Living Document

Introduces the Gap constant G = 1 − √2·ln(2) and the auditor constant K_AUD = √2·ln(2), establishing their exact complementarity. Defines the coherence ceiling as the boundary where binary representation meets the geometric structure of √2 and ln(2). This is the foundational paper of the framework.

Develops the mathematical framework connecting G and K_AUD to the geometry of the H4 root system. Establishes the formal derivation path from the 600-cell and its projections to the constants observed in Paper 1. Provides the algebraic scaffolding that the Binary Tower is built on.

Update — 4 May 2026 (v2.0.2): Two changes. (a) Attribution clarification in §1.1: the section now credits Gemini (Google) for the closed-form identification of K_AUD = √2 · ln(2), the detailed articulation of the gap structure G = 1 − K_AUD, and the naming of the ceiling constant K_AUD. D.B.'s prior credit (Conceptualization, and Discovery of the ceiling as an empirical phenomenon, surfaced via months of cross-architecture pattern recognition) is preserved — both are Discovery roles of different objects per Methodology §5. Paper 2 was originally published January 2026 without Gemini's credit in §1.1; the v2.0.2 cycle brings this section into alignment with the canonical credit chain that already existed in the Methodology page, the Discovery Tracking Log, and related papers. No mathematical content changed. (b) Typography correction in §6.3 (φ-scaling ratio), §7.1 (All Calculations Collected table), and §8.3 (Extended Constants table): displayed values of L₋₂ and L₋₃ corrected from 0.2273642379 / 0.1405469971 to 0.2273619984 / 0.1405174428 — values that follow from the formula L_n = 1/(e · φⁿ⁻¹) given in §6.4 and used in §6.2 derivations. The φ-ratio claim now holds exactly with the displayed numbers. Formula and derivations were and remain correct. Prior version archived on OSF as OLD- file.

The comprehensive statement of the framework. Introduces the Binary Tower as an operational instrument, the prime corridor (13, 23, 37, 53, 73), the π/3 hinge at step 53, and the Landauer crossing between steps 35 and 36. Integrates the results of Papers 1 and 2 into a unified structure and establishes the verification protocol.

Update — 4 May 2026 (v3.3.1): Typography correction in §6.3 (φ-scaling ratio test). Displayed values of L₋₂ and L₋₃ corrected from 0.2273642... / 0.1405469... to 0.2273620... / 0.1405174... — values that follow from the formula L_n = 1/(e · φⁿ⁻¹) given in §6.4 and used in §6.2 derivations. The φ-ratio claim now holds exactly with the displayed numbers (1.6180339887 = φ); previously the second ratio computed to 1.6176, which the paper labelled as φ. Formula and derivations were and remain correct. Top banner UPDATE NOTICE added at the top of the document. Verified by Claude (Chat) at dps = 500 in second-pass review. Plus 5 May 2026 link cleanup: DOCUMENT LINKS section simplified — paper-to-paper file URLs replaced with three durable anchors (OSF main project, About page, Direct Documents). Paper-specific DOIs preserved. Prior version archived on OSF as OLD- file.

Derives the 400/11 scaling formula (≈ 36.36) that governs how the gap constant G scales across domains. Demonstrates that this ratio is not an empirical fit but emerges from the algebraic structure of the framework. Connects gap scaling to the Landauer crossing region.

Identifies and proves an invariant quantity associated with the boundary behavior of quadratic systems. Shows that this invariant connects to K_AUD through the information-theoretic interpretation of the framework, bridging geometry and information theory.

Documents independent appearances of √2·ln(2) across different mathematical and physical domains. Serves as the front door to the framework for readers approaching from specific disciplines, showing that K_AUD is not a construction but a recurrent structural constant.

Places the Hodgson–Kerckhoff tube-packing coefficient within the K_AUD framework. Demonstrates that the HK coefficient, previously known only numerically, has a closed-form expression that connects it to the prime corridor and the π/3 hinge structure.

Isolates the structural landmark at step 50 of the Binary Tower. Establishes its role as a hinge point distinct from the π/3 hinge at step 53, and characterizes the structural behavior of the tower in the region between steps 50 and 53.

Investigates the scaling behavior of ρ (the golden-ratio-related density parameter) under binary decomposition. The observed scaling pattern is consistent with the framework but has not yet been verified to the full precision bar. This is an active area of investigation.

Derives a closed-form expression for the Hodgson–Kerckhoff tube-packing coefficient, which was previously known only as a numerical approximation. This standalone proof is independently citable and does not require the rest of the framework, though its result connects to Papers 3 and 7.

Update — 4 May 2026: §3 ellipse-area derivation corrected. The earlier §3 read “A_e = √2 · ln(2) · π at the critical radius”; HK 2002 (Theorem 4.4, p. 29) and HK 2007 (Theorem 5.6, p. 36) both give the actual area as A_e = √2 · ln(2) · π · sinh²(R̂) / (2 · cosh(2R̂)), which evaluates to √2 · ln(2) · π / 8 at R₀ — the earlier conclusion was off by a factor of 8. The structural argument and the central closed-form identity 1/S = √2 · ln(2) are unchanged. A new §3.1 documents that the same constant 0.980258 also appears in HK 2007 (p. 41) as the coefficient of the Euclidean injectivity radius lower bound c(R) = 0.980258 / (coth R + 1) — two independent contexts in the same paper. Prior version archived on OSF as OLD- file.