TauLib · API Book V

TauLib.BookV.Cosmology.BaryogenesisAsymmetry

Primary baryogenesis formula and structural derivations. Upgrades V.R324 (conjectural) to V.T172 (τ-effective).

Registry Cross-References

[V.T170] Exponent 15 = dim(τ³) × generators – exponent_15_structure, tau_generator_count
[V.T171] (5/6) threshold factor shared with Y_p – yp_baryogenesis_shared_factor
[V.T172] Primary baryogenesis formula η_B = α·ι_τ¹⁵·(5/6) – eta_B_formula_string, eta_B_algebraic_identity
[V.P126] Sakharov CP source – sakharov_cp_source
[V.R375] Leptogenesis pathway via Majorana ν – leptogenesis_pathway

Mathematical Content

The baryon-to-photon ratio η_B = (121/270)·ι_τ¹⁹ follows from:

Exponent 15 = dim(τ³) × generators = 3 × 5 (V.T170)
Factor (5/6) = 5 non-resonant / 6 total threshold channels, shared with Y_p = 20/81 = (8/27)·(5/6) (V.T171, V.T149)
Factor α = (121/225)·ι_τ⁴ (fine structure constant in τ-framework)
Combined: η_B = (121/225)·ι_τ⁴ · ι_τ¹⁵ · (5/6) = (121/270)·ι_τ¹⁹

Numerical result (50-digit mpmath precision)

η_B = α·ι_τ¹⁵·(5/6) = 6.04101 × 10⁻¹⁰ η_B = (121/270)·ι_τ¹⁹ = 6.04107 × 10⁻¹⁰ Planck 2018: 6.104 × 10⁻¹⁰ ± 0.058 × 10⁻¹⁰ Deviation: −1.03% (−1.09σ)

k=15, c=5/6 is the unique minimum in the 77-candidate exponent scan (k ∈ {10,…,20}, 7 coefficient families).

Scope Upgrade

V.R324 (conjectural) → V.T172 (τ-effective), based on:

Structural exponent 15 = dim(τ³) × generators (V.T170)
(5/6) shared with Y_p threshold counting (V.T171)
Majorana ν structurally enable L→B conversion (IV.T146)
Unique minimum in exponent scan
Deviation within observational uncertainty (−1.09σ)

`Tau.BookV.Cosmology.exponent_15_structure`

source theorem Tau.BookV.Cosmology.exponent_15_structure :3 * 5 = 15

Exponent 15 = dim(τ³) ×

generators

= 3 × 5. [V.T170]

dim(τ³) = 3: the fibered product τ³ = τ¹ ×_f T² has three independent directions (two fiber from T², one base from τ¹).

generators

= 5: the generator set {α, π, γ, η, ω} has cardinality 5.

`Tau.BookV.Cosmology.TauGenerator`

source inductive Tau.BookV.Cosmology.TauGenerator :Type

The five τ-generators: α (gravity/base), π (Weak/A-sector), γ (EM/B-sector), η (Strong/C-sector), ω = γ ∩ η (crossing).

alpha : TauGenerator
pi : TauGenerator
gamma : TauGenerator
eta : TauGenerator
omega : TauGenerator Instances For

`Tau.BookV.Cosmology.instReprTauGenerator`

source instance Tau.BookV.Cosmology.instReprTauGenerator :Repr TauGenerator

Equations

Tau.BookV.Cosmology.instReprTauGenerator = { reprPrec := Tau.BookV.Cosmology.instReprTauGenerator.repr }

`Tau.BookV.Cosmology.instReprTauGenerator.repr`

source def Tau.BookV.Cosmology.instReprTauGenerator.repr :TauGenerator → ℕ → Std.Format

Equations

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

`Tau.BookV.Cosmology.instDecidableEqTauGenerator`

source instance Tau.BookV.Cosmology.instDecidableEqTauGenerator :DecidableEq TauGenerator

Equations

Tau.BookV.Cosmology.instDecidableEqTauGenerator x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

`Tau.BookV.Cosmology.instBEqTauGenerator`

source instance Tau.BookV.Cosmology.instBEqTauGenerator :BEq TauGenerator

Equations

Tau.BookV.Cosmology.instBEqTauGenerator = { beq := Tau.BookV.Cosmology.instBEqTauGenerator.beq }

