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TauLib.BookVI.Persistence.TemporalLemniscate

TauLib.BookVI.Persistence.TemporalLemniscate

Temporal lemniscate L_T, circadian rhythms, and homochirality.

Registry Cross-References

  • [VI.D27] Temporal Lemniscate L_T — TemporalLemniscate

  • [VI.D28] Homochirality — Homochirality

  • [VI.T17] Circadian Rhythm as Poincaré Orbit — circadian_poincare_orbit

  • [VI.P09] 24-Hour Cycle as τ¹ Rotation — circadian_tau1_rotation

  • [VI.P10] L-Amino Acid Preference as Parity Shadow — homochirality_parity_shadow

Cross-Book Authority

  • Book II, Part III: Lemniscate L = S¹ ∨ S¹ construction

  • Book III, Part II: Poincaré force (periodic orbits, limit cycles)

  • Book IV, IV.T146: σ = C_τ (all neutrinos Majorana)

  • Book IV, IV.T160: θ_QCD = 0 (strong CP solved)

Ground Truth Sources

  • Book VI Chapter 15 (2nd Edition): Circadian Rhythms

  • Book VI Chapter 16 (2nd Edition): Homochirality

Tau.BookVI.TempLem.TemporalLemniscate

source structure Tau.BookVI.TempLem.TemporalLemniscate :Type

[VI.D27] Temporal Lemniscate L_T = S¹_act ∨ S¹_rest. The persistence Life loop projected onto τ¹ traces a figure-eight: active phase (S¹_act) and rest phase (S¹_rest). Inherits lemniscate topology from L = S¹ ∨ S¹ (Book II, Part III).

  • lobe_count : ℕ Number of lobes.

  • lobes_eq : self.lobe_count = 2 Exactly 2 lobes.

  • active_lobe : String Active-phase lobe.

  • rest_lobe : String Rest-phase lobe.

  • winding_number : ℕ Winding number on τ¹.

Instances For

Tau.BookVI.TempLem.instReprTemporalLemniscate

source instance Tau.BookVI.TempLem.instReprTemporalLemniscate :Repr TemporalLemniscate

Equations

  • Tau.BookVI.TempLem.instReprTemporalLemniscate = { reprPrec := Tau.BookVI.TempLem.instReprTemporalLemniscate.repr }

Tau.BookVI.TempLem.instReprTemporalLemniscate.repr

source def Tau.BookVI.TempLem.instReprTemporalLemniscate.repr :TemporalLemniscate → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.temporal_lem

source def Tau.BookVI.TempLem.temporal_lem :TemporalLemniscate

Equations

  • Tau.BookVI.TempLem.temporal_lem = { lobe_count := 2, lobes_eq := Tau.BookVI.TempLem.temporal_lem._proof_1 } Instances For

Tau.BookVI.TempLem.temporal_lemniscate_two_lobes

source theorem Tau.BookVI.TempLem.temporal_lemniscate_two_lobes :temporal_lem.lobe_count = 2

Tau.BookVI.TempLem.temporal_lemniscate_winding_one

source theorem Tau.BookVI.TempLem.temporal_lemniscate_winding_one :temporal_lem.winding_number = 1

Tau.BookVI.TempLem.CircadianPoincare

source structure Tau.BookVI.TempLem.CircadianPoincare :Type

[VI.T17] Circadian Rhythm as Poincaré Orbit Theorem. The persistence Life loop projected onto τ¹ is a Poincaré limit cycle tracing L_T = S¹_act ∨ S¹_rest with period T ≈ 24h. Authority: Book III, Part II (Poincaré force ensures periodic orbits).

  • period_hours : ℕ Period in hours.

  • period_eq : self.period_hours = 24 Period ≈ 24 hours.

  • is_limit_cycle : Bool Is a Poincaré limit cycle.

  • traces_L_T : Bool Traces temporal lemniscate.

  • winding_alpha : ℕ Winding number w_α = 1 per cycle.

  • characteristics : ℕ Three characteristics: entrainable, temperature-compensated, free-running.

