Spectral Trichotomy Lemma
The Spectral Trichotomy Lemma (III.T14) classifies every τ-spectral datum into one of exactly three sectors: B (boundary-type), I (interior-type), or S (singular-type). The trichotomy is the structural input to Book III's spectral correspondence (A02), the bridge functor (A01), and Grand GRH (A03). With 16 incoming edges, it is the framework's spectral-side classification scheme.
τ-Definition
The Spectral Trichotomy Lemma (III.T14) classifies every τ-spectral datum into one of exactly three sectors: B (boundary-type), I (interior-type), or S (singular-type). The trichotomy is the structural input to Book III's spectral correspondence (A02), the bridge functor (A01), and Grand GRH (A03). With 16 incoming edges, it is the framework's spectral-side classification scheme.
Categorical invariant. A canonical surjection τ-Spec → {B, I, S} compatible with the spectrum functor (S01) and the lemniscate boundary's polarity grading.
Primary registry anchor:
III.T14
τ-Derivation Chain
Mathematical content
There is a canonical sector function trichotomy : τ-Spec → {B, I, S} such that for every τ-object A, Spec(A) lies in exactly one of the three sectors. The function is total, surjective, and compatible with the lemniscate boundary's polarity grading from Prime Polarity (T06).
Proof sketch (expand)
Each τ-spectral datum maps to a polarity-graded position on the algebraic lemniscate (T09). Three structural types of position exist: boundary points of the lemniscate (B), interior points (I), and singular points at the lemniscate's self-intersection (S). K6 closure forbids a fourth type. The map is canonical because the polarity grading is canonical (T06).
Consequences:
- (B, I, S) sector decomposition input to spectral correspondence (A02).
- Bridge functor (A01) — translates each sector type to the corresponding classical L-function structure.
- Grand GRH adelic (A03) — uniform spectral purity across all three sectors universally.
Lean Coverage
Status: Formalized
Module: TauLib.BookIII.Spectral.Trichotomy
Lean kind: theorem
Lean symbol: Tau.BookIII.Spectral.spectralTrichotomy
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.
-
PG-C18-kappa-tauGravity-sector coupling κ_τ →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma -
PG-Q24-velocityVelocity →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma