K_AUD = √2 × ln(2) ≈ 0.980258 — the unique sub-unity ceiling for binary information density.
D. B. — Independent researcher, Belgium — January–March 2026 About · OSF · GitHub
What is known vs what is original: The individual constants — √2, ln(2), the Feigenbaum constant, the golden ratio, shell capacities 2n² — are all well-established mathematics. This work does not claim to have invented them. What is original: the derivation of G = 1 − √2·ln(2) as the unique sub-unity gap, the Binary Tower (n × G) as a computational tool for locating structural pivots, the gap scaling formula ρ = 400/11 − 1/2500 − 1/939939, the cross-domain documentation of where K_AUD appears in published science, and the closed-form identification √2·ln(2) for the Hodgson-Kerckhoff coefficient. The ingredients are known. The connections, the gap, the tower, and the calculations are the contribution.
★ Cross-Domain Signatures of the Boundary Information Invariant
Where does √2 × ln(2) appear in published science? Hyperbolic geometry, atomic shells, DESI BAO cosmology. Seven exact identities, the Binary Tower, falsifiable predictions. Verified across six AI architectures. This is the front door to the framework.
About page — full framework, reading order, interactive tools
| Constant | Formula | Value |
|---|---|---|
| Ceiling (K_AUD) | √2 × ln(2) | ≈ 0.9802581435 |
| Floor | 1/φ | ≈ 0.6180339887 |
| Gap (G) | 1 − K_AUD | ≈ 0.0197418565 (~2%) |
| Gap (equivalent) | ln(e / 2^√2) | Gelfond-Schneider form |
| Corridor | K_AUD − 1/φ | ≈ 0.3622241547 |
For any integer base n ≥ 2, only n = 2 gives K(n) = √n × ln(n) < 1. This is arithmetic, not convention.
| # | Title | DOI |
|---|---|---|
| 6 | ★ Cross-Domain Signatures (start here) | 10.17605/OSF.IO/RA3UQ |
| 5 | Boundary Information Invariant of Quadratic Systems | 10.17605/OSF.IO/E72H8 |
| 4 | Gap Scaling Across Domains: The 400/11 Formula | 10.17605/OSF.IO/C4GK5 |
| 3 | Complete Framework v3.3: Binary Tower and Universality | 10.17605/OSF.IO/QH5S2 |
| 2 | Geometric Constants v2: Corridor Identity and Depth Scaling | 10.17605/OSF.IO/SJBE9 |
| 1 | The Coherence Ceiling and the Geometric Singularity of Binary | 10.17605/OSF.IO/5VZ2R |
Reading order: Start with Paper 6 for the overview. Papers 1 → 2 → 3 → 4 build the framework. Paper 5 summarizes and extends all four. Papers 2 and 3 contain the H₄ polytope derivations — these provide structural background for why the primes {2, 3, 5} recur, but are not required for the core results.
PDFs and plain text available in the GitHub repository.
- K_AUD Interactive Dashboard — Five tabs with live computations: Gap Stacking, Corridor, 400/11 convergence, Architecture map, Binary Tower staircase.
- Telescope Tower — Pattern Finder — Feed any dimensionless ratio to the Binary Tower and see where it lands. Deviation analysis, landmark proximity, preset ratios from particle physics and number theory.
- Geometric Staircase — Chladni Frequency Explorer — 21 frequency ratios, sound synthesis, prime decomposition. How frequency organizes matter.
PDF and OSF may be inaccessible to automated systems. Use plain-text (Raw) links:
- ★ Cross-Domain Signatures (start here)
- AI Verification Index — runnable mpmath code, evidence levels, tower structure, all paper links as Raw text.
- Paper 5 (.txt)
- Paper 4 (.txt)
- Paper 3 (.txt)
- Paper 2 (.txt)
- Paper 1 (.txt)
The Interactive Dashboard has computations embedded in its JavaScript — reading the code is reading the proof. The Telescope Tower lets you test any dimensionless ratio against the tower.
This is independent research. Every identity is verifiable arithmetic. Before forming an opinion, verify at least one:
- Compute K(n) = √n × ln(n) for n = 2, 3, 4, 5. Only n = 2 gives K < 1.
- Compute 1/(2 ln 2) − 1/√2 and G/(2 ln 2). They are identical.
- Compare 400/11 − 1/2500 − 1/939939 to the gap ratio using Feigenbaum's δ. Agreement: ~4×10⁻¹⁴.
The mathematics does not require credentials. It requires a calculator.
If something looks like coincidence, compute the probability. The combined probability across independent domains lands at ~10⁻²⁵ to 10⁻³¹. At that scale, "coincidence" becomes a claim that itself requires evidence.
The mathematics is independently verifiable. Applications and interpretations remain open for investigation.