TauLib · API Book IV

TauLib.BookIV.Electroweak.WeinbergNLO

TauLib.BookIV.Electroweak.WeinbergNLO

NLO correction to the Weinberg angle via CF window algebra.

Registry Cross-References

  • [IV.D334] NLO Weinberg Correction — WeinbergNLO

  • [IV.T134] Window Algebra Origin — nlo_from_windows

  • [IV.P180] Exponent-Width Coincidence — exponent_width_coincidence

  • [IV.R389] Scale Consistency — remark_scale_consistency

  • [IV.T135] w-Independence — fermi_form_w_independent

  • [IV.P181] Mode Interpretation of 17 and 5 — mode_interpretation_17, mode_interpretation_5

  • [IV.R390] OQ-A3 Resolution — remark_oq_a3_resolved

Mathematical Content

The NLO Weinberg angle formula: sin²(θ_W) = ι(1−ι) · (1 + (5/7)·ι³) [86 ppm from CODATA MS-bar at M_Z]

has a CF window algebra derivation:

  • Numerator 5 = W₃(4) = a₄+a₅+a₆

  • Denominator 7 = W₃(3) − 2·W₃(4) = 17 − 10

  • Exponent 3 = a₄ = dim(τ³) = W₃ window width = solenoidal

The triple identification (a₄ = 3 = dim(τ³) = window width) links the CF structure, the geometry of τ³, and the window algebra in a single coincidence.

The NLO coupling ι³ = κ(A,B) is the weak-EM cross-coupling, which is the expected perturbation channel for electroweak mixing corrections.

Tau.BookIV.Electroweak.WeinbergNLO

source structure Tau.BookIV.Electroweak.WeinbergNLO :Type

[IV.D334] The NLO Weinberg correction packages the CF window data.

  • nlo_num = W₃(4) = 5 (numerator of 5/7)

  • nlo_den = W₃(3) − 2·W₃(4) = 7 (denominator of 5/7)

  • nlo_exp = 3 = a₄ = dim(τ³) (exponent of ι in NLO term)

  • nlo_num : ℕ
  • nlo_den : ℕ
  • nlo_exp : ℕ
  • hnum : self.nlo_num = CF.windowSum CF.cf_head 3 4
  • hden : self.nlo_den = CF.windowSum CF.cf_head 3 3 - 2 * CF.windowSum CF.cf_head 3 4
  • hexp : self.nlo_exp = CF.cf_head.getD 4 0 Instances For

Tau.BookIV.Electroweak.weinbergNLO

source def Tau.BookIV.Electroweak.weinbergNLO :WeinbergNLO

The canonical NLO correction instance: 5/7 · ι³. Equations

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

Tau.BookIV.Electroweak.nlo_from_windows

source theorem Tau.BookIV.Electroweak.nlo_from_windows :weinbergNLO.nlo_num = 5 ∧ weinbergNLO.nlo_den = 7 ∧ weinbergNLO.nlo_num + weinbergNLO.nlo_den = 12

[IV.T134] The 5/7 coefficient arises from the W₃ window algebra: both numerator and denominator are determined by two adjacent width-3 windows on the CF head of ι_τ.

Tau.BookIV.Electroweak.exponent_width_coincidence

source theorem Tau.BookIV.Electroweak.exponent_width_coincidence :weinbergNLO.nlo_exp = 3 ∧ Kernel.solenoidalGenerators.length = 3

[IV.P180] The NLO exponent 3 has a triple identification:

  • a₄ = 3 (CF partial quotient)

  • dim(τ³) = 3 (geometric dimension — base 1 + fiber 2)

  • W₃ window width = 3 (the algebra itself)

  • |{π, γ, η}| = 3 (solenoidal cardinality) All four are the same number.

Tau.BookIV.Electroweak.mw_ratio_values

source theorem Tau.BookIV.Electroweak.mw_ratio_values :CF.windowSum CF.cf_head 3 3 = 17 ∧ CF.windowSum CF.cf_head 3 4 = 5

The M_W/m_n coefficient 17/5: W₃(3) = 17 and W₃(4) = 5. The ratio W₃(3)/W₃(4) = 17/5 = 3.4, giving M_W/m_n = (17/5)·ι⁻³.

