TauLib · API Book VI

TauLib.BookVI.Persistence.PersistenceSector

TauLib.BookVI.Persistence.PersistenceSector

Persistence sector (Part 2): α-sector temporal stability along τ¹. Archetype: Archaea. Dominant forces: Poincaré (circulation) + Riemann (energy).

Registry Cross-References

  • [VI.D24] Persistence Sector — PersistenceSectorDef

  • [VI.D25] Temporal Stability Predicate — TemporalStabilityPredicate

  • [VI.T16] Persistence = α-Base Stability — persistence_is_alpha_stability

  • [VI.D26] Abiogenesis as First Persistence Event — AbiogenesisDef

  • [VI.P08] Thermodynamic Inevitability of Life — thermodynamic_inevitability

  • [VI.D74] Far-From-Equilibrium Regime — FarFromEquilibriumRegime

  • [VI.D75] Complexity Budget — ComplexityBudget

  • [VI.L15] Complexity Monotone — complexity_monotone

  • [VI.D76] Distinction Threshold — DistinctionThreshold

  • [VI.T44] Attractor Existence — attractor_existence

  • [VI.L16] Basin Is Absorbing — basin_is_absorbing

  • [VI.D77] Abiogenesis Timescale Bound — AbiogenesisTimescaleBound

  • [VI.T45] Timescale From Half-Life — timescale_from_half_life

  • [VI.R27] Timescale Geological Consistency — TimescaleGeologicalConsistency

  • [VI.T46] Abiogenesis Inevitability — abiogenesis_inevitability

  • [VI.R28] Abiogenesis Not Contingent — AbiogenesisNotContingent

Cross-Book Authority

  • Book I, Part I: α generator (radial, base circle τ¹)

  • Book III, Part II: Poincaré force (periodic orbits on τ³)

  • Book III, Part III: Riemann force (energy quantization)

Ground Truth Sources

  • Book VI Chapter 12 (2nd Edition): The Persistence Sector

  • Book VI Chapter 14 (2nd Edition): Thermodynamic Necessity

Tau.BookVI.Persistence.PersistenceSectorDef

source structure Tau.BookVI.Persistence.PersistenceSectorDef :Type

[VI.D24] Persistence Sector: α-sector on base circle τ¹. Life Loop restricted to base-temporal dynamics. Generator: α (radial, Book I Part I). Dominant forces: Poincaré + Riemann (Book III, Parts II–III).

  • generator : String Generator is alpha (base).

  • is_primitive : Bool Sector is primitive (single generator).

  • archetype : String Archetype organism.

  • dominant_poincare : Bool Dominant force 1: Poincaré (temporal orbits).

  • dominant_riemann : Bool Dominant force 2: Riemann (energy quanta).

Instances For

Tau.BookVI.Persistence.instReprPersistenceSectorDef

source instance Tau.BookVI.Persistence.instReprPersistenceSectorDef :Repr PersistenceSectorDef

Equations

  • Tau.BookVI.Persistence.instReprPersistenceSectorDef = { reprPrec := Tau.BookVI.Persistence.instReprPersistenceSectorDef.repr }

Tau.BookVI.Persistence.instReprPersistenceSectorDef.repr

source def Tau.BookVI.Persistence.instReprPersistenceSectorDef.repr :PersistenceSectorDef → ℕ → Std.Format

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Tau.BookVI.Persistence.persistence_def

source def Tau.BookVI.Persistence.persistence_def :PersistenceSectorDef

Equations

  • Tau.BookVI.Persistence.persistence_def = { } Instances For

Tau.BookVI.Persistence.persistence_generator_match

source theorem Tau.BookVI.Persistence.persistence_generator_match :persistence_def.generator = FourPlusOne.persistence_sector.generator

Persistence sector matches the FourPlusOne persistence_sector definition.

Tau.BookVI.Persistence.TemporalStabilityPredicate

source structure Tau.BookVI.Persistence.TemporalStabilityPredicate :Type

[VI.D25] Temporal Stability Predicate: 3 conditions for persistence. (i) Defect-functional norm bounded over τ¹ period (ii) α-flow orbit returns to ε-neighborhood (iii) Refinement tower eventually constant on base

  • condition_count : ℕ Number of conditions.

