TauLib · API Book V

TauLib.BookV.Thermodynamics.Inversion

TauLib.BookV.Thermodynamics.Inversion

The Categorical Second Law: classical second-law inversion. The arrow of time is structural (alpha-orbit on base tau^1), not thermodynamic. Holomorphic entropy vs defect entropy. Gravity-driven defect absorption.

Registry Cross-References

  • [V.T55] The Categorical Second Law – CategoricalSecondLaw

  • [V.D83] Thermodynamic Equilibrium (categorical) – CategoricalEquilibrium

  • [V.D84] Coherence Horizon – ThermalCoherenceHorizon

  • [V.P24] Defect Absorption Rate – DefectAbsorptionRate

  • [V.P25] Weak Redistribution Preserves Defect Count – WeakRedistribution

  • [V.P26] The 180-degree Inversion – inversion_180

  • [V.L02] Geometric Contraction of Defect Support – GeometricContraction

  • [V.C05] Defect Support Exhaustion – defect_support_exhaustion

  • [V.R111] The Explanatory Gap – structural remark

  • [V.R112] Pixel-Resolution Analogy – pixel_analogy

  • [V.R113] Compatibility with Book IV – structural remark

  • [V.R114] Not the Same as Thermal Equilibrium – structural remark

  • [V.R115] Role of Gravity in Ordering – structural remark

  • [V.R116] Contraction Rate is Gravitational Coupling – contraction_is_kappa_D

  • [V.R117] Circulation Not Stasis – structural remark

  • [V.R118] Orbit Steps vs Physical Time – OrbitStepsVsTime

Mathematical Content

The Categorical Second Law

Along the alpha-orbit on base tau^1, defect entropy is monotonically non-increasing: dS_def/d(alpha-orbit) <= 0. The count of structurally non-trivial holomorphic obstructions can only decrease.

Defect Absorption

The gravitational self-coupling kappa(D;1) = 1 - iota_tau controls the contraction rate: |supp(d_{n+1})| <= (1 - iota_tau) |supp(d_n)|.

The 180-degree Inversion

Classical Boltzmann: dS_class/dt >= 0 (entropy increases). Categorical: dS_def/dn <= 0 (defect entropy decreases). The two are exactly opposite under t <-> n identification.

Ground Truth Sources

  • Book V ch21: second-law inversion

  • kappa_n_closing_identity_sprint.md: gravitational ordering

Tau.BookV.Thermodynamics.contraction_numer

source def Tau.BookV.Thermodynamics.contraction_numer :ℕ

Gravitational contraction factor numerator: 1 - iota_tau. kappa(D;1) = 1 - iota_tau = 658541/1000000. This is the rate at which defect support contracts per orbit step. Equations

  • Tau.BookV.Thermodynamics.contraction_numer = Tau.Boundary.iota_tau_denom - Tau.Boundary.iota_tau_numer Instances For

Tau.BookV.Thermodynamics.contraction_denom

source def Tau.BookV.Thermodynamics.contraction_denom :ℕ

Contraction factor denominator. Equations

  • Tau.BookV.Thermodynamics.contraction_denom = Tau.Boundary.iota_tau_denom Instances For

Tau.BookV.Thermodynamics.contraction_pos

source theorem Tau.BookV.Thermodynamics.contraction_pos :contraction_numer > 0

The contraction factor is positive: 1 - iota_tau > 0.

Tau.BookV.Thermodynamics.contraction_lt_one

source theorem Tau.BookV.Thermodynamics.contraction_lt_one :contraction_numer < contraction_denom

The contraction factor is less than 1 (strict contraction).

Tau.BookV.Thermodynamics.CategoricalSecondLaw

source structure Tau.BookV.Thermodynamics.CategoricalSecondLaw :Type

[V.T55] The Categorical Second Law.

Along the alpha-orbit on base tau^1, defect entropy is monotonically non-increasing. The contraction factor is (1 - iota_tau) = kappa(D;1), the gravitational self-coupling.

This inverts the classical second law: classical entropy increases, but defect entropy (the physically meaningful component) decreases.

  • contraction_factor_numer : ℕ Contraction factor numerator (1 - iota_tau).

  • contraction_factor_denom : ℕ Contraction factor denominator.

  • denom_pos : self.contraction_factor_denom > 0 Denominator positive.

  • strict_contraction : self.contraction_factor_numer < self.contraction_factor_denom The contraction factor is strictly less than 1.

  • scope : String Scope: tau-effective.

