TauLib · API Book IV

TauLib.BookIV.Electroweak.EWSynthesis

TauLib.BookIV.Electroweak.EWSynthesis

Electroweak synthesis: the complete EW prediction table, Yukawa ordering, the sqrt(3) triad, and structural closure — nine EW quantities from iota_tau and m_n alone, zero free parameters.

Registry Cross-References

  • [IV.D143] τ-Yukawa Coupling (Full Definition) — YukawaCouplingFull

  • [IV.T66] Nine EW Quantities from ι_τ and m_n — nine_ew_quantities

  • [IV.T67] Every EW Prediction Traces to ι_τ + K0-K6 — ew_traces_to_axioms

  • [IV.T124] √3 Triad Theorem — Sqrt3Triad, sqrt3_triad_count

  • [IV.P78] Yukawa Ordering Follows Sector Hierarchy — yukawa_ordering

  • [IV.P79] Zero Free Parameters vs SM’s 19 — zero_vs_nineteen

  • [IV.P175] Three Lemniscate Supports — three_lemniscate_supports

  • [IV.R37] EW Synthesis Complete — structural remark

  • [IV.R38] No BSM Particles Required — structural remark

  • [IV.R39] Experimental Test Program — structural remark

  • [IV.R40] Connection to Book V Cosmology — structural remark

Mathematical Content

This module synthesizes the electroweak sector of Book IV. The key result (IV.T66) is that ALL nine electroweak quantities — three boson masses (W, Z, H), the VEV, the Weinberg angle, three Yukawa couplings (top, bottom, electron), and the fine structure constant — are determined by exactly two inputs: ι_τ = 2/(π+e) and the neutron mass anchor m_n.

The SM requires 19 free parameters for the same predictions. The τ-framework achieves zero free parameters by deriving everything from the 7 axioms K0-K6.

The √3 triad theorem (IV.T124) reveals that the same algebraic quantity √3 = |1 − ω| (where ω = e^{2πi/3}) appears in three independent physical contexts: the R mass ratio correction, the proton-neutron mass splitting, and the gravitational closing identity.

Ground Truth Sources

  • Chapter 35 of Book IV (2nd Edition)

  • electron_mass_first_principles.md (master synthesis)

  • holonomy_correction_sprint.md §14 (√3 triad)

Tau.BookIV.Electroweak.YukawaCouplingFull

source structure Tau.BookIV.Electroweak.YukawaCouplingFull :Type

[IV.D143] Full Yukawa coupling definition: associates each fermion flavor with its coupling strength, the sector that determines it, and the generation index.

The coupling is determined by the sector hierarchy:

  • 3rd gen (top): coupling ≈ 1 (O(ι_τ⁰))

  • 2nd gen (charm, muon): coupling ≈ ι_τ² (O(ι_τ²))

  • 1st gen (up, electron): coupling ≈ ι_τ⁴ (O(ι_τ⁴))

Each step down in generation multiplies by ι_τ².

  • flavor : String Fermion flavor label.

  • generation : ℕ Generation (1, 2, or 3).

  • coupling_numer : ℕ Coupling numerator (scaled ×10⁶).

  • coupling_denom : ℕ Coupling denominator.

  • denom_pos : self.coupling_denom > 0 Denominator positive.

  • gen_valid : self.generation ≥ 1 ∧ self.generation ≤ 3 Generation is 1, 2, or 3.

Instances For

Tau.BookIV.Electroweak.instReprYukawaCouplingFull

source instance Tau.BookIV.Electroweak.instReprYukawaCouplingFull :Repr YukawaCouplingFull

Equations

  • Tau.BookIV.Electroweak.instReprYukawaCouplingFull = { reprPrec := Tau.BookIV.Electroweak.instReprYukawaCouplingFull.repr }

Tau.BookIV.Electroweak.instReprYukawaCouplingFull.repr

source def Tau.BookIV.Electroweak.instReprYukawaCouplingFull.repr :YukawaCouplingFull → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.YukawaCouplingFull.toFloat

source def Tau.BookIV.Electroweak.YukawaCouplingFull.toFloat (y : YukawaCouplingFull) :Float

Yukawa coupling as Float. Equations

  • y.toFloat = Float.ofNat y.coupling_numer / Float.ofNat y.coupling_denom Instances For