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

  • defect_bounded : Bool (i) Defect-norm bounded.

  • alpha_flow_returns : Bool (ii) α-flow returns (Poincaré recurrence on τ¹).

  • refinement_constant : Bool (iii) Refinement eventually constant.

Instances For

Tau.BookVI.Persistence.instReprTemporalStabilityPredicate

source instance Tau.BookVI.Persistence.instReprTemporalStabilityPredicate :Repr TemporalStabilityPredicate

Equations

  • Tau.BookVI.Persistence.instReprTemporalStabilityPredicate = { reprPrec := Tau.BookVI.Persistence.instReprTemporalStabilityPredicate.repr }

Tau.BookVI.Persistence.instReprTemporalStabilityPredicate.repr

source def Tau.BookVI.Persistence.instReprTemporalStabilityPredicate.repr :TemporalStabilityPredicate → ℕ → Std.Format

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Tau.BookVI.Persistence.temporal_stability

source def Tau.BookVI.Persistence.temporal_stability :TemporalStabilityPredicate

Equations

  • Tau.BookVI.Persistence.temporal_stability = { condition_count := 3, count_eq := Tau.BookVI.Persistence.temporal_stability._proof_1 } Instances For

Tau.BookVI.Persistence.temporal_stability_three_conditions

source theorem Tau.BookVI.Persistence.temporal_stability_three_conditions :temporal_stability.condition_count = 3

Tau.BookVI.Persistence.temporal_stability_all_hold

source theorem Tau.BookVI.Persistence.temporal_stability_all_hold :temporal_stability.defect_bounded = true ∧ temporal_stability.alpha_flow_returns = true ∧ temporal_stability.refinement_constant = true

Tau.BookVI.Persistence.PersistenceStability

source structure Tau.BookVI.Persistence.PersistenceStability :Type

[VI.T16] Persistence = α-Base Stability Theorem. A Life loop restricted to τ¹ base with winding number w_α = 1 satisfies the Temporal Stability Predicate iff it is a persistence-sector Life loop.

  • winding_alpha : ℕ Winding number on base.

  • winding_eq : self.winding_alpha = 1 Winding is exactly 1.

  • temporal_stable : Bool Temporal stability holds.

  • sector_persistence : Bool Sector assignment is persistence.

Instances For

Tau.BookVI.Persistence.instReprPersistenceStability

source instance Tau.BookVI.Persistence.instReprPersistenceStability :Repr PersistenceStability

Equations

  • Tau.BookVI.Persistence.instReprPersistenceStability = { reprPrec := Tau.BookVI.Persistence.instReprPersistenceStability.repr }

Tau.BookVI.Persistence.instReprPersistenceStability.repr

source def Tau.BookVI.Persistence.instReprPersistenceStability.repr :PersistenceStability → ℕ → Std.Format

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Tau.BookVI.Persistence.pers_stability

source def Tau.BookVI.Persistence.pers_stability :PersistenceStability

Equations

  • Tau.BookVI.Persistence.pers_stability = { winding_alpha := 1, winding_eq := Tau.BookVI.Persistence.pers_stability._proof_1 } Instances For

Tau.BookVI.Persistence.persistence_is_alpha_stability

source theorem Tau.BookVI.Persistence.persistence_is_alpha_stability :pers_stability.winding_alpha = 1 ∧ pers_stability.temporal_stable = true ∧ pers_stability.sector_persistence = true

Tau.BookVI.Persistence.AbiogenesisDef

source structure Tau.BookVI.Persistence.AbiogenesisDef :Type

[VI.D26] Abiogenesis: first entry into the persistence sector. The transition from non-Life to Life (E₁ → E₂).

  • first_sector : String First sector entered.

  • transition_type : String Transition type: E₁ → E₂.

  • scope : String Scope: τ-effective (upgraded from conjectural via VI.T46 derivation chain).

