TauLib · API Book IV

TauLib.BookIV.MassDerivation.BreathingModes

Breathing operator, torus spectral modes, three-fold lemniscate, and their connection to the Epstein zeta function and √3 spectral distance.

Registry Cross-References

[IV.R336] Three-tier mass hierarchy — MassHierarchy
[IV.D309] Breathing operator — BreathingOperator
[IV.P171] Breathing spectrum discrete — breathing_spectrum_discrete
[IV.D310] Epstein zeta on T² — EpsteinZetaOnT2
[IV.R337] Toroidal modes 99.95% — toroidal_dominance
[IV.D311] Chowla-Selberg — wraps ChowlaSelbergTerms
[IV.T114] Leading term ∝ ι_τ^{-7} — leading_exponent_seven
[IV.R338] s=4 forced — s4_forced
[IV.D312] Three-fold lemniscate — ThreeFoldLemniscate
[IV.D313] Spectral distance √3 — spectral_distance_sq
[IV.T115] d²=3 — adjacent_distance_sq_is_3
[IV.R339-R343] structural remarks

Ground Truth Sources

electron_mass_first_principles.md §22-§28
sqrt3_derivation_sprint.md §11-§15

`Tau.BookIV.MassDerivation.MassHierarchy`

source structure Tau.BookIV.MassDerivation.MassHierarchy :Type

[IV.R336] Three-tier mass hierarchy: bulk ι_τ^(-7) ≈ 1848, surface √3·ι_τ^(-2) ≈ 14.9, coupling π³α²·ι_τ^(-2) ≈ 0.014.

bulk_approx : ℕ
surface_approx : ℕ
coupling_approx : ℕ
bulk_gt_surface : self.bulk_approx > self.surface_approx
surface_gt_coupling : self.surface_approx > self.coupling_approx Instances For

`Tau.BookIV.MassDerivation.instReprMassHierarchy.repr`

source def Tau.BookIV.MassDerivation.instReprMassHierarchy.repr :MassHierarchy → ℕ → Std.Format

Equations

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

`Tau.BookIV.MassDerivation.instReprMassHierarchy`

source instance Tau.BookIV.MassDerivation.instReprMassHierarchy :Repr MassHierarchy

Equations

Tau.BookIV.MassDerivation.instReprMassHierarchy = { reprPrec := Tau.BookIV.MassDerivation.instReprMassHierarchy.repr }

`Tau.BookIV.MassDerivation.mass_hierarchy`

source def Tau.BookIV.MassDerivation.mass_hierarchy :MassHierarchy

Equations

Tau.BookIV.MassDerivation.mass_hierarchy = { bulk_gt_surface := Tau.BookIV.MassDerivation.mass_hierarchy._proof_3, surface_gt_coupling := Tau.BookIV.MassDerivation.mass_hierarchy._proof_4 } Instances For

`Tau.BookIV.MassDerivation.BreathingOperator`

source structure Tau.BookIV.MassDerivation.BreathingOperator :Type

[IV.D309] Breathing operator B = (1/ι_τ²)·Δ_Hodge⁻¹ on fiber T².

inv_coeff_numer : ℕ
inv_coeff_denom : ℕ
denom_pos : self.inv_coeff_denom > 0
fiber_restricted : Bool Instances For

`Tau.BookIV.MassDerivation.instReprBreathingOperator`

source instance Tau.BookIV.MassDerivation.instReprBreathingOperator :Repr BreathingOperator

Equations

Tau.BookIV.MassDerivation.instReprBreathingOperator = { reprPrec := Tau.BookIV.MassDerivation.instReprBreathingOperator.repr }

`Tau.BookIV.MassDerivation.instReprBreathingOperator.repr`

source def Tau.BookIV.MassDerivation.instReprBreathingOperator.repr :BreathingOperator → ℕ → Std.Format

Equations

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

`Tau.BookIV.MassDerivation.breathing_operator`

source def Tau.BookIV.MassDerivation.breathing_operator :BreathingOperator

Equations

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

`Tau.BookIV.MassDerivation.breathing_is_inverse_iota_sq`

source theorem Tau.BookIV.MassDerivation.breathing_is_inverse_iota_sq :breathing_operator.inv_coeff_numer = Sectors.iota_sq_denom ∧ breathing_operator.inv_coeff_denom = Sectors.iota_sq_numer

`Tau.BookIV.MassDerivation.BreathingSpectrum`

source structure Tau.BookIV.MassDerivation.BreathingSpectrum :Type

[IV.P171] Breathing spectrum on T²: discrete positive eigenvalues.

is_discrete : Bool
all_positive : Bool
shape_numer : ℕ
shape_denom : ℕ
denom_pos : self.shape_denom > 0 Instances For

`Tau.BookIV.MassDerivation.instReprBreathingSpectrum.repr`

source def Tau.BookIV.MassDerivation.instReprBreathingSpectrum.repr :BreathingSpectrum → ℕ → Std.Format

Equations

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

`Tau.BookIV.MassDerivation.instReprBreathingSpectrum`

source instance Tau.BookIV.MassDerivation.instReprBreathingSpectrum :Repr BreathingSpectrum

Equations

Tau.BookIV.MassDerivation.instReprBreathingSpectrum = { reprPrec := Tau.BookIV.MassDerivation.instReprBreathingSpectrum.repr }

