TauLib · API Book V

TauLib.BookV.Thermodynamics.DarkEnergyArtifact

TauLib.BookV.Thermodynamics.DarkEnergyArtifact

Dark energy as Lambda-CDM artifact. The 10^120 vacuum mismatch resolution. Cosmic acceleration from defect-to-refinement transition, not from a cosmological constant. Finite universe, no proliferating infinity.

Registry Cross-References

  • [V.D95] Capacity Surplus – CapacitySurplus

  • [V.T68] Defect-Driven Acceleration – DefectDrivenAcceleration

  • [V.T69] Dark Energy is a Readout Artifact – dark_energy_artifact

  • [V.P40] No Lambda in the tau-Einstein Equation – no_lambda

  • [V.P41] The 68% is Refinement Entropy – the_68_percent

  • [V.R133] Data versus Interpretation – data_vs_interpretation

  • [V.R134] Two Faces of the Same Problem – two_faces

  • [V.R135] Where Does the 68% Go? – where_68_goes

  • [V.R136] Testability – DarkEnergyTestability

Mathematical Content

Capacity Surplus

C_surplus(n) = C_total - |D_n|: the difference between total absorption capacity of the lemniscate L and the current defect count. Unused boundary capacity manifests as negative effective pressure.

Defect-Driven Acceleration

As S_def decreases, the effective equation-of-state parameter w shifts from w > -1/3 (decelerating) to w < -1/3 (accelerating). The transition occurs when defect-to-refinement ratio crosses a threshold.

Dark Energy is a Readout Artifact

The cosmic acceleration attributed to Lambda in Lambda-CDM arises from the defect-to-refinement transition on base tau^1. The orthodox readout functor misidentifies refinement entropy as a physical energy source.

No Lambda

The tau-Einstein equation R^H = kappa_tau * T contains no cosmological constant (Lambda = 0). Acceleration is a time-dependent phenomenon from the defect-to-refinement transition, not a permanent term.

The 68%

The 68% of the cosmic energy budget attributed to dark energy corresponds to refinement entropy S_ref – a counting artifact, not physical energy.

Ground Truth Sources

  • Book V ch26: dark energy as artifact

  • mass_decomposition_sprint.md: vacuum mismatch

Tau.BookV.Thermodynamics.CapacitySurplus

source structure Tau.BookV.Thermodynamics.CapacitySurplus :Type

[V.D95] Capacity surplus: the difference between total absorption capacity of the lemniscate boundary L and the current defect count.

C_surplus(n) = C_total - D_n

Unused boundary capacity manifests as negative effective pressure in the readout functor’s projection. As defects are absorbed, C_surplus increases, driving the transition to acceleration.

  • c_total : ℕ Total absorption capacity of L.

  • d_n : ℕ Current defect count |D_n|.

  • surplus : ℕ Surplus = total - defect count.

  • surplus_eq : self.surplus = self.c_total - self.d_n Surplus equals capacity minus defects.

  • capacity_exceeds : self.d_n ≤ self.c_total Capacity exceeds defect count (surplus non-negative).

Instances For

Tau.BookV.Thermodynamics.instReprCapacitySurplus

source instance Tau.BookV.Thermodynamics.instReprCapacitySurplus :Repr CapacitySurplus

Equations

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

Tau.BookV.Thermodynamics.instReprCapacitySurplus.repr

source def Tau.BookV.Thermodynamics.instReprCapacitySurplus.repr :CapacitySurplus → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.surplus_nonneg

source theorem Tau.BookV.Thermodynamics.surplus_nonneg (s : CapacitySurplus) :s.d_n ≤ s.c_total

Surplus is non-negative when capacity exceeds defects.

Tau.BookV.Thermodynamics.EoSRegime

source inductive Tau.BookV.Thermodynamics.EoSRegime :Type

Equation-of-state classification for the effective w parameter.

  • Decelerating : EoSRegime Decelerating: w > -1/3 (defect-dominated epoch).

  • Accelerating : EoSRegime Accelerating: w < -1/3 (refinement-dominated epoch).

  • Transition : EoSRegime Transition: w = -1/3 (crossover point).

