Complemented-Seam Binary Difference Dynamics: Affine Conjugacy, Power-of-Two Collapse, Half-Turn Mismatch, and Finite State-Space Verification

This paper studies a binary cyclic absolute-difference process with a single complemented seam. The resulting affine dynamical system possesses a distinguished fixed state e_0=(1,0,...,0). Translation by this state conjugates the dynamics to the classical binary cyclic difference operator A=I+S over F_2. Several exact results follow. For widths d=2^k, every initial state reaches e_0 within d iterations. For the canonical spacetime orbit generated from e_0, the zero density is exactly 1-1/d. For every even d, cutting the canonical d×d spacetime array in half, rotating the lower half by 180 degrees, and overlaying it with the upper half produces exactly M=d mismatches, confined to the boundary. Forward and reverse orientations are additionally shown to be reflection-conjugate and commuting. The universal statements are proved algebraically and supplemented by finite computational audits. Affine conjugacy was exhaustively checked through d=12; the nilpotence condition was scanned through d=64; canonical density and half-turn results were checked for every even width through 64; and orientation identities were checked through d=64. The classical binary Ducci operator, its I+S representation, and the power-of-two nilpotence mechanism are explicitly credited to the established Ducci/Pascal-mod-2 literature. The paper instead isolates the complemented affine seam, its distinguished fixed state, its canonical spacetime observables, and their exact consequences. Research conception, construction, experimental direction, and decisions regarding what to test were supplied by Kevin Mark Schimmel. ChatGPT (OpenAI) was used as a computational assistant for implementation, finite verification, algebraic cross-checking, literature-search assistance, and manuscript preparation; it is not an author.