A Geometric Framework for Fundamental Constants: Numerical Patterns, Packing Limits, and Falsifiable Predictions (Research Program Preprint, v3.0)

This record archives a timestamped research-program preprint (Version 3.0) proposing a geometric “packing channel” perspective on the emergence of selected physical and mathematical constants.

Framework. A bounded dimensionless interval (“packing channel”) is defined with endpoints at the Kepler packing density π/(3√2) ≈ 0.7404804896 and a conjectured upper bound π/(2√2) ≈ 1.1107207345. Derived channel quantities (e.g., Ω0 = π/(2√3)) and exact interval identities are recorded for internal consistency checks.

Formula catalog. Using a primitive basis of six dimensionless constants {π, √2, √3, √11, √13, φ}, the document constructs a catalog of 30 numerical formulas that approximate selected reference quantities. The catalog is explicitly presented as numerical observations (pattern matching), not as theoretical derivations. The reported mean relative error across the catalog is on the order of 0.22% (below 0.3% in summary statements).

Prediction registry. Five falsifiable predictions are registered with stated criteria for future experimental/observational verification.

Scope disclaimer. This preprint is not peer-reviewed and does not claim solutions to any Clay Millennium Prize Problem. It is intended as a structured, timestamped program document: definitions, identities, numerical benchmarks, and testable predictions.