Instances For

Tau.BookVI.TempLem.instReprCircadianPoincare.repr

source def Tau.BookVI.TempLem.instReprCircadianPoincare.repr :CircadianPoincare → ℕ → Std.Format

Equations

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Tau.BookVI.TempLem.instReprCircadianPoincare

source instance Tau.BookVI.TempLem.instReprCircadianPoincare :Repr CircadianPoincare

Equations

  • Tau.BookVI.TempLem.instReprCircadianPoincare = { reprPrec := Tau.BookVI.TempLem.instReprCircadianPoincare.repr }

Tau.BookVI.TempLem.circadian

source def Tau.BookVI.TempLem.circadian :CircadianPoincare

Equations

  • Tau.BookVI.TempLem.circadian = { period_hours := 24, period_eq := Tau.BookVI.TempLem.circadian._proof_1 } Instances For

Tau.BookVI.TempLem.circadian_poincare_orbit

source theorem Tau.BookVI.TempLem.circadian_poincare_orbit :circadian.period_hours = 24 ∧ circadian.is_limit_cycle = true ∧ circadian.traces_L_T = true ∧ circadian.winding_alpha = 1

Tau.BookVI.TempLem.CircadianTau1

source structure Tau.BookVI.TempLem.CircadianTau1 :Type

[VI.P09] 24-Hour Cycle as τ¹ Rotation (conjectural). Molecular clock intrinsic period near 24h across all terrestrial life suggests a τ¹-derived timescale constraint.

  • scope : String Scope: conjectural.

  • tau1_locked : Bool Period locked to τ¹ rotation.

Instances For

Tau.BookVI.TempLem.instReprCircadianTau1

source instance Tau.BookVI.TempLem.instReprCircadianTau1 :Repr CircadianTau1

Equations

  • Tau.BookVI.TempLem.instReprCircadianTau1 = { reprPrec := Tau.BookVI.TempLem.instReprCircadianTau1.repr }

Tau.BookVI.TempLem.instReprCircadianTau1.repr

source def Tau.BookVI.TempLem.instReprCircadianTau1.repr :CircadianTau1 → ℕ → Std.Format

Equations

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Tau.BookVI.TempLem.circadian_tau1

source def Tau.BookVI.TempLem.circadian_tau1 :CircadianTau1

Equations

  • Tau.BookVI.TempLem.circadian_tau1 = { } Instances For

Tau.BookVI.TempLem.circadian_tau1_rotation

source theorem Tau.BookVI.TempLem.circadian_tau1_rotation :circadian_tau1.tau1_locked = true

Tau.BookVI.TempLem.Homochirality

source structure Tau.BookVI.TempLem.Homochirality :Type

[VI.D28] Homochirality: L-amino acids / D-sugars. Phenomenological shadow of the Parity Bridge (conjectural). The weak sector’s parity violation (IV.T146, IV.T160) seeds the biological chirality preference.

  • l_amino_acids : Bool L-amino acids preferred.

  • d_sugars : Bool D-sugars preferred.

  • parity_bridge_shadow : Bool Connected to Parity Bridge.

  • scope : String Scope: τ-effective (upgraded from conjectural via VI.R26 derivation chain).

Instances For

Tau.BookVI.TempLem.instReprHomochirality

source instance Tau.BookVI.TempLem.instReprHomochirality :Repr Homochirality

Equations

  • Tau.BookVI.TempLem.instReprHomochirality = { reprPrec := Tau.BookVI.TempLem.instReprHomochirality.repr }

Tau.BookVI.TempLem.instReprHomochirality.repr

source def Tau.BookVI.TempLem.instReprHomochirality.repr :Homochirality → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.HomochiralityParityShadow

source structure Tau.BookVI.TempLem.HomochiralityParityShadow :Type

[VI.P10] L-amino acid preference as Parity Shadow (conjectural). The weak sector’s chirality (IV.T146 σ=C_τ Majorana, IV.T160 θ_QCD=0) seeds the biological enantiomeric excess via the Parity Bridge.

  • iv_t146_majorana : Bool IV.T146: σ = C_τ, all neutrinos Majorana from self-adjointness.

  • iv_t160_strong_cp : Bool IV.T160: θ_QCD = 0, strong CP solved from SA-i mod-3.

  • temporal_protection : Bool Temporal stability protects chirality.

