TauLib · API Book VI

TauLib.BookVI.CosmicLife.CrossLimit

TauLib.BookVI.CosmicLife.CrossLimit

Crossing-Limit Theorem: merger-directed net converges to ι_τ = 2/(π+e). Includes ω-representative, Lift_ω constructor, primorial ladder convergence, fusion convergence, and universal BH.

Registry Cross-References

  • [VI.D60] ω-Representative of Life — OmegaRepresentative

  • [VI.D61] Lift_ω Constructor — LiftOmegaConstructor

  • [VI.L11] Primorial Ladder Convergence — primorial_convergence

  • [VI.T31] Fusion Convergence — fusion_convergence

  • [VI.T35] Crossing-Limit Theorem — crossing_limit_theorem

  • [VI.T36] Universal BH = Fully Alive — universal_bh_alive

Book V Authority

  • [V.D171] Blueprint Fusion: Fuse_ω operator

  • [V.D172] Blueprint Monoid: M_BH has no inverses

  • [V.T112] Blueprint Monoid Closure: monoid is closed under merger

  • [V.T116] Finite Motif Theorem: cofinal sequence existence

  • [V.T117] Saturation Radius Theorem: colimit existence

Ground Truth Sources

  • Book VI Chapters 45, 49 (2nd Edition): ω-Representatives, Crossing Limit

Tau.BookVI.CrossLimit.OmegaRepresentative

source structure Tau.BookVI.CrossLimit.OmegaRepresentative :Type

[VI.D60] ω-Representative: carrier at boundary of code space. Three conditions: code dominance, boundary saturation, crossing faithfulness. BHs are the unique physical ω-representatives.

  • condition_count : ℕ Number of defining conditions.

  • count_eq : self.condition_count = 3 Exactly 3 conditions.

  • code_dominance : Bool Code dominance: ω-germ determines basin.

  • boundary_saturation : Bool Boundary saturation: maximal information density.

  • crossing_faithful : Bool Crossing faithfulness: evaluator factors through ω.

Instances For

Tau.BookVI.CrossLimit.instReprOmegaRepresentative.repr

source def Tau.BookVI.CrossLimit.instReprOmegaRepresentative.repr :OmegaRepresentative → ℕ → Std.Format

Equations

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

Tau.BookVI.CrossLimit.instReprOmegaRepresentative

source instance Tau.BookVI.CrossLimit.instReprOmegaRepresentative :Repr OmegaRepresentative

Equations

  • Tau.BookVI.CrossLimit.instReprOmegaRepresentative = { reprPrec := Tau.BookVI.CrossLimit.instReprOmegaRepresentative.repr }

Tau.BookVI.CrossLimit.omega_rep

source def Tau.BookVI.CrossLimit.omega_rep :OmegaRepresentative

Equations

  • Tau.BookVI.CrossLimit.omega_rep = { condition_count := 3, count_eq := Tau.BookVI.CrossLimit.omega_rep._proof_1 } Instances For

Tau.BookVI.CrossLimit.LiftOmegaConstructor

source structure Tau.BookVI.CrossLimit.LiftOmegaConstructor :Type

[VI.D61] Lift_ω constructor: recursive builder from bipolar seed through primorial ladder P_k = 2, 6, 30, 210, 2310, … Converges superexponentially to ι_τ.

  • recursive : Bool Recursive construction.

  • primorial_ladder : Bool Uses primorial ladder.

  • converges_to_iota : Bool Converges to ι_τ = 2/(π+e).

  • superexponential : Bool Convergence rate is superexponential.

  • iota_irrational : Bool Well-definedness requires ι_τ irrational.

Instances For

Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor

source instance Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor :Repr LiftOmegaConstructor

Equations

  • Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor = { reprPrec := Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor.repr }

Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor.repr

source def Tau.BookVI.CrossLimit.instReprLiftOmegaConstructor.repr :LiftOmegaConstructor → ℕ → Std.Format

Equations

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

Tau.BookVI.CrossLimit.lift_omega

source def Tau.BookVI.CrossLimit.lift_omega :LiftOmegaConstructor

Equations

  • Tau.BookVI.CrossLimit.lift_omega = { } Instances For

Tau.BookVI.CrossLimit.primorial_approx

source def Tau.BookVI.CrossLimit.primorial_approx :List (ℕ × ℕ)

First few primorial approximations to ι_τ. P_0=2: c_0/P_0 = 1/2 = 0.500 P_1=6: c_1/P_1 = 2/6 = 0.333 P_3=210: c_3/P_3 = 72/210 = 0.342857 P_4=2310: c_4/P_4 = 789/2310 = 0.341558 Equations

  • Tau.BookVI.CrossLimit.primorial_approx = [(1, 2), (2, 6), (10, 30), (72, 210), (789, 2310)] Instances For

Tau.BookVI.CrossLimit.primorial_stage4_numer

source theorem Tau.BookVI.CrossLimit.primorial_stage4_numer :primorial_approx[4]!.1 = 789

Primorial approximation at stage 4 (c₄=789, P₄=2310). 789/2310 ≈ 0.341558, within 10⁻⁴ of ι_τ.

Tau.BookVI.CrossLimit.primorial_stage4_denom

source theorem Tau.BookVI.CrossLimit.primorial_stage4_denom :primorial_approx[4]!.2 = 2310

Tau.BookVI.CrossLimit.primorial_convergence

source theorem Tau.BookVI.CrossLimit.primorial_convergence :lift_omega.superexponential = true ∧ lift_omega.converges_to_iota = true ∧ lift_omega.iota_irrational = true

[VI.L11] Primorial ladder converges superexponentially to ι_τ. Error bound: |c_k/P_k - ι_τ| ≤ 1/(2·p_{k+1}). Coherence: c_{k+1} ≡ c_k (mod P_k) for all k.