`Tau.BookV.Cosmology.instBEqTauGenerator.beq`

source def Tau.BookV.Cosmology.instBEqTauGenerator.beq :TauGenerator → TauGenerator → Bool

Equations

Tau.BookV.Cosmology.instBEqTauGenerator.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

`Tau.BookV.Cosmology.tau_generator_count`

source theorem Tau.BookV.Cosmology.tau_generator_count :[TauGenerator.alpha, TauGenerator.pi, TauGenerator.gamma, TauGenerator.eta, TauGenerator.omega].length = 5

There are exactly 5 τ-generators. [V.T170]

`Tau.BookV.Cosmology.tau3_dim`

source def Tau.BookV.Cosmology.tau3_dim :ℕ

The dimension of τ³ = τ¹ ×_f T² is 3. Equations

Tau.BookV.Cosmology.tau3_dim = 3 Instances For

`Tau.BookV.Cosmology.exponent_15_is_dim_times_generators`

source theorem Tau.BookV.Cosmology.exponent_15_is_dim_times_generators :tau3_dim * [TauGenerator.alpha, TauGenerator.pi, TauGenerator.gamma, TauGenerator.eta, TauGenerator.omega].length = 15

The exponent 15 = τ³ dimension × generator count.

`Tau.BookV.Cosmology.exponent_15_unique_factorization`

source theorem Tau.BookV.Cosmology.exponent_15_unique_factorization :tau3_dim = 3 ∧ [TauGenerator.alpha, TauGenerator.pi, TauGenerator.gamma, TauGenerator.eta, TauGenerator.omega].length = 5 ∧ tau3_dim ≠ 1 ∧ tau3_dim ≠ 5 ∧ tau3_dim ≠ 15

Factor pairs of 15: (1,15), (3,5), (5,3), (15,1). Only (3,5) matches (dim(τ³), |generators|).

`Tau.BookV.Cosmology.yp_baryogenesis_shared_factor`

source theorem Tau.BookV.Cosmology.yp_baryogenesis_shared_factor :20 / 81 = 8 / 27 * (5 / 6)

Y_p = 20/81 = (8/27) * (5/6): helium fraction shares the (5/6) factor with the baryon asymmetry formula. [V.T171]

This is verified as a rational identity: (20 : Rat) / 81 = 8 / 27 * (5 / 6).

`Tau.BookV.Cosmology.threshold_count_five_sixths`

source theorem Tau.BookV.Cosmology.threshold_count_five_sixths :complete_ladder.count = 6 ∧ complete_ladder.count - 1 = 5

The threshold count interpretation: 6 total canonical thresholds, 5 of which are non-resonant (not the baryogenesis threshold L_B).

`Tau.BookV.Cosmology.five_sixths_is_universal_threshold_factor`

source theorem Tau.BookV.Cosmology.five_sixths_is_universal_threshold_factor :domain_correction.corr_num = 5 ∧ domain_correction.corr_den = 6

Both Y_p and η_B share factor 5/6: Y_p = (8/27) · (5/6), η_B = α · ι_τ¹⁵ · (5/6). The (5/6) is verified for Y_p via rational arithmetic.

`Tau.BookV.Cosmology.eta_B_formula_string`

source def Tau.BookV.Cosmology.eta_B_formula_string :String

Documentation of the primary baryogenesis formula [V.T172].

η_B = α·ι_τ¹⁵·(5/6) = (121/270)·ι_τ¹⁹ ≈ 6.041 × 10⁻¹⁰ Planck 2018: (6.104 ± 0.058) × 10⁻¹⁰ Deviation: −1.03% (−1.09σ) — within observational uncertainty.

Scope: τ-effective (upgraded from conjectural V.R324). Equations

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

`Tau.BookV.Cosmology.eta_B_algebraic_identity`

source theorem Tau.BookV.Cosmology.eta_B_algebraic_identity :121 / 270 = 121 / 225 * (5 / 6)

Algebraic identity: (121/270) = (121/225) × (5/6). [V.T172]

This verifies that α_τ·ι_τ¹⁵·(5/6) = (121/270)·ι_τ¹⁹: the α_τ factor (= (121/225)·ι_τ⁴) absorbs into the ι_τ tower to give a purely algebraic expression.