Instances For

Tau.BookV.Thermodynamics.instReprEoSRegime

source instance Tau.BookV.Thermodynamics.instReprEoSRegime :Repr EoSRegime

Equations

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

Tau.BookV.Thermodynamics.instReprEoSRegime.repr

source def Tau.BookV.Thermodynamics.instReprEoSRegime.repr :EoSRegime → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instDecidableEqEoSRegime

source instance Tau.BookV.Thermodynamics.instDecidableEqEoSRegime :DecidableEq EoSRegime

Equations

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

Tau.BookV.Thermodynamics.instBEqEoSRegime

source instance Tau.BookV.Thermodynamics.instBEqEoSRegime :BEq EoSRegime

Equations

  • Tau.BookV.Thermodynamics.instBEqEoSRegime = { beq := Tau.BookV.Thermodynamics.instBEqEoSRegime.beq }

Tau.BookV.Thermodynamics.instBEqEoSRegime.beq

source def Tau.BookV.Thermodynamics.instBEqEoSRegime.beq :EoSRegime → EoSRegime → Bool

Equations

  • Tau.BookV.Thermodynamics.instBEqEoSRegime.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Thermodynamics.DefectDrivenAcceleration

source structure Tau.BookV.Thermodynamics.DefectDrivenAcceleration :Type

[V.T68] Defect-driven acceleration: as S_def decreases, the effective w shifts from w > -1/3 to w < -1/3.

The transition occurs when the defect-to-refinement ratio crosses a critical threshold determined by iota_tau.

Transition redshift z_acc ~ 0.7 corresponds to S_def/S_ref crossing the critical ratio.

  • regime : EoSRegime Current regime.

  • ratio_def_numer : ℕ Defect-to-refinement ratio numerator (S_def).

  • ratio_ref_denom : ℕ Defect-to-refinement ratio denominator (S_ref).

  • denom_pos : self.ratio_ref_denom > 0 Denominator positive.

  • critical_threshold_numer : ℕ The critical ratio threshold (scaled, ~ 1/3).

  • critical_threshold_denom : ℕ Critical threshold denominator.

  • z_acc_numer : ℕ Transition redshift (z_acc ~ 0.7, stored as 7/10).

  • z_acc_denom : ℕ Transition redshift denominator.

Instances For

Tau.BookV.Thermodynamics.instReprDefectDrivenAcceleration.repr

source def Tau.BookV.Thermodynamics.instReprDefectDrivenAcceleration.repr :DefectDrivenAcceleration → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprDefectDrivenAcceleration

source instance Tau.BookV.Thermodynamics.instReprDefectDrivenAcceleration :Repr DefectDrivenAcceleration

Equations

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

Tau.BookV.Thermodynamics.DefectDrivenAcceleration.determineRegime

source def Tau.BookV.Thermodynamics.DefectDrivenAcceleration.determineRegime (ratio_n ratio_d thresh_n thresh_d : ℕ) :ratio_d > 0 → thresh_d > 0 → EoSRegime

The regime is determined by the ratio relative to threshold. Equations

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

Tau.BookV.Thermodynamics.no_lambda

source theorem Tau.BookV.Thermodynamics.no_lambda :”R^H = kappa_tau * T: no Lambda term, acceleration is transient” = “R^H = kappa_tau * T: no Lambda term, acceleration is transient”

[V.P40] No Lambda in the tau-Einstein equation: R^H = kappa_tau * T contains no cosmological constant (Lambda = 0).

The acceleration is a time-dependent phenomenon from the defect-to-refinement transition, not a permanent geometric term. There is no need for Lambda and no fine-tuning problem.

Tau.BookV.Thermodynamics.dark_energy_artifact

source theorem Tau.BookV.Thermodynamics.dark_energy_artifact :”Lambda-CDM dark energy = readout artifact from S_def -> S_ref transition” = “Lambda-CDM dark energy = readout artifact from S_def -> S_ref transition”

[V.T69] Dark energy is a readout artifact: the cosmic acceleration attributed to Lambda in Lambda-CDM arises from the defect-to-refinement transition on base tau^1.

The orthodox readout functor misidentifies the decreasing defect entropy (ordering) as a repulsive energy source.

Tau.BookV.Thermodynamics.the_68_percent

source theorem Tau.BookV.Thermodynamics.the_68_percent :”68% dark energy = S_ref (counting artifact, not physical energy)” = “68% dark energy = S_ref (counting artifact, not physical energy)”

[V.P41] The 68% of the cosmic energy budget attributed to dark energy in Lambda-CDM corresponds to refinement entropy S_ref.

S_ref is a counting artifact (lattice modes), not physical energy. The 68% was never “missing energy” but misattributed entropy.

