The Silver Exclusion Sieve: Pell Recurrence Calibration and Pell-Window Self-Similarity

We introduce the Silver Exclusion Sieve, a deterministic subset of the positive integers defined by a history-dependent shell-count rule based on the silver ratio δ=1+2.δ=1+√2. At stage vv, the sieve examines the previously admitted integers between ⌊v/δ⌋ v/δ and vv, admitting vv when their number remains below the corresponding silver-ratio capacity. The inherited shell mechanism yields an exact logarithmic-depth counting recursion and natural density 1/δ=2−1.1/δ=√2-1. The Silver-specific arithmetic is governed by a simple defect identity that leads to a unified calibration theorem for every positive integer recurrence satisfying xn+2=2xn+1+xn.xₙ₊₂=2xₙ₊₁+x_n. Consequently, every such recurrence has only finitely many admitted terms. Explicit sharp calibrations are obtained for the Pell, Companion Pell, and Pell-Lucas sequences. Around Pell landmarks, the sieve exhibits a two-sided fixed-offset self-similarity: every fixed positive offset eventually reproduces its original membership bit except on two parity-sensitive boundary orbits, identified with OEIS A048739 and A098586. The local membership word also has sharp structure: admitted runs have maximum length 33, rejected runs maximum length 44, and consecutive admitted integers have exactly the gap set {1,2,3,4,5}\{1,2,3,4,5\}. Finally, the Silver parent arithmetic closes on four mechanical types and lifts to a primitive seven-state substitution whose binary projection is the membership word. Membership is finite-state on canonical Silver-parent ancestry codes, whose place values are governed exactly by Pell numbers. Standard automaticity in a canonical Pell/Ostrowski representation remains open. The accompanying archive contains the manuscript, exact reference implementations, independent verification code, finite-state certificates and audit material, vector figures with reproducible generators, a 10,000-term b-file, membership data, and SHA-256 checksums.