From a Gravity-Calibrated Helix to Neutrino Oscillation Correspondences: A Third Cross-Sector Comparison of the Frozen TEF Shape Parameter

Abstract

This paper is the third cross-sector comparison in a TEF sequence built around the same dimensionless helical shape parameter q = a/R = 0.556324180045.... The value of q was fixed in Paper I by a gravity-calibrated one-cycle condition, before any neutrino observable entered the analysis, and was inherited without reoptimization by Paper II in a conditional fine-structure correspondence. Motivated by the electrically neutral and colorless character of active neutrinos, the present work carries the same frozen q into neutrino oscillation data. Two post-hoc but coefficient-free correspondences are recorded: q² = 0.3094965933..., compared with JUNO's sin²θ₁₂ = 0.3092 ± 0.0087, and q⁶ = 0.0296461034..., compared with rν = Δm²₂₁ / |Δm²₃ℓ|. Current JUNO and NuFIT 6.1 values give rν ≈ 0.02975 for normal ordering and rν ≈ 0.02988 for inverted ordering, both compatible with q⁶ at present precision. Eliminating q gives the prospective phenomenological relation rν ≈ sin⁶θ₁₂; this is not an independent third piece of evidence. No neutrino mechanism is claimed. For the Euclidean circular-helix ansatz, q also satisfies the exact Frenet–Serret identity q = τ/κ, giving it an intrinsic dimensionless geometric meaning. A future theory would need a common relativistic spectral operator whose eigenvectors and mass-squared eigenvalue gaps generate the two frozen correspondences without inserting them by hand. Context note. This preprint is the third paper in The Emergent Frame (TEF) research sequence built around the same frozen helical shape parameter. It directly follows: Paper I: A Helical Spacetime Ansatz Linking the Planck Scale and Electroweak Mixing. Zenodo. DOI: 10.5281/zenodo.22101000 Paper II: From a Gravity-Calibrated Helix to a Conditional Fine-Structure Correspondence. Zenodo. DOI: 10.5281/zenodo.22117153 The present work extends the same pre-existing geometric construction into neutrino oscillation phenomenology without refitting q. The reported q² and q⁶ mappings were identified after examining the neutrino sector and are therefore presented as conditional cross-sector numerical correspondences rather than preregistered predictions or a completed derivation of neutrino mixing. From this publication onward, these mappings are treated as frozen targets for prospective comparison with future data.