Rigorous Derivations of the MOND Scale and Gauge Coupling Running from the the Spectral Eigenvalues 0, 4, 7, and the Holographic Coupling Dictionary Paper LXXVI of the Brane-Bulk Octonionic Series

The two most-pressed points in a peer-review of the brane-bulk framework are the MOND acceleration scale a0 = c0H0/2π and the gauge coupling running αs (µ) = 2π/ (7 ln µ/ΛQCD). Both have appeared in earlier papers as results; neither has been given a tight derivation from a minimal set of explicitly stated assumptions. A referee reading will press three questions: (Q1) Why exactly 2π and not 4π or some other factor? (Q2) Why exactly the eigenvalues 0, 4, 7 for the three gauge sectors? (Q3) Why should inverse coupling equal the mode delocalization action? This paper provides the tightest currently defensible derivation of each result, explicitly distinguishing what is proven from stated assumptions and what remains a geometric hypothesis. For the MOND scale, we show that scale invariance of the deep-regime constitutive law, ux conservation, and the existence of a single post-electroweak cosmological acceleration scale together require a0 = c0H0/2π and imply the Tully-Fisher relation v4 c = GMba0. For the coupling running, we show that the identication of λ ∈0, 4, 7 with the real dimensions of the three canonical octonionic subspaces Re (O), H+, Im (O), combined with the standard AdS/CFT holographic coupling dictionary, gives αs (µ) = 2π/ (7 ln µ/ΛQCD) as a one- loop-exact result. For 2π, we show it is the holonomy of one complete rapidity-space U (1) cycle, appearing universally as the period normalization of the Polyakov loop in the hyperbolic bulk. Three open questions remain requiring further derivation, and are stated precisely. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.