Instances For

Tau.BookVI.Persistence.instReprAbiogenesisDef.repr

source def Tau.BookVI.Persistence.instReprAbiogenesisDef.repr :AbiogenesisDef → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprAbiogenesisDef

source instance Tau.BookVI.Persistence.instReprAbiogenesisDef :Repr AbiogenesisDef

Equations

  • Tau.BookVI.Persistence.instReprAbiogenesisDef = { reprPrec := Tau.BookVI.Persistence.instReprAbiogenesisDef.repr }

Tau.BookVI.Persistence.ThermodynamicInevitability

source structure Tau.BookVI.Persistence.ThermodynamicInevitability :Type

[VI.P08] Thermodynamic Inevitability of Life (τ-effective). Life is a thermodynamic attractor with positive-measure basin. Three-step argument: entropy production → SelfDesc attractor → speed of abiogenesis. Upgraded from conjectural via VI.T46 derivation chain: K0–K6 → V.T60 → V.T62 → VI.D75 → VI.L15 → VI.T44 → VI.L16 → VI.T46.

  • argument_steps : ℕ Number of argument steps.

  • steps_eq : self.argument_steps = 3 Exactly 3 steps.

  • entropy_maximization : Bool (i) Entropy production maximization.

  • selfdesc_attractor : Bool (ii) SelfDesc as thermodynamic attractor.

  • rapid_abiogenesis : Bool (iii) Speed of abiogenesis (~500 Myr).

  • scope : String Scope: τ-effective (upgraded via VI.T46 derivation chain).

Instances For

Tau.BookVI.Persistence.instReprThermodynamicInevitability.repr

source def Tau.BookVI.Persistence.instReprThermodynamicInevitability.repr :ThermodynamicInevitability → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprThermodynamicInevitability

source instance Tau.BookVI.Persistence.instReprThermodynamicInevitability :Repr ThermodynamicInevitability

Equations

  • Tau.BookVI.Persistence.instReprThermodynamicInevitability = { reprPrec := Tau.BookVI.Persistence.instReprThermodynamicInevitability.repr }

Tau.BookVI.Persistence.thermo_inev

source def Tau.BookVI.Persistence.thermo_inev :ThermodynamicInevitability

Equations

  • Tau.BookVI.Persistence.thermo_inev = { argument_steps := 3, steps_eq := Tau.BookVI.Persistence.temporal_stability._proof_1 } Instances For

Tau.BookVI.Persistence.thermodynamic_inevitability

source theorem Tau.BookVI.Persistence.thermodynamic_inevitability :thermo_inev.argument_steps = 3 ∧ thermo_inev.entropy_maximization = true ∧ thermo_inev.selfdesc_attractor = true ∧ thermo_inev.rapid_abiogenesis = true

Tau.BookVI.Persistence.FarFromEquilibriumRegime

source structure Tau.BookVI.Persistence.FarFromEquilibriumRegime :Type

[VI.D74] Far-From-Equilibrium Regime: pre-E₂ state where |D_n| » 0. A system in active defect decay, before coherence horizon (V.D89), with sustained dissipative energy throughput. Cross-ref: V.T62 (geometric decay), V.D89 (coherence horizon).

  • defect_above_zero : Bool Defect count significantly above zero.

  • dissipative : Bool System is dissipative (sustained energy throughput).

  • pre_coherence_horizon : Bool Pre-coherence-horizon: defect decay still active.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime.repr

source def Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime.repr :FarFromEquilibriumRegime → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime

source instance Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime :Repr FarFromEquilibriumRegime

Equations

  • Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime = { reprPrec := Tau.BookVI.Persistence.instReprFarFromEquilibriumRegime.repr }

Tau.BookVI.Persistence.far_from_equilibrium

source def Tau.BookVI.Persistence.far_from_equilibrium :FarFromEquilibriumRegime

Equations

  • Tau.BookVI.Persistence.far_from_equilibrium = { } Instances For

Tau.BookVI.Persistence.far_from_equilibrium_conditions

source theorem Tau.BookVI.Persistence.far_from_equilibrium_conditions :far_from_equilibrium.defect_above_zero = true ∧ far_from_equilibrium.dissipative = true ∧ far_from_equilibrium.pre_coherence_horizon = true

Tau.BookVI.Persistence.ComplexityBudget

source structure Tau.BookVI.Persistence.ComplexityBudget :Type

[VI.D75] Complexity Budget: C(n) = N − |D_n|, dual of defect count. As defects decay geometrically (V.T62), freed capacity increases monotonically, providing structural resources for complex configurations. Cross-ref: V.T60 (finite budget), V.T62 (geometric decay).

