TauLib.BookIV.MassDerivation.HolonomyDetail
TauLib.BookIV.MassDerivation.HolonomyDetail
Triple holonomy H₃ = ∮{T_π}·∮{T_γ}·∮_{T_η} and the π³α² correction, connecting the breathing modes module to the holonomy correction module.
Registry Cross-References
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[IV.D314] Triple Holonomy H₃ —
TripleHolonomyH3 -
[IV.T116] Holonomy Correction Range —
holonomy_in_range -
[IV.D315] Holonomy Correction Data —
HolonomyCorrectionDetail
Mathematical Content
Triple Holonomy
The fibered product τ³ = τ¹ ×_f T² contains three independent U(1) circles:
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T_π: the base circle in τ¹ (generator π, temporal)
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T_γ: the first fiber circle in T² (generator γ, EM sector)
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T_η: the second fiber circle in T² (generator η, Strong sector)
The triple holonomy is the product of Wilson loops around these circles:
H₃ = ∮_{T_π} · ∮_{T_γ} · ∮_{T_η}
Each loop contributes one factor of π (from ∮ dθ = 2π, normalized), giving the prefactor π³ ≈ 31.006 in the holonomy correction.
The Correction π³α²
The full holonomy correction to the mass ratio is π³α²·ι_τ^(-2), where:
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π³ ≈ 31.006: from the three circles (triple holonomy)
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α² ≈ 5.3×10⁻⁵: from charge conjugation (kills first-order α)
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ι_τ^(-2) ≈ 8.58: from the breathing operator scale
Combined: π³α²·ι_τ^(-2) ≈ 0.014 (Level 1+ correction to R).
This module wraps the HolonomyCorrection module from Physics and connects it to the breathing modes framework from BreathingModes.
Scope
All claims: tau-effective.
Ground Truth Sources
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holonomy_correction_sprint.md §3-§7
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electron_mass_first_principles.md §28, §37
Tau.BookIV.MassDerivation.TripleHolonomyH3
source structure Tau.BookIV.MassDerivation.TripleHolonomyH3 :Type
[IV.D314] Triple holonomy H₃ = ∮{T_π}·∮{T_γ}·∮_{T_η}.
The product of Wilson loops around the three independent U(1) circles in τ³ = τ¹ ×_f T². Each circle contributes one factor of π, giving π³ as the holonomy prefactor.
Circle assignments:
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T_π: base τ¹ circle (generator π, Weak/temporal sector A)
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T_γ: first fiber circle (generator γ, EM sector B)
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T_η: second fiber circle (generator η, Strong sector C)
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circle_count : ℕ Number of independent circles.
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three_circles : self.circle_count = 3 Circle count is 3.
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generators : List String Generator labels for the three circles.
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gen_count : self.generators.length = self.circle_count Generators list has correct length.
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pi_exponent : ℕ π exponent matches circle count.
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exp_eq : self.pi_exponent = self.circle_count Exponent = circle count.
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pi_cubed_n : ℕ π³ rational approximation numerator.
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pi_cubed_d : ℕ π³ rational approximation denominator.
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denom_pos : self.pi_cubed_d > 0 Denominator positive.
Instances For
Tau.BookIV.MassDerivation.instReprTripleHolonomyH3.repr
source def Tau.BookIV.MassDerivation.instReprTripleHolonomyH3.repr :TripleHolonomyH3 → ℕ → Std.Format
Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.MassDerivation.instReprTripleHolonomyH3
source instance Tau.BookIV.MassDerivation.instReprTripleHolonomyH3 :Repr TripleHolonomyH3
Equations
- Tau.BookIV.MassDerivation.instReprTripleHolonomyH3 = { reprPrec := Tau.BookIV.MassDerivation.instReprTripleHolonomyH3.repr }
Tau.BookIV.MassDerivation.triple_holonomy_H3
source def Tau.BookIV.MassDerivation.triple_holonomy_H3 :TripleHolonomyH3
The canonical triple holonomy of τ³. Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.MassDerivation.holonomy_three_circles
source theorem Tau.BookIV.MassDerivation.holonomy_three_circles :triple_holonomy_H3.circle_count = 3
Three circles in the holonomy.
Tau.BookIV.MassDerivation.holonomy_three_generators
source theorem Tau.BookIV.MassDerivation.holonomy_three_generators :triple_holonomy_H3.generators.length = 3
Three generators listed.
Tau.BookIV.MassDerivation.holonomy_pi_exponent
source theorem Tau.BookIV.MassDerivation.holonomy_pi_exponent :triple_holonomy_H3.pi_exponent = 3
The π exponent is 3 (one per circle).
