Euler Congruences

Abstract

We prove Nishibuchi's generalized Euler congruence conjecture in its full stated parameter range and develop sharper arithmetic consequences. For every prime-power packet width, a charged Hurwitz differential operator transfers integral difference estimates through the reciprocal generating series. In the dyadic case this gives mixed finite-difference bounds at every row, without a starting threshold. For the four addresses of width four, a divided-power argument adds the factorial gain \(v_2(k!)\), determines every repeated-step valuation in addresses zero and one, and gives a stronger address-two estimate with exact values at all power-of-two orders. The corresponding first differences yield least periods and bijective isometric continuations onto explicit dyadic residue classes. We also determine exact Taylor radii for three addresses. The manuscript retains the complete earlier arithmetic, lifting, deformation, and analytic-regularity development, with dated historical appendices. A separate appendix states the precise remaining simplicity problem for ordinary Euler tails. Formal coverage is explicit: the external conjecture and several four-address results have inherited Lean verification; the new factorial, mixed-step, all-rank, and radius arguments are written proofs with recorded audits and exact computational checks, and await dedicated Lean formalization.