TauLib · API Book IV

TauLib.BookIV.Physics.Thermodynamics

TauLib.BookIV.Physics.Thermodynamics

Entropy splitting, defect thermodynamics, and the No-Running Principle.

Registry Cross-References

  • [IV.D24] Entropy Splitting — EntropySplitting

  • [IV.D25] Defect Budget — DefectBudget

  • [IV.R05] S_def → 0 at coherence horizon — structural remark

  • [IV.R06] S_ref unbounded — structural remark

  • [IV.P04] No-Running Principle — NoRunningPrinciple, no_running_all_sectors

  • [IV.T04] Euler budget conservation — euler_budget_conserved

Mathematical Content

Entropy Splitting

Orthodox entropy S splits into two structurally distinct components:

S_total = S_def + S_ref

  • S_def (defect entropy): Tracks defect novelty events. Reaches ZERO at the coherence horizon (all defects resolved).

  • S_ref (refinement entropy): Tracks refinement depth. Grows unboundedly (refinement never terminates).

The Second-Law Inversion: In τ-cosmology, S_def reverses at the coherence horizon — a novel thermodynamic asymmetry absent in orthodox physics.

Defect Budget Conservation

In the Euler regime (inviscid), the total defect budget is conserved:

∑(mobility + vorticity + compression + topological) = const

This is the τ-native form of the Kelvin circulation theorem.

No-Running Principle

Orthodox “running couplings” (α_s(Q²), etc.) are NOT ontic:

Fixed ontic coupling + regime-dependent readout = apparent "running"

The τ-kernel coupling constants are boundary fixed-point invariants, absolutely fixed regardless of measurement regime. What orthodox QFT interprets as “running” is the regime-dependence of the readout functor projecting the fixed ontic value onto different measurement scales.

Ground Truth Sources

  • fluid-condensed-matter.json: defect-thermodynamics, defect-budget

  • particle-physics-defects.json: running-vs-regime-framework

  • holonomy-sectors.json: fixed-point-readout-machinery

Tau.BookIV.Physics.EntropySplitting

source structure Tau.BookIV.Physics.EntropySplitting :Type

[IV.D24] Entropy splitting: S = S_def + S_ref.

  • S_def (defect entropy): coherence measure of defect novelty. Reaches zero at the coherence horizon.

  • S_ref (refinement entropy): measure of refinement depth. Grows unboundedly (refinement never terminates).

The Second-Law Inversion: S_def reverses at the horizon, creating a novel thermodynamic asymmetry.

  • s_def_numer : ℕ Defect entropy numerator (→ 0 at coherence horizon).

  • s_def_denom : ℕ Defect entropy denominator.

  • s_ref_numer : ℕ Refinement entropy numerator (unbounded growth).

  • s_ref_denom : ℕ Refinement entropy denominator.

  • denom_pos_def : self.s_def_denom > 0 Defect entropy denominator positive.

  • denom_pos_ref : self.s_ref_denom > 0 Refinement entropy denominator positive.

Instances For

Tau.BookIV.Physics.instReprEntropySplitting

source instance Tau.BookIV.Physics.instReprEntropySplitting :Repr EntropySplitting

Equations

  • Tau.BookIV.Physics.instReprEntropySplitting = { reprPrec := Tau.BookIV.Physics.instReprEntropySplitting.repr }

Tau.BookIV.Physics.instReprEntropySplitting.repr

source def Tau.BookIV.Physics.instReprEntropySplitting.repr :EntropySplitting → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.EntropySplitting.totalFloat

source def Tau.BookIV.Physics.EntropySplitting.totalFloat (e : EntropySplitting) :Float

Total entropy as Float (for display). Equations

  • e.totalFloat = Float.ofNat e.s_def_numer / Float.ofNat e.s_def_denom + Float.ofNat e.s_ref_numer / Float.ofNat e.s_ref_denom Instances For

Tau.BookIV.Physics.EntropySplitting.sDefFloat

source def Tau.BookIV.Physics.EntropySplitting.sDefFloat (e : EntropySplitting) :Float

S_def as Float (for display). Equations

  • e.sDefFloat = Float.ofNat e.s_def_numer / Float.ofNat e.s_def_denom Instances For

Tau.BookIV.Physics.EntropySplitting.sRefFloat

source def Tau.BookIV.Physics.EntropySplitting.sRefFloat (e : EntropySplitting) :Float

S_ref as Float (for display). Equations

  • e.sRefFloat = Float.ofNat e.s_ref_numer / Float.ofNat e.s_ref_denom Instances For

Tau.BookIV.Physics.DefectBudget

source structure Tau.BookIV.Physics.DefectBudget :Type

[IV.D25] Defect budget: the conserved total of the 4-component defect tuple in the Euler (inviscid) regime.

In the Euler regime, the defect-budget is preserved under boundary-automorphism steps: ∑(mobility + vorticity + compression + topological) = const

This is the τ-native Kelvin circulation theorem.

