{
  "packet_id": "full_manuscript",
  "scope_summary": "The main conclusion is an exact dimension-twenty counterexample: K=T_10\u00d7T_10 has R_20(K)/c_20=22,355,476/22,020,096>1. All six numbered proof-bearing claims were audited: the facet formula, product identity, affine covariance, simplex value, counterexample theorem, and exponential-excess corollary. Their supporting arguments, hypotheses, arithmetic, and dependence were checked, including the general product test and low-dimensional cases. The six proofs are self-contained and have no circular dependency. The cited historical and external research claims were not independently verified and are recorded separately; none is needed to establish the counterexample or corollary.",
  "audited_steps": [
    {
      "anchor": "Lemma lem:facets; equations eq:facet and eq:zonotope",
      "result": "checked",
      "evidence": "For a unit u, projection of a facet with unit normal \u03bd_F has absolute Jacobian |\u03bd_F\u00b7u|. After removing the projection boundary, projections of faces of dimension at most d\u22122, and projections of facets parallel to u, every remaining fiber through P is a nondegenerate interval whose two endpoints lie in facet interiors. Thus \u03a3_F s_F|\u03bd_F\u00b7u|=2 vol_{d\u22121}(proj_{u\u22a5}P). All removed sets have (d\u22121)-dimensional measure zero. Homogeneity handles nonunit u, including u=0. The centered segment with endpoints \u00b1s_F\u03bd_F/2 has support function s_F|\u03bd_F\u00b7u|/2, proving the zonotope identity using the support-function uniqueness argument supplied immediately before the lemma. As a normalization check, P=[0,1]^d gives h_{\u03a0P}(u)=\u03a3_i|u_i| and \u03a0P=[\u22121,1]^d."
    },
    {
      "anchor": "Proposition prop:product; equations eq:product-body and eq:multiplicative; general product test in Section sec:counterexample",
      "result": "checked",
      "evidence": "Full dimensionality ensures that a nonzero functional on either factor exposes a proper face. Consequently a product facet must be F\u00d7B or A\u00d7G; a functional nonzero on both factors produces codimension at least two. Orthogonality gives area-normal vectors (|B|s_F\u03bd_F,0) and (0,|A|s_G\u03bd_G). Lemma lem:facets therefore gives h_{\u03a0(A\u00d7B)}(u,v)=|B|h_{\u03a0A}(u)+|A|h_{\u03a0B}(v), the support function of (|B|\u03a0A)\u00d7(|A|\u03a0B). Its volume is |B|^r|A|^s|\u03a0A||\u03a0B|. Dividing by |A|^{r+s\u22121}|B|^{r+s\u22121} yields exactly R_r(A)R_s(B), without a binomial factor. The stated hypotheses r,s\u22652 cover every subsequent application. For A=B=[0,1]^2, this gives R_4([0,1]^4)=16=4\u00b74. Substitution of the simplex constants also gives the displayed general ratio ((r+1)(s+1)/(r+s+1)) binom(r+s,r) r^r s^s/(r+s)^{r+s}; hence any universal upper constant must be at least c_rc_s."
    },
    {
      "anchor": "Lemma lem:affine, including the approximation paragraph beginning 'For completeness'",
      "result": "checked",
      "evidence": "For a polytope facet, the transformed normal is L^{\u2212t}\u03bd_F/||L^{\u2212t}\u03bd_F||. The image of a unit-height prism over F has perpendicular height 1/||L^{\u2212t}\u03bd_F||, so its base area is |det L| ||L^{\u2212t}\u03bd_F||s_F. Thus every area-normal vector transforms by |det L|L^{\u2212t}, and Lemma lem:facets proves covariance even when det L<0. This map has absolute determinant |det L|^{d\u22121}, exactly matching the denominator's scaling. For general bodies, the supplied increasing inscribed polytopes contain a fixed aB_2^d and satisfy P_m\u2282P\u2282\u03bb_mP_m with \u03bb_m\u21921. Their projection support functions converge uniformly, and the closure of their increasing projection bodies supplies existence of \u03a0P. The subsequent support and volume sandwiches justify passage to the limit. Applying L retains LP_m\u2282LP\u2282\u03bb_mLP_m, so the same argument proves covariance for LP without presupposing it. Translation invariance follows directly from projected volumes. The cases L=tI with t>0 and a coordinate reflection agree with the formula."