  • initial_defects : ℕ Initial defect count N (finite by V.T60).

  • freed_capacity_monotone : Bool Freed capacity increases monotonically as defects decay.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprComplexityBudget.repr

source def Tau.BookVI.Persistence.instReprComplexityBudget.repr :ComplexityBudget → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprComplexityBudget

source instance Tau.BookVI.Persistence.instReprComplexityBudget :Repr ComplexityBudget

Equations

  • Tau.BookVI.Persistence.instReprComplexityBudget = { reprPrec := Tau.BookVI.Persistence.instReprComplexityBudget.repr }

Tau.BookVI.Persistence.ComplexityMonotone

source structure Tau.BookVI.Persistence.ComplexityMonotone :Type

[VI.L15] Complexity Monotone Lemma: C(n) ≤ C(n+1). Defects decrease geometrically (V.T62), so freed capacity C(n) = N − |D_n| increases monotonically. Proof: |D_{n+1}| ≤ (1−ι_τ)|D_n| < |D_n| ⟹ C(n+1) > C(n).

  • decay_rate_factor : String Defect decay rate per step: (1−ι_τ).

  • monotone_increasing : Bool C(n) ≤ C(n+1) for all n.

  • derived_from_geometric_decay : Bool Derived from V.T62 geometric decay.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprComplexityMonotone

source instance Tau.BookVI.Persistence.instReprComplexityMonotone :Repr ComplexityMonotone

Equations

  • Tau.BookVI.Persistence.instReprComplexityMonotone = { reprPrec := Tau.BookVI.Persistence.instReprComplexityMonotone.repr }

Tau.BookVI.Persistence.instReprComplexityMonotone.repr

source def Tau.BookVI.Persistence.instReprComplexityMonotone.repr :ComplexityMonotone → ℕ → Std.Format

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Tau.BookVI.Persistence.complexity_monotone_def

source def Tau.BookVI.Persistence.complexity_monotone_def :ComplexityMonotone

Equations

  • Tau.BookVI.Persistence.complexity_monotone_def = { } Instances For

Tau.BookVI.Persistence.complexity_monotone

source theorem Tau.BookVI.Persistence.complexity_monotone :complexity_monotone_def.monotone_increasing = true ∧ complexity_monotone_def.derived_from_geometric_decay = true

Tau.BookVI.Persistence.DistinctionThreshold

source structure Tau.BookVI.Persistence.DistinctionThreshold :Type

[VI.D76] Distinction Threshold: minimum complexity for life. Distinction requires 5 conditions, SelfDesc requires 3 conditions, giving threshold = 8. When C(n) ≥ 8, the system has sufficient freed capacity to instantiate Distinction + SelfDesc simultaneously.

  • threshold_conditions : ℕ Total threshold conditions.

  • threshold_eq : self.threshold_conditions = 8 Threshold is exactly 8.

  • distinction_count : ℕ Distinction contributes 5 conditions.

  • selfdesc_count : ℕ SelfDesc contributes 3 conditions.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprDistinctionThreshold.repr

source def Tau.BookVI.Persistence.instReprDistinctionThreshold.repr :DistinctionThreshold → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprDistinctionThreshold

source instance Tau.BookVI.Persistence.instReprDistinctionThreshold :Repr DistinctionThreshold

Equations

  • Tau.BookVI.Persistence.instReprDistinctionThreshold = { reprPrec := Tau.BookVI.Persistence.instReprDistinctionThreshold.repr }