Tau.BookIV.Electroweak.yukawa_full_top

source def Tau.BookIV.Electroweak.yukawa_full_top :YukawaCouplingFull

Top quark: generation 3, y_t ≈ 0.995. Equations

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

Tau.BookIV.Electroweak.yukawa_full_bottom

source def Tau.BookIV.Electroweak.yukawa_full_bottom :YukawaCouplingFull

Bottom quark: generation 3 (down-type), y_b ≈ 0.024. Equations

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

Tau.BookIV.Electroweak.yukawa_full_charm

source def Tau.BookIV.Electroweak.yukawa_full_charm :YukawaCouplingFull

Charm quark: generation 2, y_c ≈ 0.0072. Equations

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

Tau.BookIV.Electroweak.yukawa_full_electron

source def Tau.BookIV.Electroweak.yukawa_full_electron :YukawaCouplingFull

Electron: generation 1, y_e ≈ 2.9 × 10⁻⁶. Equations

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

Tau.BookIV.Electroweak.EWSynthesisPrediction

source structure Tau.BookIV.Electroweak.EWSynthesisPrediction :Type

An EW prediction entry: name, τ-value, experimental value, deviation.

  • name : String Quantity name.

  • tau_numer : ℕ τ-predicted value numerator.

  • tau_denom : ℕ τ-predicted value denominator.

  • exp_numer : ℕ Experimental value numerator.

  • exp_denom : ℕ Experimental value denominator.

  • deviation_ppm : ℕ Approximate deviation in parts per million.

Instances For

Tau.BookIV.Electroweak.instReprEWSynthesisPrediction

source instance Tau.BookIV.Electroweak.instReprEWSynthesisPrediction :Repr EWSynthesisPrediction

Equations

  • Tau.BookIV.Electroweak.instReprEWSynthesisPrediction = { reprPrec := Tau.BookIV.Electroweak.instReprEWSynthesisPrediction.repr }

Tau.BookIV.Electroweak.instReprEWSynthesisPrediction.repr

source def Tau.BookIV.Electroweak.instReprEWSynthesisPrediction.repr :EWSynthesisPrediction → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.EWSynthesisPrediction.tauFloat

source def Tau.BookIV.Electroweak.EWSynthesisPrediction.tauFloat (p : EWSynthesisPrediction) :Float

τ-predicted value as Float. Equations

  • p.tauFloat = Float.ofNat p.tau_numer / Float.ofNat p.tau_denom Instances For

Tau.BookIV.Electroweak.EWSynthesisPrediction.expFloat

source def Tau.BookIV.Electroweak.EWSynthesisPrediction.expFloat (p : EWSynthesisPrediction) :Float

Experimental value as Float. Equations

  • p.expFloat = Float.ofNat p.exp_numer / Float.ofNat p.exp_denom Instances For

Tau.BookIV.Electroweak.ew_prediction_table

source def Tau.BookIV.Electroweak.ew_prediction_table :List EWSynthesisPrediction

[IV.T66] The nine EW quantities determined by ι_τ and m_n. Equations

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

Tau.BookIV.Electroweak.nine_ew_quantities

source theorem Tau.BookIV.Electroweak.nine_ew_quantities :ew_prediction_table.length = 9

The table has exactly 9 entries.

Tau.BookIV.Electroweak.EWAxiomTrace

source structure Tau.BookIV.Electroweak.EWAxiomTrace :Type

[IV.T67] The derivation chain for every EW quantity passes through at most two fundamental inputs:

  • ι_τ = 2/(π+e) — the master constant from K0-K6.

  • m_n — the neutron mass anchor (a single measured input).

All coupling constants, mixing angles, and masses are rational functions of ι_τ evaluated at the neutron anchor.

  • input_count : ℕ Number of fundamental inputs.

  • input_1 : String Input 1: master constant.

  • input_2 : String Input 2: neutron mass anchor.

  • all_trace : Bool All 9 quantities trace to these.

Instances For

Tau.BookIV.Electroweak.instReprEWAxiomTrace.repr

source def Tau.BookIV.Electroweak.instReprEWAxiomTrace.repr :EWAxiomTrace → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprEWAxiomTrace

source instance Tau.BookIV.Electroweak.instReprEWAxiomTrace :Repr EWAxiomTrace

Equations

  • Tau.BookIV.Electroweak.instReprEWAxiomTrace = { reprPrec := Tau.BookIV.Electroweak.instReprEWAxiomTrace.repr }

Tau.BookIV.Electroweak.ew_traces_to_axioms

source def Tau.BookIV.Electroweak.ew_traces_to_axioms :EWAxiomTrace

Equations

  • Tau.BookIV.Electroweak.ew_traces_to_axioms = { } Instances For

Tau.BookIV.Electroweak.ew_two_inputs

source theorem Tau.BookIV.Electroweak.ew_two_inputs :ew_traces_to_axioms.input_count = 2

Tau.BookIV.Electroweak.Sqrt3Triad

source structure Tau.BookIV.Electroweak.Sqrt3Triad :Type

[IV.T124] The √3 triad: the same algebraic quantity √3 = |1 − ω| (where ω = e^{2πi/3} is a primitive cube root of unity) appears in three independent physical contexts:

  • R mass ratio correction: √3 · ι_τ^{-2} (spectral distance on L)

  • Proton-neutron mass splitting: δ_A/m_n ≈ (√3/2) · ι_τ⁶ (isospin)

  • Gravitational closing identity: α_G = α¹⁸ · √3 (bi-rotation)

All three appearances trace to the SAME geometric origin: the three-fold structure of the lemniscate L = S¹ ∨ S¹ with its three sectors {Lobe₁, Lobe₂, Crossing}.