    },
    {
      "anchor": "Proposition prop:simplex, especially the facet-vector calculation and the fiber identity |Q+[0,w]|=1+h_{\u03a0Q}(w)",
      "result": "checked",
      "evidence": "The coordinate facets have area-normal vectors \u2212e_i/(d\u22121)!. For the remaining facet, the coordinate-dropping projection has Jacobian 1/\u221ad and image volume 1/(d\u22121)!, giving area-normal vector w/(d\u22121)!. Lemma lem:facets therefore makes \u03a0T_d a translate of ([0,1]^d+[0,w])/(d\u22121)!. Along lines parallel to w, adding [0,w] leaves the base projection unchanged and increases every nonempty fiber length by ||w||; Fubini yields volume 1+h_{\u03a0Q}(w). The cube facet formula gives h_{\u03a0Q}(w)=d. Hence |\u03a0T_d|=(d+1)/((d\u22121)!)^d and R_d(T_d)=(d+1)(d!)^{d\u22121}/((d\u22121)!)^d=(d+1)d^d/d!. Lemma lem:affine proves the assertion for all full-dimensional simplices. Local checks give |\u03a0T_2|=3 and R_2(T_2)=6, while |\u03a0T_3|=1/2 and R_3(T_3)=18."
    },
    {
      "anchor": "Theorem thm:counterexample and its proof in Section sec:counterexample",
      "result": "checked",
      "evidence": "The point (1/11,\u2026,1/11) is interior to T_10, so both factors and their product satisfy the full-dimensional polytope hypotheses. The checked product and simplex propositions give R_20(K)=c_10^2. Thus R_20(K)/c_20=121\u00b720!/(21\u00b72^20\u00b7(10!)^2)=121 binom(20,10)/(21\u00b72^20). Independently checking the integers gives binom(20,10)=184,756, numerator 22,355,476, denominator 22,020,096, and difference 335,380>0. The inequality is therefore a strict violation of the explicitly stated simplex benchmark. The proof depends on the product and standard-simplex calculations, not on any cited counterexample theorem."
    },
    {
      "anchor": "Corollary cor:exponential-excess; representation n=10k+d_r, exponential-series bound, and exact value of A",
      "result": "checked",
      "evidence": "Writing n=10q+r with 0\u2264r\u22649 gives the unique representation k=q\u22121\u22651 and d_r=10+r for every n\u226520. Every factor and intermediate product is full-dimensional with dimension at least ten, so the product proposition applies repeatedly and yields R_n(K_n)=c_10^k c_{d_r}. The factorial tail after 1/3! has successive ratios at most 1/4, hence strictly less than 1/3; therefore e<5/2+(1/6)\u03a3_{j\u22650}3^{\u2212j}=11/4=\u03b2. The positive Taylor term n^n/n! is less than e^n, giving c_n<(n+1)\u03b2^n. Exact arithmetic verifies c_10=17,187,500/567 and A=c_10/\u03b2^10=1,638,400,000,000/1,336,956,340,797; the numerator exceeds the denominator by 301,443,659,203. For any 1<\u03bb<A^{1/10}, division gives R_n(K_n)/(c_n\u03bb^n)>c_{d_r}(\u03b2\u03bb)^{\u2212d_r}(A/\u03bb^10)^k/(10k+d_r+1). The exponential base exceeds one, so this tends to infinity for each of the ten residues. Taking a dimension threshold valid for all ten residues proves the quantified conclusion. This deduction uses checked constants and product identities, not an unverified asymptotic prerequisite."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "Introduction: Brannen citation and the statement that simplices were conjectured to maximize R_d",
      "obligation": "Verify the exact formulation, normalization, and historical attribution in Brannen's cited paper. The present manuscript independently proves violation of its displayed bound, but attribution of that bound was not externally audited."
    },
    {
      "anchor": "Introduction: ChenEtAl, Theorems 1.1 and 1.2",
      "obligation": "Inspect the cited external proofs and hypotheses for the three-dimensional minimum result and universal inequality R_3(K)\u226418. The manuscript's calculation of the tetrahedral value 18 was checked; the universal inequality and minimum classification were not. Neither is a prerequisite for the six proofs."
    },
    {
      "anchor": "Introduction: FengEtAl, Theorem 1.3 and Section 4",
      "obligation": "Verify the asserted counterexamples in every dimension d\u22659, the at-most-d+2-facets condition, and the description involving perturbed cube projections and Minkowski's existence theorem. Those external constructions and proofs were not audited and are not used in the manuscript's construction."
    },
    {
      "anchor": "Introduction: Henk citation and the lower bound 2^d(9/8)^{floor(d/3)}",
      "obligation": "Check the exact dimensional scope, normalization, and centrally symmetric hypotheses of the cited lower bound. Its external proof was not inspected, and the exponential-excess corollary does not depend on it."
    },
    {
      "anchor": "Introduction and Section sec:facets: Petty, Ludwig Equation (1), and Saroglou Section 1",
      "obligation": "Verify the historical covariance attribution and the cited descriptions distinguishing projection-body volume, polar projection inequalities, and the minimum problem. The covariance and polytopal facet identities themselves were independently checked from the supplied proofs; the unseen cited proofs and historical descriptions were not audited."
    }
  ],
  "assessment": "no_demonstrated_error_found"
}