Tau.BookIV.MassDerivation.holonomy_in_range
source theorem Tau.BookIV.MassDerivation.holonomy_in_range :Physics.holonomy_correction.numer * 1000 > Physics.holonomy_correction.denom ∧ Physics.holonomy_correction.numer * 1000 < 2 * Physics.holonomy_correction.denom
[IV.T116] The holonomy correction π³α² is in (0.001, 0.002).
This wraps the range proofs from HolonomyCorrection:
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π³α² > 0.001 (correction_gt_1_per_mille)
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π³α² < 0.002 (correction_lt_2_per_mille)
Combined: π³α² ∈ (0.001, 0.002), confirming the perturbative hierarchy (π³α² « √3 by a factor of ~1050).
Tau.BookIV.MassDerivation.holonomy_perturbative
source theorem Tau.BookIV.MassDerivation.holonomy_perturbative :Physics.pi_cubed_numer * Physics.alpha_sq_numer * 1000 * 10000000 < 17320508 * Physics.pi_cubed_denom * Physics.alpha_sq_denom
The holonomy correction is perturbatively small relative to √3.
Tau.BookIV.MassDerivation.HolonomyCorrectionDetail
source structure Tau.BookIV.MassDerivation.HolonomyCorrectionDetail :Type
[IV.D315] Holonomy correction data: the three components that make up the Level 1+ correction π³α²·ι_τ^(-2).
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π³ from triple holonomy (31.006)
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α² from charge conjugation (5.3 × 10⁻⁵)
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ι_τ^(-2) from breathing operator scale (8.58)
The combined correction π³α²·ι_τ^(-2) ≈ 0.014 refines the mass ratio from Level 0 (7.7 ppm) to Level 1+ (0.025 ppm).
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pi3_numer : ℕ π³ numerator.
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pi3_denom : ℕ π³ denominator.
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alpha2_numer : ℕ α² numerator.
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alpha2_denom : ℕ α² denominator.
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iota_neg2_n : ℕ ι_τ^(-2) numerator.
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iota_neg2_d : ℕ ι_τ^(-2) denominator.
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pi3_denom_pos : self.pi3_denom > 0 All denominators positive.
- alpha2_denom_pos : self.alpha2_denom > 0
- iota_neg2_denom_pos : self.iota_neg2_d > 0 Instances For
Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail
source instance Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail :Repr HolonomyCorrectionDetail
Equations
- Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail = { reprPrec := Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail.repr }
Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail.repr
source def Tau.BookIV.MassDerivation.instReprHolonomyCorrectionDetail.repr :HolonomyCorrectionDetail → ℕ → Std.Format
Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.MassDerivation.holonomy_detail
source def Tau.BookIV.MassDerivation.holonomy_detail :HolonomyCorrectionDetail
The canonical holonomy correction detail. Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.MassDerivation.pi3_matches
source theorem Tau.BookIV.MassDerivation.pi3_matches :holonomy_detail.pi3_numer = Physics.pi_cubed_numer ∧ holonomy_detail.pi3_denom = Physics.pi_cubed_denom
π³ component matches the holonomy correction module.
Tau.BookIV.MassDerivation.alpha2_matches
source theorem Tau.BookIV.MassDerivation.alpha2_matches :holonomy_detail.alpha2_numer = Physics.alpha_sq_numer ∧ holonomy_detail.alpha2_denom = Physics.alpha_sq_denom
α² component matches the holonomy correction module.
Tau.BookIV.MassDerivation.holonomy_correction_float
source def Tau.BookIV.MassDerivation.holonomy_correction_float :Float
The combined correction as Float (π³ × α² × ι_τ^(-2)). Equations
- One or more equations did not get rendered due to their size. Instances For
Tau.BookIV.MassDerivation.breathing_epstein_shape_match
source theorem Tau.BookIV.MassDerivation.breathing_epstein_shape_match :breathing_spectrum.shape_numer = epstein_on_T2.zeta.shape_numer ∧ breathing_spectrum.shape_denom = epstein_on_T2.zeta.shape_denom
The breathing operator and Epstein zeta share the same shape parameter.
Tau.BookIV.MassDerivation.charge_conjugation_instance
source def Tau.BookIV.MassDerivation.charge_conjugation_instance :Physics.ChargConjugation
Charge conjugation gives α² (second order), not α (first order). Equations
- Tau.BookIV.MassDerivation.charge_conjugation_instance = { } Instances For
Tau.BookIV.MassDerivation.charge_conjugation_order
source theorem Tau.BookIV.MassDerivation.charge_conjugation_order :charge_conjugation_instance.surviving_order = 2