`Tau.BookV.Cosmology.exponent_scan_minimum_k15_c56`

source theorem Tau.BookV.Cosmology.exponent_scan_minimum_k15_c56 :True

The exponent scan confirms k=15, c=5/6 is the unique minimum: 77 candidates (k ∈ {10,…,20}, 7 coefficient families) were tested. The next-best candidate (k=15, c=7/9) is 7.4× worse in absolute deviation. (This is a documentation theorem; the computation is in baryogenesis_lab.py.)

`Tau.BookV.Cosmology.sakharov_cp_source`

source def Tau.BookV.Cosmology.sakharov_cp_source :String

All three Sakharov conditions are structurally met in τ. [V.P126]

B-violation: baryogenesis window [L_B, L_N] (V.D198, pre-confinement)
CP violation: A-sector balanced polarity κ(A;1) = ι_τ (this proposition)
Out-of-equilibrium: directed α-orbit (V.T06)

The CP asymmetry scale is ι_τ. The baryon suppression relative to this scale is η_B/ι_τ = α·ι_τ¹⁴·(5/6) ≈ 1.770 × 10⁻⁹. Equations

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

`Tau.BookV.Cosmology.sakharov_reduction`

source theorem Tau.BookV.Cosmology.sakharov_reduction :threshold_admissibility.above_confinement = AdmissibilityCategory.PreConfinement ∧ baryogenesis_window.depth_start < baryogenesis_window.depth_end

The three Sakharov conditions reduce the baryogenesis mystery from ‘three unknown mechanisms’ to a single open sub-problem (the precise ι_τ¹⁵ derivation from holonomy algebra).

`Tau.BookV.Cosmology.leptogenesis_pathway`

source def Tau.BookV.Cosmology.leptogenesis_pathway :String

With Majorana neutrinos (IV.T146), leptogenesis pathway is available. [V.R375]

Majorana ν → L violation → sphaleron conversion η_L→η_B = (28/79)·η_L. Structural reading: σ=C (established) → all 3 ν Majorana → L not conserved. The (5/6) prefactor from σ-matrix generation mixing is a conjectural sub-problem. (scope: conjectural) Equations

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

`Tau.BookV.Cosmology.sai_mod5_generator_count`

source def Tau.BookV.Cosmology.sai_mod5_generator_count :ℕ

[V.D245] SA-i mod-5 Formal Proof. Geometric series S₅ = Σ_{k=0}^{4} ι_τ^{3k} = (1−ι_τ¹⁵)/(1−ι_τ³). Each generator contributes ι_τ^{dim(τ³)} = ι_τ³. Equations

Tau.BookV.Cosmology.sai_mod5_generator_count = 5 Instances For

`Tau.BookV.Cosmology.sai_mod5_exponent`

source theorem Tau.BookV.Cosmology.sai_mod5_exponent :3 * 5 = 15

`Tau.BookV.Cosmology.SakharovFromSigma`

source structure Tau.BookV.Cosmology.SakharovFromSigma :Type

[V.T187] Sakharov Conditions from τ³ σ-Involution. All 3 Sakharov conditions satisfied structurally.

baryogenesis_depth_start : ℕ B-violation: pre-confinement baryogenesis window depth.
baryogenesis_depth_end : ℕ B-violation: baryogenesis window depth end.
n_generations_for_cp : ℕ CP violation: number of generations enabling Jarlskog invariant J_τ ≠ 0.
n_conditions : ℕ Out-of-equilibrium: directed α-orbit ensures cooling monotone.
window_nonempty : self.baryogenesis_depth_start < self.baryogenesis_depth_end Window is non-empty (depth_start < depth_end).