Tau.BookVI.Persistence.distinction_threshold

source def Tau.BookVI.Persistence.distinction_threshold :DistinctionThreshold

Equations

  • Tau.BookVI.Persistence.distinction_threshold = { threshold_conditions := 8, threshold_eq := Tau.BookVI.Persistence.distinction_threshold._proof_1 } Instances For

Tau.BookVI.Persistence.threshold_is_sum

source theorem Tau.BookVI.Persistence.threshold_is_sum :distinction_threshold.distinction_count + distinction_threshold.selfdesc_count = distinction_threshold.threshold_conditions

Tau.BookVI.Persistence.AttractorExistence

source structure Tau.BookVI.Persistence.AttractorExistence :Type

[VI.T44] Attractor Existence Theorem: under 3 conditions, Distinction+SelfDesc basin entry is forced. C1: finite defect budget (V.T60) C2: polarity seed (VI.T01) C3: temporal stability (VI.D25) Proof: C(n) increases monotonically (VI.L15), threshold is finite (VI.D76), so ∃ n₀: C(n₀) ≥ threshold; SelfDesc closure (VI.T03) makes basin absorbing.

  • condition_count : ℕ Number of structural conditions.

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

  • finite_budget : Bool C1: Finite defect budget (V.T60).

  • polarity_seed : Bool C2: Polarity seed exists (VI.T01).

  • temporal_stability : Bool C3: Temporal stability predicate satisfiable (VI.D25).

  • entry_forced : Bool Basin entry is forced (threshold crossing guaranteed).

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprAttractorExistence

source instance Tau.BookVI.Persistence.instReprAttractorExistence :Repr AttractorExistence

Equations

  • Tau.BookVI.Persistence.instReprAttractorExistence = { reprPrec := Tau.BookVI.Persistence.instReprAttractorExistence.repr }

Tau.BookVI.Persistence.instReprAttractorExistence.repr

source def Tau.BookVI.Persistence.instReprAttractorExistence.repr :AttractorExistence → ℕ → Std.Format

Equations

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

Tau.BookVI.Persistence.attractor_existence_def

source def Tau.BookVI.Persistence.attractor_existence_def :AttractorExistence

Equations

  • Tau.BookVI.Persistence.attractor_existence_def = { condition_count := 3, conditions_eq := Tau.BookVI.Persistence.temporal_stability._proof_1 } Instances For

Tau.BookVI.Persistence.attractor_existence

source theorem Tau.BookVI.Persistence.attractor_existence :attractor_existence_def.condition_count = 3 ∧ attractor_existence_def.finite_budget = true ∧ attractor_existence_def.polarity_seed = true ∧ attractor_existence_def.temporal_stability = true ∧ attractor_existence_def.entry_forced = true

Tau.BookVI.Persistence.BasinAbsorbing

source structure Tau.BookVI.Persistence.BasinAbsorbing :Type

[VI.L16] Basin Is Absorbing Lemma: once entered, the system stays. SelfDesc closure (VI.T03) provides an internal evaluator that actively reconstructs the distinction after perturbation, making the basin absorbing. Proof: SelfDesc evaluator reads code and returns system to basin (VI.T03).

  • selfdesc_closure : Bool SelfDesc closure guarantees return.

  • absorbing : Bool Basin is absorbing (no escape under bounded perturbation).

  • derived_from_selfdesc : Bool Derived from VI.T03.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprBasinAbsorbing.repr

source def Tau.BookVI.Persistence.instReprBasinAbsorbing.repr :BasinAbsorbing → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprBasinAbsorbing

source instance Tau.BookVI.Persistence.instReprBasinAbsorbing :Repr BasinAbsorbing

Equations

  • Tau.BookVI.Persistence.instReprBasinAbsorbing = { reprPrec := Tau.BookVI.Persistence.instReprBasinAbsorbing.repr }

Tau.BookVI.Persistence.basin_absorbing_def

source def Tau.BookVI.Persistence.basin_absorbing_def :BasinAbsorbing

Equations

  • Tau.BookVI.Persistence.basin_absorbing_def = { } Instances For

Tau.BookVI.Persistence.basin_is_absorbing

source theorem Tau.BookVI.Persistence.basin_is_absorbing :basin_absorbing_def.selfdesc_closure = true ∧ basin_absorbing_def.absorbing = true ∧ basin_absorbing_def.derived_from_selfdesc = true

