TauLib · API Book IV

TauLib.BookIV.Electroweak.WeakChirality2

TauLib.BookIV.Electroweak.WeakChirality2

Chirality selection theorem, parity bridge recall, polarity-switching suppression, SU(2)_L identification, and structural remarks on right-handed invisibility and parity preservation.

Registry Cross-References

  • [IV.C01] Right-Handed Configurations Invisible — right_handed_invisible

  • [IV.D319] Polarity-Switching Transition in A-Sector — PolaritySwitching

  • [IV.T122] Parity Bridge Recall (III.T07) — parity_bridge_recall

  • [IV.T123] Chirality Selection Theorem — chirality_selection

  • [IV.P54] EM, Strong, Gravity Are Parity-Preserving — non_weak_parity_preserving

  • [IV.P55] SU(2)_L As Automorphism Group — su2l_identification

  • [IV.P174] Polarity-Switching Suppressed by exp(-pol(X)) — switching_suppression

  • [IV.R28] Chirality is boundary-character property — (structural remark)

  • [IV.R380] Parity violation is a tau-effective prediction — (structural remark)

  • [IV.R382] No right-handed neutrinos in minimal tau — (structural remark)

  • [IV.R383] CP violation from sigma-polarity involution — (structural remark)

Ground Truth Sources

  • Chapter 30 of Book IV (2nd Edition)

Tau.BookIV.Electroweak.right_handed_invisible

source theorem Tau.BookIV.Electroweak.right_handed_invisible :sigma_a_admissible ChiralityType.Right = false

[IV.C01] Right-handed fermion configurations are invisible to the weak force. This is a structural consequence: sigma_A-inadmissible states do not couple to A-sector transport modes. Formalized as: sigma_a_admissible Right = false.

Tau.BookIV.Electroweak.right_handed_no_weak

source theorem Tau.BookIV.Electroweak.right_handed_no_weak :sigma_a_admissible ChiralityType.Right = false ∧ sigma_a_admissible ChiralityType.Left = true

Right-handed states have zero weak coupling.

Tau.BookIV.Electroweak.PolaritySwitching

source structure Tau.BookIV.Electroweak.PolaritySwitching :Type

[IV.D319] A polarity-switching transition in the A-sector: a process that changes the chi_plus/chi_minus composition of a state. The W boson mediates such transitions (e.g., up-quark to down-quark). The transition amplitude depends on the sector polarity measure.

  • source : ChiralityType Source chirality.

  • target : ChiralityType Target chirality.

  • source_left : self.source = ChiralityType.Left Source must be left-handed (sigma_A-admissible).

  • target_left : self.target = ChiralityType.Left Target must be left-handed.

  • mediator : String The mediating boson name.

Instances For

Tau.BookIV.Electroweak.instReprPolaritySwitching.repr

source def Tau.BookIV.Electroweak.instReprPolaritySwitching.repr :PolaritySwitching → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprPolaritySwitching

source instance Tau.BookIV.Electroweak.instReprPolaritySwitching :Repr PolaritySwitching

Equations

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

Tau.BookIV.Electroweak.w_transition

source def Tau.BookIV.Electroweak.w_transition :PolaritySwitching

Canonical polarity-switching transition via W boson. Equations

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

Tau.BookIV.Electroweak.ParityBridgeRecall

source structure Tau.BookIV.Electroweak.ParityBridgeRecall :Prop

[IV.T122] Parity Bridge Recall (III.T07): the A-sector is the unique sector where the sigma-involution acts non-trivially on chirality. This is the physical consequence of the Parity Bridge theorem from Book III. The balanced polarity (chi_plus = chi_minus = 50) implies maximal parity violation.

  • a_balanced : pol_A.chi_plus = pol_A.chi_minus The A-sector has balanced polarity.

  • d_not_balanced : pol_D.chi_plus ≠ pol_D.chi_minus Only A is balanced among primitive sectors.

  • b_not_balanced : pol_B.chi_plus ≠ pol_B.chi_minus
  • c_not_balanced : pol_C.chi_plus ≠ pol_C.chi_minus Instances For

Tau.BookIV.Electroweak.parity_bridge_recall

source def Tau.BookIV.Electroweak.parity_bridge_recall :ParityBridgeRecall

The parity bridge recall is verified by computation. Equations

  • Tau.BookIV.Electroweak.parity_bridge_recall = ⋯ Instances For

Tau.BookIV.Electroweak.chirality_selection

source theorem Tau.BookIV.Electroweak.chirality_selection (c : ChiralityType) :sigma_a_admissible c = true ↔ c = ChiralityType.Left

[IV.T123] Chirality Selection Theorem: the weak interaction selects exactly the left-handed projection. This follows from: (1) A-sector has balanced polarity (Parity Bridge), (2) sigma_A acts non-trivially (flips chirality), (3) only sigma_A-admissible states couple to W/Z.

