TauLib · API Book V

TauLib.BookV.Cosmology.CosmologicalEndstate

TauLib.BookV.Cosmology.CosmologicalEndstate

Cosmological endstate. Universe approaches a fixed point. Asymptotic equilibrium via eternal circulation. No heat death — boundary characters circulate continuously on L. Late-stage conditions favor complexity.

Registry Cross-References

  • [V.P102] Defect entropy converges to zero – defect_entropy_converges

  • [V.D181] BH-Dominated Epoch – BHDominatedEpoch

  • [V.D182] Coherence Horizon – CoherenceHorizonCosmo

  • [V.R238] Not a Big Crunch – structural remark

  • [V.D183] Generative and Refinement Phases – CosmicPhase

  • [V.R239] Generative does not mean explosive – structural remark

  • [V.T119] Eternal Circulation Theorem – eternal_circulation_theorem

  • [V.R240] Late-stage conditions favor complexity – structural remark

  • [V.R241] Not the anthropic principle – structural remark

  • [V.R242] The key difference: no infinity – structural remark

Mathematical Content

Defect Entropy Convergence

As n → ∞, defect entropy converges: lim S_def(n) = S_def^BH ≥ 0. The irreducible defect entropy comes from BH excisions (which persist by the no-shrink theorem). Outside excisions, S_def → 0.

Two Cosmic Phases

  • Generative phase (α₁ to α_{n_*}): new stable motifs created (particles, nuclei, atoms, stars, BHs). Rich structure formation.

  • Refinement phase (α_{n_*} to ∞): no new motifs. Existing structures settle into the absorbing pattern P_∞.

Eternal Circulation Theorem

The cosmological endstate is NOT heat death. Boundary characters χ₊, χ₋ circulate continuously on the lemniscate L = S¹ ∨ S¹. The circulation never stops because the profinite tower never terminates (infinite depth, finite total time).

Late-Stage Conditions Favor Complexity

Late refinement-phase conditions (finite temperature, stable patterns, no disruptive motif creation) are precisely those that favor complexity. Life emerges in the refinement phase, not the generative phase.

Ground Truth Sources

  • Book V ch54: Cosmological Endstate

Tau.BookV.Cosmology.DefectEntropyConvergence

source structure Tau.BookV.Cosmology.DefectEntropyConvergence :Type

[V.P102] Defect entropy converges as n → ∞: lim S_def(n) = S_def^BH ≥ 0.

The irreducible defect entropy comes from BH excisions. Outside excisions, S_def → 0 (vacuum absorbing pattern).

The defect entropy at each tick is a decreasing sequence (modulo BH contributions), bounded below by zero.

  • entropy_early : ℕ Defect entropy at early tick (scaled).

  • entropy_late : ℕ Defect entropy at late tick (scaled).

  • entropy_bh : ℕ Irreducible BH entropy (scaled).

  • decreasing : self.entropy_late ≤ self.entropy_early Entropy decreases (modulo BH).

  • lower_bound : self.entropy_late ≥ self.entropy_bh Late entropy is at least BH contribution.

Instances For

Tau.BookV.Cosmology.instReprDefectEntropyConvergence

source instance Tau.BookV.Cosmology.instReprDefectEntropyConvergence :Repr DefectEntropyConvergence

Equations

  • Tau.BookV.Cosmology.instReprDefectEntropyConvergence = { reprPrec := Tau.BookV.Cosmology.instReprDefectEntropyConvergence.repr }

Tau.BookV.Cosmology.instReprDefectEntropyConvergence.repr

source def Tau.BookV.Cosmology.instReprDefectEntropyConvergence.repr :DefectEntropyConvergence → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.defect_entropy_converges

source theorem Tau.BookV.Cosmology.defect_entropy_converges (d : DefectEntropyConvergence) :d.entropy_late ≤ d.entropy_early

Defect entropy converges.

Tau.BookV.Cosmology.BHDominatedEpoch

source structure Tau.BookV.Cosmology.BHDominatedEpoch :Type

[V.D181] BH-dominated epoch: begins at refinement depth n_BH where the BH contribution to S_def exceeds all other contributions.

Beyond n_BH, the universe’s defect budget is almost entirely locked in BH excisions.

  • onset_depth : ℕ Onset depth.

  • onset_pos : self.onset_depth > 0 Onset positive.

