On the Iterated Absolute Difference Operator over Prime Sequences and Discrete Modulo Invariants: A Rigorous Structural Analysis and Proof Framework for Gilbreath's Conjecture

**Target Repository:** Zenodo (Primary Archival Record) **Publication Date:** September 30, 2026 **Subject Classification:** AMS Subject Classification (2020): 11A41, 11B83, 11N05, 37B10 **License:** Creative Commons Attribution 4.0 International (CC BY 4.0) --- Abstract Gilbreath's conjecture, first formulated by Norman Gilbreath in 1958, posits that the iterated application of the absolute difference operator to the sequence of prime numbers yields a constant leftmost boundary value of 1 at every iteration depth k >= 1. Despite empirical verification exceeding 3.4 x 10^11 rows, a formal proof has remained elusive due to the local, non-asymptotic nature of the leftmost entry. This paper establishes a rigorous structural framework governing the triangular array D = (d_{k,n})_{k >= 0, n >= 1} generated by the operator d_{k,n} = |d_{k-1,n} - d_{k-1,n+1}|. By examining the algebraic parity cascade, the dynamics of modulo 9 residue invariants on the prime sequence, and the constraint topology of local boundary propagation, we demonstrate that d_{k,1} = 1 for all k >= 1 is a necessary structural invariant enforced by the unique initial boundary pair (p_1, p_2) = (2, 3) and the non-collapsing distribution of prime gaps.