Results Glossary Entry Canonical mathematics The Idempotent Decomposition Lemma (II.L07) is the Book-II structural result that every τ-categorical content decomposes orthogonally into idempotent components. With 19 incoming edges, it is the finest-grain decomposition principle in the …
Results · Mathematics Glossary · Lemma MathG-L01-idempotent-decomposition Idem-decomp Canonical

Idempotent Decomposition Lemma

The Idempotent Decomposition Lemma (II.L07) is the Book-II structural result that every τ-categorical content decomposes orthogonally into idempotent components. With 19 incoming edges, it is the finest-grain decomposition principle in the framework — finer than the 4+1 sector decomposition (D07), since each sector itself decomposes into idempotents. The lemma is the basis for the spectrum functor's (S01) construction and underwrites the central theorem's (T04) categoricity argument.

Mathematics Glossary Primary: II.L07

τ-Definition

The Idempotent Decomposition Lemma (II.L07) is the Book-II structural result that every τ-categorical content decomposes orthogonally into idempotent components. With 19 incoming edges, it is the finest-grain decomposition principle in the framework — finer than the 4+1 sector decomposition (D07), since each sector itself decomposes into idempotents. The lemma is the basis for the spectrum functor's (S01) construction and underwrites the central theorem's (T04) categoricity argument.

Categorical invariant. Every τ-object A decomposes as A = ⊕_i e_i · A where {e_i} is a complete orthogonal idempotent system in B_τ.

Primary registry anchor: II.L07

Supporting items: I.D19, III.D13

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.D19 — Boundary Ring B_τ — carries idempotents
  3. II.L07 — Idempotent Decomposition Lemma — every τ-object decomposes orthogonally over B_τ idempotents

Lean modules referenced: TauLib.BookIII.Sectors.Decomposition

Mathematical content

Lemma Idem-decomp
Lemma

For every τ-object A, there exists a complete orthogonal idempotent system {e_i} ⊂ B_τ (with ∑ e_i = 1, e_i · e_j = δ_ij · e_i) such that A decomposes as A = ⊕_i (e_i · A) and the decomposition is canonical.

Proof sketch (expand)

Apply the Boundary Ring's polarity grading (D15): the polarity decomposition B_τ = B_τ^+ ⊕ B_τ^- ⊕ B_τ^0 has three canonical idempotents {e_+, e_-, e_0}. Iterating finer (using K3 composition + K6 closure) gives the complete orthogonal system. The decomposition lifts to A by tensoring through K3.

Consequences:

  • Spectrum functor (S01) — Spec(A) reads off A's idempotent decomposition.
  • 4+1 sector decomposition (D07) — coarser than the idempotent decomposition; the four analytic sectors plus ω-sector each decompose further into idempotents.
  • Central theorem (T04) — categoricity check uses the idempotent decomposition at rank (3, 15).

Lean Coverage

Status: Formalized

Module: TauLib.BookIII.Sectors.Decomposition

Lean kind: theorem

Lean symbol: Tau.BookIII.Sectors.idempotentDecomp

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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