  • tuple : DefectTuple The defect tuple.

  • total : ℕ The conserved total budget.

  • budget_eq : self.total = self.tuple.mobility + self.tuple.vorticity + self.tuple.compression + self.tuple.topological Budget equals sum of components.

Instances For

Tau.BookIV.Physics.instReprDefectBudget.repr

source def Tau.BookIV.Physics.instReprDefectBudget.repr :DefectBudget → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprDefectBudget

source instance Tau.BookIV.Physics.instReprDefectBudget :Repr DefectBudget

Equations

  • Tau.BookIV.Physics.instReprDefectBudget = { reprPrec := Tau.BookIV.Physics.instReprDefectBudget.repr }

Tau.BookIV.Physics.NoRunningPrinciple

source structure Tau.BookIV.Physics.NoRunningPrinciple :Type

[IV.P04] No-Running Principle: ontic coupling constants do NOT run.

Orthodox “running couplings” (α_s(Q²), α_EM(Q²), etc.) are NOT ontic. The τ-kernel coupling constants are boundary fixed-point invariants:

  • Fixed ontic value (determined by ι_τ alone)

  • Regime-dependent readout functor

  • Apparent “running” = projection drift from different measurement scales

The coupling value in the registry IS the fixed ontic value.

  • sector : BookIII.Sectors.Sector Which sector this principle applies to.

  • ontic_coupling_numer : ℕ The fixed ontic coupling for this sector.

  • ontic_coupling_denom : ℕ The fixed ontic coupling denominator.

  • denom_pos : self.ontic_coupling_denom > 0 Denominator positive.

  • regime_independent : Bool The coupling is regime-independent (fixed).

Instances For

Tau.BookIV.Physics.instReprNoRunningPrinciple.repr

source def Tau.BookIV.Physics.instReprNoRunningPrinciple.repr :NoRunningPrinciple → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprNoRunningPrinciple

source instance Tau.BookIV.Physics.instReprNoRunningPrinciple :Repr NoRunningPrinciple

Equations

  • Tau.BookIV.Physics.instReprNoRunningPrinciple = { reprPrec := Tau.BookIV.Physics.instReprNoRunningPrinciple.repr }

Tau.BookIV.Physics.no_running_em

source def Tau.BookIV.Physics.no_running_em :NoRunningPrinciple

No-Running for EM sector: α_EM is fixed at ι_τ². Equations

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

Tau.BookIV.Physics.no_running_weak

source def Tau.BookIV.Physics.no_running_weak :NoRunningPrinciple

No-Running for Weak sector: g_w is fixed at ι_τ. Equations

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

Tau.BookIV.Physics.no_running_strong

source def Tau.BookIV.Physics.no_running_strong :NoRunningPrinciple

No-Running for Strong sector: α_s is fixed at ι_τ³/(1−ι_τ). Equations

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

Tau.BookIV.Physics.no_running_gravity

source def Tau.BookIV.Physics.no_running_gravity :NoRunningPrinciple

No-Running for Gravity sector: κ_GR is fixed at 1−ι_τ. Equations

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

Tau.BookIV.Physics.no_running_higgs

source def Tau.BookIV.Physics.no_running_higgs :NoRunningPrinciple

No-Running for Higgs/crossing sector: coupling fixed at ι_τ³/(1+ι_τ). Equations

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

Tau.BookIV.Physics.all_no_running

source def Tau.BookIV.Physics.all_no_running :List NoRunningPrinciple

All 5 sectors obey the No-Running Principle. Equations

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

Tau.BookIV.Physics.euler_budget_conserved

source theorem Tau.BookIV.Physics.euler_budget_conserved (b : DefectBudget) :b.total = b.tuple.total

[IV.T04] Euler budget conservation: the total defect budget is well-defined (equals sum of components).

Tau.BookIV.Physics.no_running_all_sectors

source theorem Tau.BookIV.Physics.no_running_all_sectors :all_no_running.length = 5

All 5 sectors have no-running entries.

Tau.BookIV.Physics.all_regime_independent

source theorem Tau.BookIV.Physics.all_regime_independent :((List.map NoRunningPrinciple.regime_independent all_no_running).all fun (x : Bool) => x == true) = true

All no-running entries are regime-independent.

Tau.BookIV.Physics.s_def_zero_at_horizon

source **theorem Tau.BookIV.Physics.s_def_zero_at_horizon (e : EntropySplitting)

(h : e.s_def_numer = 0) :e.s_def_numer = 0**

[IV.R05] S_def = 0 at coherence horizon: when s_def_numer = 0, the defect entropy numerator vanishes.

Tau.BookIV.Physics.budget_nonneg

source theorem Tau.BookIV.Physics.budget_nonneg (b : DefectBudget) :b.total ≥ 0

The budget total is non-negative (Nat).