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

Xiaodan Wu

doi:10.5281/zenodo.22117153

Abstract

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, $R_{G} = h G / (8 π c^{3}) = ℓ_{P} /2$ , and then fixes the dimensionless shape by one-cycle phase closure, $q 1 + q^{2} = 2/ π$ , giving $q = 0.5563241800 \dots$ without electroweak input. The present paper introduces the scale-free helical path excess $δ_{h} = 1 + q^{2} - 1$ and explores a conditional spinorial completion of a framed state by $SU (2) ≃ S^{3}$ . We explicitly separate this completion, the minimal Hopf-coordinate identification $η = β$ with $q = tan β$ , and the choice of a unit round $S^{3}$ with its unnormalized Riemann volume measure. Under these assumptions, the Hopf-band volume fraction is exactly $sin^{2} β$ , while the unit-sphere volume is $2 π^{2}$ . We then postulate, not derive, the electromagnetic correspondence $α_{TEF} = δ_{h} / (2 π^{2})$ , obtaining $α_{TEF}^{- 1} = 136.7621663 \dots$ , about $0.2002%$ from the 2022 CODATA recommended low-energy value. The factor $1/ (2 π^{2})$ 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 $G \to R_{G} \to q \to α_{TEF}$ and, equivalently after calibration, a simple expression of the fine-structure candidate in terms of the inherited $q$ .

Research Context

This preprint asks whether the dimensionless helical parameter developed in the preceding TEF spacetime construction can also support a conditional geometric correspondence with electromagnetic coupling.

The construction combines the scale-free excess of a helical path with a conditional $SU (2) ≃ S^{3}$ completion. Under an explicit metric and measure convention, it yields $α_{TEF}^{- 1} = 136.7621663 \dots$ , approximately $0.2002%$ from the 2022 CODATA recommended low-energy value. The residual is retained rather than removed through fitting or a post-hoc scale choice.

The factor $1/ (2 π^{2})$ is postulated rather than derived and is not interpreted as a physical consequence of probability normalization. The result is presented as a frozen, gravity-anchored correspondence with explicit failure conditions, not as a derivation or precision prediction of quantum electrodynamics.

Publication Status

Version 4.7 is publicly archived on Zenodo as a preprint under the Creative Commons Attribution 4.0 International license. Its version-specific DOI is 10.5281/zenodo.22117153; the concept DOI for the complete version history is 10.5281/zenodo.22117152. The preprint has not undergone peer review.

Relationship to the Companion Manuscript

The paper inherits its helical parameter from A Helical Spacetime Ansatz Linking the Planck Scale and Electroweak Mixing. The two preprints should be read as linked constructions: the first develops the gravity-calibrated spacetime ansatz and weak-mixing candidate, while this paper tests a narrower, explicitly conditional fine-structure correspondence using the inherited geometry.

Citation

Zenodo citation

Wu, X. (2026). From a Gravity-Calibrated Helix to a Conditional Fine-Structure Correspondence (Version 4.7). Zenodo. https://doi.org/10.5281/zenodo.22117153

@misc{wu2026gravitycalibratedhelix,
  author = {Wu, Xiaodan},
  title = {From a Gravity-Calibrated Helix to a Conditional Fine-Structure Correspondence},
  year = {2026},
  publisher = {Zenodo},
  version = {4.7},
  doi = {10.5281/zenodo.22117153},
  url = {https://doi.org/10.5281/zenodo.22117153}
}

Version History

v4.7 - August 26, 2026

Public preprint release on Zenodo. Version DOI: 10.5281/zenodo.22117153.

v3.3 - August 17, 2026

Previous public working manuscript.

@unpublished{wu2026helical, author = {Xiaodan Wu}, title = {From a Gravity-Calibrated Helix to a Conditional Fine-Structure Correspondence}, year = {2026}, note = {TEF-2026-002, version v4.7}, url = {https://theemergentframe.org/papers/helical-geometry-fine-structure-constant/}, doi = {10.5281/zenodo.22117153}, }