The Mathematics of ONE — V1.4: Testing the Boundary ε. Observational Bounds on the Photon Mass and the Speed of Light

Abstract

Version 1.4 — Falsifiable test (5 languages). A stand-alone paper continuing V1.2 (Section 9) and V1.3 (Section 10). In V1.2 the boundaries of the UNU interval are α = ε and Ω = 2 − ε, with ε a hyperreal infinitesimal. This note tests what happens if ε is instead a positive real number. Two physically distinct readings are identified. In Model A (constant ε, no dispersion, Lorentz-violating), the GW170817/GRB 170817A observation gives ε ≤ 7×10⁻¹⁶. In Model B (fixed Proca photon mass), ε(2 − ε) = (m_γ/E)², exactly the relation R(2 − R) = 1/γ² of V1.2, with ε ≈ (m_γ²/8π²)e^(2s) on the scale axis of V1.3. Published photon-mass bounds (FRB: 2.1×10⁻¹⁵ eV; PDG: 10⁻¹⁸ eV) require ε ≤ 3.6×10⁻³¹ and ε ≤ 8.1×10⁻³⁸ for visible light. The real-ε hypothesis is not excluded by current data, but it is constrained to extremely small values. The infinitesimal form of V1.2 is compatible with all current bounds. The test does not prove the infinitesimal; it shows that any real ε would have to be extraordinarily small. AI collaborators: Claude (Anthropic); Perplexity and Gemini (Google), independent numerical verification.