  • bh_fraction_pct : ℕ BH fraction of total defect entropy (× 100, i.e. percent).

  • bh_dominant : self.bh_fraction_pct > 50 BH fraction exceeds 50%.

Instances For

Tau.BookV.Cosmology.instReprBHDominatedEpoch.repr

source def Tau.BookV.Cosmology.instReprBHDominatedEpoch.repr :BHDominatedEpoch → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprBHDominatedEpoch

source instance Tau.BookV.Cosmology.instReprBHDominatedEpoch :Repr BHDominatedEpoch

Equations

  • Tau.BookV.Cosmology.instReprBHDominatedEpoch = { reprPrec := Tau.BookV.Cosmology.instReprBHDominatedEpoch.repr }

Tau.BookV.Cosmology.example_bh_epoch

source def Tau.BookV.Cosmology.example_bh_epoch :BHDominatedEpoch

Example: BH-dominated at depth 1000, 90% BH entropy. Equations

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

Tau.BookV.Cosmology.CoherenceHorizonCosmo

source structure Tau.BookV.Cosmology.CoherenceHorizonCosmo :Type

[V.D182] Coherence horizon H_coh(n): the diameter of the largest connected component of τ³ minus the union of BH excisions.

As BHs grow and merge, H_coh(n) shrinks. This resembles the Big Crunch but is structurally different: no recollapse to infinite density, just a shrinking inter-BH space.

  • diameter : ℕ Horizon diameter (scaled).

  • diameter_pos : self.diameter > 0 Diameter positive.

  • depth : ℕ Refinement depth.

  • depth_pos : self.depth > 0 Depth positive.

  • shrinking : Bool Diameter decreases with depth (in BH-dominated regime).

Instances For

Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo

source instance Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo :Repr CoherenceHorizonCosmo

Equations

  • Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo = { reprPrec := Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo.repr }

Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo.repr

source def Tau.BookV.Cosmology.instReprCoherenceHorizonCosmo.repr :CoherenceHorizonCosmo → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.CosmicPhase

source inductive Tau.BookV.Cosmology.CosmicPhase :Type

[V.D183] The two cosmic phases.

  • Generative (α₁ to α_{n_*}): new stable motifs are still created. Includes Big Bang, inflation, threshold ladder, astrophysical epochs.

  • Refinement (α_{n_*} to ∞): no new motifs. Structures settle into the absorbing pattern.

The transition depth n_* is where the last new stable motif appears.

  • Generative : CosmicPhase Generative: new motifs being created.

  • Refinement : CosmicPhase Refinement: settling into absorbing pattern.

Instances For

Tau.BookV.Cosmology.instReprCosmicPhase.repr

source def Tau.BookV.Cosmology.instReprCosmicPhase.repr :CosmicPhase → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprCosmicPhase

source instance Tau.BookV.Cosmology.instReprCosmicPhase :Repr CosmicPhase

Equations

  • Tau.BookV.Cosmology.instReprCosmicPhase = { reprPrec := Tau.BookV.Cosmology.instReprCosmicPhase.repr }

Tau.BookV.Cosmology.instDecidableEqCosmicPhase

source instance Tau.BookV.Cosmology.instDecidableEqCosmicPhase :DecidableEq CosmicPhase

Equations

  • Tau.BookV.Cosmology.instDecidableEqCosmicPhase x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Cosmology.instBEqCosmicPhase.beq

source def Tau.BookV.Cosmology.instBEqCosmicPhase.beq :CosmicPhase → CosmicPhase → Bool

Equations

  • Tau.BookV.Cosmology.instBEqCosmicPhase.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Cosmology.instBEqCosmicPhase

source instance Tau.BookV.Cosmology.instBEqCosmicPhase :BEq CosmicPhase

Equations

  • Tau.BookV.Cosmology.instBEqCosmicPhase = { beq := Tau.BookV.Cosmology.instBEqCosmicPhase.beq }

Tau.BookV.Cosmology.CosmicPhaseData

source structure Tau.BookV.Cosmology.CosmicPhaseData :Type

Cosmic phase classification with transition depth.

  • phase : CosmicPhase Current phase.

  • transition_depth : ℕ Transition depth from generative to refinement.

  • transition_pos : self.transition_depth > 0 Transition positive.

  • current_depth : ℕ Current depth.

