Five Generators (definition)
The Five Generators definition (I.D01) names the five canonical generators of the τ-kernel as a single bundled definition, complementing the K01–K06 axiomatization of MathG-K02. While K02 frames the generators as a postulate (the kernel HAS five generators), D01 frames them as the definitional building blocks (the kernel IS what these five generate, closed under K6). With 28 incoming edges, the definitional perspective is the structural input to the orbit machinery (I.D08 rank-transfer) and the iteration ladder (I.D? — supporting).
τ-Definition
The Five Generators definition (I.D01) names the five canonical generators of the τ-kernel as a single bundled definition, complementing the K01–K06 axiomatization of MathG-K02. While K02 frames the generators as a postulate (the kernel HAS five generators), D01 frames them as the definitional building blocks (the kernel IS what these five generate, closed under K6). With 28 incoming edges, the definitional perspective is the structural input to the orbit machinery (I.D08 rank-transfer) and the iteration ladder (I.D? — supporting).
Categorical invariant. A 5-tuple (g_1, g_2, g_3, g_4, g_5) of distinguished morphisms in τ such that K1–K5 axiomatize their joint algebraic structure.
Primary registry anchor:
I.D01
Supporting items:
I.K1
τ-Derivation Chain
Mathematical content
The Five Generators of τ are a 5-tuple (g_1, g_2, g_3, g_4, g_5) of distinguished τ-morphisms such that: (a) g_i satisfies the K_i axiom for i = 1, …, 5; (b) K6 closes the structure; (c) every τ-morphism is reachable from {g_1, …, g_5} by finite K3 composition.
Relationship to K02. MathG-K02 (postulate) asserts that the kernel HAS five generators. MathG-D08 (this definition) treats the five as a bundled definitional unit, which is the form most theorems consume them in (e.g. the Hyperfactorization theorem T01 uses the bundled tuple, not the individual axioms).
Consequences:
- Every τ-categorical morphism reduces to a finite K3-composition of the bundled tuple — the decidability backbone of the framework.
- Hyperfactorization theorem (T01) operates on the bundled tuple.
- Orbit closure (I.T01 — the registry's 'Ontic Closure' result) follows from K6 applied to the bundled tuple.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Orbit.Generation
Lean kind: def
Lean symbol: Tau.BookI.Orbit.fiveGenerators
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-C05-fine-structure-alphaFine-structure constant α →MathG-D08-five-generators-defFive Generators (definition) -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-D08-five-generators-defFive Generators (definition) -
PG-P04-photonτ-Photon →MathG-D08-five-generators-defFive Generators (definition) -
PG-Q10-proper-timeProper Time →MathG-D08-five-generators-defFive Generators (definition) -
MathG-D08-five-generators-defFive Generators (definition) →PG-C02-iota-tauMaster constant ι_τ