Tau.BookVI.Persistence.AbiogenesisTimescaleBound

source structure Tau.BookVI.Persistence.AbiogenesisTimescaleBound :Type

[VI.D77] Abiogenesis Timescale Bound: upper bound in orbit steps. T_abio ≤ n₁/₂ · ⌈ln(N/threshold)⌉ where n₁/₂ ≈ 1.66 (V.D90). Geometric decay with half-life n₁/₂ gives time to reach threshold from initial N defects. Cross-ref: V.D90 (defect half-life n₁/₂ ≈ 1.66).

  • half_life_steps : ℕ Half-life in orbit steps (scaled: 166 = 1.66 × 100).

  • threshold : ℕ Threshold conditions to cross.

  • derived_from_half_life : Bool Bound is derived from half-life.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound

source instance Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound :Repr AbiogenesisTimescaleBound

Equations

  • Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound = { reprPrec := Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound.repr }

Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound.repr

source def Tau.BookVI.Persistence.instReprAbiogenesisTimescaleBound.repr :AbiogenesisTimescaleBound → ℕ → Std.Format

Equations

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

Tau.BookVI.Persistence.TimescaleFromHalfLife

source structure Tau.BookVI.Persistence.TimescaleFromHalfLife :Type

[VI.T45] Timescale From Half-Life Theorem: T_abio ≤ n₁/₂ · ⌈ln(N/threshold)⌉. Proof: geometric decay rate (1−ι_τ)^n with half-life n₁/₂ gives |D_n| = N·(1−ι_τ)^n. Threshold crossing at C(n₀) = N − |D_{n₀}| ≥ 8 requires |D_{n₀}| ≤ N − 8, giving n₀ ≤ n₁/₂ · ⌈log₂(N/8)⌉.

  • decay_factor : String Decay factor per orbit step.

  • half_life : String Half-life n₁/₂ ≈ 1.66 orbit steps (V.D90).

  • logarithmic_bound : Bool Upper bound is logarithmic in initial defect count.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprTimescaleFromHalfLife.repr

source def Tau.BookVI.Persistence.instReprTimescaleFromHalfLife.repr :TimescaleFromHalfLife → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprTimescaleFromHalfLife

source instance Tau.BookVI.Persistence.instReprTimescaleFromHalfLife :Repr TimescaleFromHalfLife

Equations

  • Tau.BookVI.Persistence.instReprTimescaleFromHalfLife = { reprPrec := Tau.BookVI.Persistence.instReprTimescaleFromHalfLife.repr }

Tau.BookVI.Persistence.timescale_half_life_def

source def Tau.BookVI.Persistence.timescale_half_life_def :TimescaleFromHalfLife

Equations

  • Tau.BookVI.Persistence.timescale_half_life_def = { } Instances For

Tau.BookVI.Persistence.timescale_from_half_life

source theorem Tau.BookVI.Persistence.timescale_from_half_life :timescale_half_life_def.logarithmic_bound = true

Tau.BookVI.Persistence.TimescaleGeologicalConsistency

source structure Tau.BookVI.Persistence.TimescaleGeologicalConsistency :Type

[VI.R27] Timescale Geological Consistency: orbit-step → physical-time mapping gives ~500 Myr, consistent with geological evidence (3.8–4.1 Gya). The logarithmic bound (VI.T45) with characteristic step time ~10⁻¹³ s and 10¹⁵–10²¹ correlated steps gives τ_origin 10²–10⁸ years. Scope note: structural bound (τ-effective), physical mapping (remark).

  • geological_window_myr : String Geological window: ~100–300 Myr.

  • predicted_bound_myr : String Predicted bound: ~500 Myr.

  • consistent : Bool Consistent with observation.

  • scope : String Scope: remark (supporting).

