Rank coordinates (n, k)
Rank coordinates (n, k) are the indexing scheme used throughout Books I–II to locate τ-categorical content along two structural axes: n is the prime-rank index (which prime-numbered window of the ABCD chart), and k is the depth (how many K1 strict-order steps deep into that window). Together they index the window-algebra integers W_n(k), the central theorem rank (3, 15), and most theorem statements in Book II.
τ-Definition
Rank coordinates (n, k) are the indexing scheme used throughout Books I–II to locate τ-categorical content along two structural axes: n is the prime-rank index (which prime-numbered window of the ABCD chart), and k is the depth (how many K1 strict-order steps deep into that window). Together they index the window-algebra integers W_n(k), the central theorem rank (3, 15), and most theorem statements in Book II.
Categorical invariant. A pair (n, k) ∈ ℕ_+ × ℕ identifying a unique window-depth pair on the ABCD coordinate chart; the indexing scheme of the τ-rank-transfer machinery.
Primary registry anchor:
I.D08
τ-Derivation Chain
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I.K0— Universe Postulate -
I.K1— K1 strict order — depth coordinate k indexes K1-iteration depth -
I.D17— ABCD chart — the chart whose windows are indexed -
I.D08— Rank-transfer maps — formal definition of (n, k) and the transfer rules between adjacent rank coordinates
Lean modules referenced:
TauLib.BookI.Coordinates.PrimeEnumeration,
TauLib.BookI.Coordinates.HyperfactIsomorphism
Mathematical content
A rank coordinate is a pair (n, k) ∈ ℕ_+ × ℕ where n ∈ {1, 2, 3, …} is the prime-rank index (the n-th prime-numbered window of the ABCD coordinate chart, indexed by the K1 strict order) and k ∈ {0, 1, 2, …} is the depth coordinate (the number of K1-iteration steps taken into the n-th window).
Rank transfer. Rank-transfer maps (I.D08) provide canonical isomorphisms between bounded-window content at different (n, k) pairs, making the framework's algebra of windows manageable as a single graded structure.
Load-bearing rank pairs:
(3, 4)— first non-trivial window identity — W₃(4) = 5(5, 3)— second window identity — W₅(3) = 19(3, 15)— central theorem rank — categoricity check
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Coordinates.PrimeEnumeration
Lean kind: structure
Lean symbol: Tau.BookI.Coordinates.RankCoord
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-D05-rank-coordinatesRank coordinates (n, k) -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-D05-rank-coordinatesRank coordinates (n, k) -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-D05-rank-coordinatesRank coordinates (n, k) -
PG-L12-tau-gravitational-waveτ-Gravitational Wave →MathG-D05-rank-coordinatesRank coordinates (n, k) -
PG-Q10-proper-timeProper Time →MathG-D05-rank-coordinatesRank coordinates (n, k)