Classification: Theoretical Physics / Closure Geometry / Constants Ontology / Gauge-Sector Interpretation This paper proposes a closure-geometric interpretation of several striking numerical correspondences involving Euler’s identity, the fine-structure constant, fractional-dimensional gauge sectors, and the proton–electron mass ratio. Rather than treating the constants of nature as arbitrary inputs or isolated numerical coincidences, the manuscript explores the possibility that they arise as residues of a deeper transcendental closure architecture. The central proposal is a π-mirror cascade in which curvature saturation near π discloses into an open macroscopic photon sector at 3. 0D, a weak closure sector at 6 − π, and a strong confinement sector at 9 − 2π. Within this framework, the fine-structure constant is interpreted as a boundary impedance coefficient governing transmission between constrained and unconstrained coherence domains. A related mass-ratio correspondence, mₚ/mₑ ≈ 6π⁵, is retained as a candidate geometric residue, not as a completed proof. The paper does not claim to have established a completed derivation of the Standard Model constants. Instead, it presents a structured mathematical-ontological map showing how π, e, α, and key particle-scale ratios may become mutually intelligible within a fractional-dimensional closure framework. The aim is to convert apparent numerical coincidence into a disciplined candidate theory of closure residues, suitable for further mathematical development, numerical testing, and physical interpretation. Keywords Euler identity; fine-structure constant; alpha; proton–electron mass ratio; fractional dimensions; gauge closure; π-mirror cascade; coherence physics; closure geometry; PSOC4 (3) ; transcendental constants.
Philip Lilien (Sat,) studied this question.