{
  "packet_id": "foulkes-howe-stabilization",
  "scope_summary": "The manuscript proves that the canonical Foulkes-Howe map is surjective for a >= 2 and b >= a(a-1). The proof relies on an annihilator criterion, a differential injectivity lemma using the Fischer inner product, and a four-step induction on a. All sections and numbered claims have been audited.",
  "audited_steps": [
    {
      "anchor": "Section 2",
      "result": "checked",
      "evidence": "The criterion for surjectivity correctly reduces the problem to the vanishing of symmetric a-linear forms whose diagonal vanishes on products of b vectors, using the fact that products span the symmetric algebra."
    },
    {
      "anchor": "Lemma 1",
      "result": "checked",
      "evidence": "The injectivity of the differential operator D_{Y,U} for p > q is correctly proved using the Fischer inner product and the sl_2 commutator relation ED - DE = N_U - N_Y."
    },
    {
      "anchor": "eq:binary",
      "result": "checked",
      "evidence": "Step 1 correctly uses the splitting of X^r + c Y^r over C to construct a product of b vectors and extracts the coefficient of c to obtain T(x^r Q, ..., x^r Q, t^r Q) = 0."
    },
    {
      "anchor": "eq:first-composition",
      "result": "checked",
      "evidence": "Step 2 correctly applies D_{y_k, t} to H_x. The degree of t before the i-th operator is b-i+1 and the degree of y_i is r. Since b-i >= r, the strict inequality p > q holds, so Lemma 1 applies and proves H_x = 0."
    },
    {
      "anchor": "eq:inductive-vanishing",
      "result": "checked",
      "evidence": "Step 3 correctly applies the induction hypothesis to the symmetric r-linear form S_{x,t} on Sym^m V, since the degree condition m >= r^2 > r(r-1) is satisfied."
    },
    {
      "anchor": "eq:second-composition",
      "result": "checked",
      "evidence": "Step 4 correctly applies D_{x, v_i}^r to f. The degree of x before any operator is at most r^2-1 and the degree of v_i is at least m+1. Since m >= r^2, the strict inequality p > q holds, so Lemma 1 applies and proves f = 0."
    },
    {
      "anchor": "Theorem 1",
      "result": "checked",
      "evidence": "Theorem 1 follows directly from Proposition 2 and the criterion in Section 2."
    },
    {
      "anchor": "Corollary 3",
      "result": "checked",
      "evidence": "Corollary 3 correctly uses complete reducibility of finite-dimensional representations over C to split the equivariant surjection and obtain an embedding."
    }
  ],
  "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": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21",
    "started_at": "2026-10-13T02:18:51.150020+00:00",
    "finished_at": "2026-10-13T02:22:12.668039+00:00",
    "elapsed_seconds": 201.519,
    "selected_attempt": "panel",
    "source_only_independent_first_pass": true
  }
}