Remark. Why algebraic periodicity is a separate branch
[chevalley1954algebraic, II.2.1, II.2.5, and II.2.9, pp. 42--46, 65--66];
[lawson2016spin, I.4, Theorems 4.1 and 4.3, pp. 25--29] [fcap-001Y]
Remark. Why algebraic periodicity is a separate branch [chevalley1954algebraic, II.2.1, II.2.5, and II.2.9, pp. 42--46, 65--66]; [lawson2016spin, I.4, Theorems 4.1 and 4.3, pp. 25--29] [fcap-001Y]
The recurrences in Theorem [fcap-0013], Lemma [fcap-0014], and Theorem [fcap-001X] use Clifford universal properties, orthogonal sums, tensor products, coordinate isometries, and the real base entries. They do not use the Spin representation, the structure theorem obtained from a spinor module, or the Pin and Spin double covers. This is the algebraic branch of Layer 7.
The real groups \(\operatorname {Pin}(p,q)\) and \(\operatorname {Spin}(p,q)\) form a different branch. Their actions, kernels, and algebraic extensions specialize the group theory developed in § [fcap-0007], and therefore genuinely consume the Layer-2 Pin/Spin work. The distinction between these branches is the dependency correction proposed in TauCetiRoadmap PR 225. It changes the order in which the mathematics can be built; it does not turn the roadmap proposal itself into a formal theorem.