Block Supercongruences, Connected Cumulants, and p-Adic Interpolation for the Riemann–Xi Center Transport

This preprint studies arithmetic relations among complete transport blocks arising from the center expansion of the standard Riemann–Xi kernel. The characteristic polynomials of the transport form a specific continuous dual Hahn family. Building on the arithmetic structure of a single prime-sized block, the paper develops congruence laws for products of several complete blocks and determines how the failure of exact block multiplicativity is organized across successive prime-power levels. A central result is a uniform block supercongruence valid for arbitrary integral block lengths. For prime-sized cells, a simultaneous scaling of the block index and spectral variable leads to a Frobenius-type contraction for an auxiliary continuous dual Hahn family. This yields a Newton filtration for the zero-primary factor and substantially improves the general bound for the preperiod of the associated finite dynamics. The minimal preperiod is determined exactly through the first three prime-power levels, while higher-depth cases are reduced to explicit residual data. The paper also derives exact finite-field trace and determinant formulas for shifted prime cells. These formulas control the first higher-order corrections to block multiplicativity and reveal an additional factor in the normalized two-cell defect. Using the rank-one structure of the base cell, the higher connected corrections are described by an all-order cumulant law. The odd principal corrections are governed by explicit Fuss–Catalan coefficients, while the even corrections acquire additional prime-adic divisibility. This produces a uniform valuation bound for the centered Bell coefficients and, consequently, a finite Mahler filtration at every prime-power precision. On the unit locus, the normalized block sequence admits a locally analytic interpolation in the number of blocks. The paper also gives an exact criterion for when the unnormalized sequence extends continuously to the prime-adic integers. All principal results are proved algebraically. The accompanying source archive contains exact-arithmetic verification code for the block congruences, shifted-cell identities, connected-cumulant formulas, Mahler filtration, and preperiod calculations.