From Residue Dynamics to Arithmetic Fibers: The Pronic Structure of A392243 and Its Elliptic Sister Fiber

Abstract

This paper develops the exact arithmetic structure behind a modulo-threshold pseudo-Recamán recurrence and the signed square-block sequence OEIS A392243. At the critical threshold, the canonical orbit is proved to be the absolute value of the signed sum, with an exact folded pronic-distance formula that reconciles square-shell and pronic-cell coordinates. The paper classifies the adjacent-equal-magnitude seam, develops the legal signed-offset Pell surface for square values, proves boundary and Fibonacci sixth-power families, establishes finiteness of each fixed square-magnitude class, and records collision provenance through certificate-typed edges. It also gives the exact Jacobi-theta encoding of the square-shell sign mask. For the weight-two sister sum, the paper derives the fixed-offset law and two genus-one twists. The noncentral d=-18 uniqueness result is explicitly computer-assisted: PARI/GP 2.17.2 and Magma V2.29-9 supply the archived elliptic-curve certificate, and the reproducibility supplement carries the five theorem-frozen artifacts and hashes. Finite atlas computations are explicitly labeled finite evidence and are not presented as exhaustion or asymptotic theorems.