  • context_R : String Context 1: R mass ratio.

  • context_delta_A : String Context 2: proton-neutron splitting.

  • context_alpha_G : String Context 3: gravitational closing.

  • origin : String Geometric origin.

  • appearance_count : ℕ Number of independent appearances.

Instances For

Tau.BookIV.Electroweak.instReprSqrt3Triad.repr

source def Tau.BookIV.Electroweak.instReprSqrt3Triad.repr :Sqrt3Triad → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprSqrt3Triad

source instance Tau.BookIV.Electroweak.instReprSqrt3Triad :Repr Sqrt3Triad

Equations

  • Tau.BookIV.Electroweak.instReprSqrt3Triad = { reprPrec := Tau.BookIV.Electroweak.instReprSqrt3Triad.repr }

Tau.BookIV.Electroweak.sqrt3_triad

source def Tau.BookIV.Electroweak.sqrt3_triad :Sqrt3Triad

Equations

  • Tau.BookIV.Electroweak.sqrt3_triad = { } Instances For

Tau.BookIV.Electroweak.sqrt3_triad_count

source theorem Tau.BookIV.Electroweak.sqrt3_triad_count :sqrt3_triad.appearance_count = 3

Tau.BookIV.Electroweak.yukawa_ordering

source theorem Tau.BookIV.Electroweak.yukawa_ordering :yukawa_full_top.coupling_numer * yukawa_full_bottom.coupling_denom > yukawa_full_bottom.coupling_numer * yukawa_full_top.coupling_denom ∧ yukawa_full_bottom.coupling_numer * yukawa_full_charm.coupling_denom > yukawa_full_charm.coupling_numer * yukawa_full_bottom.coupling_denom ∧ yukawa_full_charm.coupling_numer * yukawa_full_electron.coupling_denom > yukawa_full_electron.coupling_numer * yukawa_full_charm.coupling_denom

[IV.P78] Yukawa couplings are ordered by generation, and the ordering follows the sector coupling hierarchy: y_top > y_bottom > y_charm > y_electron.

Each generation step down multiplies the coupling by approximately ι_τ², reflecting the spectral gap of the torus T².

Tau.BookIV.Electroweak.ZeroVsNineteen

source structure Tau.BookIV.Electroweak.ZeroVsNineteen :Type

[IV.P79] The τ-framework determines all 9 EW quantities with zero free parameters, compared to the Standard Model’s 19.

SM free parameters include: 3 gauge couplings, 6 quark masses, 3 lepton masses, 4 CKM parameters, 1 QCD vacuum angle, 1 Higgs mass, 1 Higgs VEV = 19 total.

The τ-framework replaces all 19 with derivations from ι_τ.

  • tau_params : ℕ τ free parameters.

  • sm_params : ℕ SM free parameters.

  • reduction : String Reduction factor.

Instances For

Tau.BookIV.Electroweak.instReprZeroVsNineteen

source instance Tau.BookIV.Electroweak.instReprZeroVsNineteen :Repr ZeroVsNineteen

Equations

  • Tau.BookIV.Electroweak.instReprZeroVsNineteen = { reprPrec := Tau.BookIV.Electroweak.instReprZeroVsNineteen.repr }

Tau.BookIV.Electroweak.instReprZeroVsNineteen.repr

source def Tau.BookIV.Electroweak.instReprZeroVsNineteen.repr :ZeroVsNineteen → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.zero_vs_nineteen

source def Tau.BookIV.Electroweak.zero_vs_nineteen :ZeroVsNineteen

Equations

  • Tau.BookIV.Electroweak.zero_vs_nineteen = { } Instances For

Tau.BookIV.Electroweak.tau_zero_params

source theorem Tau.BookIV.Electroweak.tau_zero_params :zero_vs_nineteen.tau_params = 0

Tau.BookIV.Electroweak.sm_nineteen_params

source theorem Tau.BookIV.Electroweak.sm_nineteen_params :zero_vs_nineteen.sm_params = 19

Tau.BookIV.Electroweak.LemniscateSupport

source structure Tau.BookIV.Electroweak.LemniscateSupport :Type

[IV.P175] The three structural supports of the lemniscate L = S¹ ∨ S¹ in the EW context:

  • B-lobe (EM sector): photon propagation, α determination

  • C-lobe (Strong sector): confinement, mass gap

  • Crossing point (ω-sector): Higgs mechanism, mass assignment

Each support corresponds to a distinct physical mechanism.

