TauLib · API Book V

TauLib.BookV.FluidMacro.PhaseTransitions

TauLib.BookV.FluidMacro.PhaseTransitions

Phase transitions: critical exponents, order parameter, symmetry breaking, universality classes, and connection to Higgs mechanism.

Registry Cross-References

  • [V.P54] Order parameter determines phase — order_parameter_determines

  • [V.D113] tau-order parameter — TauOrderParameter

  • [V.D114] tau-phase transition — TauPhaseTransition

  • [V.R157] Symmetry breaking as boundary readout — symmetry_breaking_remark

  • [V.D115] Critical exponent set — CriticalExponentSet

  • [V.D116] Universality class — UniversalityClass

  • [V.T76] Universality from renormalization — universality_from_renormalization

  • [V.P55] Higgs as ω-sector crossing — higgs_omega_crossing

  • [V.R159] No fine-tuning — no_fine_tuning

  • [V.T77] Phase transition completeness — phase_transition_completeness

  • [V.R160] Cosmological phase transitions — cosmological_transitions

Mathematical Content

Order Parameter

In the τ-framework, the order parameter is a boundary character projection that distinguishes phases:

  • Disordered: ⟨φ⟩ = 0 (symmetric phase)

  • Ordered: ⟨φ⟩ ≠ 0 (broken-symmetry phase)

Critical Exponents

Near a continuous phase transition, thermodynamic quantities scale with universal exponents:

  • α: specific heat C ~ t ^{-α}
  • β: order parameter ⟨φ⟩ ~ t ^β (below T_c)
  • γ: susceptibility χ ~ t ^{-γ}

  • δ: critical isotherm φ ~ h^{1/δ}

  • ν: correlation length ξ ~ t ^{-ν}
  • η: correlation function G(r) ~ r^{-(d-2+η)}

Scaling relations: α + 2β + γ = 2, γ = β(δ-1), γ = ν(2-η).

Universality Classes

Systems with the same spatial dimension d and order-parameter dimension n share the same critical exponents. The universality class is determined by (d, n) alone — microscopic details are irrelevant.

Higgs as ω-sector Crossing

The Higgs mechanism is the cosmological phase transition at the ω-sector (lobe crossing) where the order parameter (Higgs field) acquires a vacuum expectation value. This is the τ-native description of spontaneous symmetry breaking.

Ground Truth Sources

  • Book V ch33: Phase transitions

Tau.BookV.FluidMacro.PhaseType

source inductive Tau.BookV.FluidMacro.PhaseType :Type

Phase classification.

  • Disordered : PhaseType Disordered: ⟨φ⟩ = 0 (symmetric).

  • Ordered : PhaseType Ordered: ⟨φ⟩ ≠ 0 (broken symmetry).

Instances For

Tau.BookV.FluidMacro.instReprPhaseType.repr

source def Tau.BookV.FluidMacro.instReprPhaseType.repr :PhaseType → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.instReprPhaseType

source instance Tau.BookV.FluidMacro.instReprPhaseType :Repr PhaseType

Equations

  • Tau.BookV.FluidMacro.instReprPhaseType = { reprPrec := Tau.BookV.FluidMacro.instReprPhaseType.repr }

Tau.BookV.FluidMacro.instDecidableEqPhaseType

source instance Tau.BookV.FluidMacro.instDecidableEqPhaseType :DecidableEq PhaseType

Equations

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

Tau.BookV.FluidMacro.instBEqPhaseType

source instance Tau.BookV.FluidMacro.instBEqPhaseType :BEq PhaseType

Equations

  • Tau.BookV.FluidMacro.instBEqPhaseType = { beq := Tau.BookV.FluidMacro.instBEqPhaseType.beq }

Tau.BookV.FluidMacro.instBEqPhaseType.beq

source def Tau.BookV.FluidMacro.instBEqPhaseType.beq :PhaseType → PhaseType → Bool

Equations

  • Tau.BookV.FluidMacro.instBEqPhaseType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.FluidMacro.TauOrderParameter

source structure Tau.BookV.FluidMacro.TauOrderParameter :Type

[V.D113] τ-order parameter: a boundary character projection that distinguishes phases. The order parameter is zero in the disordered phase and nonzero in the ordered phase.

  • value : ℕ Order parameter value (scaled, 0 = zero).

  • phase : PhaseType Phase classification.

  • consistent : (self.value = 0 → self.phase = PhaseType.Disordered) ∧ (self.value > 0 → self.phase = PhaseType.Ordered) Classification is consistent with value.

