Hyperfactorization theorem
The Hyperfactorization theorem (I.T04) is the Book-I structural result that every τ-categorical morphism admits a unique hyperfactorization: a canonical decomposition through the rank-coordinate machinery into a (prime-shift) × (depth-shift) × (boundary-extension) chain. It is the theorem that makes the rank coordinates (n, k) tractable and the windows W_n(k) finite-dimensional.
τ-Definition
The Hyperfactorization theorem (I.T04) is the Book-I structural result that every τ-categorical morphism admits a unique hyperfactorization: a canonical decomposition through the rank-coordinate machinery into a (prime-shift) × (depth-shift) × (boundary-extension) chain. It is the theorem that makes the rank coordinates (n, k) tractable and the windows W_n(k) finite-dimensional.
Categorical invariant. An orthogonal-factorization system on τ given by (prime-shift, depth-shift, boundary-extension) — every τ-morphism factors uniquely as the composition of these three classes.
Primary registry anchor:
I.T04
τ-Derivation Chain
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I.K0— Universe Postulate -
I.K3— K3 composition law -
I.D17— ABCD coordinate chart -
I.D08— Rank-transfer maps — the three factor classes are read off the chart -
I.T04— Hyperfactorization theorem — every τ-morphism factors uniquely through the three classes
Lean modules referenced:
TauLib.BookI.Coordinates.HyperfactIsomorphism,
TauLib.BookI.Coordinates.Hyperfact
Mathematical content
Every τ-categorical morphism f : A → B admits a unique decomposition f = h ∘ g ∘ p where p is a prime-shift, g is a depth-shift, and h is a boundary-extension. Equivalently, (prime-shift, depth-shift, boundary-extension) is an orthogonal factorization system on τ.
Proof sketch (expand)
By induction on the K1 strict order on A: for each generator step in the K1-trace of f, the K3 composition law commutes with the rank-transfer maps (I.D08), so f's trace can be reordered as (all prime-shifts) ∘ (all depth-shifts) ∘ (all boundary-extensions). Uniqueness follows from the orthogonality property of the three factor classes.
Consequences:
- Window-algebra integers W_n(k) are finite-dimensional (each window admits only finitely many hyperfactor decompositions of bounded depth).
- Rank-coordinate transfer is well-defined (the rank-transfer maps I.D08 are canonical because the factorization is unique).
- Central theorem at rank (3, 15) is reducible to a finite check (the categoricity check enumerates hyperfactorizations of bounded length).
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Coordinates.HyperfactIsomorphism
Lean kind: theorem
Lean symbol: Tau.BookI.Coordinates.hyperfactorization
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-T01-hyperfactorizationHyperfactorization theorem -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-T01-hyperfactorizationHyperfactorization theorem -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T01-hyperfactorizationHyperfactorization theorem -
PG-L12-tau-gravitational-waveτ-Gravitational Wave →MathG-T01-hyperfactorizationHyperfactorization theorem -
PG-Q10-proper-timeProper Time →MathG-T01-hyperfactorizationHyperfactorization theorem