{
  "packet_id": "foulkes-howe-stabilization",
  "scope_summary": "Main conclusion is Theorem 1: for every integer a\u22652, every integer b\u2265a(a-1), and every finite-dimensional complex V, the multiplication/averaging map \u03bc_{a,b,V}: Sym^b(Sym^a V)\u2192Sym^a(Sym^b V) is surjective. The full internal argument was audited: polarization and image span, the diagonal annihilator criterion, the Fischer/sl_2 injectivity lemma, all four steps of the vanishing induction, the theorem deduction, and the equivariant-splitting corollary. Historical comparisons with Brion, RSW, Landsberg, CIM, and Ikenmeyer are not proof dependencies.",
  "audited_steps": [
    {
      "anchor": "Section 2 polarization identity and pure-power spanning",
      "result": "checked",
      "evidence": "For k=1 the identity is u_1=u_1. For k=2 the RHS is -(u_1)^2-(u_2)^2+(u_1+u_2)^2=2u_1u_2. In general the square-free monomial u_1\u2026u_k occurs only for J full, with coefficient k!, and inclusion-exclusion cancels monomials missing an index. Since k!\u22600 in C, pure powers span Sym^k V for k\u22651."
    },
    {
      "anchor": "Section 2 image description A_b=span{P^{\u2297a}}",
      "result": "checked",
      "evidence": "Averaging sends u^a to the already-invariant tensor u^{\u2297a}, so pure powers spanning Sym^a V imply that these tensors span R_1. Componentwise multiplication gives z(u_1)\u2026z(u_b)=(u_1\u2026u_b)^{\u2297a}. Multilinearity of the product of b factors then spans A_b by such tensors as the u_i vary."
    },
    {
      "anchor": "Section 2 italicized surjectivity criterion",
      "result": "checked",
      "evidence": "T_\u03bb(w_1,\u2026,w_a)=\u03bb(Av(w_1\u2297\u22ef\u2297w_a)) bijects R_b^* with symmetric a-linear forms on E=Sym^b V, because averages of simple tensors span the invariants and symmetry makes averaging redundant. For a product P, Av(P\u2297\u22ef\u2297P)=P^{\u2297a}, so \u03bb annihilates A_b iff the diagonal of T_\u03bb vanishes on products. Finite-dimensionality gives A_b=R_b iff that annihilator is zero."
    },
    {
      "anchor": "Lemma 3 (lem:shift) and equation (eq:norm)",
      "result": "checked",
      "evidence": "For one pair, ED-DE acts on U^p Y^q by p(q+1)-q(p+1)=p-q. Distinct coordinate pairs commute, so ED-DE=N_U-N_Y in every dimension. The stated monomial Hermitian product makes \u2202_{U_k} adjoint to multiplication by U_k and multiplication by Y_k adjoint to \u2202_{Y_k}, hence D^*=E even after reversing composition. Thus ||Dh||^2-||Eh||^2=(p-q)||h||^2. If Dh=0 and p>q, both sides vanish, so positive-definiteness forces h=0. Extra variables are spectators."
    },
    {
      "anchor": "Proposition 4, Step 1, equation (eq:binary)",
      "result": "checked",
      "evidence": "If \u03b1^r=-c\u22600 and \u03b6=exp(2\u03c0i/r), then X^r-(\u03b1Y)^r=\u220f(X-\u03b6^j \u03b1 Y) equals X^r+cY^r with no extra scalar; c=0 is X^r. The polynomial identity transfers to Sym V for arbitrary, possibly dependent, x,t. Thus P=(x^r+ct^r)Q is a product of r+m=b vectors. Expanding T(A+cB,\u2026,A+cB), the coefficient of c is a T(A,\u2026,A,B) with r=a-1 copies of A. It vanishes for all c, and a\u22600, so (eq:binary) holds."
    },
    {
      "anchor": "Proposition 4, Step 2, equation (eq:first-composition)",
      "result": "checked",
      "evidence": "H_x depends on t only through the last slot, and D_{y,t}(t^b)=b t^{b-1}y, so k applications contribute b!/(b-k)! and insert y_1\u2026y_k only in that slot. At k=m, b-m=r and (eq:binary) makes the composition identically zero as a polynomial. Before the i-th factor, t-degree is b-i+1 and y_i-degree remains r, since earlier operators use other destinations; the difference is m-i+1\u22651. Each ambient bihomogeneous operator is injective, so the composition is injective and H_x=0."
    },
    {
      "anchor": "Proposition 4, Step 3, equation (eq:inductive-vanishing)",
      "result": "checked",
      "evidence": "S_{x,t}(Q_1,\u2026,Q_r)=T(x^r Q_1,\u2026,x^r Q_r,t^b) is symmetric r-linear on Sym^m V because multiplication by the fixed x^r is linear and T is symmetric. Step 2 gives vanishing of its diagonal on products of m vectors. From b\u2265a(a-1)=r(r+1) one gets m=b-r\u2265r^2\u2265r(r-1), and m\u22651. The induction is on a, simultaneously for every admissible degree and every finite-dimensional V, so the hypothesis for a-1 applies, including the a=1 spanning base when r=1."
    },
    {
      "anchor": "Proposition 4, Step 4, equation (eq:second-composition)",
      "result": "checked",
      "evidence": "D_{x,v}^r(v^b)=b!/(b-r)! x^r v^m, and distinct slots do not interfere, giving the scalar (b!/m!)^r. The resulting evaluation vanishes by Step 3 for all vector values, hence as a polynomial. The r^2 operators are applied in blocks D_{x,v_i}^r. Before any one, x-degree is at most r^2-1, while its source has been differentiated at most r-1 times, so source degree is at least b-r+1=m+1\u2265r^2+1. Thus source degree strictly exceeds destination degree at every factor, the composition is injective, and f=0. Pure powers span, so multilinearity yields T=0. Equality b=a(a-1) still gives source degree \u2265r^2+1>r^2-1."
    },
    {
      "anchor": "Proof of Theorem 1",
      "result": "checked",
      "evidence": "For a\u22652 and b\u2265a(a-1), Proposition 4 is exactly the vanishing half of the Section 2 criterion, which is equivalent to surjectivity of \u03bc_{a,b,V} and to A_b=R_b. No additional hypothesis is used."
    },
    {
      "anchor": "Corollary 5",
      "result": "conditional",
      "evidence": "Averaging and multiplication commute with GL(V), so Theorem 1 supplies an equivariant surjection. An invariant complement to the kernel exists only after complete reducibility; restriction is then an equivariant isomorphism onto the target, and inverse followed by inclusion is an embedding. The arithmetic b\u226530 for a=6 is 6\u00b75=30. The embedding is not identified with a prescribed canonical map, as stated."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "Corollary 5, citation to Macdonald Chapter I, Appendix A, Sections 7\u20138",
      "obligation": "Audit the external complete-reducibility theorem for finite-dimensional polynomial representations of GL(V) over C; the written splitting deduction was checked only conditionally on that theorem."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "grok-4.7",
    "model_key": "grok-4.7",
    "provider": "xai",
    "effort": "high",
    "packet_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21",
    "started_at": "2026-10-13T02:19:21.453648+00:00",
    "finished_at": "2026-10-13T02:26:21.504111+00:00",
    "elapsed_seconds": 420.051,
    "selected_attempt": "panel",
    "source_only_independent_first_pass": true
  }
}