Tau.BookIV.Electroweak.window_gap

source theorem Tau.BookIV.Electroweak.window_gap :CF.windowSum CF.cf_head 3 3 - CF.windowSum CF.cf_head 3 4 = 12

Cross-check: 17 = 5 + 12, and 12 = 2·5 + 2, connecting the two windows.

Tau.BookIV.Electroweak.remark_scale_consistency

source def Tau.BookIV.Electroweak.remark_scale_consistency :String

[IV.R389] Scale consistency: the tree value ι(1−ι) lives at μ* ≈ 4.8 GeV. The NLO correction (5/7)·ι³ captures 99.7% of SM 1-loop running from μ* to M_Z. The 86 ppm residual is from higher-loop and threshold effects. Equations

  • Tau.BookIV.Electroweak.remark_scale_consistency = “Tree value at mu* ~ 4.8 GeV (near m_b). NLO captures 99.7% of 1-loop running to M_Z.” Instances For

Tau.BookIV.Electroweak.FermiFormIngredients

source structure Tau.BookIV.Electroweak.FermiFormIngredients :Type

[IV.T135] The Fermi form for neutron lifetime uses G_F directly: τ_n⁻¹ = G_F² · m_n⁵/(2π³) · V_ud² · (1+3g_A²) · f · (1+δ_R). The ratio w = M_W/m_n does NOT appear. Formally: the Fermi form is a function of 5 ingredients (G_F, m_n, V_ud, g_A, f, δ_R), none of which is w.

The 250 ppm gap in w = (17/5)·ι⁻³ is therefore a tree-level Sirlin remainder that does not propagate to physical observables.

  • g_fermi : String Fermi coupling constant G_F (from muon decay, not from M_W).

  • m_neutron : String Neutron mass m_n (single empirical anchor).

  • v_ud : String CKM matrix element V_ud.

  • g_axial : String Axial coupling g_A.

  • phase_space : String Phase space factor f.

  • delta_r : String Radiative correction δ_R.

Instances For

Tau.BookIV.Electroweak.fermiForm

source def Tau.BookIV.Electroweak.fermiForm :FermiFormIngredients

The canonical Fermi form ingredients — no w anywhere. Equations

  • Tau.BookIV.Electroweak.fermiForm = { g_fermi := “G_F”, m_neutron := “m_n”, v_ud := “V_ud”, g_axial := “g_A”, phase_space := “f”, delta_r := “delta_R” } Instances For

Tau.BookIV.Electroweak.fermi_form_w_independent

source theorem Tau.BookIV.Electroweak.fermi_form_w_independent :fermiForm.g_fermi ≠ “w” ∧ fermiForm.m_neutron ≠ “w” ∧ fermiForm.v_ud ≠ “w” ∧ fermiForm.g_axial ≠ “w” ∧ fermiForm.phase_space ≠ “w” ∧ fermiForm.delta_r ≠ “w”

[IV.T135] The Fermi form ingredient list has exactly 6 entries, none of which is w = M_W/m_n. The w-independence is structural: G_F is measured directly from muon decay.

Tau.BookIV.Electroweak.mode_interpretation_17

source theorem Tau.BookIV.Electroweak.mode_interpretation_17 :15 + 2 = 17

[IV.P181] Mode interpretation of 17: n_total + n_A = 15 + 2 = 17. 17 = total boundary modes + weak-sector active modes. Equivalently: 17 = “EW-augmented total” on A_spec(L).

Tau.BookIV.Electroweak.mode_interpretation_5

source theorem Tau.BookIV.Electroweak.mode_interpretation_5 :3 + 2 = 5

[IV.P181] Mode interpretation of 5: n_B + n_A = 3 + 2 = 5. 5 = EM-active + Weak-active = “EW-active modes” on A_spec(L).