Instances For

Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw

source instance Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw :Repr CategoricalSecondLaw

Equations

  • Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw = { reprPrec := Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw.repr }

Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw.repr

source def Tau.BookV.Thermodynamics.instReprCategoricalSecondLaw.repr :CategoricalSecondLaw → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.categorical_second_law

source def Tau.BookV.Thermodynamics.categorical_second_law :CategoricalSecondLaw

The canonical Categorical Second Law instance. Equations

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

Tau.BookV.Thermodynamics.CategoricalEquilibrium

source structure Tau.BookV.Thermodynamics.CategoricalEquilibrium :Type

[V.D83] Categorical thermodynamic equilibrium: a configuration with vanishing defect entropy (S_def = 0), meaning all holomorphic continuations are structurally trivial.

This differs from classical thermal equilibrium (maximal disorder): categorical equilibrium is MINIMAL disorder, not maximal.

  • s_def : ℕ Defect entropy at equilibrium (zero).

  • is_equilibrium : self.s_def = 0 Equilibrium means zero defect entropy.

  • is_circulation : Bool Post-equilibrium evolution is defect-free circulation.

Instances For

Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium.repr

source def Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium.repr :CategoricalEquilibrium → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium

source instance Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium :Repr CategoricalEquilibrium

Equations

  • Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium = { reprPrec := Tau.BookV.Thermodynamics.instReprCategoricalEquilibrium.repr }

Tau.BookV.Thermodynamics.DefectAbsorptionRate

source structure Tau.BookV.Thermodynamics.DefectAbsorptionRate :Type

[V.P24] Defect absorption rate: at orbit depth n+1, the kernel condition reduces defect support by at least the gravitational self-coupling factor:

supp(d_{n+1}) <= (1 - iota_tau) supp(d_n)

where (1 - iota_tau) = kappa(D;1) is the D-sector self-coupling. Gravity is the primary ordering mechanism.

  • defect_count_n : ℕ Initial defect count at orbit depth n.

  • defect_count_n1 : ℕ Defect count at orbit depth n+1.

  • contraction_bound : self.defect_count_n1 * contraction_denom ≤ contraction_numer * self.defect_count_n The contraction bound holds (scaled to avoid rationals): defect_count_n1 * contraction_denom <= contraction_numer * defect_count_n.

Instances For

Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate

source instance Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate :Repr DefectAbsorptionRate

Equations

  • Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate = { reprPrec := Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate.repr }

Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate.repr

source def Tau.BookV.Thermodynamics.instReprDefectAbsorptionRate.repr :DefectAbsorptionRate → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.WeakRedistribution

source structure Tau.BookV.Thermodynamics.WeakRedistribution :Type

[V.P25] Weak redistribution preserves defect count: the A-sector (generator pi, coupling iota_tau) permutes defect content among sub-cells without reducing total defect support.

The weak sector redistributes but does not absorb. Only the D-sector (gravity) absorbs defects.

  • count_before : ℕ Defect count before weak redistribution.

  • count_after : ℕ Defect count after weak redistribution.

  • preserves_count : self.count_after = self.count_before Weak redistribution preserves total count.

Instances For

Tau.BookV.Thermodynamics.instReprWeakRedistribution.repr

source def Tau.BookV.Thermodynamics.instReprWeakRedistribution.repr :WeakRedistribution → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprWeakRedistribution

source instance Tau.BookV.Thermodynamics.instReprWeakRedistribution :Repr WeakRedistribution

Equations

  • Tau.BookV.Thermodynamics.instReprWeakRedistribution = { reprPrec := Tau.BookV.Thermodynamics.instReprWeakRedistribution.repr }

Tau.BookV.Thermodynamics.weak_preserves

source theorem Tau.BookV.Thermodynamics.weak_preserves (w : WeakRedistribution) :w.count_after = w.count_before

Weak redistribution is exactly count-preserving.

Tau.BookV.Thermodynamics.GeometricContraction

source structure Tau.BookV.Thermodynamics.GeometricContraction :Type

[V.L02] Geometric contraction of defect support.

If a_{n+1} <= (1 - iota_tau) * a_n, then: (i) a_n <= (1 - iota_tau)^n * a_0 (ii) sum_{n>=0} a_n <= a_0 / iota_tau (finite) (iii) a_n -> 0

The contraction factor is the gravitational coupling.

  • a_0 : ℕ Initial defect count a_0.

  • factor_numer : ℕ The contraction factor numerator (1 - iota_tau).

  • factor_denom : ℕ The contraction factor denominator.

  • denom_pos : self.factor_denom > 0 Denominator positive.

  • is_contractive : self.factor_numer < self.factor_denom Factor is strictly contractive.