  • label : String Support label.

  • sector : BookIII.Sectors.Sector Associated sector.

  • mechanism : String Physical mechanism.

Instances For

Tau.BookIV.Electroweak.instReprLemniscateSupport

source instance Tau.BookIV.Electroweak.instReprLemniscateSupport :Repr LemniscateSupport

Equations

  • Tau.BookIV.Electroweak.instReprLemniscateSupport = { reprPrec := Tau.BookIV.Electroweak.instReprLemniscateSupport.repr }

Tau.BookIV.Electroweak.instReprLemniscateSupport.repr

source def Tau.BookIV.Electroweak.instReprLemniscateSupport.repr :LemniscateSupport → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.support_B_lobe

source def Tau.BookIV.Electroweak.support_B_lobe :LemniscateSupport

Equations

  • Tau.BookIV.Electroweak.support_B_lobe = { label := “B-lobe”, sector := Tau.BookIII.Sectors.Sector.B, mechanism := “photon propagation, alpha” } Instances For

Tau.BookIV.Electroweak.support_C_lobe

source def Tau.BookIV.Electroweak.support_C_lobe :LemniscateSupport

Equations

  • Tau.BookIV.Electroweak.support_C_lobe = { label := “C-lobe”, sector := Tau.BookIII.Sectors.Sector.C, mechanism := “confinement, mass gap” } Instances For

Tau.BookIV.Electroweak.support_crossing

source def Tau.BookIV.Electroweak.support_crossing :LemniscateSupport

Equations

  • Tau.BookIV.Electroweak.support_crossing = { label := “Crossing”, sector := Tau.BookIII.Sectors.Sector.Omega, mechanism := “Higgs mechanism, mass assignment” } Instances For

Tau.BookIV.Electroweak.three_lemniscate_supports

source def Tau.BookIV.Electroweak.three_lemniscate_supports :List LemniscateSupport

All three lemniscate supports. Equations

  • Tau.BookIV.Electroweak.three_lemniscate_supports = [Tau.BookIV.Electroweak.support_B_lobe, Tau.BookIV.Electroweak.support_C_lobe, Tau.BookIV.Electroweak.support_crossing] Instances For

Tau.BookIV.Electroweak.three_supports_count

source theorem Tau.BookIV.Electroweak.three_supports_count :three_lemniscate_supports.length = 3

Tau.BookIV.Electroweak.remark_ew_complete

source def Tau.BookIV.Electroweak.remark_ew_complete :String

[IV.R37] EW synthesis is complete: all quantities determined, no free parameters, all scope labels assigned. Equations

  • Tau.BookIV.Electroweak.remark_ew_complete = “EW synthesis complete: 9 quantities, 2 inputs (iota_tau, m_n), 0 free parameters” Instances For

Tau.BookIV.Electroweak.remark_no_bsm

source def Tau.BookIV.Electroweak.remark_no_bsm :String

[IV.R38] No BSM (Beyond Standard Model) particles are required. The τ-framework reproduces all EW physics without supersymmetry, extra dimensions, or additional gauge groups. Equations

  • Tau.BookIV.Electroweak.remark_no_bsm = “No BSM particles required: no SUSY, no extra dimensions, no extended gauge groups” Instances For

Tau.BookIV.Electroweak.remark_test_program

source def Tau.BookIV.Electroweak.remark_test_program :String

[IV.R39] Experimental test program: the τ-framework makes specific predictions testable at current and future colliders:

  • Weinberg angle: 2.7% tree-level deviation

  • Higgs branching ratios: percent-level corrections

  • W mass: sub-MeV prediction

  • No proton decay from GUT-type mechanisms

Equations

  • Tau.BookIV.Electroweak.remark_test_program = “Testable: sin2(theta_W) 2.7%, Higgs BRs ~1%, M_W sub-MeV, no proton decay” Instances For

Tau.BookIV.Electroweak.remark_book_v_connection

source def Tau.BookIV.Electroweak.remark_book_v_connection :String

[IV.R40] Book V (Categorical Macrocosm) extends the EW synthesis to the gravitational sector. The closing identity α_G = α¹⁸ · √3 · (1 − (3/π)·α) connects the EW fine structure constant to Newton’s gravitational constant G via 18 powers of α. Equations

  • Tau.BookIV.Electroweak.remark_book_v_connection = “Book V extends to gravity: alpha_G = alpha^18 * sqrt(3) * (1 - (3/pi)*alpha)” Instances For