{
  "packet_id": "full_manuscript",
  "scope_summary": "The manuscript claims to provide a counterexample to Brannen's simplex maximum conjecture for projection-body volume, using the Cartesian product of two 10-dimensional simplices. It also claims an exponential excess over the simplex bound in high dimensions. All six proof-bearing claims (Lemmas 1 and 3, Propositions 2 and 4, Theorem 5, and Corollary 6) were audited.",
  "audited_steps": [
    {
      "anchor": "Lemma 1",
      "result": "checked",
      "evidence": "The facet formula for the projection body of a polytope is a standard consequence of Cauchy's projection formula. The proof correctly identifies that almost every line parallel to the projection direction intersects exactly two facet interiors, leading to the factor of 1/2."
    },
    {
      "anchor": "Proposition 2",
      "result": "checked",
      "evidence": "The facets of a Cartesian product A x B are of the form F x B and A x G. The area-normal vectors are correctly computed as (|B|s_F\\nu_F, 0) and (0, |A|s_G\\nu_G). The support function of the projection body is then shown to be the sum of the support functions of the scaled projection bodies of the factors, which establishes the product formula for the projection body and the multiplicativity of the functional R_d."
    },
    {
      "anchor": "Lemma 3",
      "result": "checked",
      "evidence": "The affine covariance of the projection body is derived by tracking the transformation of the area-normal vectors under an invertible linear map L. The area scales by |det L| ||L^{-t}\\nu_F||, and the normal transforms to L^{-t}\\nu_F / ||L^{-t}\\nu_F||. The resulting projection body scales by |det L| L^{-t}, which exactly cancels the volume scaling in the functional R_d. The limit argument for arbitrary convex bodies is standard and correct."
    },
    {
      "anchor": "Proposition 4",
      "result": "checked",
      "evidence": "The projection body of the standard simplex T_d is computed by identifying its d+1 facets. The area-normal vectors are -e_i/(d-1)! and w/(d-1)!, where w is the all-ones vector. The projection body is a translate of the Minkowski sum of a scaled cube and a segment. The volume of this sum is correctly computed using Fubini's theorem as 1 + h_{\\Pi Q}(w) = d+1, yielding the simplex value c_d = (d+1)d^d/d!."
    },
    {
      "anchor": "Theorem 5",
      "result": "checked",
      "evidence": "The counterexample is K = T_{10} x T_{10} in dimension 20. Using the product formula, R_{20}(K) = c_{10}^2. The ratio R_{20}(K)/c_{20} is exactly computed as (121 * 184756) / (21 * 1048576) = 22355476 / 22020096, which is strictly greater than 1. The arithmetic is exact and correct."
    },
    {
      "anchor": "Corollary 6",
      "result": "checked",
      "evidence": "The exponential excess is established by considering products of k copies of T_{10} and one factor T_{d_r} to reach any dimension n >= 20. The simplex value c_n is bounded by (n+1)\\beta^n where \\beta = 11/4. The ratio c_{10}/\\beta^{10} is exactly computed as 1638400000000 / 1336956340797 > 1, which guarantees exponential growth of the ratio R_n(K_n)/c_n."
    }
  ],
  "objections": [],
  "unverified_obligations": [],
  "assessment": "no_demonstrated_error_found"
}