We report numerical coincidences between simple π-expressions and three fundamental constants, each at a specific scale and scheme: α(0) from atomic recoil, sin²θ_eff^lept at the Z-pole, and α_s(M_Z) in MS-bar. The geometric constant G_π ≡ 4π³ + π² + π = 137.036 is numerically close to α⁻¹. (The raw difference G_π − α⁻¹ ≈ 3.05×10⁻⁴, or ~1.45×10⁴σ relative to atomic-recoil precision; the self-consistency equation accounts for this gap.) The equation 1/α + α/D = G_π with D = 24 − 1/(8π) yields α⁻¹ = 137.035999215, matching Rb 2020 to 0.8σ. The algebraic identity G_π = π[π²/sin²θ + 1] is exact when sin²θ = π/(4π+1) = 0.231572, matching sin²θ_eff^lept to 0.26σ. Additionally, 1/(3π−1) = 0.1187 matches α_s(M_Z) MS-bar to 0.8σ. We present conjectured physical interpretations involving electromagnetic phase-space structure, while clearly distinguishing mathematical facts, empirical fits, and mechanism conjectures. We conjecture that a rigorous derivation from QED will show that the balance equation emerges from the requirement that stable localized electromagnetic configurations exist, with the coefficients following from gauge field mode counting and vacuum polarization. Three sub-σ matches across independent constants at specific scales/schemes demand explanation. Whether these regularities reflect deep structure or coincidence will be determined by theoretical progress and precision measurements.

Version 3.3 (Final) This preprint establishes empirical regularities linking π-expressions to three fundamental constants:- α(0)⁻¹ = 137.035999215 (matches Rb 2020 to 0.8σ)- sin²θ_eff^lept = 0.231572 (matches PDG 2024 to 0.26σ) - α_s(M_Z) MS-bar = 0.1187 (matches PDG 2024 to 0.8σ) All three derive from a single geometric constant G_π = 4π³ + π² + π = 137.036 via specific equations. Key contributions:- Exact algebraic identity: G_π = π[π²/sin²θ + 1] ⟺ sin²θ = π/(4π+1)- Self-consistency equation: 1/α + α/D = G_π with D = 24 − 1/(8π)- Conjectured mechanism: electromagnetic vortex stability / phase-space mode counting- Explicit falsification criteria for each prediction- Predicted form of QED derivation (stakes claim to derivation pathway) Claim taxonomy clearly distinguishes:1. Exact identities (algebraically proven)2. Empirical fits (falsifiable by measurement)3. Mechanism conjectures (require derivation) Related work by the same author:- Gravitational collapse / bounce paper: DOI 10.5281/zenodo.18452211 Suggested citation:Graham, J. (2026). Geometric Regularities in Fundamental Constants: Empirical Fits and Conjectured Mechanisms (Version 3.4). Zenodo. https://doi.org/10.5281/zenodo.10.5281/zenodo.18489736