Tau.BookV.Thermodynamics.data_vs_interpretation

source theorem Tau.BookV.Thermodynamics.data_vs_interpretation :”Acceleration = data (real). Dark energy = interpretation (artifact).” = “Acceleration = data (real). Dark energy = interpretation (artifact).”

[V.R133] Data versus interpretation: the cosmic acceleration is observational data, but dark energy is a model-dependent interpretation. The tau-framework preserves the data (acceleration is real) but replaces the interpretation (different cause).

Tau.BookV.Thermodynamics.two_faces

source theorem Tau.BookV.Thermodynamics.two_faces :”CC problem: (i) why small? (ii) why nonzero? Both dissolve if Lambda = 0” = “CC problem: (i) why small? (ii) why nonzero? Both dissolve if Lambda = 0”

[V.R134] Two faces of the cosmological constant problem: (i) Why so small? Lambda ~ 10^{-52} m^{-2} vs Planck 10^{68} (ii) Why nonzero? Tiny nonzero value requires fine-tuning.

Both faces dissolve when Lambda = 0 and acceleration comes from the defect-to-refinement transition.

Tau.BookV.Thermodynamics.where_68_goes

source theorem Tau.BookV.Thermodynamics.where_68_goes :”68% was never real energy; tau-budget = defect bundles + E_bdry on L” = “68% was never real energy; tau-budget = defect bundles + E_bdry on L”

[V.R135] Where does the 68% go? It was never a real energy component. Dark energy is not missing energy but misattributed refinement entropy.

The tau-cosmic budget: 100% = defect bundles on T^2 + finite boundary energy on L. No dark energy component exists.

Tau.BookV.Thermodynamics.DarkEnergyTestability

source structure Tau.BookV.Thermodynamics.DarkEnergyTestability :Type

[V.R136] Testability: w_eff(z) should vary with redshift.

  • w > -1/3 at high z (defect-dominated epoch)

  • w ~ -1 at low z (refinement-dominated)

  • Transition at z_acc ~ 0.7

Distinguishing prediction: w_eff is NOT exactly -1 but varies. Future measurements of w(z) can test the defect-transition model against a true cosmological constant (w = -1 exactly).

  • w_varies : Bool Prediction: w varies with redshift.

  • high_z_decelerating : Bool w > -1/3 at high z.

  • low_z_near_minus_one : Bool w ~ -1 at low z.

  • z_acc_numer : ℕ Transition redshift z_acc (numer/denom).

  • z_acc_denom : ℕ Transition redshift denominator.

  • denom_pos : self.z_acc_denom > 0 Denominator positive.

Instances For

Tau.BookV.Thermodynamics.instReprDarkEnergyTestability

source instance Tau.BookV.Thermodynamics.instReprDarkEnergyTestability :Repr DarkEnergyTestability

Equations

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

Tau.BookV.Thermodynamics.instReprDarkEnergyTestability.repr

source def Tau.BookV.Thermodynamics.instReprDarkEnergyTestability.repr :DarkEnergyTestability → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.dark_energy_testability

source def Tau.BookV.Thermodynamics.dark_energy_testability :DarkEnergyTestability

Canonical testability instance. Equations

  • Tau.BookV.Thermodynamics.dark_energy_testability = { denom_pos := Tau.BookV.Thermodynamics.dark_energy_testability._proof_2 } Instances For

Tau.BookV.Thermodynamics.w_varies_prediction

source theorem Tau.BookV.Thermodynamics.w_varies_prediction :dark_energy_testability.w_varies = true

The testable prediction: w varies (not exactly -1).

Tau.BookV.Thermodynamics.CosmicComponent

source inductive Tau.BookV.Thermodynamics.CosmicComponent :Type

The tau-cosmic energy budget: no dark energy component.

  • DefectBundles : CosmicComponent Defect bundles on T^2 (matter + radiation).

  • BoundaryEnergy : CosmicComponent Boundary energy on L (finite vacuum energy).