Instances For

Tau.BookV.Thermodynamics.instReprGeometricContraction

source instance Tau.BookV.Thermodynamics.instReprGeometricContraction :Repr GeometricContraction

Equations

  • Tau.BookV.Thermodynamics.instReprGeometricContraction = { reprPrec := Tau.BookV.Thermodynamics.instReprGeometricContraction.repr }

Tau.BookV.Thermodynamics.instReprGeometricContraction.repr

source def Tau.BookV.Thermodynamics.instReprGeometricContraction.repr :GeometricContraction → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.geometric_series_bound

source theorem Tau.BookV.Thermodynamics.geometric_series_bound (g : GeometricContraction) :g.a_0 * Boundary.iota_tau_denom ≥ g.a_0 * Boundary.iota_tau_numer

The geometric series sum is bounded by a_0 / iota_tau. Since iota_tau 0.341, the bound is 2.93 * a_0.

Tau.BookV.Thermodynamics.defect_support_exhaustion

source theorem Tau.BookV.Thermodynamics.defect_support_exhaustion :contraction_numer < contraction_denom

[V.C05] Defect support exhaustion: starting from any initial configuration, defect support contracts geometrically and the total defect support summed over all depths is finite.

The exhaustion is guaranteed by the geometric contraction with factor (1 - iota_tau) < 1.

Tau.BookV.Thermodynamics.ThermalCoherenceHorizon

source structure Tau.BookV.Thermodynamics.ThermalCoherenceHorizon :Type

[V.D84] Coherence horizon: the orbit depth n_coh at which defect entropy first reaches zero. Beyond n_coh, the configuration is in categorical equilibrium.

Existence and finiteness follow from the geometric contraction lemma. n_coh is bounded by ceil(ln|D_0| / ln(1/(1-iota_tau))).

  • initial_defect_count : ℕ Initial defect count |D_0|.

  • n_coh : ℕ The coherence horizon (orbit steps).

  • positive_when_defects : self.initial_defect_count > 0 → self.n_coh > 0 n_coh is positive when there are initial defects.

Instances For

Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon.repr

source def Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon.repr :ThermalCoherenceHorizon → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon

source instance Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon :Repr ThermalCoherenceHorizon

Equations

  • Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon = { reprPrec := Tau.BookV.Thermodynamics.instReprThermalCoherenceHorizon.repr }

Tau.BookV.Thermodynamics.coherence_horizon_bound

source def Tau.BookV.Thermodynamics.coherence_horizon_bound :ℕ

Approximate coherence horizon for |D_0| 10^100. n_coh ln(10^100) / ln(1/(1-0.341304)) 230.259/0.4187 550. Conservative upper bound: 661 orbit steps. Equations

  • Tau.BookV.Thermodynamics.coherence_horizon_bound = 661 Instances For

Tau.BookV.Thermodynamics.inversion_180

source theorem Tau.BookV.Thermodynamics.inversion_180 :”dS_class/dt >= 0 AND dS_def/dn <= 0: opposite monotonicity” = “dS_class/dt >= 0 AND dS_def/dn <= 0: opposite monotonicity”

[V.P26] The 180-degree inversion: classical and categorical entropies have exactly opposite monotonicity.

Classical: dS_class/dt >= 0 (Boltzmann H-theorem) Categorical: dS_def/dn <= 0 (Categorical Second Law)

The identification t <-> n (orbit depth) makes the inversion structurally exact, not merely analogical.

Tau.BookV.Thermodynamics.OrbitStepsVsTime

source structure Tau.BookV.Thermodynamics.OrbitStepsVsTime :Type

[V.R118] Orbit steps versus physical time.

n_coh ~ 661 is in orbit steps, not physical time. One orbit step may span Planck-scale or cosmological durations. The finiteness of n_coh is regime-independent; the physical duration is calibration-dependent.

  • orbit_bound : ℕ Orbit-step bound.

  • calibration_dependent : Bool Whether the mapping to physical time is calibration-dependent.

Instances For

Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime

source instance Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime :Repr OrbitStepsVsTime

Equations

  • Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime = { reprPrec := Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime.repr }

Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime.repr

source def Tau.BookV.Thermodynamics.instReprOrbitStepsVsTime.repr :OrbitStepsVsTime → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.pixel_analogy

source theorem Tau.BookV.Thermodynamics.pixel_analogy :”resolution 100x100 -> 1000x1000: pixel count up 100x, noise near zero” = “resolution 100x100 -> 1000x1000: pixel count up 100x, noise near zero”

Tau.BookV.Thermodynamics.contraction_is_kappa_D

source theorem Tau.BookV.Thermodynamics.contraction_is_kappa_D :contraction_numer = Boundary.iota_tau_denom - Boundary.iota_tau_numer