Part V: Hyperfactorization
Part IV defined the ABCD coordinate chart: a total map Φ : Obj(τ) → τ-Idx⁴ that assigns four typed coordinates to every object. Existence of the chart was proved; the dimension dim_τ = 4 was established. But a coordinate system that merely exists is not yet a coordinate system that works: the chart must be injective — distinct objects must receive distinct addresses.
This Part proves the Hyperfactorization Theorem, the first of the two hinge theorems that anchor the entire Panta Rhei series. The theorem states that the ABCD encoding is unique: every X ∈ τ-Idx with X ≥ 2 has exactly one decomposition X = ((A ↑↑ C)^{B}) · D satisfying the greedy-peel constraints.
The proof rests on three critical lemmas: enumerate
- Tetration injectivity (already proved in Part III, the relevant chapter): a ↑↑ c₁ = a ↑↑ c₂ implies c₁ = c₂.
- No-tie determinism : the greedy peel’s choice is deterministic at every step.
- Strict remainder descent : the remainder D is strictly less than X. enumerate
The chapter sequence: the relevant chapter frames the uniqueness problem and recalls the proof strategy. the relevant chapter proves the no-tie lemma. the relevant chapter proves the descent lemma. the relevant chapter assembles the full proof. the relevant chapter derives the constructive consequences, including earned Cantor pairing and sequence encoding without importing set theory.
With the Hyperfactorization Theorem in hand, the ABCD chart becomes a faithful coordinate system: shadow equality collapses to ontic identity , and every object of Category τ has exactly one canonical address.