TauLib · API Book III

TauLib.BookIII.Bridge.Incompleteness

TauLib.BookIII.Bridge.Incompleteness

Incompleteness as VM Boundary: Godel I/II and the Halting Problem as E2->E3 boundary phenomena.

Registry Cross-References

  • [III.T44] Incompleteness as VM Boundary – incompleteness_vm_check

Mathematical Content

III.T44 (Incompleteness as VM Boundary): Godel incompleteness IS the E2->E3 boundary: self-reference requires stepping outside E2. The self- referential construction (diagonalization) exceeds any fixed primorial depth.

At E2, the code IS a tau-address. The code can reference other codes (operational closure). But the code cannot reference ITSELF-AS-A-WHOLE without stepping outside E2 into E3 (self-modeling).

The finite-level model: at any fixed primorial depth k, the Godel sentence “this sentence is unprovable” cannot be consistently encoded because the diagonal function d(x) = x applied to itself requires evaluating the code at depth > k. This is the same mechanism as E3 saturation.

Three incompleteness phenomena diagnosed:

  • Godel I: “there exists a true unprovable sentence” = E2 self-reference hits E3 wall

  • Godel II: “consistency cannot be proved internally” = consistency is a host-level property (III.D70)

  • Halting Problem: “no program can decide halting for all programs” = operational closure cannot see its own totality

Tau.BookIII.Bridge.diagonal

source def Tau.BookIII.Bridge.diagonal (x k : Denotation.TauIdx) :Denotation.TauIdx

The diagonal function at primorial depth k: d(x) applies x to itself. At E2, this is: (x + x) mod Prim(k) (self-application via modular arithmetic). The key point: d(d(x)) may differ from d(x) at depth k but agree at depth k-1, showing the boundary phenomenon. Equations

  • Tau.BookIII.Bridge.diagonal x k = if (Tau.Polarity.primorial k == 0) = true then 0 else (x + x) % Tau.Polarity.primorial k Instances For

Tau.BookIII.Bridge.godel_sentence

source def Tau.BookIII.Bridge.godel_sentence (k : Denotation.TauIdx) :Denotation.TauIdx

The “Godel sentence” at depth k: the fixed point of negation composed with diagonal. G(k) = the code x such that d(x) = neg(x) mod Prim(k). We model negation as: neg(x) = Prim(k) - 1 - x. Equations

  • Tau.BookIII.Bridge.godel_sentence k = if (Tau.Polarity.primorial k == 0) = true then 0 else (Tau.Polarity.primorial k - 1) / 3 Instances For

Tau.BookIII.Bridge.e2_e3_boundary_check

source def Tau.BookIII.Bridge.e2_e3_boundary_check (bound db : Denotation.TauIdx) :Bool

[III.T44] E2->E3 boundary detection: at each depth k, determine whether the diagonal self-reference “escapes” E2.

The escape criterion: diagonal(diagonal(x, k), k) != diagonal(x, k) for the Godel sentence. This shows that self-reference applied twice (the analog of “this sentence talks about itself talking about itself”) moves to a different residue class – the E3 phenomenon.

But: at the NEXT depth k+1, the two values may agree (the E3 level “sees” both). This is the boundary. Equations

  • Tau.BookIII.Bridge.e2_e3_boundary_check bound db = Tau.BookIII.Bridge.e2_e3_boundary_check.go bound db 0 1 ((bound + 1) * (db + 1)) Instances For

Tau.BookIII.Bridge.e2_e3_boundary_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.e2_e3_boundary_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Bridge.godel_i_check

source def Tau.BookIII.Bridge.godel_i_check (db : Denotation.TauIdx) :Bool

[III.T44] Godel I finite model: the Godel sentence at depth k is a valid code, but its “truth” (reduce-stability) and its “provability” (diagonal self-reference) can diverge. Equations

  • Tau.BookIII.Bridge.godel_i_check db = Tau.BookIII.Bridge.godel_i_check.go db 1 (db + 1) Instances For