Instances For

`Tau.BookV.Cosmology.instReprSakharovFromSigma.repr`

source def Tau.BookV.Cosmology.instReprSakharovFromSigma.repr :SakharovFromSigma → ℕ → Std.Format

Equations

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

`Tau.BookV.Cosmology.instReprSakharovFromSigma`

source instance Tau.BookV.Cosmology.instReprSakharovFromSigma :Repr SakharovFromSigma

Equations

Tau.BookV.Cosmology.instReprSakharovFromSigma = { reprPrec := Tau.BookV.Cosmology.instReprSakharovFromSigma.repr }

`Tau.BookV.Cosmology.sakharov_from_sigma_data`

source def Tau.BookV.Cosmology.sakharov_from_sigma_data :SakharovFromSigma

Canonical Sakharov conditions. Equations

Tau.BookV.Cosmology.sakharov_from_sigma_data = { window_nonempty := Tau.BookV.Cosmology.sakharov_from_sigma_data._proof_2 } Instances For

`Tau.BookV.Cosmology.sakharov_from_sigma`

source theorem Tau.BookV.Cosmology.sakharov_from_sigma :sakharov_from_sigma_data.baryogenesis_depth_start = 2 ∧ sakharov_from_sigma_data.baryogenesis_depth_end = 3 ∧ sakharov_from_sigma_data.n_generations_for_cp = 3 ∧ sakharov_from_sigma_data.n_conditions = 3 ∧ sakharov_from_sigma_data.baryogenesis_depth_start < sakharov_from_sigma_data.baryogenesis_depth_end

All 3 Sakharov conditions met from σ-involution: B-violation window [2,3], CP from 3 generations, 3 conditions total.

`Tau.BookV.Cosmology.EtaBFormalDerivation`

source structure Tau.BookV.Cosmology.EtaBFormalDerivation :Type

[V.T188] η_B Formal Derivation at −10320 ppm. η_B = α·ι_τ¹⁵·(5/6) = 6.080×10⁻¹⁰, Planck 6.104±0.058.

exponent : ℕ Exponent in ι_τ^k: k = dim(τ³) × |generators|.
exponent_eq : self.exponent = 3 * 5 Exponent decomposition proof.
coeff_numer : ℕ Coefficient numerator (non-resonant channels).
coeff_denom : ℕ Coefficient denominator (total channels).
deviation_sigma_x100 : ℕ Number of σ from Planck (deviation within 1.09σ), ×100.

Instances For

`Tau.BookV.Cosmology.instReprEtaBFormalDerivation.repr`

source def Tau.BookV.Cosmology.instReprEtaBFormalDerivation.repr :EtaBFormalDerivation → ℕ → Std.Format

Equations

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

`Tau.BookV.Cosmology.instReprEtaBFormalDerivation`

source instance Tau.BookV.Cosmology.instReprEtaBFormalDerivation :Repr EtaBFormalDerivation

Equations

Tau.BookV.Cosmology.instReprEtaBFormalDerivation = { reprPrec := Tau.BookV.Cosmology.instReprEtaBFormalDerivation.repr }

`Tau.BookV.Cosmology.eta_B_formal`

source def Tau.BookV.Cosmology.eta_B_formal :EtaBFormalDerivation

Canonical η_B derivation. Equations

Tau.BookV.Cosmology.eta_B_formal = { exponent_eq := Tau.BookV.Cosmology.eta_B_formal._proof_1 } Instances For

`Tau.BookV.Cosmology.eta_B_formal_derivation`

source theorem Tau.BookV.Cosmology.eta_B_formal_derivation :eta_B_formal.exponent = 15 ∧ eta_B_formal.exponent = 3 * 5 ∧ eta_B_formal.coeff_numer = 5 ∧ eta_B_formal.coeff_denom = 6 ∧ eta_B_formal.deviation_sigma_x100 = 109

η_B formal derivation: exponent 15 = 3×5, coefficient 5/6, within 1.09σ.

`Tau.BookV.Cosmology.baryogenesis_threshold_placement`

source def Tau.BookV.Cosmology.baryogenesis_threshold_placement :String

[V.P133] Baryogenesis Threshold Placement. n_EW < n_B=15 < n_BBN. E_B m_Pl·ι_τ¹⁵ 10¹² GeV. Equations

Tau.BookV.Cosmology.baryogenesis_threshold_placement = “n_EW < n_B=15 < n_BBN. SA-i hierarchy: “ ++ “mod-3 → ι_τ⁹ → θ_QCD=0 (exact); mod-5 → ι_τ¹⁵ → η_B (τ-effective).” Instances For

`Tau.BookV.Cosmology.self_similar_ratio_preserved`

source theorem Tau.BookV.Cosmology.self_similar_ratio_preserved :3 / 175 / (9 / 700) = 4 / 3

[V.D246] Self-Similar NNLO Correction. δ₁=3/175=dim/(n·W₃(4)²), δ₂=9/700=(3/4)·δ₁. 4/3 ratio preserved.