Tau.BookVI.CrossLimit.FusionConvergence

source structure Tau.BookVI.CrossLimit.FusionConvergence :Type

[VI.T31] Fusion Convergence: BH merger monotonically converges codes. (i) Monotone: d_k(code_f) ≤ max{d_k(code_1), d_k(code_2)} (ii) Strict improvement for distinct codes at ∞-many levels (iii) Limit: merger net → ι_τ Authority: V.D171 (Blueprint Fusion), V.T112 (Monoid Closure).

  • monotone : Bool Fusion never increases ι_τ-distance.

  • strict_improvement : Bool Distinct codes yield strict improvement.

  • converges_to_iota : Bool Net converges to ι_τ.

  • no_inverses : Bool Blueprint monoid has no inverses (irreversibility).

Instances For

Tau.BookVI.CrossLimit.instReprFusionConvergence

source instance Tau.BookVI.CrossLimit.instReprFusionConvergence :Repr FusionConvergence

Equations

  • Tau.BookVI.CrossLimit.instReprFusionConvergence = { reprPrec := Tau.BookVI.CrossLimit.instReprFusionConvergence.repr }

Tau.BookVI.CrossLimit.instReprFusionConvergence.repr

source def Tau.BookVI.CrossLimit.instReprFusionConvergence.repr :FusionConvergence → ℕ → Std.Format

Equations

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

Tau.BookVI.CrossLimit.fusion_conv

source def Tau.BookVI.CrossLimit.fusion_conv :FusionConvergence

Equations

  • Tau.BookVI.CrossLimit.fusion_conv = { } Instances For

Tau.BookVI.CrossLimit.fusion_convergence

source theorem Tau.BookVI.CrossLimit.fusion_convergence :fusion_conv.monotone = true ∧ fusion_conv.strict_improvement = true ∧ fusion_conv.converges_to_iota = true

Tau.BookVI.CrossLimit.CrossingLimitTheorem

source structure Tau.BookVI.CrossLimit.CrossingLimitTheorem :Type

[VI.T35] Crossing-Limit Theorem: merger-directed net → ι_τ. Three-step proof: (1) monotonicity from VI.T31, (2) strict improvement from primorial ladder, (3) standard net convergence. Cofinal sequence authority: V.T116 (Finite Motif), V.T117 (Saturation Radius).

  • target : String Target value is ι_τ = 2/(π+e).

  • monotone_fusion : Bool Monotone fusion (from VI.T31).

  • strictly_contracting : Bool Strictly contracting along primorial ladder.

  • maximal_aliveness : Bool Convergence to maximal aliveness.

  • cofinal_from_bookV : Bool Cofinal sequence via V.T116 + V.T117.

Instances For

Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem.repr

source def Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem.repr :CrossingLimitTheorem → ℕ → Std.Format

Equations

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Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem

source instance Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem :Repr CrossingLimitTheorem

Equations

  • Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem = { reprPrec := Tau.BookVI.CrossLimit.instReprCrossingLimitTheorem.repr }

Tau.BookVI.CrossLimit.crossing_limit

source def Tau.BookVI.CrossLimit.crossing_limit :CrossingLimitTheorem

Equations

  • Tau.BookVI.CrossLimit.crossing_limit = { } Instances For

Tau.BookVI.CrossLimit.crossing_limit_theorem

source theorem Tau.BookVI.CrossLimit.crossing_limit_theorem :crossing_limit.monotone_fusion = true ∧ crossing_limit.strictly_contracting = true ∧ crossing_limit.maximal_aliveness = true ∧ crossing_limit.cofinal_from_bookV = true

Tau.BookVI.CrossLimit.UniversalBH

source structure Tau.BookVI.CrossLimit.UniversalBH :Type

[VI.T36] Universal BH: colimit of merger net. (i) code = ι_τ exactly (ii) All defect functionals vanish (iii) 7/7 hallmarks at terminal values Colimit existence: V.T117 (Saturation Radius Theorem).

  • code_is_iota : Bool ω-germ code equals ι_τ exactly.

  • all_defects_zero : Bool All defect functionals (frame + strong) vanish.

  • hallmark_count : ℕ All 7 hallmarks satisfied at terminal values.

  • count_eq : self.hallmark_count = 7 Exactly 7 hallmarks.

Instances For

Tau.BookVI.CrossLimit.instReprUniversalBH

source instance Tau.BookVI.CrossLimit.instReprUniversalBH :Repr UniversalBH

Equations

  • Tau.BookVI.CrossLimit.instReprUniversalBH = { reprPrec := Tau.BookVI.CrossLimit.instReprUniversalBH.repr }

Tau.BookVI.CrossLimit.instReprUniversalBH.repr

source def Tau.BookVI.CrossLimit.instReprUniversalBH.repr :UniversalBH → ℕ → Std.Format

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

Tau.BookVI.CrossLimit.universal_bh

source def Tau.BookVI.CrossLimit.universal_bh :UniversalBH

Equations

  • Tau.BookVI.CrossLimit.universal_bh = { hallmark_count := 7, count_eq := Tau.BookVI.CrossLimit.universal_bh._proof_1 } Instances For

Tau.BookVI.CrossLimit.universal_bh_alive

source theorem Tau.BookVI.CrossLimit.universal_bh_alive :universal_bh.code_is_iota = true ∧ universal_bh.all_defects_zero = true ∧ universal_bh.hallmark_count = 7