The Complement of the Silver Exclusion Sieve: Complementary Mechanical Words and Boundary Orbits

This paper studies the exact complement of the Silver Exclusion Sieve as an autonomous self-referential process. With δ=1+2\\delta=1+\\sqrt2 and complementary threshold β=δ/(δ−1)=(2+2)/2\\beta=\\delta/(\\delta-1)=(2+\\sqrt2)/2, the complement is generated directly from its own earlier terms by a strict shell-count inequality. Classical Beatty complementarity makes the lower mechanical words associated with δ\\delta and β\\beta exact binary complements from index 11 onward, collapsing the generic two-source shell transfer to a single complementary mechanical word. The main new structure comes from comparing this strict complement with the corresponding weak threshold-β\\beta sieve. Their sole initial seed discrepancy propagates through Silver-parent ancestry generations: their rank functions differ by one exactly on intervals bounded by the two classical first-child orbits A048739 and A098586, while their membership words differ only at those boundary points, with opposite orientations. The generation widths are exactly the Pell numbers, ct−at=Pt+1c_t-a_t=P_{t+1}. The paper also proves the corresponding seed-propagation theorem for every irrational shell scale s>2s>2, and records inherited Pell-family, Pell-window, local-run, and finite-state consequences from the parent Silver sieve. Exact integer implementations, a 10,000-term b-file, finite membership data, and regression checks through 10610^6 accompany the paper.