Part III: The Spectral Algebra
The algebraic vocabulary for everything that follows. We earn number theory—primes, residue rings, p-adic fields, ad`eles—from the τ kernel, not by importing classical algebraic number theory.
Enrichment level. Part III operates at E₀—the same algebraic substrate as Books I and II. The constructions are number-theoretic: finite cyclic groups, modular arithmetic, inverse limits, and split-complex extensions. No enriched Hom spaces are needed; the self-enrichment functor F_E (Part I) does not enter. Part III builds the spectral toolkit at E₀ that the Millennium clusters of Parts IV–VI will apply at E₁.
Relationship to Book I. Book I earned the primorial presheaf (I.D83), the CRT decomposition (I.D29), the profinite boundary ring ℤ_τ (I.D19), the split-complex scalars (I.D20), and the spectral decomposition (I.T12). Part III extends these foundations: the primorial ladder gains a cofinality theorem (III.T09), the CRT is re-derived constructively via Fermat’s method as a cross-validation and lifted to the module level (III.D20), Hensel lifting and τ-native local fields are constructed for the first time (III.T11, III.D21), and the informal B/C lobe language is replaced by the computable classifier Label_n (III.D23) and the spectral trichotomy (III.T14).
The CRT Decomposition Theorem gives a τ-native Chinese Remainder Theorem via modular B'ezout without signed arithmetic—earned-language discipline in action. Hensel lifting is performed constructively in residue carriers. The primorial ladder Prim(k) = p₁ ⋯ p_k emerges as the canonical cofinal filtration that unifies finite-level verification across all Millennium Problems. The internal bipolar classifier Label_n replaces informal lobe language with computable predicates: every boundary character decomposes uniquely into B-supported, C-supported, and X-mixing components via the spectral trichotomy.