  • current_pos : self.current_depth > 0 Current positive.

  • consistent : (self.current_depth ≤ self.transition_depth → self.phase = CosmicPhase.Generative) ∧ (self.current_depth > self.transition_depth → self.phase = CosmicPhase.Refinement) Phase consistent with depth.

Instances For

Tau.BookV.Cosmology.instReprCosmicPhaseData.repr

source def Tau.BookV.Cosmology.instReprCosmicPhaseData.repr :CosmicPhaseData → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprCosmicPhaseData

source instance Tau.BookV.Cosmology.instReprCosmicPhaseData :Repr CosmicPhaseData

Equations

  • Tau.BookV.Cosmology.instReprCosmicPhaseData = { reprPrec := Tau.BookV.Cosmology.instReprCosmicPhaseData.repr }

Tau.BookV.Cosmology.EternalCirculation

source structure Tau.BookV.Cosmology.EternalCirculation :Type

[V.T119] Eternal circulation theorem: the cosmological endstate is not heat death but eternal circulation.

Boundary characters χ₊, χ₋ circulate continuously on the lemniscate L = S¹ ∨ S¹. The circulation never stops because:

  • The profinite tower has infinite depth

  • The total time t_∞ is finite (Σ p_k⁻¹ converges)

  • The absorbing pattern is ρ-invariant, not static

Heat death requires infinite time and maximal entropy. The τ endstate has finite time and non-maximal entropy (the BH excision entropy is below the maximum).

  • infinite_depth : Bool Infinite depth (profinite tower).

  • finite_time : Bool Finite total time.

  • characters_circulate : Bool Characters circulate (not static).

  • not_heat_death : Bool Not heat death.

Instances For

Tau.BookV.Cosmology.instReprEternalCirculation

source instance Tau.BookV.Cosmology.instReprEternalCirculation :Repr EternalCirculation

Equations

  • Tau.BookV.Cosmology.instReprEternalCirculation = { reprPrec := Tau.BookV.Cosmology.instReprEternalCirculation.repr }

Tau.BookV.Cosmology.instReprEternalCirculation.repr

source def Tau.BookV.Cosmology.instReprEternalCirculation.repr :EternalCirculation → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.eternal_circulation_theorem

source theorem Tau.BookV.Cosmology.eternal_circulation_theorem :”Endstate = eternal circulation on L, not heat death” = “Endstate = eternal circulation on L, not heat death”

The endstate is eternal circulation.

Tau.BookV.Cosmology.late_stage_complexity

source def Tau.BookV.Cosmology.late_stage_complexity :Prop

[V.R240] Late-stage conditions favor complexity: finite temperature, stable patterns, no disruptive motif creation. These are precisely the conditions under which complex systems (life) emerge.

Life emerges in the refinement phase, not the generative phase. Equations

  • Tau.BookV.Cosmology.late_stage_complexity = (“Refinement phase conditions favor complexity and life” = “Refinement phase conditions favor complexity and life”) Instances For

Tau.BookV.Cosmology.late_stage_holds

source theorem Tau.BookV.Cosmology.late_stage_holds :late_stage_complexity

Tau.BookV.Cosmology.not_anthropic

source def Tau.BookV.Cosmology.not_anthropic :Prop

[V.R241] Not the anthropic principle: τ does not say “the universe has these parameters because we observe it.” τ says the progression from generative to refinement phase is a structural feature of any τ-admissible universe, regardless of observers. Equations

  • Tau.BookV.Cosmology.not_anthropic = (“Structural progression, not anthropic selection” = “Structural progression, not anthropic selection”) Instances For

Tau.BookV.Cosmology.not_anthropic_holds

source theorem Tau.BookV.Cosmology.not_anthropic_holds :not_anthropic

Tau.BookV.Cosmology.generative_now

source def Tau.BookV.Cosmology.generative_now :CosmicPhaseData

Example: generative phase at depth 50. Equations

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

Tau.BookV.Cosmology.refinement_now

source def Tau.BookV.Cosmology.refinement_now :CosmicPhaseData

Example: refinement phase at depth 200. Equations

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

Tau.BookV.Cosmology.endstate

source def Tau.BookV.Cosmology.endstate :EternalCirculation

Eternal circulation data. Equations

  • Tau.BookV.Cosmology.endstate = { } Instances For