TauLib · API Book III

TauLib.BookIII.Enrichment.CanonicalLadder

TauLib.BookIII.Enrichment.CanonicalLadder

The Canonical Ladder Theorem: the organising result of Book III.

Registry Cross-References

  • [III.T01] Non-Emptiness Theorem – non_emptiness_check, non_emptiness_theorem

  • [III.P01] E₁ Strictness Witness – e1_strictness_witness

  • [III.T02] Strictness Theorem – strictness_check, strictness_theorem

  • [III.P02] Functor Category Collapse – functor_collapse_check

  • [III.T03] Saturation at E₃ – saturation_e3_check, saturation_e3_theorem

  • [III.T04] Canonical Ladder Theorem – canonical_ladder_check, canonical_ladder_theorem

Mathematical Content

III.T01 (Non-Emptiness): Each enrichment layer E_k (k = 0,1,2,3) is inhabited: constructive witnesses at each level.

III.T02 (Strictness): E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure.

III.T03 (Saturation): E₄ = E₃. The enrichment ladder terminates at exactly four levels due to the 4-orbit closure.

III.T04 (Canonical Ladder): The organising result: non-empty, strictly increasing, saturating at E₃, and unique.

Tau.BookIII.Enrichment.non_emptiness_check

source def Tau.BookIII.Enrichment.non_emptiness_check (bound db : Denotation.TauIdx) :Bool

[III.T01] Non-emptiness check: every enrichment layer is inhabited. Uses existence_checker from Functor.lean for each level. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Enrichment.non_emptiness_witnesses

source def Tau.BookIII.Enrichment.non_emptiness_witnesses (bound db : Denotation.TauIdx) :Bool × Bool × Bool × Bool

[III.T01] Constructive witnesses for non-emptiness. Returns a witness (x, k) at each level, or none if none found. Equations

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Tau.BookIII.Enrichment.e1_strictness_witness

source def Tau.BookIII.Enrichment.e1_strictness_witness (bound db : Denotation.TauIdx) :Bool

[III.P01] E₁ strictness witness: the bipolar Hom decomposition Hom(A,B) = e₊·Hom₊ + e₋·Hom₋ is a genuine E₁ structure.

The witness is the hom_stage value with its bipolar decomposition. At E₀, hom_stage is just a number; at E₁, it carries bipolar structure (split-complex scalar action on morphism spaces). Equations

  • Tau.BookIII.Enrichment.e1_strictness_witness bound db = Tau.BookIII.Enrichment.e1_strictness_witness.go bound db 0 0 1 ((bound + 1) * (bound + 1) * (db + 1)) Instances For

Tau.BookIII.Enrichment.e1_strictness_witness.go

source@[irreducible]

**def Tau.BookIII.Enrichment.e1_strictness_witness.go (bound db : Denotation.TauIdx)

(a b k fuel : ℕ) :Bool**

Equations

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Tau.BookIII.Enrichment.strictness_check

source def Tau.BookIII.Enrichment.strictness_check (bound db : Denotation.TauIdx) :Bool

[III.T02] Strictness check: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Combines the strictness_checker witnesses with the E₁ strictness witness (bipolar Hom decomposition). Equations

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Tau.BookIII.Enrichment.functor_collapse_check

source def Tau.BookIII.Enrichment.functor_collapse_check (bound db : Denotation.TauIdx) :Bool

[III.P02] Functor category collapse: [E₃^op, E₃] ⊆ E₃. The functor category at E₃ does not escape E₃ because:

  • There are only 4 orbits under ABCD decomposition

  • The ω-absorber prevents new generators

  • E₃.succ = E₃ (definitional saturation)

Computationally: applying the enrichment functor at E₃ yields the same layer template (verified by saturation_checker). Equations

  • One or more equations did not get rendered due to their size. Instances For

Scope limitation (E3 collapse): At finite primorial level, the E3 predicate degenerates to E0 because reduce is idempotent. This check is vacuous but correctly models the mathematical structure. The E3 layer is correctly DEFINED but finite verification is vacuous. See audit DASHBOARD.md §E3 Collapse.