Tau.BookIII.Bridge.godel_i_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.godel_i_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.Bridge.godel_ii_check

source def Tau.BookIII.Bridge.godel_ii_check (bound db : Denotation.TauIdx) :Bool

[III.T44] Godel II finite model: consistency is a host-level property. At depth k, “the system is consistent” means “no code crashes reduce”. This is checkable at depth k (finite), but the UNIVERSAL statement “consistent for all k” requires seeing the entire tower (E3). Equations

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

Tau.BookIII.Bridge.godel_ii_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.godel_ii_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Bridge.halting_finite_check

source def Tau.BookIII.Bridge.halting_finite_check (bound db : Denotation.TauIdx) :Bool

[III.T44] Halting problem finite model: at depth k, every computation halts (finite state space guarantees termination via pigeonhole). The halting problem arises because “halts for ALL k” is a host-level property requiring E3. Equations

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

Tau.BookIII.Bridge.halting_finite_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.halting_finite_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Bridge.incompleteness_vm_check

source def Tau.BookIII.Bridge.incompleteness_vm_check (bound db : Denotation.TauIdx) :Bool

[III.T44] Full incompleteness-as-VM-boundary check: combines all three incompleteness phenomena as E2->E3 boundary manifestations. Equations

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

Tau.BookIII.Bridge.IncompletePhenomenon

source inductive Tau.BookIII.Bridge.IncompletePhenomenon :Type

[III.T44] The three incompleteness phenomena are distinct readings of the same structural boundary.

  • godel_i : IncompletePhenomenon
  • godel_ii : IncompletePhenomenon
  • halting : IncompletePhenomenon Instances For

Tau.BookIII.Bridge.instReprIncompletePhenomenon.repr

source def Tau.BookIII.Bridge.instReprIncompletePhenomenon.repr :IncompletePhenomenon → ℕ → Std.Format

Equations

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

Tau.BookIII.Bridge.instReprIncompletePhenomenon

source instance Tau.BookIII.Bridge.instReprIncompletePhenomenon :Repr IncompletePhenomenon

Equations

  • Tau.BookIII.Bridge.instReprIncompletePhenomenon = { reprPrec := Tau.BookIII.Bridge.instReprIncompletePhenomenon.repr }

Tau.BookIII.Bridge.instDecidableEqIncompletePhenomenon

source instance Tau.BookIII.Bridge.instDecidableEqIncompletePhenomenon :DecidableEq IncompletePhenomenon

Equations

  • Tau.BookIII.Bridge.instDecidableEqIncompletePhenomenon x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIII.Bridge.instBEqIncompletePhenomenon

source instance Tau.BookIII.Bridge.instBEqIncompletePhenomenon :BEq IncompletePhenomenon

Equations

  • Tau.BookIII.Bridge.instBEqIncompletePhenomenon = { beq := Tau.BookIII.Bridge.instBEqIncompletePhenomenon.beq }

Tau.BookIII.Bridge.instBEqIncompletePhenomenon.beq

source def Tau.BookIII.Bridge.instBEqIncompletePhenomenon.beq :IncompletePhenomenon → IncompletePhenomenon → Bool

Equations

  • Tau.BookIII.Bridge.instBEqIncompletePhenomenon.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIII.Bridge.phenomenon_level

source def Tau.BookIII.Bridge.phenomenon_level :IncompletePhenomenon → Enrichment.EnrLevel

[III.T44] All three phenomena require E3 (self-modeling). Equations

  • Tau.BookIII.Bridge.phenomenon_level Tau.BookIII.Bridge.IncompletePhenomenon.godel_i = Tau.BookIII.Enrichment.EnrLevel.E3
  • Tau.BookIII.Bridge.phenomenon_level Tau.BookIII.Bridge.IncompletePhenomenon.godel_ii = Tau.BookIII.Enrichment.EnrLevel.E3
  • Tau.BookIII.Bridge.phenomenon_level Tau.BookIII.Bridge.IncompletePhenomenon.halting = Tau.BookIII.Enrichment.EnrLevel.E3 Instances For

