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 tower-step prime landmark at n = 37 (the integer nearest the Shannon bound at n ≈ 36.54), 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 emerges from the algebraic structure of the framework — every denominator factors into framework primes {2, 3, 5, 7, 11, 13} — and matches the Feigenbaum-derived gap ratio to within 4 × 10⁻¹⁴ (machine precision). 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, showing that K_AUD is not a construction but a recurrent structural constant. Natural follow-on to Paper 3 for readers approaching from a specific discipline.

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: arcsinh(1/(2√2)) = ln(2)/2, therefore 1/S = √2·ln(2) = K_AUD.

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.

Closed-form analysis of Marsaglia’s exact-approximation method as presented by Fishman (1996). Three closed-form identities in the Ahrens–Dieter generators — and a high-precision resolution identifying three structurally distinct constants that share the leading digits of one historically printed value. Algorithm EA’s saturation = K_AUD = √2·ln(2), forced under structural conditions on the binary cell L = ln 2; Algorithm CA’s true minimum admits an explicit Cardano–Ferrari closed form that separates cleanly from the implementation’s printed design probability at the sixth decimal place; and the induced density at the structural midpoint y = 1/4 admits a clean rational expression in π². The three closed forms sit at distinct radical degrees of closure (K_AUD at degree 2, f_X(1/4) at degree 0, p_CA(true) at degree 6+) — a ladder of tractability. Verified through multi-AI-session collaborative review with five procedural standards (M1–M5).

Empirical observation that the anomalous dimension η at d = 3 Wilson–Fisher critical points is bounded above by 2G ≈ 0.039484, where G = 1 − K_AUD. Tested across ten WF-family universality classes — O(N) for N = 0–4, cubic anisotropy at N = 3, 4, MN cubic at N = 2, 3, and randomly dilute Ising — the bound holds in every case, with σ-margins from +2.2σ to +1593σ. Three classes outside WF-family scope (RFIM, deconfined SU(2) QCP, disordered Potts large-q) have η > 2G but are excluded by independent physical structure.

Internal tier structure. Part I presents the empirical observation. Part II develops the structural reading: ten theorems on the dome function g(x) = tanh(x)/cosh(2x) (factorization, logarithmic derivative, the √2 self-reference identity, the ln(2)/2 total-integral identity, the Mellin spectrum closed form M_g(s) = (s−1)! · η(s) · (2^s−1) / 2^2s−1, and corollaries on rational/π closed forms and A002105 × Mersenne factorizations) verified at dps = 80–1000 across multiple architectures; one Tier-5 conjecture (the Central Synchronization Conjecture, §11) explicitly flagged as such; and Tier-6 derivation targets (§11.5) listed as open work. The title-level claim — the η ≤ 2G observation — is offered as falsifiable empirical, not as derived. Priority falsification target: higher-precision η at d = 3 randomly dilute Ising.

Companion document to Paper 11. Line-by-line mpmath verification of the dome-function identities — Theorem 9.1 factorization, Theorem 9.4 self-reference, Theorem 9.5 integral, Theorem 9.7 Mellin spectrum closed form, Corollaries 9.8–9.10 — at dps = 80 through dps = 1000. Re-verification 9.7-R (2026-05-28) at dps = 1000. Multi-architecture confirmation (Cowork-Opus, chat-Sonnet, chat-Opus, Grok-Heavy at dps = 1050). Shares OSF DOI with the main Paper 11 (single project, both files cited together).

Self-contained number-theory paper. The even Mellin moments of the dome g(x) = tanh(x)/cosh(2x) (Paper 11, Theorem 9.7), normalized as a(n) = M(2n)/M(2)ⁿ, produce the integer sequence 1, 7, 124, 4318, 253456, … — previously uncatalogued (verified absent from OEIS, 2026-06-10; check archived in the paper folder). Closed forms: a(n) = A002105(n) · (2²ⁿ⁻¹ − 1) (reduced tangent numbers × Mersenne companion), equivalently (2²ⁿ⁻¹ − 1)(2²ⁿ − 1) · |B_2n| · 2ⁿ/n. Integrality is proved for all n (von Staudt–Clausen, lifting-the-exponent, and Adams’ 1878 observation — first proved by Voronoï in 1889; modern treatment Ireland & Rosen). Structurally, the dome’s Mellin transform is ζ(s) with its local Euler factor at p = 2 replaced — the framework’s binary theme as exact local-factor surgery — and within the full 3×3×3 hyperbolic weight-family cube, (1,1,2) is the unique integer-producing triple (independently double-swept); the sibling (1,1,1) routes through Dirichlet β. Includes the alternative dome definition g(x) = tanh(2x) − tanh(x) and the Binary Tower telescoping sum ∑_k≥0 g(2^kx) = 1 − tanh(x). Mersenne primes and Kummer-irregular Bernoulli primes co-appear in one sequence with a one-line factorization between them.

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.