The Gap Geometry Framework
An independent mathematical research project investigating structural constants that emerge from binary information density — and their exact appearances across independent domains of mathematics and physics.
What This Work Is
This framework begins with a single observation: for any integer base n, define K(n) = √n × ln(n). Only n = 2 produces K(n) < 1. The resulting constant KAUD = √2 × ln(2) ≈ 0.980 defines a ceiling, and the gap G = 1 − KAUD ≈ 0.0197 is the subject of this research.
The framework investigates what follows from this gap: a computable structure called the Binary Tower, a set of exact algebraic identities, a prime corridor, and connections to independently published results across multiple domains.
What is known vs what is original. The individual constants in this framework — √2, ln(2), the Feigenbaum constant, the golden ratio, shell capacities 2n², the Madelung constant — are all well-established mathematics. This work does not claim to have discovered them.
What is original: the derivation of G as the unique sub-unity gap of binary information density. The Binary Tower as a computational instrument. The gap scaling formula connecting the geometric gap to the Feigenbaum constant. The cross-domain documentation of where KAUD appears independently in published science. And the closed-form identification of √2·ln(2) as the Hodgson–Kerckhoff tube-packing coefficient. The ingredients are known. The connections, the gap, the tower, and the calculations are the contribution.
Scope & Method
The framework spans ten published documents with DOIs, archived on OSF and GitHub. Four foundation papers build the framework from first principles. Five companion papers develop specific aspects. One standalone paper provides an independent proof in hyperbolic geometry.
Access architecture: GitHub Pages and OSF
GitHub Pages provides the living access layer: current navigation, update notes, AI-readable routes, and the verification entry point. OSF provides the archival layer: DOI-linked releases, fixed PDFs and text files, datasets, and older versions. The two are complementary rather than redundant — readers wanting the most current state of any document use GitHub Pages; readers wanting a citable, immutable reference use OSF.
The Binary Tower
The core instrument of the framework is the Binary Tower: a computable staircase (n × G, for n = 1 to 64+) that maps where known mathematical constants appear as structural landmarks. The tower is not a theoretical claim — it is a calculation tool. Each step adds one gap. The landmarks it hits are independently known constants. The deviations are small, constant, and algebraically forced. The tower can be computed and inspected by anyone with a calculator.
Cross-domain structure
The framework documents exact and near-exact appearances of KAUD = √2·ln(2) across independent domains:
Verification protocol
All numerical claims are verified using mpmath (Python arbitrary-precision library). Serious verification of algebraic identities is performed at ≥500 decimal digits; lighter sanity checks use ≥100 decimal digits, never below that. The framework distinguishes exact algebraic identities (which agree to the full working precision) from observational matches (with quantified deviations). The protocol is pre-committed: the bar does not relax to accommodate a claim that fails to meet it. Every identity is independently verifiable. The mathematics does not require credentials. It requires a calculator.
Living document model
This framework follows a living document model. The Living Document page tracks the current state of all claims, their verification status, and the interlaced structure of the work. It evolves as the framework evolves, with older versions archived for transparency. This approach resolves the mismatch between static publication and active research: the reference evolves at the pace the work actually evolves, and the evolution is recorded rather than hidden.
Reading Guidance
The core framework — the gap, the Binary Tower, the identities, and the gap scaling formula — requires only standard calculus and arithmetic. Some sections reference the H₄ polytope (the 120-cell, a 4D regular solid). This connection provides structural context but is not required for understanding the core results. Readers unfamiliar with polytope geometry can treat these references as background.
Where to start. For a quick overview with formulas, see the Living Document. For direct access to all papers, see Documents. For hands-on exploration, try the Visualisations. Among the papers themselves, the Cross-Domain Signatures document (Paper 6) is the recommended front door to the framework.
About This Project
This is independent research — no institution, no team, no prior reputation. The work was built by a single researcher and verified across six independent AI architectures (Claude, Grok, GPT, Gemini, Perplexity, DeepSeek). All materials are published under CC BY 4.0 and archived with DOIs on OSF.
The mathematics is independently verifiable. Applications and interpretations remain open for investigation.