Instances For

Tau.BookVI.TempLem.instReprHomochiralityParityShadow

source instance Tau.BookVI.TempLem.instReprHomochiralityParityShadow :Repr HomochiralityParityShadow

Equations

  • Tau.BookVI.TempLem.instReprHomochiralityParityShadow = { reprPrec := Tau.BookVI.TempLem.instReprHomochiralityParityShadow.repr }

Tau.BookVI.TempLem.instReprHomochiralityParityShadow.repr

source def Tau.BookVI.TempLem.instReprHomochiralityParityShadow.repr :HomochiralityParityShadow → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.homo_parity

source def Tau.BookVI.TempLem.homo_parity :HomochiralityParityShadow

Equations

  • Tau.BookVI.TempLem.homo_parity = { } Instances For

Tau.BookVI.TempLem.homochirality_parity_shadow

source theorem Tau.BookVI.TempLem.homochirality_parity_shadow :homo_parity.iv_t146_majorana = true ∧ homo_parity.iv_t160_strong_cp = true ∧ homo_parity.temporal_protection = true

Tau.BookVI.TempLem.EnantiomericExcess

source structure Tau.BookVI.TempLem.EnantiomericExcess :Type

[VI.D73] Enantiomeric Excess at refinement level n. ee(n) = |[L] - [R]| / ([L] + [R]) measures chirality purity. Seeded by ChiralitySeed (VI.D72) at n=0 with ee ≈ 10⁻¹⁷, amplified by SelfDesc closure at each refinement level.

  • refinement_level : ℕ Refinement level (0 = initial seed).

  • converges_to_one : Bool ee converges to 1 (100% homochiral).

  • monotone : Bool Monotonically increasing with refinement.

  • seed_source : String Seeded by VI.D72 ChiralitySeed.

Instances For

Tau.BookVI.TempLem.instReprEnantiomericExcess

source instance Tau.BookVI.TempLem.instReprEnantiomericExcess :Repr EnantiomericExcess

Equations

  • Tau.BookVI.TempLem.instReprEnantiomericExcess = { reprPrec := Tau.BookVI.TempLem.instReprEnantiomericExcess.repr }

Tau.BookVI.TempLem.instReprEnantiomericExcess.repr

source def Tau.BookVI.TempLem.instReprEnantiomericExcess.repr :EnantiomericExcess → ℕ → Std.Format

Equations

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Tau.BookVI.TempLem.StereochemicalSelection

source structure Tau.BookVI.TempLem.StereochemicalSelection :Type

[VI.T42] Stereochemical Selection Theorem: SelfDesc closure (VI.T03) amplifies the chirality seed (VI.D72) to full enantiomeric excess. The polarity propagation (VI.D71) provides the initial asymmetry; SelfDesc closure drives ee(n) → 1 monotonically.

  • selfdesc_amplification : Bool SelfDesc closure amplifies chirality.

  • seed_from_parity_bridge : Bool Chirality seed source: VI.D72.

  • exponential_gain : Bool Amplification is exponential (gain g per level).

  • final_ee_is_one : Bool Result: ee = 1 at convergence.

Instances For

Tau.BookVI.TempLem.instReprStereochemicalSelection.repr

source def Tau.BookVI.TempLem.instReprStereochemicalSelection.repr :StereochemicalSelection → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.instReprStereochemicalSelection

source instance Tau.BookVI.TempLem.instReprStereochemicalSelection :Repr StereochemicalSelection

Equations

  • Tau.BookVI.TempLem.instReprStereochemicalSelection = { reprPrec := Tau.BookVI.TempLem.instReprStereochemicalSelection.repr }

Tau.BookVI.TempLem.stereochemical_sel

source def Tau.BookVI.TempLem.stereochemical_sel :StereochemicalSelection

Equations

  • Tau.BookVI.TempLem.stereochemical_sel = { } Instances For

Tau.BookVI.TempLem.stereochemical_selection

source theorem Tau.BookVI.TempLem.stereochemical_selection :stereochemical_sel.selfdesc_amplification = true ∧ stereochemical_sel.seed_from_parity_bridge = true ∧ stereochemical_sel.exponential_gain = true ∧ stereochemical_sel.final_ee_is_one = true

Tau.BookVI.TempLem.EeMonotoneConvergence

source structure Tau.BookVI.TempLem.EeMonotoneConvergence :Type

[VI.P21] ee(n) → 1 monotonically: enantiomeric excess increases at every refinement level and converges to 1. The double-well potential (Hodge stabilization) prevents regression, and Poincaré topological lock-in on L = S¹ ∨ S¹ provides additional protection beyond energetic barriers.

  • monotone_increasing : Bool Monotone increasing.

  • limit_is_one : Bool Converges to ee = 1.

  • hodge_stabilization : Bool Double-well barrier prevents regression.

  • poincare_lockin : Bool Topological lock-in on L.

