From a Gravity-Calibrated Helix to a Conditional Fine-Structure Correspondence

We report a frozen numerical correspondence generated from the dimensionless helix parameter established in the preceding TEF preprint. That construction uses the measured Newton constant to calibrate a primitive radius, R_G = √[hG/(8πc³)] = ℓ_P/2, and then fixes the dimensionless shape by one-cycle phase closure, q√(1+q²) = 2/π, giving q = 0.5563241800... without electroweak input. The present paper introduces the scale-free helical path excess δ_h = √(1+q²) − 1 and explores a conditional spinorial completion of a framed state by SU(2) ≃ S³. We explicitly separate this completion, the minimal Hopf-coordinate identification η = β with q = tan β, and the choice of a unit round S³ with its unnormalized Riemann volume measure. Under these assumptions, the Hopf-band volume fraction is exactly sin²β, while the unit-sphere volume is 2π². We then postulate, not derive, the electromagnetic correspondence α_TEF = δ_h/(2π²), obtaining α_TEF⁻¹ = 136.7621663..., about 0.2002% from the 2022 CODATA recommended low-energy value. The factor 1/(2π²) is not claimed to follow physically from probability normalization: it depends on the metric scale and measure convention unless a future TEF action fixes them. The result is therefore presented as a frozen, gravity-anchored geometric correspondence with explicit theoretical failure conditions, rather than a derivation or precision prediction of QED. Its main purpose is to expose a concrete possible map G → R_G → q → α_TEF and, equivalently after calibration, a simple expression of the fine-structure candidate in terms of the inherited q. Context note. This preprint is part of The Emergent Frame (TEF), an independent research program exploring whether spacetime structure and dimensionless physical parameters may admit a common geometric origin. It directly continues the preceding TEF preprint, A Helical Spacetime Ansatz Linking the Planck Scale and Electroweak Mixing (Zenodo, DOI: 10.5281/zenodo.22101000), from which the dimensionless helical parameter q used in the present work is inherited. The present result is intended as a conditional geometric correspondence and research target, not as a completed derivation of the fine-structure constant or of QED.