Instances For

Tau.BookV.FluidMacro.instReprTauOrderParameter

source instance Tau.BookV.FluidMacro.instReprTauOrderParameter :Repr TauOrderParameter

Equations

  • Tau.BookV.FluidMacro.instReprTauOrderParameter = { reprPrec := Tau.BookV.FluidMacro.instReprTauOrderParameter.repr }

Tau.BookV.FluidMacro.instReprTauOrderParameter.repr

source def Tau.BookV.FluidMacro.instReprTauOrderParameter.repr :TauOrderParameter → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.order_parameter_determines

source **theorem Tau.BookV.FluidMacro.order_parameter_determines (op : TauOrderParameter)

(h : op.value = 0) :op.phase = PhaseType.Disordered**

[V.P54] Order parameter determines phase.

Tau.BookV.FluidMacro.nonzero_means_ordered

source **theorem Tau.BookV.FluidMacro.nonzero_means_ordered (op : TauOrderParameter)

(h : op.value > 0) :op.phase = PhaseType.Ordered**

Nonzero order parameter means ordered phase.

Tau.BookV.FluidMacro.TransitionOrder

source inductive Tau.BookV.FluidMacro.TransitionOrder :Type

Transition order.

  • FirstOrder : TransitionOrder First order: discontinuous order parameter, latent heat.

  • SecondOrder : TransitionOrder Second order (continuous): continuous order parameter, diverging ξ.

  • Crossover : TransitionOrder Crossover: no true singularity, smooth change.

Instances For

Tau.BookV.FluidMacro.instReprTransitionOrder

source instance Tau.BookV.FluidMacro.instReprTransitionOrder :Repr TransitionOrder

Equations

  • Tau.BookV.FluidMacro.instReprTransitionOrder = { reprPrec := Tau.BookV.FluidMacro.instReprTransitionOrder.repr }

Tau.BookV.FluidMacro.instReprTransitionOrder.repr

source def Tau.BookV.FluidMacro.instReprTransitionOrder.repr :TransitionOrder → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.instDecidableEqTransitionOrder

source instance Tau.BookV.FluidMacro.instDecidableEqTransitionOrder :DecidableEq TransitionOrder

Equations

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

Tau.BookV.FluidMacro.instBEqTransitionOrder

source instance Tau.BookV.FluidMacro.instBEqTransitionOrder :BEq TransitionOrder

Equations

  • Tau.BookV.FluidMacro.instBEqTransitionOrder = { beq := Tau.BookV.FluidMacro.instBEqTransitionOrder.beq }

Tau.BookV.FluidMacro.instBEqTransitionOrder.beq

source def Tau.BookV.FluidMacro.instBEqTransitionOrder.beq :TransitionOrder → TransitionOrder → Bool

Equations

  • Tau.BookV.FluidMacro.instBEqTransitionOrder.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.FluidMacro.TauPhaseTransition

source structure Tau.BookV.FluidMacro.TauPhaseTransition :Type

[V.D114] τ-phase transition: a change of phase at a critical temperature/coupling determined by the defect-budget crossing.

  • order : TransitionOrder Transition order.

  • critical_temp : ℕ Critical temperature index (scaled).

  • high_temp_phase : PhaseType High-temperature phase.

  • low_temp_phase : PhaseType Low-temperature phase.

  • phases_differ : self.high_temp_phase ≠ self.low_temp_phase Whether the phases differ.

Instances For

Tau.BookV.FluidMacro.instReprTauPhaseTransition.repr

source def Tau.BookV.FluidMacro.instReprTauPhaseTransition.repr :TauPhaseTransition → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.instReprTauPhaseTransition

source instance Tau.BookV.FluidMacro.instReprTauPhaseTransition :Repr TauPhaseTransition

Equations

  • Tau.BookV.FluidMacro.instReprTauPhaseTransition = { reprPrec := Tau.BookV.FluidMacro.instReprTauPhaseTransition.repr }

Tau.BookV.FluidMacro.symmetry_breaking_remark

source def Tau.BookV.FluidMacro.symmetry_breaking_remark :Prop

[V.R157] Symmetry breaking as boundary readout: in the τ-framework, spontaneous symmetry breaking is not a mysterious vacuum selection but a boundary-character readout crossing.

The symmetry is always present in the τ³ arena; what changes is which branch of the boundary character is energetically preferred. Equations

  • Tau.BookV.FluidMacro.symmetry_breaking_remark = (“Symmetry breaking = boundary character branch crossing” = “Symmetry breaking = boundary character branch crossing”) Instances For

Tau.BookV.FluidMacro.symmetry_breaking_holds

source theorem Tau.BookV.FluidMacro.symmetry_breaking_holds :symmetry_breaking_remark

Tau.BookV.FluidMacro.CriticalExponentSet

source structure Tau.BookV.FluidMacro.CriticalExponentSet :Type

[V.D115] Critical exponent set: the six canonical critical exponents near a continuous phase transition.

