Gauge couplings of the Standard Model in the octonionic framework: invariant normalisations, a conditional weak-sector completion, and a scale-explicit matching ledger

Abstract

We present a major revision of the gauge-sector account of the $E_8\otimes E_8$ 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 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 holds only if the Hilbert-Schmidt norm coincides with the norm induced by the projected gauge kinetic term. This gives the conditional boundary value $[{\alpha }_{\gamma }^{(0)}{]}^{-1}=137.04006064$, while comparison with CODATA requires the calculable factor $Z_{\gamma }=1.0000296379$. We also repair the two-U(1) matter-current normalisation: for parent charges $c_{1}Y$, $c_{2}Y$ with $c_1+c_2=1$, a link field leaves the diagonal connection coupled to exactly the Standard-Model $Y$. Equal connection coefficients and zero kinetic mixing give the illustrative benchmark $g_Y^2/g_2^2=3/10$, ${\sin }^2{\theta }_W=3/13$; trace-induced normalisation of the scaled generators gives a different result, so 3/13 is not an embedding-independent prediction; treated as exact, it lies about $13\sigma$ below the quoted $\overline{\mathrm{MS}}$ value. The colour bound ${\alpha }_s/{\alpha }_{\mathrm{em}}\le 8$ is restricted to colour profiles individually commuting with the complex structure. Measured couplings enter only as inputs to scale and running diagnostics. The result is a falsifiable conditional programme and a normalization audit, not a parameter-free derivation of Standard-Model gauge couplings.

Comments: v5. 25 pages