Instances For

Tau.BookVI.TempLem.instReprEeMonotoneConvergence.repr

source def Tau.BookVI.TempLem.instReprEeMonotoneConvergence.repr :EeMonotoneConvergence → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.instReprEeMonotoneConvergence

source instance Tau.BookVI.TempLem.instReprEeMonotoneConvergence :Repr EeMonotoneConvergence

Equations

  • Tau.BookVI.TempLem.instReprEeMonotoneConvergence = { reprPrec := Tau.BookVI.TempLem.instReprEeMonotoneConvergence.repr }

Tau.BookVI.TempLem.ee_convergence

source def Tau.BookVI.TempLem.ee_convergence :EeMonotoneConvergence

Equations

  • Tau.BookVI.TempLem.ee_convergence = { } Instances For

Tau.BookVI.TempLem.ee_monotone_convergence

source theorem Tau.BookVI.TempLem.ee_monotone_convergence :ee_convergence.monotone_increasing = true ∧ ee_convergence.limit_is_one = true ∧ ee_convergence.hodge_stabilization = true ∧ ee_convergence.poincare_lockin = true

Tau.BookVI.TempLem.HomochiralityUniversality

source structure Tau.BookVI.TempLem.HomochiralityUniversality :Type

[VI.T43] Homochirality Universality: all persistence-sector entries inherit the same chirality from the unique polarity propagation path. Since the Parity Bridge (VI.T01) is the unique factorization and the chirality seed (VI.D72) has definite sign, every carrier satisfying Distinction + SelfDesc must exhibit the same enantiomeric preference.

  • universal_chirality : Bool All persistence-sector entries share same chirality.

  • from_unique_path : Bool Derived from unique propagation path (VI.L14).

  • applies_to_all_carriers : Bool Applies to all carriers satisfying Distinction + SelfDesc.

  • scope : String Scope: τ-effective (derived from Parity Bridge chain).

Instances For

Tau.BookVI.TempLem.instReprHomochiralityUniversality.repr

source def Tau.BookVI.TempLem.instReprHomochiralityUniversality.repr :HomochiralityUniversality → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.instReprHomochiralityUniversality

source instance Tau.BookVI.TempLem.instReprHomochiralityUniversality :Repr HomochiralityUniversality

Equations

  • Tau.BookVI.TempLem.instReprHomochiralityUniversality = { reprPrec := Tau.BookVI.TempLem.instReprHomochiralityUniversality.repr }

Tau.BookVI.TempLem.homochirality_universality_inst

source def Tau.BookVI.TempLem.homochirality_universality_inst :HomochiralityUniversality

Equations

  • Tau.BookVI.TempLem.homochirality_universality_inst = { } Instances For

Tau.BookVI.TempLem.homochirality_universality

source theorem Tau.BookVI.TempLem.homochirality_universality :homochirality_universality_inst.universal_chirality = true ∧ homochirality_universality_inst.from_unique_path = true ∧ homochirality_universality_inst.applies_to_all_carriers = true

Tau.BookVI.TempLem.HomochiralityScopeUpgrade

source structure Tau.BookVI.TempLem.HomochiralityScopeUpgrade :Type

[VI.R26] Homochirality Scope Upgrade: documents the complete derivation chain that upgrades homochirality from conjectural to τ-effective. Chain: K0-K6 → ι_τ → holonomy sectors → σ_A-admissibility (IV.D112) → σ = C_τ Majorana (IV.T146) → Parity Bridge (VI.T01) → Polarity Propagation (VI.D71) → Chirality Seed (VI.D72) → Propagation Preserves Chirality (VI.T41) → Stereochemical Selection (VI.T42) → ee → 1 (VI.P21) → Homochirality Universality (VI.T43). Every link is τ-effective; no conjectural step remains.

  • previous_scope : String Previous scope.

  • new_scope : String New scope.

  • chain_length : ℕ Derivation chain length.

  • chain_complete : self.chain_length = 12
  • op9_status : String VI.OP9 status upgrade.

Instances For

Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade

source instance Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade :Repr HomochiralityScopeUpgrade

Equations

  • Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade = { reprPrec := Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade.repr }

Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade.repr

source def Tau.BookVI.TempLem.instReprHomochiralityScopeUpgrade.repr :HomochiralityScopeUpgrade → ℕ → Std.Format

Equations

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

Tau.BookVI.TempLem.scope_upgrade

source def Tau.BookVI.TempLem.scope_upgrade :HomochiralityScopeUpgrade

Equations

  • Tau.BookVI.TempLem.scope_upgrade = { chain_length := 12, chain_complete := Tau.BookVI.TempLem.scope_upgrade._proof_1 } Instances For