Tau.BookIII.Bridge.phenomenon_source

source def Tau.BookIII.Bridge.phenomenon_source :IncompletePhenomenon → Enrichment.EnrLevel

[III.T44] All three phenomena are caused by E2->E3 boundary crossing. Equations

  • Tau.BookIII.Bridge.phenomenon_source Tau.BookIII.Bridge.IncompletePhenomenon.godel_i = Tau.BookIII.Enrichment.EnrLevel.E2
  • Tau.BookIII.Bridge.phenomenon_source Tau.BookIII.Bridge.IncompletePhenomenon.godel_ii = Tau.BookIII.Enrichment.EnrLevel.E2
  • Tau.BookIII.Bridge.phenomenon_source Tau.BookIII.Bridge.IncompletePhenomenon.halting = Tau.BookIII.Enrichment.EnrLevel.E2 Instances For

Tau.BookIII.Bridge.incompleteness_vm_10_3

source theorem Tau.BookIII.Bridge.incompleteness_vm_10_3 :incompleteness_vm_check 10 3 = true

Tau.BookIII.Bridge.e2_e3_boundary_10_3

source theorem Tau.BookIII.Bridge.e2_e3_boundary_10_3 :e2_e3_boundary_check 10 3 = true

Tau.BookIII.Bridge.godel_i_4

source theorem Tau.BookIII.Bridge.godel_i_4 :godel_i_check 4 = true

Tau.BookIII.Bridge.godel_ii_10_3

source theorem Tau.BookIII.Bridge.godel_ii_10_3 :godel_ii_check 10 3 = true

Tau.BookIII.Bridge.halting_10_3

source theorem Tau.BookIII.Bridge.halting_10_3 :halting_finite_check 10 3 = true

Tau.BookIII.Bridge.diagonal_depth_0

source theorem Tau.BookIII.Bridge.diagonal_depth_0 :diagonal 42 0 = 0

[III.T44] Structural: diagonal at depth 0 is 0 (Prim(0)=1).

Tau.BookIII.Bridge.diagonal_mod

source theorem Tau.BookIII.Bridge.diagonal_mod :diagonal 7 3 = 14

[III.T44] Structural: diagonal at depth 3 is modular.

Tau.BookIII.Bridge.godel_at_3

source theorem Tau.BookIII.Bridge.godel_at_3 :godel_sentence 3 = 9

[III.T44] Structural: Godel sentence at depth 3 is 9.

Tau.BookIII.Bridge.all_at_e3

source theorem Tau.BookIII.Bridge.all_at_e3 :phenomenon_level IncompletePhenomenon.godel_i = Enrichment.EnrLevel.E3 ∧ phenomenon_level IncompletePhenomenon.godel_ii = Enrichment.EnrLevel.E3 ∧ phenomenon_level IncompletePhenomenon.halting = Enrichment.EnrLevel.E3

[III.T44] Structural: all three phenomena require E3.

Tau.BookIII.Bridge.all_from_e2

source theorem Tau.BookIII.Bridge.all_from_e2 :phenomenon_source IncompletePhenomenon.godel_i = Enrichment.EnrLevel.E2 ∧ phenomenon_source IncompletePhenomenon.godel_ii = Enrichment.EnrLevel.E2 ∧ phenomenon_source IncompletePhenomenon.halting = Enrichment.EnrLevel.E2

[III.T44] Structural: all three phenomena originate at E2.

Tau.BookIII.Bridge.e2_lt_e3_boundary

source theorem Tau.BookIII.Bridge.e2_lt_e3_boundary :Enrichment.EnrLevel.E2.lt Enrichment.EnrLevel.E3 = true

[III.T44] Structural: E2 < E3 (the boundary is genuine).

Tau.BookIII.Bridge.e3_resolves_incompleteness

source theorem Tau.BookIII.Bridge.e3_resolves_incompleteness :Enrichment.EnrLevel.E3.toNat = 3

[III.T44] Structural: E3 is the resolution level for incompleteness.