**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 = (phi²/2) (1 + 3 alpha² + (phi/2) alpha³) alpha²1 hbar c / mₑ². This predicts G = 6. 674298 x 10^-11 m³ 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 S² 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 (zetaS2 (-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; zetaS2 (-1) = -17/480; the charged-tower +1/10 sign flip; Deltaₜot = 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.
Jeremy Jacala (Wed,) studied this question.