The 69% Golden Shell Sieve: A Gamma-Threshold Deformation of the Golden Exclusion Sieve
Year: 2026
Type: preprint
Abstract: This paper introduces a Gamma-threshold deformation of the Golden Exclusion Sieve. The golden shell is retained while its capacity threshold is replaced by beta = Gamma(phi^2) = phi!. A pinned-shell invariant converts the apparently self-referential definition into an exact counting recursion and establishes density 1/beta. For this Gamma threshold, the paper proves sharp admitted and rejected run bounds, exact gap sets, Fibonacci and Lucas sampling laws, and a shell-transfer theorem for positive Fibonacci-type recurrence tails. The paper also develops a golden-ancestry discrepancy theory, including an exact carry-count normal form, an inverse golden-parent tree, and an integer discrepancy cocycle. Interval-certified computations exhibit discrepancy levels H = 3, 4, 5, and 6 without claiming unboundedness. The accompanying package contains the proof-audited paper, AI-readable manuscript, figures and source files, two independent sequence generators, a frozen 10,000-term b-file, regression scripts, reproduction instructions, and SHA-256 checksums.
Author: Foth, Jake
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204160377
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