Distance-Shell Tomography: Geometric Certificates for Fault-Tolerant Sensing, Moment Compression, and Hypercube Equalization

Distance-shell tomography represents anonymous populations on a graph by integer arrays and their observational ambiguities by signed trades. This preprint develops geometric certificates for recovery after sensor-report erasure, scalar compression, and hypercube equalization. On the n by n grid, the exact four-corner-constrained sensor minimum for one-erasure recovery of populations of mass at most two is ceil(3(n+1)/2) for every n >= 3. A stronger unrestricted lower bound is attained by explicitly verified placements for 5 <= n <= 20. For every positive dimension n divisible by four, the equidistant dimension of Q_n lies between 2^(n-1)+n/2-1 and 2^(n-1)+n/2; the upper value is exact in dimensions 8, 12, and 16. The paper also proves the even bishop-board two-domination assertion and gives explicit counterexamples to three further sourced conjectured statements. Sensing corollaries of classical Hamming-scheme and trade theory give single-moment capacity 2^k-1 for k =2k>=2 and 2^(k-1)<=h<2^k. The supporting material includes editable LaTeX and figures, a pinned Lean 4.35.0-rc2 project with 33 independently compared statements, exact verification programs and data, source and novelty audits, and verification records. Paper proofs, formal endpoints, and exhaustive computations are explicitly distinguished.