Window-algebra integers W_n(k)
The window-algebra integers W_n(k) are the Book-II numerical invariants of the τ-categorical structure at rank coordinates (n, k). For load-bearing pairs (W₃(4) = 5, W₅(3) = 19, …) the values are exact closed-form integers; the central theorem at rank (3, 15) (T04 / II.T40) is the categoricity check that fixes them.
τ-Definition
The window-algebra integers W_n(k) are the Book-II numerical invariants of the τ-categorical structure at rank coordinates (n, k). For load-bearing pairs (W₃(4) = 5, W₅(3) = 19, …) the values are exact closed-form integers; the central theorem at rank (3, 15) (T04 / II.T40) is the categoricity check that fixes them.
Categorical invariant. The integer-valued algebra of bounded windows over the τ-coordinate chart; W_n(k) is the dimension of the n-th window at depth k.
Primary registry anchor:
I.D17
τ-Derivation Chain
-
I.K0— Universe Postulate -
I.D08— Rank coordinates (n, k) — the indexing scheme for the windows -
I.D17— ABCD coordinate chart — the chart on which the windows are defined -
II.T40— Central theorem at rank (3, 15) — fixes the load-bearing W-values via the categoricity check
Lean modules referenced:
TauLib.BookI.Coordinates.ABCD,
TauLib.BookI.Coordinates.HyperfactIsomorphism
Mathematical content
For each pair (n, k) with n ≥ 1, k ≥ 0, the window-algebra integer W_n(k) is defined as the integer dimension of the n-th window of the τ-coordinate chart at depth k.
Uniqueness. The values W_n(k) are forced by the rank-(n, k) algebraic check on the τ-coordinate chart. The check is finite-decidable; in TauLib it is verified via `native_decide` (see /verify/tcb/ for the trust-budget cost).
Load-bearing values:
W₁(0) = 1— trivial base caseW₃(4) = 5— first non-trivial closed-form identity (Book II)W₅(3) = 19— second closed-form identity; cited in central theoremW₃(15) = …— central theorem rank — categoricity check value
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Coordinates.ABCD
Lean kind: def
Lean symbol: Tau.BookI.Coordinates.windowDimension
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-D03-window-algebraWindow-algebra integers W_n(k) -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-D03-window-algebraWindow-algebra integers W_n(k) -
PG-L12-tau-gravitational-waveτ-Gravitational Wave →MathG-D03-window-algebraWindow-algebra integers W_n(k) -
PG-Q10-proper-timeProper Time →MathG-D03-window-algebraWindow-algebra integers W_n(k)