Spectrum functor
The spectrum functor (III.D81) is the τ-internal functor sending each τ-categorical object to its associated spectral data — the analogue of the algebraic-geometric Spec functor adapted to the τ-kernel signature. It is the structural input to all of Book III's spectral results, including the spectral correspondence (T18), the Critical Line theorem (T19), and the Prime Polarity Scaling Theorem (T20).
τ-Definition
The spectrum functor (III.D81) is the τ-internal functor sending each τ-categorical object to its associated spectral data — the analogue of the algebraic-geometric Spec functor adapted to the τ-kernel signature. It is the structural input to all of Book III's spectral results, including the spectral correspondence (T18), the Critical Line theorem (T19), and the Prime Polarity Scaling Theorem (T20).
Categorical invariant. A functor Spec : τ → τ-Spec sending each object to its spectral classification on the lemniscate boundary, equipped with the trichotomy (B, I, S) sector decomposition.
Primary registry anchor:
III.D81
τ-Derivation Chain
Mathematical content
Spec : τ → τ-Spec is a functor from the τ-categorical kernel to the category of τ-spectral data. For each τ-object A, Spec(A) is A's spectral classification on the lemniscate boundary, equipped with the trichotomy (B, I, S) decomposition.
Role. structural-transit
Lean Coverage
Status: Formalized
Module: TauLib.BookIII.Doors.SpectralDecomp
Lean kind: def
Lean symbol: Tau.BookIII.Doors.spectrumFunctor