Central theorem at rank (3, 15)
The Central theorem at rank (3, 15) (II.T40) is the Book-II structural categoricity result that pins down the master constant ι_τ. The theorem asserts that the τ-categorical structure at rank coordinate (3, 15) admits a unique algebraic invariant — and that invariant is precisely 2/(π+e) = ι_τ. The proof is a finite-decidable check verified via native_decide; this places ι_τ outside any free-parameter freedom.
τ-Definition
The Central theorem at rank (3, 15) (II.T40) is the Book-II structural categoricity result that pins down the master constant ι_τ. The theorem asserts that the τ-categorical structure at rank coordinate (3, 15) admits a unique algebraic invariant — and that invariant is precisely 2/(π+e) = ι_τ. The proof is a finite-decidable check verified via native_decide; this places ι_τ outside any free-parameter freedom.
Categorical invariant. The unique τ-categorical algebraic invariant at rank (3, 15), forced to equal ι_τ = 2/(π+e) by the rank-(3, 15) categoricity check.
Primary registry anchor:
II.T40
τ-Derivation Chain
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I.K0— Universe Postulate -
I.D34— Master constant ι_τ defined as 2/(π+e) -
II.T36— Yoneda enrichment ladder closes — invariant well-defined -
II.T40— Central theorem at rank (3, 15) — finite-decidable categoricity check fixes the invariant
Lean modules referenced:
TauLib.BookII.CentralTheorem.CentralTheorem,
TauLib.BookII.CentralTheorem.Categoricity
Mathematical content
The τ-categorical structure at rank coordinate (3, 15) admits a unique algebraic invariant. This invariant equals ι_τ = 2/(π+e).
Proof method: Finite-decidable categoricity check verified via Lean's native_decide. The check enumerates the bounded state space at rank (3, 15) — finite by the Hyperfactorization theorem (T01) — and verifies the categoricity property for every state. The state space is large but finite; decidability is automatic, completion is empirical.
TCB cost. The native_decide call brings the trust-budget cost of Lean.ofReduceBool + Lean.trustCompiler beyond the kernel baseline. See /verify/tcb/ for full disclosure. A reviewer who does not accept this TCB extension can rerun the same check by kernel-only `decide` at substantial build-time cost.
Consequences:
- ι_τ is fixed at 2/(π+e) — no free parameter freedom.
- All downstream τ-categorical invariants depend on ι_τ via the Yoneda enrichment ladder.
- The framework's master schema (V.T142) — every physical constant cascading from ι_τ + m_n — has its mathematical-side anchor here.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.CentralTheorem.CentralTheorem
Lean kind: theorem
Lean symbol: Tau.BookII.CentralTheorem.centralTheorem_3_15
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-T04-central-theoremCentral theorem at rank (3, 15) -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T04-central-theoremCentral theorem at rank (3, 15) -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-T04-central-theoremCentral theorem at rank (3, 15) -
PG-Q24-velocityVelocity →MathG-T04-central-theoremCentral theorem at rank (3, 15) -
MathG-T04-central-theoremCentral theorem at rank (3, 15) →PG-C02-iota-tauMaster constant ι_τ