Mutual Determination (5-Way Equivalence)
The Mutual Determination theorem (II.T27) is the Book II equivalence statement: five distinct presentations of τ-categorical content — boundary, interior, spectrum, lemniscate, and split-complex algebra — are mutually determining. Knowing any one of the five determines the other four. With 19 incoming edges, it is the framework's structural unification: the same τ-content is visible from five different angles, and the angles are interchangeable.
τ-Definition
The Mutual Determination theorem (II.T27) is the Book II equivalence statement: five distinct presentations of τ-categorical content — boundary, interior, spectrum, lemniscate, and split-complex algebra — are mutually determining. Knowing any one of the five determines the other four. With 19 incoming edges, it is the framework's structural unification: the same τ-content is visible from five different angles, and the angles are interchangeable.
Categorical invariant. A 5-way equivalence τ-Boundary ≃ τ-Interior ≃ τ-Spec ≃ τ-Lem ≃ τ-SplitComplex.
Primary registry anchor:
II.T27
τ-Derivation Chain
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I.K0— Universe Postulate -
I.T10— Split-Complex Forced — algebra side -
I.D18— Algebraic Lemniscate — geometry side -
III.D81— Spectrum functor — spectrum side -
II.T27— Mutual Determination — 5-way equivalence (this entry)
Lean modules referenced:
TauLib.BookII.Mirror.Inventory,
TauLib.BookIII.Doors.SplitComplexZeta
Mathematical content
There exist canonical equivalences between the five presentations of τ-categorical content: τ-Boundary ≃ τ-Interior ≃ τ-Spec ≃ τ-Lem ≃ τ-SplitComplex. Each pair of presentations is equivalent via an explicit functor; the five form a fully connected equivalence graph.
Proof sketch (expand)
Five pairwise equivalences are established: (Boundary ≃ Interior) by K4 boundary identification; (Boundary ≃ SplitComplex) by T10; (Boundary ≃ Lem) by I.D18; (SplitComplex ≃ Spec) by the spectrum functor (S01) restricted to scalars; (Lem ≃ Spec) by the spectral correspondence (T18). Composing gives 5-way mutual determination; coherence verified via K3 composition.
Consequences:
- The Central theorem at rank (3, 15) (T04) is verifiable through any of the five presentations — the choice is operational, not structural.
- Mathlib bridges into multiple analytic frameworks (split-complex algebra, plane-curve geometry, spectral theory) all access the same τ-content.
- The 'mirror inventory' of Book II (II.D? — supporting) catalogs the five presentations with their canonical equivalences.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Mirror.Inventory
Lean kind: theorem
Lean symbol: Tau.BookII.Mirror.mutualDetermination
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.
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PG-C18-kappa-tauGravity-sector coupling κ_τ →MathG-T11-mutual-determinationMutual Determination (5-Way Equivalence) -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T11-mutual-determinationMutual Determination (5-Way Equivalence) -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T11-mutual-determinationMutual Determination (5-Way Equivalence)