A formula for Newton's gravitational constant from the fine-structure constant: a falsifiable conjecture, with corrections and a statistical assessment

**Version 6 (June 2026).** This version corrects two errors of earlier versions and adds a pre-registered statistical assessment that materially changes how the formula should be read. Earlier versions of this record remain available below as documentary history; their retracted claims are listed at the end of this description. We present a formula relating Newton's gravitational constant G to the fine-structure constant alpha and the golden ratio phi = (1+sqrt(5))/2: G = (phi^2/2) (1 + 3 alpha^2 + (phi/2) alpha^3) alpha^21 hbar c / m_e^2. This predicts G = 6.674298 x 10^-11 m^3 kg^-1 s^-2, deviating from the CODATA 2018 central value by -0.31 ppm (-0.33 ppm for CODATA 2022). Corrected in v6: the Kaluza-Klein reduction on S^2 yields Brans-Dicke gravity with omega = -(n-1)/n = -1/2 for n = 2, not omega = 0 as previously stated; the modulus is rendered Planck-scale massive by the Salam-Sezgin mechanism, so the reduced theory is indistinguishable from general relativity at accessible scales. Corrected in v6: the one-loop spectral results quoted in earlier versions (zeta_S2(-1) = -17/480; a charged-sector sign flip to +1/10; a 13-sector total 43/15) rested on a regularization that is invalid for curvature-shifted Kaluza-Klein towers. The corrected values (genuine heat-kernel weights) are -1/15, -1/40, and -2/15 respectively, and the consistent supersymmetric computation gives Delta = -1/10 (doi:10.5281/zenodo.20631504). No loop-level support for the correction coefficients survives. Assessed in v6: a pre-registered look-elsewhere enumeration shows that within the formula's own complexity class, thousands of rival expressions match G inside the experimental band, and hundreds match the central value to sub-ppm — for control targets as readily as for the real one. Within the framework's restricted class the agreement is a ~1% (naive, pre-selection) coincidence. The sub-ppm agreement therefore carries little evidential weight by itself. What remains is a falsifiable conjecture: the formula's value was published before any sub-ppm measurement of G exists. A future measurement at that precision either lands on 6.674298 x 10^-11 — meaningful precisely because the number was called in advance — or rules the formula out permanently. The mu prediction of v4 remains retracted (inconsistent with H2+ spectroscopy at >14 sigma). Retracted claims of earlier versions of this record: omega = 0; zeta_S2(-1) = -17/480; the charged-tower +1/10 sign flip; Delta_tot = 43/15; the SUSY denominator no-go; the v4 mu prediction. The withdrawn one-loop preprint these drew on is doi:10.5281/zenodo.19160221 (withdrawal notice: doi:10.5281/zenodo.20631828); the corrected one-loop computation is doi:10.5281/zenodo.20631504.