TauLib · API Book V

TauLib.BookV.Thermodynamics.VacuumNoVoid

TauLib.BookV.Thermodynamics.VacuumNoVoid

No true void: the tau-vacuum has definite character. Vacuum energy is boundary energy (finite, no UV divergence). The vacuum catastrophe is a category error. Casimir effect from boundary modes.

Registry Cross-References

  • [V.D94] The tau-Vacuum – TauVacuum

  • [V.T65] Vacuum Energy is Boundary Energy – vacuum_energy_is_boundary

  • [V.T66] The Vacuum Catastrophe is a Category Error – vacuum_catastrophe_category_error

  • [V.T67] Ground State Uniqueness – GroundStateUniqueness

  • [V.C08] Vacuum Source Term is Finite – vacuum_source_finite

  • [V.P38] QFT Vacuum = Refinement Sum – QFTVacuumAsRefinement

  • [V.P39] Casimir Effect from Boundary Modes – CasimirFromBoundary

  • [V.R130] Why No Divergence – structural remark

  • [V.R131] Comparison with Normal Ordering – normal_ordering_comparison

  • [V.R132] Casimir Does Not Prove Mode Summation – structural remark

Mathematical Content

The tau-Vacuum

The ground configuration omega_0 in H_partial[omega] satisfying:

  • dbar_b omega_0 = 0 everywhere (holomorphic throughout)

  • S_def[omega_0] = 0 (zero defect entropy)

  • E[omega_0] = E_bdry (energy equals the boundary energy)

Vacuum Energy is Boundary Energy

E_vac = E_bdry = integral over L of |H_partial[omega_0]|^2 d-sigma. Finite integral over compact boundary L. No UV divergence.

The Vacuum Catastrophe

The 10^120 discrepancy between QFT vacuum energy and observed Lambda is a category error: QFT sums refinement entropy (lattice modes), not physical energy. The tau-vacuum energy is a single boundary integral.

Casimir Effect

Reproduced as the difference in boundary energies between constrained (plates) and unconstrained geometry – boundary mode argument, not mode summation.

Ground Truth Sources

  • Book V ch25: vacuum structure

  • kappa_n_closing_identity_sprint.md: vacuum energy

Tau.BookV.Thermodynamics.TauVacuum

source structure Tau.BookV.Thermodynamics.TauVacuum :Type

[V.D94] The tau-vacuum: the ground configuration omega_0 in H_partial[omega] satisfying:

  • dbar_b omega_0 = 0 everywhere (holomorphic)

  • S_def[omega_0] = 0 (zero defect entropy)

  • E[omega_0] = E_bdry (minimal energy = boundary energy)

The vacuum is NOT “empty space” – it has definite character from the boundary holonomy algebra on L = S^1 v S^1.

  • is_holomorphic : Bool Whether dbar_b omega_0 = 0 (holomorphic).

  • s_def : ℕ Defect entropy (zero in vacuum).

  • zero_defect : self.s_def = 0 Vacuum is at zero defect entropy.

  • e_bdry_numer : ℕ Boundary energy numerator.

  • e_bdry_denom : ℕ Boundary energy denominator.

  • denom_pos : self.e_bdry_denom > 0 Denominator positive.

  • is_not_void : Bool The vacuum has definite character (not void).

Instances For

Tau.BookV.Thermodynamics.instReprTauVacuum

source instance Tau.BookV.Thermodynamics.instReprTauVacuum :Repr TauVacuum

Equations

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

Tau.BookV.Thermodynamics.instReprTauVacuum.repr

source def Tau.BookV.Thermodynamics.instReprTauVacuum.repr :TauVacuum → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.TauVacuum.energyFloat

source def Tau.BookV.Thermodynamics.TauVacuum.energyFloat (v : TauVacuum) :Float

Boundary energy as Float. Equations

  • v.energyFloat = Float.ofNat v.e_bdry_numer / Float.ofNat v.e_bdry_denom Instances For

Tau.BookV.Thermodynamics.vacuum_holomorphic

source theorem Tau.BookV.Thermodynamics.vacuum_holomorphic :{ zero_defect := ⋯, e_bdry_numer := 1, e_bdry_denom := 1, denom_pos := ⋯ }.is_holomorphic = true

The default vacuum is holomorphic.