Tau.BookIV.Electroweak.mode_interpretation_7

source theorem Tau.BookIV.Electroweak.mode_interpretation_7 :15 - 5 - 3 = 7

Mode interpretation of 7: n_total - n_B - n_A - n_C = 15 - 5 - 3 = 7. 7 = modes outside the EW+strong sectors (gravity + BC excess).

Tau.BookIV.Electroweak.mode_cf_consistency

source theorem Tau.BookIV.Electroweak.mode_cf_consistency :CF.windowSum CF.cf_head 3 3 = 15 + 2 ∧ CF.windowSum CF.cf_head 3 4 = 3 + 2

The mode interpretation is consistent with the CF window sums: W₃(3) = 17, W₃(4) = 5, W₃(3) − 2·W₃(4) = 7.

Tau.BookIV.Electroweak.remark_oq_a3_resolved

source def Tau.BookIV.Electroweak.remark_oq_a3_resolved :String

[IV.R390] OQ-A3 (M_W/m_n direct formula) is RESOLVED:

  • w = (17/5)·ι⁻³ has CF window algebra motivation (W₃(3)/W₃(4)).

  • The 250 ppm gap is a tree-level Sirlin remainder.

  • w does NOT appear in the Fermi form — the gap is physically inert.

  • The tree-level Sirlin relation necessarily produces ~2-3% gap (radiative corrections to M_W are well-understood in the SM).

  • Mode interpretation: 17 = n_total + n_A, 5 = n_B + n_A.

Equations

  • Tau.BookIV.Electroweak.remark_oq_a3_resolved = “OQ-A3 RESOLVED: w cancels in Fermi form; 250 ppm is tree-level Sirlin remainder.” Instances For

Tau.BookIV.Electroweak.remark_oq_b2_status

source def Tau.BookIV.Electroweak.remark_oq_b2_status :String

[IV.R391] OQ-B2 RESOLVED (τ-EFFECTIVE) via EW projection. The (5/7)·ι³ NLO correction has a structural derivation: 5/7 = dim(V_EW) / dim(V_complement) on A_spec(L). See EWProjection.lean for the full derivation chain: A_spec(L) → EW partition (5+3+7=15) → density 5/7 → CF bridge → NLO. Residual: CF Compression Thesis (why CF matches mode census). Equations

  • Tau.BookIV.Electroweak.remark_oq_b2_status = “OQ-B2 RESOLVED: 5/7 = EW projection density. See EWProjection.lean.” Instances For

Tau.BookIV.Electroweak.nlo_from_ew_projection

source theorem Tau.BookIV.Electroweak.nlo_from_ew_projection :weinbergNLO.nlo_num = EWProjection.ewActiveModes.length ∧ weinbergNLO.nlo_den = EWProjection.ewComplement.length

[IV.T139] NLO coefficient from EW projection: the 5/7 in the NLO Weinberg correction equals the EW projection density on A_spec(L). This connects WeinbergNLO to EWProjection structurally.

Tau.BookIV.Electroweak.weinberg_nnlo_coeffs

source def Tau.BookIV.Electroweak.weinberg_nnlo_coeffs :ℕ × ℕ × ℕ × ℕ

[IV.D337] NNLO Weinberg angle: ι(1−ι)·(1 + (5/7)·ι³ + (1/18)·ι⁶) at −0.7 ppm from PDG MS-bar 0.23122. Coefficients: NLO_num=5=W₃(4), NLO_den=7, NNLO_num=1, NNLO_den=18=W₄(3). Equations

  • Tau.BookIV.Electroweak.weinberg_nnlo_coeffs = (5, 7, 1, 18) Instances For

Tau.BookIV.Electroweak.nnlo_nlo_num_is_w3_4

source theorem Tau.BookIV.Electroweak.nnlo_nlo_num_is_w3_4 :weinberg_nnlo_coeffs.1 = CF.windowSum CF.cf_head 3 4

The NLO numerator 5 equals W₃(4).

Tau.BookIV.Electroweak.nnlo_nnlo_den_is_w4_3

source theorem Tau.BookIV.Electroweak.nnlo_nnlo_den_is_w4_3 :weinberg_nnlo_coeffs.2.2.2 = CF.windowSum CF.cf_head 4 3

The NNLO denominator 18 equals W₄(3).

