A Helical Spacetime Ansatz Linking the Planck Scale and Electroweak Mixing

We examine a minimal helical ansatz for microscopic spatial structure in which a primitive transverse radius is calibrated from the measured gravitational constant and the helix shape is subsequently fixed by a one-cycle phase-closure condition. The construction originates in the broader Emergent Frame (TEF) proposal, which treats spacetime as an emergent physical entity rather than a structureless pre-existing background, but the present paper is deliberately restricted to one concrete geometry and its quantitative consequence. Inspired mathematically by Euler's representation of planar rotation, a spatial line is modeled as a right-handed helix with radius R, axial advance per radian a, and dimensionless pitch ratio q = a/R = tan beta. A gravity-geometry calibration relation, using G as an empirical input, gives R_G = sqrt(hG/(8 pi c^3)) = l_P/2. The model then distinguishes the axial extension generated in one turn from the helical arc length traversed internally: the former sets the cycle energy through the gravitational line-tension scale c^4/G, while the latter sets the cycle time. Requiring one complete primitive cycle to close its quantum phase, E_cyc T_cyc/hbar = 2 pi, yields q sqrt(1 + q^2) = 2/pi, so that q approximately 0.556324, beta_G approximately 29.088 degrees, and sin^2(beta_G) approximately 0.236348. No electroweak observable enters this calculation. The resulting angle is therefore proposed only as a candidate primitive geometric reference for weak mixing, not as a completed derivation of the Weinberg angle. Its numerical proximity to the observed running weak-mixing parameter is presented as the reason this otherwise speculative spacetime ansatz merits further quantitative investigation.Release note: TEF Research Release 2026-001, version 5.1. This is a publicly released preprint and has not undergone peer review. Project website: https://theemergentframe.org