The 69% Golden Shell Sieve: A Gamma-Threshold Deformation of the Golden Exclusion Sieve

This paper introduces the 69% Golden Shell Sieve, a Gamma-threshold deformation of the Golden Exclusion Sieve. It retains the same moving golden-ratio shell as the parent sequence A395718 but replaces its capacity threshold φφ by β=Γ(φ2)=φ!β=Γ(φ^2)=φ!. The title’s “69%” is rounded shorthand for the exact natural density 1Γ(φ2)≈0.690021787047,1/Γ(φ^2)≈0.690021787047, compared with 1/φ≈0.6180339887491/φ≈0.618033988749 for the parent sieve. A pinned-shell invariant converts the apparently self-referential definition into an exact counting recursion. For the Gamma threshold, the paper proves sharp admitted and rejected run bounds, exact gap sets, Fibonacci and Lucas threshold-sampling laws, and a shell-transfer theorem for positive Fibonacci-type recurrence tails. It also develops a golden-ancestry discrepancy theory using an exact carry-count normal form, inverse-parent tree, and integer cocycle. Interval-certified computations exhibit discrepancy levels H=3,4,5,6H=3,4,5,6 without asserting unboundedness. The deposit contains the proof-audited paper, synchronized Markdown and LaTeX sources, publication PDF, figures and source files, two independent sequence generators, a frozen 10,000-term b-file, regression scripts, reproduction instructions, bibliography, OEIS submission packet, and SHA-256 manifest.