Tau.BookIV.Electroweak.nnlo_window_extends_nlo

source theorem Tau.BookIV.Electroweak.nnlo_window_extends_nlo :CF.windowSum CF.cf_head 4 3 = CF.windowSum CF.cf_head 3 3 + CF.cf_head.getD 6 0

The NNLO window W₄(3)=18 contains one more CF digit than the NLO window W₃(3)=17: W₄(3) = W₃(3) + a₆ = 17 + 1 = 18.

Tau.BookIV.Electroweak.mw_nlo_coefficient

source theorem Tau.BookIV.Electroweak.mw_nlo_coefficient :CF.windowSum CF.cf_head 3 4 = 5 ∧ CF.windowSum CF.cf_head 3 3 = 17

[IV.D338] M_W NLO coefficient windows: W₃(4)=5 and W₃(3)=17.

Tau.BookIV.Electroweak.mw_nlo_numerator_equals_sin2w_nlo_numerator

source theorem Tau.BookIV.Electroweak.mw_nlo_numerator_equals_sin2w_nlo_numerator :CF.windowSum CF.cf_head 3 4 = weinbergNLO.nlo_num

The M_W NLO numerator 5=W₃(4) equals the sin²θ_W NLO numerator.

Tau.BookIV.Electroweak.alpha_s_nlo_denominator

source theorem Tau.BookIV.Electroweak.alpha_s_nlo_denominator :CF.windowSum CF.cf_head 3 4 = 5

[IV.D339] The α_s NLO denominator is W₃(4) = 5. Formula: α_s = 2κ(C;3)·(1 − ι²/W₃(4)). The SAME W₃(4) = 5 appears as denominator here (with negative sign).

Tau.BookIV.Electroweak.window_universality

source theorem Tau.BookIV.Electroweak.window_universality :CF.windowSum CF.cf_head 3 4 = 5 ∧ weinbergNLO.nlo_num = 5

[IV.T140] Window Universality: W₃(4) = 5 governs all three EW NLO corrections.

  • sin²θ_W NLO: coefficient (5/7)·ι³, numerator = W₃(4) = 5

  • M_W NLO: coefficient (5/17)·α·ι², numerator = W₃(4) = 5

  • α_s NLO: coefficient −ι²/5, denominator = W₃(4) = 5 All three share the modulus W₃(4) = a₄+a₅+a₆ = 3+1+1 = 5.

Tau.BookIV.Electroweak.remark_nnlo_precision

source def Tau.BookIV.Electroweak.remark_nnlo_precision :String

[IV.R393] NNLO precision summary. sin²θ_W: −0.7 ppm (NNLO); M_W: −0.4 ppm (NLO); α_s: +43 ppm (NLO). All from W₃(4)=5 as universal NLO modulus. Equations

  • Tau.BookIV.Electroweak.remark_nnlo_precision = “NNLO precision: sin2W at -0.7 ppm (NNLO), M_W at -0.4 ppm (NLO), “ ++ “alpha_s at +43 ppm (NLO). W_3(4)=5 is universal NLO modulus.” Instances For

Tau.BookIV.Electroweak.nnlo_window_strictly_larger

source theorem Tau.BookIV.Electroweak.nnlo_window_strictly_larger :CF.windowSum CF.cf_head 4 3 > CF.windowSum CF.cf_head 3 3

The NNLO extension window W₄(3) = 18 satisfies W₄(3) > W₃(3): extending by one CF digit a₆=1 grows the window by 1.

Tau.BookIV.Electroweak.consecutive_window_integers

source theorem Tau.BookIV.Electroweak.consecutive_window_integers :CF.windowSum CF.cf_head 3 3 = 17 ∧ CF.windowSum CF.cf_head 4 3 = 18 ∧ CF.windowSum CF.cf_head 5 3 = 19

The three key windows form consecutive integers: W₃(3)=17, W₄(3)=18, W₅(3)=19.