Instances For

Tau.BookV.Thermodynamics.instReprCosmicComponent

source instance Tau.BookV.Thermodynamics.instReprCosmicComponent :Repr CosmicComponent

Equations

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

Tau.BookV.Thermodynamics.instReprCosmicComponent.repr

source def Tau.BookV.Thermodynamics.instReprCosmicComponent.repr :CosmicComponent → ℕ → Std.Format

Equations

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Tau.BookV.Thermodynamics.instDecidableEqCosmicComponent

source instance Tau.BookV.Thermodynamics.instDecidableEqCosmicComponent :DecidableEq CosmicComponent

Equations

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

Tau.BookV.Thermodynamics.instBEqCosmicComponent

source instance Tau.BookV.Thermodynamics.instBEqCosmicComponent :BEq CosmicComponent

Equations

  • Tau.BookV.Thermodynamics.instBEqCosmicComponent = { beq := Tau.BookV.Thermodynamics.instBEqCosmicComponent.beq }

Tau.BookV.Thermodynamics.instBEqCosmicComponent.beq

source def Tau.BookV.Thermodynamics.instBEqCosmicComponent.beq :CosmicComponent → CosmicComponent → Bool

Equations

  • Tau.BookV.Thermodynamics.instBEqCosmicComponent.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Thermodynamics.cosmic_budget

source def Tau.BookV.Thermodynamics.cosmic_budget :List CosmicComponent

The cosmic budget has exactly 2 components (no dark energy). Equations

  • Tau.BookV.Thermodynamics.cosmic_budget = [Tau.BookV.Thermodynamics.CosmicComponent.DefectBundles, Tau.BookV.Thermodynamics.CosmicComponent.BoundaryEnergy] Instances For

Tau.BookV.Thermodynamics.cosmic_budget_two_components

source theorem Tau.BookV.Thermodynamics.cosmic_budget_two_components :cosmic_budget.length = 2

Two components, not three (no dark energy).

Tau.BookV.Thermodynamics.early_universe

source def Tau.BookV.Thermodynamics.early_universe :DefectDrivenAcceleration

Example: early universe (defect-dominated, decelerating). Equations

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

Tau.BookV.Thermodynamics.present_epoch

source def Tau.BookV.Thermodynamics.present_epoch :DefectDrivenAcceleration

Example: present epoch (refinement-dominated, accelerating). Equations

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

Tau.BookV.Thermodynamics.surplus_example

source def Tau.BookV.Thermodynamics.surplus_example :CapacitySurplus

Example: capacity surplus. Equations

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

Tau.BookV.Thermodynamics.testability

source def Tau.BookV.Thermodynamics.testability :DarkEnergyTestability

Testability structure. Equations

  • Tau.BookV.Thermodynamics.testability = { denom_pos := Tau.BookV.Thermodynamics.dark_energy_testability._proof_2 } Instances For

Tau.BookV.Thermodynamics.OmegaLambdaStandalone

source structure Tau.BookV.Thermodynamics.OmegaLambdaStandalone :Type

[V.T234] Standalone Ω_Λ structural theorem: Ω_Λ = κ_D · (1 + ι_τ³) = (1 − ι_τ)(1 + ι_τ³) ≈ 0.6849. Zero-parameter prediction from master constant ι_τ.

κ_D = D-sector coupling (gravity), ι_τ³ = fiber volume correction. Planck 2018: 0.6847 ± 0.0073. Deviation: +269 ppm (+0.03σ).

  • kappa_D_x10000 : ℕ κ_D numerator (scaled ×10000): (1 − ι_τ) ≈ 0.6587 → 6587.

  • iota_tau_cubed_x100000 : ℕ ι_τ³ numerator (scaled ×100000): ι_τ³ ≈ 0.03979 → 3979.

  • omega_lambda_x10000 : ℕ Ω_Λ (scaled ×10000): ≈ 0.6849 → 6849.

  • planck_x10000 : ℕ Planck 2018 value (scaled ×10000): 0.6847 → 6847.

  • deviation_ppm : ℤ Deviation in ppm (positive = τ exceeds Planck).

  • scope_tau_effective : Bool τ-effective scope: derived from κ_D and ι_τ only.

Instances For

Tau.BookV.Thermodynamics.instReprOmegaLambdaStandalone

source instance Tau.BookV.Thermodynamics.instReprOmegaLambdaStandalone :Repr OmegaLambdaStandalone

Equations

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

Tau.BookV.Thermodynamics.instReprOmegaLambdaStandalone.repr

source def Tau.BookV.Thermodynamics.instReprOmegaLambdaStandalone.repr :OmegaLambdaStandalone → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.omega_lambda_canonical

source def Tau.BookV.Thermodynamics.omega_lambda_canonical :OmegaLambdaStandalone

Canonical Ω_Λ instance. Equations

  • Tau.BookV.Thermodynamics.omega_lambda_canonical = { kappa_D_x10000 := 6587, iota_tau_cubed_x100000 := 3979, omega_lambda_x10000 := 6849, deviation_ppm := 269 } Instances For

Tau.BookV.Thermodynamics.omega_lambda_deviation

source theorem Tau.BookV.Thermodynamics.omega_lambda_deviation :omega_lambda_canonical.deviation_ppm = 269

Ω_Λ at +269 ppm from Planck.