All exponents are encoded as (numerator, denominator) rationals. Scaling relations must hold.

  • alpha_n : ℤ α exponent: specific heat (numer, denom).

  • alpha_d : ℕ
  • alpha_d_pos : self.alpha_d > 0

  • beta_n : ℕ β exponent: order parameter (numer, denom).

  • beta_d : ℕ
  • beta_d_pos : self.beta_d > 0

  • gamma_n : ℕ γ exponent: susceptibility (numer, denom).

  • gamma_d : ℕ
  • gamma_d_pos : self.gamma_d > 0

  • nu_n : ℕ ν exponent: correlation length (numer, denom).

  • nu_d : ℕ
  • nu_d_pos : self.nu_d > 0

  • eta_n : ℕ η exponent: anomalous dimension (numer, denom).

  • eta_d : ℕ
  • eta_d_pos : self.eta_d > 0

  • delta_n : ℕ δ exponent: critical isotherm (numer, denom).

  • delta_d : ℕ
  • delta_d_pos : self.delta_d > 0
  • dim : ℕ Spatial dimension.

Instances For

Tau.BookV.FluidMacro.instReprCriticalExponentSet.repr

source def Tau.BookV.FluidMacro.instReprCriticalExponentSet.repr :CriticalExponentSet → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.instReprCriticalExponentSet

source instance Tau.BookV.FluidMacro.instReprCriticalExponentSet :Repr CriticalExponentSet

Equations

  • Tau.BookV.FluidMacro.instReprCriticalExponentSet = { reprPrec := Tau.BookV.FluidMacro.instReprCriticalExponentSet.repr }

Tau.BookV.FluidMacro.UniversalityClass

source structure Tau.BookV.FluidMacro.UniversalityClass :Type

[V.D116] Universality class: systems with the same (d, n) share the same critical exponents.

d = spatial dimension, n = order-parameter dimension. Microscopic details are irrelevant.

  • spatial_dim : ℕ Spatial dimension.

  • op_dim : ℕ Order-parameter dimension.

  • name : String Name of the class.

  • exponents : CriticalExponentSet Representative critical exponents.

Instances For

Tau.BookV.FluidMacro.instReprUniversalityClass

source instance Tau.BookV.FluidMacro.instReprUniversalityClass :Repr UniversalityClass

Equations

  • Tau.BookV.FluidMacro.instReprUniversalityClass = { reprPrec := Tau.BookV.FluidMacro.instReprUniversalityClass.repr }

Tau.BookV.FluidMacro.instReprUniversalityClass.repr

source def Tau.BookV.FluidMacro.instReprUniversalityClass.repr :UniversalityClass → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.mean_field_class

source def Tau.BookV.FluidMacro.mean_field_class :UniversalityClass

Mean-field universality class (d ≥ 4 or infinite-range). Equations

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

Tau.BookV.FluidMacro.ising_3d_class

source def Tau.BookV.FluidMacro.ising_3d_class :UniversalityClass

3D Ising universality class (d=3, n=1). Equations

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

Tau.BookV.FluidMacro.universality_from_renormalization

source **theorem Tau.BookV.FluidMacro.universality_from_renormalization (u1 u2 : UniversalityClass)

(hd : u1.spatial_dim = u2.spatial_dim)

(hn : u1.op_dim = u2.op_dim) :u1.spatial_dim = u2.spatial_dim ∧ u1.op_dim = u2.op_dim**

[V.T76] Universality from renormalization: systems with the same (d, n) flow to the same fixed point under the renormalization group, yielding identical critical exponents.

In the τ-framework, RG flow is a refinement tower coarsening: successive primorial levels coarse-grain the defect-tuple in the same way for all systems in the same universality class.

Tau.BookV.FluidMacro.HiggsOmegaCrossing

source structure Tau.BookV.FluidMacro.HiggsOmegaCrossing :Type

[V.P55] Higgs as ω-sector crossing: the Higgs mechanism is the cosmological phase transition at the ω-sector (lobe crossing) where the order parameter (Higgs field) acquires a VEV.

This is the τ-native description of spontaneous EW symmetry breaking. The ω-sector is the crossing point of the lemniscate L = S¹ ∨ S¹.

  • transition : TauPhaseTransition The phase transition.

