Gauge couplings of the Standard Model in the octonionic framework: invariant normalisations, current-consistent Abelian matching, and a scale-explicit matching ledger

Abstract

We present a normalisation audit of the gauge-sector account of the $E_8\otimes \omega E_8$ octonionic unification programme. Exact identities within the specified algebraic construction, consequences of an effective support model, and microscopic hypotheses are kept separate. The recurrent $3/8$ is the squared half-spacing of the adopted exceptional-Jordan family spectrum, while the exponential charge dependence is a multiplicative-character hypothesis motivated by the trace-dynamics action; the action parameter ${\alpha }_{\mathrm{TD}}$ is not the electromagnetic fine-structure constant. On the real ladder space $H_6$, the real Schur lemma fixes the shape of a colour-singlet self-adjoint photon profile to be proportional to the identity. The resulting $1/6$ dilution follows only if the Hilbert-Schmidt norm used in the support model is the physical norm induced by the projected gauge kinetic term. It therefore gives the conditional boundary value $[{\alpha }_{\gamma }^{(0)}{]}^{-1}=137.04006064\dots$, whereas comparison with CODATA requires the factor $Z_{\gamma }=1.0000296379\dots$, which remains to be calculated independently. [abstract truncated].

arXiv commentv6. 27 pages