{
  "packet_id": "full_manuscript",
  "scope_summary": "Main claim: for K=T_10\u00d7T_10 in R^20, R_20(K)/c_20=121*binom(20,10)/(21*2^20)=22355476/22020096>1, so Brannen's proposed bound |\u03a0K|\u2264c_d|K|^{d-1} fails. All six proof-bearing results were audited from the source: facet formula, product identity, affine covariance, simplex evaluation, the dimension-20 comparison (and the displayed general product test), and the exponential-excess corollary. The proofs are self-contained; cited external papers are context, not premises.",
  "audited_steps": [
    {
      "anchor": "Lemma lem:facets, equations (eq:facet) and (eq:zonotope)",
      "result": "checked",
      "evidence": "For unit u, projection from the facet hyperplane has Jacobian |\u27e8\u03bd_F,u\u27e9|=|det(v_1,...,v_{d-1},u)|. Almost every point of the projection has exactly two preimages in facet interiors, forcing the factor 1/2; homogeneity extends off the sphere. The support function of [-v/2,v/2] is (1/2)|\u27e8v,u\u27e9|, so the centered zonotope representation follows."
    },
    {
      "anchor": "Proposition prop:product, equations (eq:product-body) and (eq:multiplicative)",
      "result": "checked",
      "evidence": "Facets of an orthogonal product are F\u00d7B and A\u00d7G, with (r+s-1)-volumes |B|s_F and |A|s_G and unit normals (\u03bd_F,0) and (0,\u03bd_G). Lemma lem:facets gives h_\u03a0(A\u00d7B)(u,v)=|B|h_\u03a0A(u)+|A|h_\u03a0B(v), the support function of (|B|\u03a0A)\u00d7(|A|\u03a0B). Then |B|^r|A|^s|\u03a0A||\u03a0B|/(|A||B|)^{r+s-1} equals R_r(A)R_s(B), since the |A|-exponent is s-(r+s-1)=1-r and the |B|-exponent is 1-s. Hypotheses r,s\u22652 and full-dimensional polytopes are the ones later used."
    },
    {
      "anchor": "Lemma lem:affine",
      "result": "checked",
      "evidence": "A height-1 prism over F has volume s_F; its image has volume |det L|s_F and height 1/||L^{-t}\u03bd_F||, so the area-normal becomes |det L|L^{-t}(s_F \u03bd_F), including det L<0. Thus \u03a0(LP)=|det L|L^{-t}\u03a0P. This map has absolute determinant |det L|^{d-1}, matching |LP|^{d-1}/|P|^{d-1}, so R_d is invariant on polytopes. Translations preserve projection volumes. The increasing inscribed sandwich P_m\u2282P\u2282\u03bb_m P_m, \u03bb_m\u21921, gives uniform support convergence and R_d(P)=lim R_d(P_m); the polytope case alone suffices for the counterexample."
    },
    {
      "anchor": "Proposition prop:simplex",
      "result": "checked",
      "evidence": "|T_d|=1/d! by iterated base-height. Coordinate area-normals are -e_i/(d-1)!. Dropping one coordinate projects the opposite facet onto T_{d-1} with Jacobian |\u27e8w/\u221ad,e_d\u27e9|=1/\u221ad, so its area is \u221ad/(d-1)! and its area-normal is w/(d-1)!. Sign flips do not change centered segments, so \u03a0T_d is a translate of (Q+[0,w])/(d-1)!. Fibers of Q parallel to w are intervals and lengthen by ||w|| with unchanged base, hence |Q+[0,w]|=1+h_\u03a0Q(w). Opposite unit-cube facets give h_\u03a0Q(w)=\u03a3|w_i|=d, so the volume is d+1. Scaling yields |\u03a0T_d|=(d+1)/((d-1)!)^d and R_d=(d+1)(d!)^{d-1}/((d-1)!)^d=(d+1)d^d/d!=c_d. The d=2 and d=3 cases recover c_2=6 and c_3=18. Every full-dimensional simplex is an invertible affine image of T_d."
    },
    {
      "anchor": "Theorem thm:counterexample and the following general product test",
      "result": "checked",
      "evidence": "Propositions prop:product and prop:simplex give R_20(T_10\u00d7T_10)=c_10^2=(11\u00b710^{10}/10!)^2. Division by c_20=21\u00b720^{20}/20! produces 121\u00b710^{20}\u00b720!/((10!)^2\u00b721\u00b720^{20}). Since 20^{20}=2^{20}\u00b710^{20}, this equals 121\u00b7binom(20,10)/(21\u00b72^{20}). Independently, binom(20,10)=184756, 2^{20}=1048576, 121\u00b7184756=22355476, 21\u00b71048576=22020096, and 22355476-22020096=335380>0. The same algebra gives the displayed general ratio (r+1)(s+1)/(r+s+1)\u00b7binom(r+s,r)\u00b7r^r s^s/(r+s)^{r+s}. Thus K strictly violates the proposed bound in dimension 20."
    },
    {
      "anchor": "Corollary cor:exponential-excess",
      "result": "checked",
      "evidence": "Every n\u226520 is uniquely n=10k+d_r with k\u22651 and d_r\u2208{10,...,19}. All factor dimensions are at least 10, so iterative application of prop:product and prop:simplex gives R_n(K_n)=c_10^k c_{d_r}. For j\u22653 the ratios 1/(j+1)<1/3, so e<5/2+(1/6)\u03a3_{j\u22650}3^{-j}=5/2+1/4=11/4=\u03b2, and the k=n term of e^n gives c_n<(n+1)e^n<(n+1)\u03b2^n. Prime cancellation gives c_10=11\u00b72^2\u00b75^8/(3^4\u00b77)=17187500/567 and A=c_10/\u03b2^{10}=2^{14}\u00b710^8/(567\u00b711^9)=1638400000000/1336956340797>1. For 1<\u03bb<A^{1/10}, R_n(K_n)/(R_n(T_n)\u03bb^n) exceeds the positive quantity c_{d_r}(\u03b2\u03bb)^{-d_r}(A/\u03bb^{10})^k/(10k+d_r+1), which tends to infinity in each of the ten residue classes, so one finite N\u226520 exists."
    }
  ],
  "objections": [],
  "unverified_obligations": [],
  "assessment": "no_demonstrated_error_found"
}