Prime Polarity Theorem
The Prime Polarity Theorem (I.T05) is the most-referenced kernel result in Books I–III: it classifies every prime p with respect to the τ-categorical kernel into a polarity sector, establishing the bipolar algebra (positive/negative polarity) on which Books III's spectral correspondence is built. With 47 incoming dependency edges, it is the single most load-bearing theorem of the framework's foundational layer.
τ-Definition
The Prime Polarity Theorem (I.T05) is the most-referenced kernel result in Books I–III: it classifies every prime p with respect to the τ-categorical kernel into a polarity sector, establishing the bipolar algebra (positive/negative polarity) on which Books III's spectral correspondence is built. With 47 incoming dependency edges, it is the single most load-bearing theorem of the framework's foundational layer.
Categorical invariant. A canonical polarity classification function pol : Primes → {+1, −1} compatible with the K1 strict order and the K2 boundary structure.
Primary registry anchor:
I.T05
τ-Derivation Chain
-
I.K0— Universe Postulate -
I.K1— K1 strict order on kernel atoms -
I.D18— Algebraic lemniscate — boundary structure on which polarity is read -
I.T05— Prime Polarity Theorem — every prime classified into ±-polarity sector
Lean modules referenced:
TauLib.BookI.Polarity.PrimePolarityClassifier,
TauLib.BookI.Polarity.PrimePolarityIsomorphism
Mathematical content
There exists a canonical polarity function pol : Primes → {+1, −1} such that, for every prime p, pol(p) is determined by p's image under the K2 boundary map. The function is well-defined, total, and respects the K1 strict order.
Proof sketch (expand)
By K2 (labelled boundary): every kernel atom maps to a labelled boundary point. For p prime, the boundary point's label takes one of exactly two structural types (positive- or negative-polarity), determined by p's residue under the algebraic lemniscate (I.D18). The well-definedness follows from K6 (no-ω closure forbids ambiguous polarity); the total claim follows from K2 (every kernel atom has a boundary image).
Consequences:
- Bipolar algebra on Primes (the algebraic structure underlying Book I.D19 boundary ring).
- Foundation for Book III spectral correspondence (the (B, I, S) trichotomy reads off polarity at higher orders).
- 47 transitive dependents — most-referenced kernel theorem.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Polarity.PrimePolarityClassifier
Lean kind: theorem
Lean symbol: Tau.BookI.Polarity.primePolarityTheorem
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-T06-prime-polarityPrime Polarity Theorem -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-L12-tau-gravitational-waveτ-Gravitational Wave →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-Q10-proper-timeProper Time →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-Q24-velocityVelocity →MathG-T06-prime-polarityPrime Polarity Theorem -
MathG-T06-prime-polarityPrime Polarity Theorem →PG-C02-iota-tauMaster constant ι_τ