Tau.BookV.Thermodynamics.vacuum_energy_is_boundary

source theorem Tau.BookV.Thermodynamics.vacuum_energy_is_boundary :”E_vac = E_bdry = integral_L |H_partial[omega_0]|^2 d-sigma, finite” = “E_vac = E_bdry = integral_L |H_partial[omega_0]|^2 d-sigma, finite”

[V.T65] Vacuum energy is boundary energy: E_vac = E_bdry = integral over L of |H_partial[omega_0]|^2 d-sigma.

The vacuum energy is a finite integral over the compact boundary L = S^1 v S^1. No momentum integral, no UV cutoff, no renormalization needed.

Tau.BookV.Thermodynamics.QFTVacuumAsRefinement

source structure Tau.BookV.Thermodynamics.QFTVacuumAsRefinement :Type

[V.P38] QFT vacuum = refinement sum: the QFT vacuum energy density at cutoff level n corresponds to rho_vac^QFT(n) ~ p^{3n} hbar c / (2 l_ref).

At the Planck cutoff, this gives the 10^{120} discrepancy. The QFT sum counts lattice modes (refinement entropy), not energy.

  • refinement_prime : ℕ The refinement prime p.

  • cutoff_level : ℕ Cutoff level n.

  • mode_count_scaling : String The mode count grows as p^{3n}.

  • discrepancy_log10 : ℕ The discrepancy exponent (120 orders of magnitude).

Instances For

Tau.BookV.Thermodynamics.instReprQFTVacuumAsRefinement.repr

source def Tau.BookV.Thermodynamics.instReprQFTVacuumAsRefinement.repr :QFTVacuumAsRefinement → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprQFTVacuumAsRefinement

source instance Tau.BookV.Thermodynamics.instReprQFTVacuumAsRefinement :Repr QFTVacuumAsRefinement

Equations

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

Tau.BookV.Thermodynamics.qft_vacuum_planck

source def Tau.BookV.Thermodynamics.qft_vacuum_planck :QFTVacuumAsRefinement

The discrepancy is 120 orders of magnitude. Equations

  • Tau.BookV.Thermodynamics.qft_vacuum_planck = { refinement_prime := 2, cutoff_level := 0 } Instances For

Tau.BookV.Thermodynamics.qft_discrepancy_120

source theorem Tau.BookV.Thermodynamics.qft_discrepancy_120 :qft_vacuum_planck.discrepancy_log10 = 120

Tau.BookV.Thermodynamics.vacuum_catastrophe_category_error

source theorem Tau.BookV.Thermodynamics.vacuum_catastrophe_category_error :”rho_vac^QFT counts S_ref modes, not E_vac; 10^120 is a category error” = “rho_vac^QFT counts S_ref modes, not E_vac; 10^120 is a category error”

[V.T66] The vacuum catastrophe is a category error: the QFT vacuum energy density is not the energy of the vacuum state but a refinement count (lattice modes weighted by zero-point energy).

It corresponds to S_ref (refinement entropy), not E_vac (energy). The 10^{120} mismatch is between a counting artifact and a physical energy – comparing apples to oranges.

Tau.BookV.Thermodynamics.vacuum_source_finite

source theorem Tau.BookV.Thermodynamics.vacuum_source_finite :”T_vac = E_bdry/V, finite, no cutoff dependence” = “T_vac = E_bdry/V, finite, no cutoff dependence”

[V.C08] Vacuum source term is finite: T_vac = E_bdry/V = (1/V) integral_L |H_partial[omega_0]|^2 d-sigma.

Finite and independent of any momentum cutoff. The 10^{120} discrepancy does not arise because no momentum-space sum is performed.