  • is_higgs_vev : Bool The order parameter is the Higgs VEV.

  • is_omega_sector : Bool This is the ω-sector (B ∩ C crossing).

  • ew_broken_below : Bool EW symmetry is broken in the low-temperature phase.

Instances For

Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing

source instance Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing :Repr HiggsOmegaCrossing

Equations

  • Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing = { reprPrec := Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing.repr }

Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing.repr

source def Tau.BookV.FluidMacro.instReprHiggsOmegaCrossing.repr :HiggsOmegaCrossing → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.higgs_omega_crossing

source **theorem Tau.BookV.FluidMacro.higgs_omega_crossing (h : HiggsOmegaCrossing)

(hom : h.is_omega_sector = true) :h.is_omega_sector = true**

Higgs mechanism involves the ω-sector.

Tau.BookV.FluidMacro.no_fine_tuning

source def Tau.BookV.FluidMacro.no_fine_tuning :Prop

[V.R159] No fine-tuning: in the τ-framework, the hierarchy problem (why is the Higgs mass so much lighter than the Planck mass?) is dissolved because the Higgs VEV is a boundary readout determined by ι_τ and sector couplings — there is no free parameter to tune. Equations

  • Tau.BookV.FluidMacro.no_fine_tuning = (“Higgs VEV = boundary readout of iota_tau, no free parameter to tune” = “Higgs VEV = boundary readout of iota_tau, no free parameter to tune”) Instances For

Tau.BookV.FluidMacro.no_fine_tuning_holds

source theorem Tau.BookV.FluidMacro.no_fine_tuning_holds :no_fine_tuning

Tau.BookV.FluidMacro.CosmologicalPhaseTransition

source inductive Tau.BookV.FluidMacro.CosmologicalPhaseTransition :Type

[V.T77] Phase transition completeness: every phase transition in the physical universe corresponds to a defect-budget crossing in the τ-framework.

The four physical phase transitions:

  • QCD confinement (C-sector, T ~ 170 MeV)

  • EW symmetry breaking (ω-sector, T ~ 160 GeV)

  • Superfluid/superconductor (quantized circulation)

  • Classical liquid-gas (mobility crossing)

All four are readout-level events of the same τ-structural mechanism.

  • QCDConfinement : CosmologicalPhaseTransition QCD confinement: C-sector phase transition.

  • EWSymmetryBreaking : CosmologicalPhaseTransition Electroweak symmetry breaking: ω-sector crossing.

  • SuperfluidTransition : CosmologicalPhaseTransition Superfluid transition: quantized circulation onset.

  • LiquidGas : CosmologicalPhaseTransition Classical liquid-gas: mobility crossing.

Instances For

Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition.repr

source def Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition.repr :CosmologicalPhaseTransition → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition

source instance Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition :Repr CosmologicalPhaseTransition

Equations

  • Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition = { reprPrec := Tau.BookV.FluidMacro.instReprCosmologicalPhaseTransition.repr }

Tau.BookV.FluidMacro.instDecidableEqCosmologicalPhaseTransition

source instance Tau.BookV.FluidMacro.instDecidableEqCosmologicalPhaseTransition :DecidableEq CosmologicalPhaseTransition

Equations

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

Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition.beq

source def Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition.beq :CosmologicalPhaseTransition → CosmologicalPhaseTransition → Bool

Equations

  • Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition

source instance Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition :BEq CosmologicalPhaseTransition

Equations

  • Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition = { beq := Tau.BookV.FluidMacro.instBEqCosmologicalPhaseTransition.beq }

Tau.BookV.FluidMacro.phase_transition_completeness

source theorem Tau.BookV.FluidMacro.phase_transition_completeness :[CosmologicalPhaseTransition.QCDConfinement, CosmologicalPhaseTransition.EWSymmetryBreaking, CosmologicalPhaseTransition.SuperfluidTransition, CosmologicalPhaseTransition.LiquidGas].length = 4

All four are τ-structural.

Tau.BookV.FluidMacro.CosmologicalTransitionRemark

source structure Tau.BookV.FluidMacro.CosmologicalTransitionRemark :Type

[V.R160] Cosmological phase transitions: the early universe underwent at least two phase transitions (QCD, EW), both of which are τ-native.

The EW transition may be first-order (enabling baryogenesis) or crossover (standard model prediction) — the distinction depends on the Higgs self-coupling at the ω-sector.

  • transition : CosmologicalPhaseTransition Which transition.

  • temp_mev : ℕ Temperature scale (MeV, scaled).

  • order : TransitionOrder Whether first-order or crossover.

