2026/08/06 by Omri Abas
Mathematics · #math.CO #math.MG #math.PR #math.SP #msc:35P20 #msc:52A22 #msc:52B11 #msc:52C35 #msc:60D05
paper · pdf · doi:10.5281/zenodo.21827694
68 pages, 4 figures
We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a d-dimensional orthotope, we prove an exact parity law for the maximal π-1-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first d scalar width moments recover the unordered side vector, and d-1 moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies Mn,a = λn(a)1/2/π. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.