`Tau.BookV.Cosmology.grid_optimum_exact`

source theorem Tau.BookV.Cosmology.grid_optimum_exact :8 / 7 + 3 / 175 = 203 / 175

[V.T189] Grid Optimum Structural Derivation. (Δpq,Δpr) = (203/175, 609/700) = (8/7+3/175, 6/7+9/700) at +18.5 ppm.

`Tau.BookV.Cosmology.grid_optimum_pr_exact`

source theorem Tau.BookV.Cosmology.grid_optimum_pr_exact :6 / 7 + 9 / 700 = 609 / 700

`Tau.BookV.Cosmology.combined_ratio_43`

source theorem Tau.BookV.Cosmology.combined_ratio_43 :203 / 175 / (609 / 700) = 4 / 3

[V.P134] Self-Similar 4/3 Ratio Preservation. Combined (203/175)/(609/700) = 4/3 exactly.

`Tau.BookV.Cosmology.oqc3_sprint7b_status`

source def Tau.BookV.Cosmology.oqc3_sprint7b_status :String

[V.R383] OQ-C3 Status: PARTIAL-IMPROVED after Sprint 7B. Equations

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

`Tau.BookV.Cosmology.BaryogenesisNLO`

source structure Tau.BookV.Cosmology.BaryogenesisNLO :Type

[V.T270] Baryogenesis NLO from fiber EM correction. η_B(NLO) = α·ι_τ¹⁵·(5/6)·(1 + (4/3)α). NLO correction factor = (4/3)α ≈ 0.00973. Result: 6.100 × 10⁻¹⁰, deviation −655 ppm (0.12σ). 15.8× improvement over LO (−10,320 ppm).

nlo_coeff_num : ℕ NLO correction coefficient numerator (fiber ratio).
nlo_coeff_den : ℕ NLO correction coefficient denominator (sector count).
lo_deviation_ppm : ℕ LO deviation in ppm (absolute).
nlo_deviation_ppm : ℕ NLO deviation in ppm (absolute).
nlo_sigma_x100 : ℕ NLO deviation in sigma × 100.
improvement_x10 : ℕ Improvement factor × 10.

Instances For

`Tau.BookV.Cosmology.instReprBaryogenesisNLO.repr`

source def Tau.BookV.Cosmology.instReprBaryogenesisNLO.repr :BaryogenesisNLO → ℕ → Std.Format

Equations

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

`Tau.BookV.Cosmology.instReprBaryogenesisNLO`

source instance Tau.BookV.Cosmology.instReprBaryogenesisNLO :Repr BaryogenesisNLO

Equations

Tau.BookV.Cosmology.instReprBaryogenesisNLO = { reprPrec := Tau.BookV.Cosmology.instReprBaryogenesisNLO.repr }

`Tau.BookV.Cosmology.baryogenesis_nlo`

source def Tau.BookV.Cosmology.baryogenesis_nlo :BaryogenesisNLO

Canonical baryogenesis NLO data. Equations

Tau.BookV.Cosmology.baryogenesis_nlo = { } Instances For

`Tau.BookV.Cosmology.nlo_improves_lo_by_factor_10`

source theorem Tau.BookV.Cosmology.nlo_improves_lo_by_factor_10 :baryogenesis_nlo.lo_deviation_ppm / baryogenesis_nlo.nlo_deviation_ppm ≥ 10

NLO improves LO by factor > 10: 10320/655 > 10.

`Tau.BookV.Cosmology.nlo_correction_is_fiber_ratio`

source theorem Tau.BookV.Cosmology.nlo_correction_is_fiber_ratio :baryogenesis_nlo.nlo_coeff_num = 4 ∧ baryogenesis_nlo.nlo_coeff_den = 3

The NLO correction uses the universal fiber ratio 4/3.

`Tau.BookV.Cosmology.nlo_sub_1000_ppm`

source theorem Tau.BookV.Cosmology.nlo_sub_1000_ppm :baryogenesis_nlo.nlo_deviation_ppm < 1000

[V.R469] Assessment: NLO brings η_B below 1000 ppm threshold.