unnumbered annihilator criterion",
      "result": "checked",
      "evidence": "R_b^* is identified with symmetric a-linear forms on E=Sym^b V via lambda -> lambda\u2218Av. Averages of simple tensors span R_b, which gives uniqueness. Since P^{\u2297a}=Av(P\u2297...\u2297P), we get lambda(P^{\u2297a})=T_lambda(P,...,P). In finite dimension, A_b=R_b if and only if Ann(A_b)=0. Surjectivity is unaffected by the scaling of the averaging identifications."
    },
    {
      "anchor": "Lemma 2 and eq. (norm)",
      "result": "checked",
      "evidence": "In the Fischer inner product, \u2202_k is adjoint to multiplication by X_k, and D=Y\u00b7\u2202_U, E=U\u00b7\u2202_Y satisfy D^*=E. ED-DE=N_U-N_Y gives ||Dh||^2=<h,EDh>=||Eh||^2+(p-q)||h||^2, so Dh=0 with p>q forces h=0. Monomial check with h=u^p y^q: ||Dh||^2-||Eh||^2 = p!q![p(q+1)-q(p+1)] = (p-q)p!q!. Extra variables are harmless."
    },
    {
      "anchor": "Prop. 3, eqs. (rm),(range)",
      "result": "checked",
      "evidence": "a(a-1)=(r+1)r, so b>=r(r+1) and m=b-r>=r^2>=1."
    },
    {
      "anchor": "Prop. 3, Step 1, eq. (binary)",
      "result": "checked",
      "evidence": "X^r+cY^r splits over C, so (x^r+ct^r)Q is a product of b vectors. With P=A+cB, A=x^rQ, B=t^rQ, symmetry gives T(P,...,P)=sum_k C(a,k) c^k T(A^{a-k},B^k). The c-coefficient is a\u00b7T(A,...,A,B), which must vanish."
    },
    {
      "anchor": "Prop. 3, Step 2, eq. (first-composition)",
      "result": "checked",
      "evidence": "D_{y_k,t} acts only on t, which occurs only in the last slot. Each application gives (b-k+1)t^{b-k}y_1...y_k, so the constant is b!/(b-k)!. At k=m the last slot is t^rQ, which vanishes by (binary) pointwise and hence as a polynomial. Before operator i, the t-degree is b-i+1 and the y_i-degree is r (from the r slots containing Q), so p-q=m-i+1>=1. Backward application of Lemma 2 gives H_x=0."
    },
    {
      "anchor": "Prop. 3, Step 3, eq. (inductive-vanishing)",
      "result": "checked",
      "evidence": "S_{x,t} is symmetric r-linear on Sym^m V with diagonal vanishing on products, by Step 2. The induction hypothesis needs m>=r(r-1) and m>=1, and both follow from m>=r^2. The induction is over all V and b simultaneously, as stated."
    },
    {
      "anchor": "Prop. 3, Step 4, eq. (second-composition) and degree count",
      "result": "checked",
      "evidence": "D_{x,v}^r v^b = (b!/m!) x^r v^m, and the operators act only on their own v_i. Before the j-th application to v_i (j=0..r-1), the x-degree is (i-1)r+j and the v_i-degree is b-j. The requirement b > (i-1)r+2j is at worst b > r^2+r-2, and b>=r^2+r holds. All intermediates are bihomogeneous, so Lemma 2 applies at each step, giving f=0. Pure powers span Sym^b V, so T=0."
    },
    {
      "anchor": "Prop. 3, base case a=1 and V=0",
      "result": "checked",
      "evidence": "A linear form vanishing on products, which span Sym^b V for b>=1, is zero. For V=0 both sides vanish."
    },
    {
      "anchor": "Proof of Theorem 1",
      "result": "checked",
      "evidence": "This is a direct combination of Proposition 3 with the Sec. 2 criterion."
    },
    {
      "anchor": "Corollary 4",
      "result": "checked",
      "evidence": "mu is GL(V)-equivariant. Complete reducibility of polynomial representations gives an invariant complement M to ker mu, and (mu|_M)^{-1} composed with inclusion is the embedding. For a=6, a(a-1)=30."
    },
    {
      "anchor": "Boundary sanity checks",
      "result": "checked",
      "evidence": "For a=2, the argument yields T(xQ,t^b)=0, hence T=0, consistent with known normality for rank<=2 quadrics. At (5,5): r=4, m=1, Steps 2-3 go through (the Step 3 base works for m=1). Step 4 needs b > r^2+r-2 = 18 and fails at b=5. The proof therefore does not contradict the cited (5,5) failure, and the bottleneck lies exactly where the hypothesis is used."
    }
  ],
  "objections": [
    {
      "anchor": "Introduction, 'History and significance' (citations to CIM Thm 6(b),(c)/Remark 7, Landsberg Problem 7.19, Brion Thm 3.3, RSW Thm 3.9)",
      "classification": "unverified_step",
      "claim": "The bibliographic characterizations were not checked against the sources: kernels at (5,5) and (6,6), the parameter dictionary d=b, n=a, w=dim V, and the claim that Problem 7.19 is answered.",
      "evidence": "These are statements about external texts. No proof in the manuscript depends on them.",
      "derivation": "None of Theorem 1, Lemma 2, Proposition 3 or Corollary 4 uses these citations. The only cited result used internally is complete reducibility, in Corollary 4, which is standard.",
      "hypotheses_checked": [
        "no proof step invokes CIM, Landsberg, Brion or RSW",
        "the direction convention is consistent: non-injectivity of the dual Hadamard\u2013Howe map at (5,5) equals non-surjectivity of mu_{5,5}"
      ],
      "possible_repair": "Verify the quoted locators and notation against the cited versions.",
      "dependency_impact": "None on the correctness of Theorem 1. It affects only the significance claims."
    }
  ],
  "unverified_obligations": [
    {
      "anchor": "Introduction, Landsberg Problem 7.19",
      "obligation": "Confirm that the problem asks for a polynomial bound for this canonical map, and that the stated dictionary d=b, n=a, w=dim V matches the published version."
    },
    {
      "anchor": "Introduction, CIM Theorem 6(b),(c), Remark 7",
      "obligation": "Confirm that the cited arXiv version shows nonzero kernels of the canonical map at (5,5) and (6,6), and in which direction and dimension."
    },
    {
      "anchor": "Corollary 4, Macdonald Ch. I App. A \u00a7\u00a77-8",
      "obligation": "Standard complete reducibility of polynomial GL(V)-representations; the exact locator was not checked."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "claude-opus-5-5",
    "model_key": "claude-opus-5.5",
    "provider": "anthropic",
    "effort": "high",
    "packet_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21",
    "started_at": "2026-10-13T02:28:26.741725+00:00",
    "finished_at": "2026-10-13T02:30:05.827824+00:00",
    "elapsed_seconds": 99.087,
    "selected_attempt": "panel-high",
    "source_only_independent_first_pass": true
  }
}