13.6. Consistency of the octonionic interpretation

The colour-block embedding $\mathrm{SU}(3)\subset\mathrm{SU}(4)=\mathrm{Spin}(6)$ and the octonionic embedding $\mathrm{SU}(3)\subset G_{2}\subset\mathrm{Spin}(6)$ are two distinct subgroups of the same spin group inside PST’s substrate; the choice of which one is the physical colour does not extend the framework assumption. This section shows that the octonionic interpretation is consistent with every other major structural result of PST.

13.6.1. Embedding-independence of the structural results

Computation 12 (together with Computation 13) compares the canonical and octonionic embeddings side-by-side under each of the verified results listed in Chapter 19. Computations 4, 5, 6, 8, 9, 10, 11 each produce identical conclusions under either embedding: the spin structure on $\mathbb{R}\times S^{3}$ is unique in both cases (Computation 4); the Einstein–Hilbert action and Newton’s constant emerge from the spectral action with the same coefficient in both cases (Computation 5); the gauge group is the unimodular unitary group of $A_{F}$ in both cases (Computation 6); the Higgs sector is unchanged (Computation 8); the Dirac-convergence prerequisites of §18.3 hold in both cases (Computation 9); the tree-level custodial protection $\Delta T=0$ of §15.11 holds in both cases (Computation 10); and the chiral-bimodule selection of §14.4 holds in both cases (Computation 11).

13.6.2. Unitary covariance of the order-one selection

The order-one condition $[[D_{F},a],Jb^{*}J^{-1}]=0$ that selects the Yukawa form of $D_{F}$ (Computation 7, §14.4) is unitarily covariant: under any unitary embedding change $D_{F}\mapsto UD_{F}U^{-1}$ , $a\mapsto UaU^{-1}$ , the order-one residue transforms covariantly and its vanishing pattern is preserved (Computation 15). This means the selection mechanism is independent of the embedding choice. Numerical verification on the lepton-sector finite triple gives a residue

	$\displaystyle\bigl{\\|}[[D_{F},a],\,Jb^{*}J^{-1}]\bigr{\\|}_{F}$	$\displaystyle\;\sim\;2\times 10^{-5}$		(168)
		(Yukawa $D_{F}$ , both embeddings)		(168)

versus

\bigl{\|}[[D_{F},a],\,Jb^{*}J^{-1}]\bigr{\|}_{F}\;\sim\;10\quad\text{(generic % $D_{F}$, both embeddings),}

(169)

a selection ratio of $\sim 5\times 10^{5}$ in either case. The Yukawa form is therefore picked out by the order-one condition under both embedding interpretations, and the choice of canonical versus octonionic SU(3) does not affect the chiral-bimodule story established in §14.4.

13.6.3. Structural scope under the octonionic embedding

Computation 1’s substrate-level structural-scope result, sections 1–6 of the script (Yukawa eigenvalues are $S_{3}$ -symmetric in leading order under the Bernoulli measure) is unaffected by the choice of embedding: it depends only on the substrate’s $S_{3}$ symmetry, not on the embedding of $A_{F}$ into $\mathrm{Cl}(0,6)$ . A side-by-side comparison of the canonical and octonionic embeddings on the load-bearing observables is recorded in Computation 13. Computation 1, section 7 of the script (the inner-fluctuation extension) holds under the octonionic embedding (Computation 14): individual fluctuations under the octonionic embedding can mix generation copies, but the substrate-level $S_{3}$ invariance from §1–6 (upstream of any $A_{F}$ embedding choice) constrains the coefficient structure of any physical observable to remain $S_{3}$ -symmetric in leading order. The Yukawa-contingency conclusion of §14.3 is preserved.

Under the octonionic interpretation, four things hold. The gauge structure of the octonionic-SU(3) closure is structurally derived (§13.5). Every established structural result of PST that depends on the embedding choice is verified under both canonical and octonionic embeddings. The order-one condition selects the Yukawa form identically. And the Yukawa-contingency reading of §14.3 is preserved. The structural derivation of $N_{\mathrm{gen}}=3$ on top of the verified octonionic embedding follows from the $V_{7}$ -valued asymmetric tension and the $\hat{\tau}$ -selection at modal sublimation, closed in §14.2 (Computations 46–48).