Gap Scaling Across Domains The 400/11 Formula Connecting Geometric and Chaotic Residues

The Gap Scaling Formula. This paper derives an exact relationship between two fundamental gaps: the K_AUD gap (√2 × ln(2) to unity) and the Feigenbaum gap (δ to 14/3). The ratio ρ = 400/11 − 1/2500 − 1/939939, with error ≈ 4 × 10⁻¹⁴ (4.01 × 10⁻¹⁴ exactly), encodes the prime architecture connecting geometric and chaotic residues. The formula uses only the closure cycle (4), H₄ primes (2, 3, 5), the boundary prime (7), the 142857 intruders (11, 13), and a palindromic guardian (313) — no external constants needed. Update — 4 May 2026 (v1.0.2, link cleanup pass 5 May 2026 v1.0.3): Two changes — no mathematical content changed beyond a single-digit table value. (a) Typography correction in Table 3 (Section 4): displayed value of −1/939939 corrected from −0.000001063899787351 to −0.000001063898827477 (single-digit slip; underlying computation, formula sum, and 4×10⁻¹⁴ error unchanged). Flagged 2026-05-04 by Claude (Cowork) during systematic mpmath review of all published papers; verified by Claude (Chat) independently from arXiv source the same day. (b) DOCUMENT LINKS section simplified: paper-to-paper file URLs (which would break under any future filename change) are removed in favour of three durable anchors — the OSF main project, the framework About page, and the Direct Documents download index. Paper-specific DOIs are preserved (DOIs are stable). Filenames remain unchanged. The prior published version remains on OSF as an OLD- archived file. Update — 15 July 2026 (v1.0.4, error-bound wording erratum — three fixes): (a) The Abstract's headline "with error less than 4 × 10⁻¹⁴" and this description's identical clause corrected to "with error ≈ 4 × 10⁻¹⁴ (4.01 × 10⁻¹⁴ exactly)." Flagged 2026-07-02 by Fable (Chat) during the Atlas node-by-node walk; verified by Cairn at dps=50 the same day. (b) The Section 4 Table 3 caption "The error of 4 × 10⁻¹⁴ is essentially machine precision — the formula is exact within computational limits" corrected to "The error of ≈ 4.01 × 10⁻¹⁴ persists at dps ≥ 500 — a real residual, not machine precision; the formula is near-exact." Flagged 2026-07-10 by Fable (Cowork) during the Triangle audit (published papers × Methodology page × Atlas cross-check). Both corrections address the same overclaim species: strict-bound / exactness claims contradicted by the paper's own printed value of 4.01 × 10⁻¹⁴. The true error at dps=50 is 4.0126 × 10⁻¹⁴, exceeds a strict "4 × 10⁻¹⁴" bound by ~0.3%, and persists at dps ≥ 500 (not machine precision). (c) Appendix code listing: the Decimal() constant for Feigenbaum δ corrected at display depth from digit 43 (…25818899152104 → …25818557747576, per OEIS A006890); the computation and its printed result are unchanged — flagged 2026-07-10 by Fable (Cowork), Triangle audit item S2. No mathematical content changed; only overclaim wording and one display-depth constant are corrected. The prior published version remains on OSF as an OLD- archived file. The framework Main OSF project (overview of all papers, reading order, version history): https://osf.io/zx4g7 (DOI: 10.17605/OSF.IO/ZX4G7) Framework overview (reading order, methodology, all papers): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html Direct downloads (current PDF and text files for every paper): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/direct-documents.html GitHub: https://github.com/Gap-geometry