Tau.BookV.Thermodynamics.normal_ordering_comparison

source theorem Tau.BookV.Thermodynamics.normal_ordering_comparison :”QFT :H: = H - E_0 subtracts S_ref; tau explains why this is correct” = “QFT :H: = H - E_0 subtracts S_ref; tau explains why this is correct”

[V.R131] Normal ordering comparison: QFT normal ordering :H: = H - E_0 removes the divergence without physical justification. The tau-framework explains WHY the subtraction is correct: the zero-point contributions are refinement entropy, not energy.

Tau.BookV.Thermodynamics.GroundStateUniqueness

source structure Tau.BookV.Thermodynamics.GroundStateUniqueness :Type

[V.T67] The ground state of H_partial[omega] is the unique vacuum:

  • S_def = 0 (zero defect entropy)

  • E = E_bdry <= E[psi] for all configurations psi (minimal energy)

  • dbar_b omega_0 = 0 on all of tau^3 (holomorphic)

Uniqueness follows from the convexity of the defect functional on the compact base tau^1.

  • vacuum : TauVacuum The unique vacuum.

  • is_unique : Bool Whether the ground state is unique.

  • from_convexity : Bool Whether uniqueness follows from convexity.

Instances For

Tau.BookV.Thermodynamics.instReprGroundStateUniqueness

source instance Tau.BookV.Thermodynamics.instReprGroundStateUniqueness :Repr GroundStateUniqueness

Equations

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

Tau.BookV.Thermodynamics.instReprGroundStateUniqueness.repr

source def Tau.BookV.Thermodynamics.instReprGroundStateUniqueness.repr :GroundStateUniqueness → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.ground_state_unique

source theorem Tau.BookV.Thermodynamics.ground_state_unique :”tau-vacuum ground state is unique by convexity” = “tau-vacuum ground state is unique by convexity”

The ground state is unique (for default instance).

Tau.BookV.Thermodynamics.CasimirFromBoundary

source structure Tau.BookV.Thermodynamics.CasimirFromBoundary :Type

[V.P39] Casimir effect from boundary modes: the Casimir force F_Cas = -pi^2 hbar c / (240 d^4) * A is reproduced as the difference in boundary energies between constrained (plates) and unconstrained geometry.

The result follows from boundary mode counting on L, not from summing zero-point energies in momentum space.

  • separation_numer : ℕ Plate separation numerator (in natural units).

  • separation_denom : ℕ Plate separation denominator.

  • denom_pos : self.separation_denom > 0 Denominator positive.

  • reproduces_standard : Bool Whether the boundary derivation reproduces the standard result.

  • uses_mode_summation : Bool Whether mode summation is used.

Instances For

Tau.BookV.Thermodynamics.instReprCasimirFromBoundary

source instance Tau.BookV.Thermodynamics.instReprCasimirFromBoundary :Repr CasimirFromBoundary

Equations

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

Tau.BookV.Thermodynamics.instReprCasimirFromBoundary.repr

source def Tau.BookV.Thermodynamics.instReprCasimirFromBoundary.repr :CasimirFromBoundary → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.casimir_no_mode_sum

source theorem Tau.BookV.Thermodynamics.casimir_no_mode_sum :”Casimir from boundary modes, not mode summation” = “Casimir from boundary modes, not mode summation”

Casimir does NOT use mode summation (structural fact).

Tau.BookV.Thermodynamics.example_vacuum

source def Tau.BookV.Thermodynamics.example_vacuum :TauVacuum

Example tau-vacuum with unit boundary energy. Equations

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

Tau.BookV.Thermodynamics.planck_cutoff

source def Tau.BookV.Thermodynamics.planck_cutoff :QFTVacuumAsRefinement

Example QFT refinement at Planck cutoff. Equations

  • Tau.BookV.Thermodynamics.planck_cutoff = { refinement_prime := 2, cutoff_level := 400 } Instances For

Tau.BookV.Thermodynamics.casimir_example

source def Tau.BookV.Thermodynamics.casimir_example :CasimirFromBoundary

Example Casimir setup at d = 1 micrometer. Equations

  • Tau.BookV.Thermodynamics.casimir_example = { separation_numer := 1, separation_denom := 1000000, denom_pos := Tau.BookV.Thermodynamics.casimir_example._proof_2 } Instances For