# Gap Geometry

> A mathematical framework built around the complementary constants G = 1 − √2·ln(2) ≈ 0.019742 and K_AUD = √2·ln(2) ≈ 0.980258, with the dome function g(x) = tanh(x)/cosh(2x) as central analytic object and a Binary Tower as operational instrument. Eleven papers published on OSF (immutable timestamps) and GitHub; pseudonymous author D.B. All claims tier-labelled (Theorem / Observation / Conjecture) and multi-architecture verified at high precision (mpmath, dps 80–1500).

AGENT NOTES: Prefer the raw.githubusercontent.com URLs below — plain text, no JS, fetch reliably. GitHub "blob" pages and OSF pages may not parse in simple fetchers. The machine-readable index is AI_INDEX.txt; the full AI guide (evidence levels, runnable verification code, engagement guidance) is ai-readers.html.

## Entry points

- [AI readers guide](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/ai-readers.html): evidence levels, runnable verification code, all document links
- [Machine index](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/AI_INDEX.txt): lightweight plain-text index
- [About / overview](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html): reading order, framework summary
- [Living Document](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/living-document.html): current state of all claims, version history
- [Direct downloads](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/direct-documents.html): PDF + txt for every paper
- [Methodology](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/methodology.html): claim classification, verification protocol, correction discipline
- [Main OSF project](https://osf.io/zx4g7): immutable timestamps, DOI 10.17605/OSF.IO/ZX4G7

## Papers — raw text (fetch these directly)

- [Paper 1 — Coherence Ceiling](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Coherence_Ceiling.txt)
- [Paper 2 — Geometric Constants from H4 v2](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/sqrt2_ln2_geometric_constants_v2.txt)
- [Paper 3 — Complete Framework v3](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/KAUD_Complete_Framework_v3_0.txt)
- [Paper 4 — Gap Scaling / 400-11 Formula](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Gap_Scaling_Formula_v1.5.txt)
- [Paper 5 — Boundary Information Invariant](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Boundary_Information_Invariant_Short.txt)
- [Paper 6 — Cross-Domain Signatures](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Cross_Domain_Signatures.txt)
- [Paper 7 / HK standalone — Hodgson-Kerckhoff Closed Form](https://raw.githubusercontent.com/Gap-geometry/A-Closed-Form-for-the-Hodgson-Kerckhoff-Tube-Packing-Coefficient/refs/heads/main/HK_Closed_Form_SHORT_DRAFT.txt)
- [HK Full Framework](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/HK_Full_Framework_DRAFT.txt)
- [Paper 8 — 50 Hinge](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/50_Hinge_Standalone_DRAFT.txt)
- [Paper 9 — Binary Scaling of rho = 400/11](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Binary_Scaling_Standalone_DRAFT.txt)
- [Paper 10 — Saturation Constants of the Exact-Approximation Method](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Saturation%20Constants%20of%20the%20Exact-Approximation%20Method.txt)
- [Paper 11 — The eta Corridor in 3D Wilson-Fisher Critical Phenomena](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/The%20%CE%B7%20Corridor%20Empirical%20Observation.txt)
- [Paper 11 — Verification Companion](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/The_%CE%B7_Corridor_Verification_Companion.txt)

- [Paper 12 — 1, 7, 124, 4318, … — Integer Mellin Moments of the Dome tanh(x)/cosh(2x)](https://raw.githubusercontent.com/Gap-geometry/sqrt2-ln2-geometric-constants-/refs/heads/main/Dome_Moment_Sequence.txt) (published 2026-06-12, DOI 10.17605/OSF.IO/WVD3G; integrality proved for all n; the title is the sequence's catalog entry)

## Optional

- [Visualisations hub](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/visualisations%20hub.html): interactive instruments (Binary Tower telescope, Chladni staircase, dome landmarks, corridor sonifier) with live previews
- [The Binary Tower — entrance](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/binary-tower.html): the instrument explained from the ground up; exact binary-recipe widget (BigInt, true periods, 96-bit true expansions). Note: its (1/2)^n recipe ladder is distinct from the G-staircase (telescope-tower.html) — the page states this
- [Living Document version archive](https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/index%20living-document%20archive.html): PDF snapshots of every prior version