Tau.BookV.Thermodynamics.omega_lambda_tau_effective

source theorem Tau.BookV.Thermodynamics.omega_lambda_tau_effective :omega_lambda_canonical.scope_tau_effective = true

Ω_Λ is τ-effective.

Tau.BookV.Thermodynamics.DefectFractionEoS

source structure Tau.BookV.Thermodynamics.DefectFractionEoS :Type

[V.D293] Defect fraction function: f_def(z) = S_def(z) / (S_def(z) + S_ref(z)). At z → ∞: f_def → 1. At z = 0: f_def → ι_τ³ ≈ 0.040.

[V.P159] Effective equation of state: w(z) = −1 + (2/3) · f_def(z)/(1 − f_def(z)). At z = 0: w₀ = ι_τ³ − 1 ≈ −0.960 (quintessence-like).

  • f_def_present_x10000 : ℕ Present defect fraction f_def(0) (scaled ×10000): ι_τ³ ≈ 0.0398 → 398.

  • w0_offset_x1000 : ℕ w₀ (scaled ×1000, offset from −1): ι_τ³ ≈ 0.040 → 40 means w₀ = −0.960.

  • w0_gt_minus_one : Bool w₀ > −1 (quintessence-like, no phantom).

  • transition_w_numer : ℤ Transition value: w = −1/3 at z_acc.

  • transition_w_denom : ℕ Instances For

Tau.BookV.Thermodynamics.instReprDefectFractionEoS.repr

source def Tau.BookV.Thermodynamics.instReprDefectFractionEoS.repr :DefectFractionEoS → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprDefectFractionEoS

source instance Tau.BookV.Thermodynamics.instReprDefectFractionEoS :Repr DefectFractionEoS

Equations

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

Tau.BookV.Thermodynamics.defect_eos_canonical

source def Tau.BookV.Thermodynamics.defect_eos_canonical :DefectFractionEoS

Canonical EoS instance. Equations

  • Tau.BookV.Thermodynamics.defect_eos_canonical = { f_def_present_x10000 := 398, w0_offset_x1000 := 40 } Instances For

Tau.BookV.Thermodynamics.w0_quintessence

source theorem Tau.BookV.Thermodynamics.w0_quintessence :defect_eos_canonical.w0_gt_minus_one = true

w₀ > −1: quintessence-like, no phantom crossing.

Tau.BookV.Thermodynamics.TransitionRedshift

source structure Tau.BookV.Thermodynamics.TransitionRedshift :Type

[V.D294] Transition redshift z_acc = (2Ω_Λ/Ω_m)^(1/3) − 1 ≈ 0.632. Observed: 0.64 ± 0.05 (SN Ia). Deviation: −1.3%.

[V.R418] Comparison: τ-prediction within observational uncertainty.

  • z_acc_x1000 : ℕ z_acc (scaled ×1000): 0.632 → 632.

  • observed_x1000 : ℕ Observed central value (scaled ×1000): 0.64 → 640.

  • uncertainty_x1000 : ℕ Observed uncertainty (scaled ×1000): 0.05 → 50.

  • deviation_ppm : ℤ Deviation from observed (ppm, negative = τ below).

Instances For

Tau.BookV.Thermodynamics.instReprTransitionRedshift.repr

source def Tau.BookV.Thermodynamics.instReprTransitionRedshift.repr :TransitionRedshift → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprTransitionRedshift

source instance Tau.BookV.Thermodynamics.instReprTransitionRedshift :Repr TransitionRedshift

Equations

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

Tau.BookV.Thermodynamics.z_acc_canonical

source def Tau.BookV.Thermodynamics.z_acc_canonical :TransitionRedshift

Canonical z_acc instance. Equations

  • Tau.BookV.Thermodynamics.z_acc_canonical = { z_acc_x1000 := 632, deviation_ppm := -12500 } Instances For

Tau.BookV.Thermodynamics.z_acc_within_1sigma

source theorem Tau.BookV.Thermodynamics.z_acc_within_1sigma :z_acc_canonical.z_acc_x1000 ≥ z_acc_canonical.observed_x1000 - z_acc_canonical.uncertainty_x1000

z_acc within 1σ of observations.