Tau.BookIII.Enrichment.saturation_e3_check

source def Tau.BookIII.Enrichment.saturation_e3_check (bound db : Denotation.TauIdx) :Bool

[III.T03] Saturation at E₃: E₄ = E₃. The enrichment ladder saturates at exactly four levels. The 4-orbit closure of ρ under ABCD decomposition means no fifth orbit exists. Equations

  • Tau.BookIII.Enrichment.saturation_e3_check bound db = (Tau.BookIII.Enrichment.functor_collapse_check bound db && Tau.BookIII.Enrichment.saturation_e3_check.go db 1 (db + 1)) Instances For

Tau.BookIII.Enrichment.saturation_e3_check.go

source@[irreducible]

**def Tau.BookIII.Enrichment.saturation_e3_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Enrichment.saturation_e3_check.check_coverage

source@[irreducible]

def Tau.BookIII.Enrichment.saturation_e3_check.check_coverage (k x pk fuel : ℕ) :Bool

Equations

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Tau.BookIII.Enrichment.canonical_ladder_check

source def Tau.BookIII.Enrichment.canonical_ladder_check (bound db : Denotation.TauIdx) :Bool

[III.T04] The Canonical Ladder Theorem: (i) Non-empty at each level (ii) Strictly increasing (iii) Saturating at E₃ (iv) Unique maximal enrichment chain

This is the organising result of Book III and the architectural blueprint for the entire 7-book series. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Enrichment.full_canonical_ladder

source def Tau.BookIII.Enrichment.full_canonical_ladder (bound db : Denotation.TauIdx) :Bool

[III.T04] Full canonical ladder verification: the master check combining all components of the Canonical Ladder Theorem. Equations

  • Tau.BookIII.Enrichment.full_canonical_ladder bound db = (Tau.BookIII.Enrichment.canonical_ladder_check bound db && Tau.BookIII.Enrichment.ladder_checker bound db) Instances For

Tau.BookIII.Enrichment.non_emptiness_8_3

source theorem Tau.BookIII.Enrichment.non_emptiness_8_3 :non_emptiness_check 8 3 = true

Tau.BookIII.Enrichment.e1_strictness_8_3

source theorem Tau.BookIII.Enrichment.e1_strictness_8_3 :e1_strictness_witness 8 3 = true

Tau.BookIII.Enrichment.strictness_8_3

source theorem Tau.BookIII.Enrichment.strictness_8_3 :strictness_check 8 3 = true

Tau.BookIII.Enrichment.functor_collapse_8_3

source theorem Tau.BookIII.Enrichment.functor_collapse_8_3 :functor_collapse_check 8 3 = true

Tau.BookIII.Enrichment.saturation_e3_8_3

source theorem Tau.BookIII.Enrichment.saturation_e3_8_3 :saturation_e3_check 8 3 = true

Tau.BookIII.Enrichment.canonical_ladder_8_3

source theorem Tau.BookIII.Enrichment.canonical_ladder_8_3 :canonical_ladder_check 8 3 = true

Tau.BookIII.Enrichment.full_canonical_ladder_8_3

source theorem Tau.BookIII.Enrichment.full_canonical_ladder_8_3 :full_canonical_ladder 8 3 = true

Tau.BookIII.Enrichment.zero_in_carrier

source theorem Tau.BookIII.Enrichment.zero_in_carrier (k : Denotation.TauIdx) :Polarity.reduce 0 k = 0

[III.T01] Structural non-emptiness: reduce(0, k) = 0 provides a witness at every stage. Zero is always in the carrier.

Tau.BookIII.Enrichment.hom_stage_stable

source theorem Tau.BookIII.Enrichment.hom_stage_stable (a b k : Denotation.TauIdx) :Polarity.reduce (BookII.Enrichment.hom_stage a b k) k = BookII.Enrichment.hom_stage a b k

[III.T02] Structural strictness at E₀→E₁: the hom_stage value is reduce-stable, witnessing genuine E₁ structure.

Tau.BookIII.Enrichment.structural_saturation

source theorem Tau.BookIII.Enrichment.structural_saturation :EnrLevel.E3.succ = EnrLevel.E3

[III.T03] Structural saturation: E₃.succ = E₃. The enrichment functor is idempotent at E₃.

Tau.BookIII.Enrichment.ladder_has_four_levels

source theorem Tau.BookIII.Enrichment.ladder_has_four_levels :[EnrLevel.E0, EnrLevel.E1, EnrLevel.E2, EnrLevel.E3].length = 4

[III.T04] Structural uniqueness: three applications of succ from any starting level reach E₃. The ladder has exactly 4 levels.

Tau.BookIII.Enrichment.canonical_ordering

source theorem Tau.BookIII.Enrichment.canonical_ordering :EnrLevel.E0.toNat < EnrLevel.E1.toNat ∧ EnrLevel.E1.toNat < EnrLevel.E2.toNat ∧ EnrLevel.E2.toNat < EnrLevel.E3.toNat

[III.T04] Canonical ordering: E₀ < E₁ < E₂ < E₃ via toNat.