What question did this study set out to answer?

This paper aims to explore the deeper mathematical structure behind fundamental physical constants like the fine-structure constant and mass ratios.

June 22, 2026Open Access

Euler, 137, and the Transcendental Cascade A Closure-Geometric Interpretation of Gauge Coupling and Fractional-Dimensional Structure

Key Points

This paper aims to explore the deeper mathematical structure behind fundamental physical constants like the fine-structure constant and mass ratios.
Proposed a mathematical mapping of constants involving Euler's identity and gauge sectors.
Introduced the concept of the π-mirror cascade to explain curvature saturation and coherence domains.
Examined numerical correspondences within a closure framework.
Identified the fine-structure constant as a boundary impedance in the proposed framework.
Highlighted the mass ratio m_p/m_e ≈ 6π^5 as a potential geometric residue.
Established a conceptual basis for further exploration of constants through closure geometry.

Abstract

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.

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