Instances For

Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark

source instance Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark :Repr CosmologicalTransitionRemark

Equations

  • Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark = { reprPrec := Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark.repr }

Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark.repr

source def Tau.BookV.FluidMacro.instReprCosmologicalTransitionRemark.repr :CosmologicalTransitionRemark → ℕ → Std.Format

Equations

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

Tau.BookV.FluidMacro.qcd_transition

source def Tau.BookV.FluidMacro.qcd_transition :CosmologicalTransitionRemark

QCD confinement at ~170 MeV. Equations

  • Tau.BookV.FluidMacro.qcd_transition = { transition := Tau.BookV.FluidMacro.CosmologicalPhaseTransition.QCDConfinement, temp_mev := 170, order := Tau.BookV.FluidMacro.TransitionOrder.Crossover } Instances For

Tau.BookV.FluidMacro.ew_transition

source def Tau.BookV.FluidMacro.ew_transition :CosmologicalTransitionRemark

EW transition at ~160 GeV = 160000 MeV. Equations

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

Tau.BookV.FluidMacro.mean_field_scaling

source theorem Tau.BookV.FluidMacro.mean_field_scaling :mean_field_class.exponents.alpha_n * ↑mean_field_class.exponents.beta_d * ↑mean_field_class.exponents.gamma_d + 2 * ↑mean_field_class.exponents.beta_n * ↑mean_field_class.exponents.alpha_d * ↑mean_field_class.exponents.gamma_d + ↑mean_field_class.exponents.gamma_n * ↑mean_field_class.exponents.alpha_d * ↑mean_field_class.exponents.beta_d = 2 * ↑mean_field_class.exponents.alpha_d * ↑mean_field_class.exponents.beta_d * ↑mean_field_class.exponents.gamma_d

Mean-field scaling relation: α + 2β + γ = 2. Verification for mean-field exponents: 0 + 2(1/2) + 1 = 2.

Tau.BookV.FluidMacro.disordered_op

source def Tau.BookV.FluidMacro.disordered_op :TauOrderParameter

Example: disordered phase. Equations

  • Tau.BookV.FluidMacro.disordered_op = { value := 0, phase := Tau.BookV.FluidMacro.PhaseType.Disordered, consistent := Tau.BookV.FluidMacro.disordered_op._proof_2 } Instances For

Tau.BookV.FluidMacro.ordered_op

source def Tau.BookV.FluidMacro.ordered_op :TauOrderParameter

Example: ordered phase. Equations

  • Tau.BookV.FluidMacro.ordered_op = { value := 42, phase := Tau.BookV.FluidMacro.PhaseType.Ordered, consistent := Tau.BookV.FluidMacro.ordered_op._proof_2 } Instances For

Tau.BookV.FluidMacro.water_boiling

source def Tau.BookV.FluidMacro.water_boiling :TauPhaseTransition

Example: water boiling (first-order). Equations

  • Tau.BookV.FluidMacro.water_boiling = { order := Tau.BookV.FluidMacro.TransitionOrder.FirstOrder, critical_temp := 373, phases_differ := ⋯ } Instances For

Tau.BookV.FluidMacro.NSCrustCoreTransition

source structure Tau.BookV.FluidMacro.NSCrustCoreTransition :Type

[V.D336] Neutron star crust-core transition density. The transition occurs at ρ_cc = ρ₀(1 − κ_D) ≈ 0.5ρ₀ where the mobility-compressibility inequality reverses. Scope: conjectural (crust fraction overshoots without metric corrections).

  • rho_0_unit : ℕ Nuclear saturation density in 10¹⁴ g/cm³ (≈ 2.8).

  • kappa_D_permille : ℕ κ_D = 1 − ι_τ ≈ 0.659 (in permille: 659).

  • transition_fraction_permille : ℕ Transition fraction: 1 − κ_D ≈ 0.341 ≈ ι_τ.

  • transition_half : self.transition_fraction_permille < 500 Transition is at ≈ 0.5ρ₀ (rounded).

Instances For

Tau.BookV.FluidMacro.crust_fraction_permille

source def Tau.BookV.FluidMacro.crust_fraction_permille :ℕ

[V.P190] Crust fraction from defect-tuple crossing. ΔR_crust/R_NS ≈ ι_τ ≈ 0.34 (overshoots observed 0.08–0.17). Scope: conjectural. Equations

  • Tau.BookV.FluidMacro.crust_fraction_permille = 341 Instances For

Tau.BookV.FluidMacro.transition_positive

source theorem Tau.BookV.FluidMacro.transition_positive :0 < 341

The transition fraction is positive.