Instances For

Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency.repr

source def Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency.repr :TimescaleGeologicalConsistency → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency

source instance Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency :Repr TimescaleGeologicalConsistency

Equations

  • Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency = { reprPrec := Tau.BookVI.Persistence.instReprTimescaleGeologicalConsistency.repr }

Tau.BookVI.Persistence.AbiogenesisInevitability

source structure Tau.BookVI.Persistence.AbiogenesisInevitability :Type

[VI.T46] Abiogenesis Inevitability Theorem: first persistence-sector entry is structurally inevitable. Derivation chain: K0–K6 → defect budget (V.T60) → exhaustion (V.T62) → complexity budget (VI.D75) → monotone increase (VI.L15) → threshold crossing (VI.T44) → absorbing basin (VI.L16) → first entry inevitable within timescale bound (VI.T45). This combines attractor existence + timescale bound + SelfDesc closure to establish that abiogenesis is not contingent but structurally forced.

  • chain_length : ℕ Number of links in derivation chain.

  • chain_eq : self.chain_length = 7 Chain has 7 links.

  • attractor_exists : Bool Attractor existence established (VI.T44).

  • timescale_bounded : Bool Timescale is bounded (VI.T45).

  • basin_absorbing : Bool Basin is absorbing (VI.L16).

  • inevitable : Bool Conclusion: first entry inevitable.

  • scope : String Scope: τ-effective.

Instances For

Tau.BookVI.Persistence.instReprAbiogenesisInevitability.repr

source def Tau.BookVI.Persistence.instReprAbiogenesisInevitability.repr :AbiogenesisInevitability → ℕ → Std.Format

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Tau.BookVI.Persistence.instReprAbiogenesisInevitability

source instance Tau.BookVI.Persistence.instReprAbiogenesisInevitability :Repr AbiogenesisInevitability

Equations

  • Tau.BookVI.Persistence.instReprAbiogenesisInevitability = { reprPrec := Tau.BookVI.Persistence.instReprAbiogenesisInevitability.repr }

Tau.BookVI.Persistence.abiogenesis_inevitability_def

source def Tau.BookVI.Persistence.abiogenesis_inevitability_def :AbiogenesisInevitability

Equations

  • Tau.BookVI.Persistence.abiogenesis_inevitability_def = { chain_length := 7, chain_eq := Tau.BookVI.Persistence.abiogenesis_inevitability_def._proof_1 } Instances For

Tau.BookVI.Persistence.abiogenesis_inevitability

source theorem Tau.BookVI.Persistence.abiogenesis_inevitability :abiogenesis_inevitability_def.chain_length = 7 ∧ abiogenesis_inevitability_def.attractor_exists = true ∧ abiogenesis_inevitability_def.timescale_bounded = true ∧ abiogenesis_inevitability_def.basin_absorbing = true ∧ abiogenesis_inevitability_def.inevitable = true

Tau.BookVI.Persistence.AbiogenesisNotContingent

source structure Tau.BookVI.Persistence.AbiogenesisNotContingent :Type

[VI.R28] Abiogenesis Not Contingent: philosophical consequence. Life is not an accident requiring explanation but an inevitable structural feature of any τ-governed universe with sustained energy gradient. The derivation chain (VI.T46) shows that from K0–K6 alone, given dissipative conditions, persistence-sector entry is forced.

  • not_contingent : Bool Life is structurally necessary, not contingent.

  • requires_energy_gradient : Bool Requires sustained energy gradient.

  • derived_from_inevitability : Bool Follows from VI.T46.

  • scope : String Scope: τ-effective (remark).

Instances For

Tau.BookVI.Persistence.instReprAbiogenesisNotContingent

source instance Tau.BookVI.Persistence.instReprAbiogenesisNotContingent :Repr AbiogenesisNotContingent

Equations

  • Tau.BookVI.Persistence.instReprAbiogenesisNotContingent = { reprPrec := Tau.BookVI.Persistence.instReprAbiogenesisNotContingent.repr }

Tau.BookVI.Persistence.instReprAbiogenesisNotContingent.repr

source def Tau.BookVI.Persistence.instReprAbiogenesisNotContingent.repr :AbiogenesisNotContingent → ℕ → Std.Format

Equations

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