Results Glossary Entry Canonical mathematics A window-algebra object is the categorical presentation of a window in the ABCD coordinate chart at rank coordinate (n, k). It is a τ-object (O01) carrying the K1-iteration depth k and the prime-window index n as structural data. The object…
Results · Mathematics Glossary · Object MathG-O02-window-object W_n(k)-obj Canonical

Window-algebra object

A window-algebra object is the categorical presentation of a window in the ABCD coordinate chart at rank coordinate (n, k). It is a τ-object (O01) carrying the K1-iteration depth k and the prime-window index n as structural data. The object's hom-set encodes the W_n(k) integer dimension; its endomorphism algebra is the τ-internal model of the n-th window's dynamics at depth k. Window-objects are the categorical entities the central theorem (T04) operates on at rank (3, 15).

Mathematics Glossary Primary: II.D54

τ-Definition

A window-algebra object is the categorical presentation of a window in the ABCD coordinate chart at rank coordinate (n, k). It is a τ-object (O01) carrying the K1-iteration depth k and the prime-window index n as structural data. The object's hom-set encodes the W_n(k) integer dimension; its endomorphism algebra is the τ-internal model of the n-th window's dynamics at depth k. Window-objects are the categorical entities the central theorem (T04) operates on at rank (3, 15).

Categorical invariant. A τ-object W_n(k)-obj parametrized by rank coordinate (n, k) ∈ ℕ_+ × ℕ; its hom-internal structure is the W-algebra W_n(k).

Primary registry anchor: II.D54

Supporting items: I.D17, I.D08

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.D17 — ABCD coordinate chart
  3. I.D08 — Rank coordinates (n, k)
  4. II.D54 — Hom-object construction — categorical presentation of the window structure

Lean modules referenced: TauLib.BookII.Enrichment.Homological, TauLib.BookI.Coordinates.ABCD

Mathematical content

Object W_n(k)-obj
Object

For each rank coordinate (n, k), the window-algebra object W_n(k)-obj is the τ-categorical object presenting the window of the ABCD coordinate chart at that rank. Its hom-internal structure encodes the W_n(k) integer dimension; its endomorphism algebra is the τ-internal model of the window's dynamics.

Construction. W_n(k)-obj is constructed as the hom-object hom_τ(prime-shift_n, depth-shift_k) in the self-enriched τ (S03), where prime-shift and depth-shift are the corresponding morphism classes in the Hyperfactorization OFS (T01).

Consequence. The window objects are the categorical entities the central theorem (T04) operates on. The categoricity check at rank (3, 15) verifies that End(W_3(15)-obj) — the endomorphism algebra of the W_3(15)-object — has the unique structure forcing ι_τ = 2/(π+e).

Load-bearing objects:

  • W_3(4)-obj — first non-trivial window object — endomorphism dim = 5
  • W_5(3)-obj — second window object — endomorphism dim = 19
  • W_3(15)-obj — central theorem rank — categoricity check operates on this object

Lean Coverage

Status: Formalized

Module: TauLib.BookII.Enrichment.Homological

Lean kind: def

Lean symbol: Tau.BookII.Enrichment.windowObject

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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