`Tau.BookIV.MassDerivation.breathing_spectrum`

source def Tau.BookIV.MassDerivation.breathing_spectrum :BreathingSpectrum

Equations

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

`Tau.BookIV.MassDerivation.breathing_spectrum_discrete`

source theorem Tau.BookIV.MassDerivation.breathing_spectrum_discrete :breathing_spectrum.is_discrete = true

`Tau.BookIV.MassDerivation.EpsteinZetaOnT2`

source structure Tau.BookIV.MassDerivation.EpsteinZetaOnT2 :Type

[IV.D310] Epstein zeta Z(s; iι_τ) as spectral zeta of B on T².

zeta : Calibration.EpsteinZetaStructure
interpretation : String Instances For

`Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2.repr`

source def Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2.repr :EpsteinZetaOnT2 → ℕ → Std.Format

Equations

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

`Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2`

source instance Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2 :Repr EpsteinZetaOnT2

Equations

Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2 = { reprPrec := Tau.BookIV.MassDerivation.instReprEpsteinZetaOnT2.repr }

`Tau.BookIV.MassDerivation.epstein_on_T2`

source def Tau.BookIV.MassDerivation.epstein_on_T2 :EpsteinZetaOnT2

Equations

Tau.BookIV.MassDerivation.epstein_on_T2 = { zeta := Tau.BookIV.Calibration.epstein_at_T2 } Instances For

`Tau.BookIV.MassDerivation.epstein_shape_is_iota`

source theorem Tau.BookIV.MassDerivation.epstein_shape_is_iota :epstein_on_T2.zeta.shape_numer = Boundary.iota_tau_numer ∧ epstein_on_T2.zeta.shape_denom = Boundary.iota_tau_denom

`Tau.BookIV.MassDerivation.toroidal_dominance`

source theorem Tau.BookIV.MassDerivation.toroidal_dominance :Calibration.n_axis_dominant.dominance_lower_bound > 9900

[IV.R337] n-axis modes contribute 99.95% of Z(4).

`Tau.BookIV.MassDerivation.chowla_selberg_data`

source def Tau.BookIV.MassDerivation.chowla_selberg_data :Calibration.ChowlaSelbergTerms

[IV.D311] Chowla-Selberg data at s=4. Equations

Tau.BookIV.MassDerivation.chowla_selberg_data = Tau.BookIV.Calibration.chowla_selberg_s4 Instances For

`Tau.BookIV.MassDerivation.leading_exponent_seven`

source theorem Tau.BookIV.MassDerivation.leading_exponent_seven :Calibration.chowla_selberg_s4.leading_exp = -7

[IV.T114] Leading term ∝ ι_τ^{-7} (exponent = 1−2×4 = −7).

`Tau.BookIV.MassDerivation.s4_forced`

source theorem Tau.BookIV.MassDerivation.s4_forced (s : ℕ) :1 - 2 * ↑s = -7 → s = 4

[IV.R338] s=4 forced by mass operator: 1−2s = −7 → s = 4.

`Tau.BookIV.MassDerivation.ThreeFoldLemniscate`

source structure Tau.BookIV.MassDerivation.ThreeFoldLemniscate :Type

[IV.D312] Three-fold lemniscate: Lobe_B, Lobe_C, Crossing_ω.

three_fold_data : Physics.LemniscateThreeFold
sector_B : String
sector_C : String
sector_omega : String Instances For

`Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate`

source instance Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate :Repr ThreeFoldLemniscate

Equations

Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate = { reprPrec := Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate.repr }

`Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate.repr`

source def Tau.BookIV.MassDerivation.instReprThreeFoldLemniscate.repr :ThreeFoldLemniscate → ℕ → Std.Format

Equations

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

`Tau.BookIV.MassDerivation.three_fold_lemniscate`

source def Tau.BookIV.MassDerivation.three_fold_lemniscate :ThreeFoldLemniscate

Equations

Tau.BookIV.MassDerivation.three_fold_lemniscate = { three_fold_data := Tau.BookIV.Physics.three_fold } Instances For

`Tau.BookIV.MassDerivation.three_supports`

source theorem Tau.BookIV.MassDerivation.three_supports :three_fold_lemniscate.three_fold_data.supports.length = 3

`Tau.BookIV.MassDerivation.spectral_distance_sq`

source theorem Tau.BookIV.MassDerivation.spectral_distance_sq :three_fold_lemniscate.three_fold_data.distance_sq = 3

[IV.D313] Spectral distance² = 3 (distance = √3).

`Tau.BookIV.MassDerivation.adjacent_distance_sq_is_3`

source theorem Tau.BookIV.MassDerivation.adjacent_distance_sq_is_3 :Physics.omega_real_sq + Physics.omega_imag_sq = 3 * Physics.omega_denom

[IV.T115] d²=3 from

1 − e^{2πi/3}

² = (3/2)² + (√3/2)² = 3.

`Tau.BookIV.MassDerivation.bulk_dominates_surface`

source theorem Tau.BookIV.MassDerivation.bulk_dominates_surface :mass_hierarchy.bulk_approx > 100 * mass_hierarchy.surface_approx

`Tau.BookIV.MassDerivation.surface_dominates_coupling`

source theorem Tau.BookIV.MassDerivation.surface_dominates_coupling :mass_hierarchy.surface_approx > 100 * mass_hierarchy.coupling_approx