Formally: for all chirality types c, c couples to weak iff c = Left.

Tau.BookIV.Electroweak.chirality_selection_complete

source theorem Tau.BookIV.Electroweak.chirality_selection_complete :sigma_a_admissible ChiralityType.Left = true

The selection is complete: every left-handed state couples.

Tau.BookIV.Electroweak.chirality_selection_exclusive

source theorem Tau.BookIV.Electroweak.chirality_selection_exclusive :sigma_a_admissible ChiralityType.Right = false

The selection is exclusive: no right-handed state couples.

Tau.BookIV.Electroweak.non_weak_parity_preserving

source theorem Tau.BookIV.Electroweak.non_weak_parity_preserving :parity_em.preserves = true ∧ parity_strong.preserves = true ∧ parity_gravity.preserves = true

[IV.P54] EM, Strong, and Gravity are parity-preserving: they do not distinguish chirality. Their parity violation measure is zero.

Tau.BookIV.Electroweak.weak_unique_non_preserving

source theorem Tau.BookIV.Electroweak.weak_unique_non_preserving :parity_weak.preserves = false ∧ parity_em.preserves = true ∧ parity_strong.preserves = true ∧ parity_gravity.preserves = true ∧ parity_higgs.preserves = true

Weak is the unique non-preserving sector.

Tau.BookIV.Electroweak.SU2LIdentification

source structure Tau.BookIV.Electroweak.SU2LIdentification :Type

[IV.P55] SU(2)_L as the automorphism group of the A-sector left-handed doublet structure. The subscript L means “left”: only left-handed fermions transform under SU(2)_L.

Structural encoding: gauge_group_dim = 3 (dim of SU(2)), chirality = Left (only left-handed transforms).

  • gauge_dim : ℕ Gauge group dimension: dim SU(2) = 3.

  • gauge_dim_eq : self.gauge_dim = 3

  • num_generators : ℕ Number of generators = 3 (Pauli matrices).

  • num_gen_eq : self.num_generators = 3

  • chirality : ChiralityType Only left-handed chirality.

  • chirality_left : self.chirality = ChiralityType.Left Instances For

Tau.BookIV.Electroweak.instReprSU2LIdentification.repr

source def Tau.BookIV.Electroweak.instReprSU2LIdentification.repr :SU2LIdentification → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprSU2LIdentification

source instance Tau.BookIV.Electroweak.instReprSU2LIdentification :Repr SU2LIdentification

Equations

  • Tau.BookIV.Electroweak.instReprSU2LIdentification = { reprPrec := Tau.BookIV.Electroweak.instReprSU2LIdentification.repr }

Tau.BookIV.Electroweak.su2l_identification

source def Tau.BookIV.Electroweak.su2l_identification :SU2LIdentification

The canonical SU(2)_L identification. Equations

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

Tau.BookIV.Electroweak.polarity_difference

source def Tau.BookIV.Electroweak.polarity_difference (p : PolarityIndex) :ℤ

[IV.P174] Polarity-switching transitions are suppressed in sectors with unbalanced polarity. The suppression factor is exp(-|pol(X)|) where pol(X) = chi_plus - chi_minus. For unbalanced sectors, |pol| > 0, so transitions are exponentially suppressed. For the balanced A-sector, |pol| = 0 and there is no suppression.

Structural: we verify that only A has pol = 0. Equations

  • Tau.BookIV.Electroweak.polarity_difference p = Int.ofNat p.chi_plus - Int.ofNat p.chi_minus Instances For

Tau.BookIV.Electroweak.switching_suppression_a_zero

source theorem Tau.BookIV.Electroweak.switching_suppression_a_zero :polarity_difference pol_A = 0

Tau.BookIV.Electroweak.switching_suppression_d_nonzero

source theorem Tau.BookIV.Electroweak.switching_suppression_d_nonzero :polarity_difference pol_D ≠ 0

Tau.BookIV.Electroweak.switching_suppression_b_nonzero

source theorem Tau.BookIV.Electroweak.switching_suppression_b_nonzero :polarity_difference pol_B ≠ 0

Tau.BookIV.Electroweak.switching_suppression_c_nonzero

source theorem Tau.BookIV.Electroweak.switching_suppression_c_nonzero :polarity_difference pol_C ≠ 0