TauLib · API Book IV

TauLib.BookIV.Electroweak.WeakChirality

TauLib.BookIV.Electroweak.WeakChirality

Polarity indices, parity operations, chirality, and the structural origin of maximal parity violation in the weak (A) sector.

Registry Cross-References

  • [IV.D107] Polarity Index chi(X) — PolarityIndex

  • [IV.D108] Parity-Preserving Interaction — ParityPreserving

  • [IV.D109] W Boson as Polarity-Switching Transport — WBosonMode

  • [IV.D110] Z Boson as Neutral Weak Transport — ZBosonMode

  • [IV.D111] Chirality (Left/Right-Handedness) — ChiralityType

  • [IV.D112] sigma_A-Admissibility (Left-Handed Only) — sigma_a_admissible

  • [IV.D113] Parity Transformation — ParityTransformation

  • [IV.D114] Parity Violation Measure — ParityViolation

  • [IV.L05] sigma_A-Admissible iff Left-Handed — sigma_a_iff_left

  • [IV.T51] Weak Interaction Maximally Violates Parity — weak_max_parity_violation

  • [IV.P52] Polarity Signatures of All Sectors — all_sector_polarities

  • [IV.P53] W/Z Massive Because omega Is Conical Singularity — wz_massive_from_omega

Ground Truth Sources

  • Chapter 30 of Book IV (2nd Edition)

Tau.BookIV.Electroweak.PolarityIndex

source structure Tau.BookIV.Electroweak.PolarityIndex :Type

[IV.D107] Polarity index chi(X) for a sector X: the pair (chi_plus, chi_minus) encoding the relative weight of multiplicative vs additive boundary characters. Values scaled to denom = 100.

  • sector : BookIII.Sectors.Sector The sector.

  • chi_plus : ℕ chi_plus weight (scaled, 0-100).

  • chi_minus : ℕ chi_minus weight (scaled, 0-100).

  • sum_eq : self.chi_plus + self.chi_minus = 100 Weights sum to 100 (full allocation).

Instances For

Tau.BookIV.Electroweak.instReprPolarityIndex.repr

source def Tau.BookIV.Electroweak.instReprPolarityIndex.repr :PolarityIndex → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprPolarityIndex

source instance Tau.BookIV.Electroweak.instReprPolarityIndex :Repr PolarityIndex

Equations

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

Tau.BookIV.Electroweak.pol_D

source def Tau.BookIV.Electroweak.pol_D :PolarityIndex

Polarity index for D (Gravity): chi_plus-dominant. Equations

  • Tau.BookIV.Electroweak.pol_D = { sector := Tau.BookIII.Sectors.Sector.D, chi_plus := 66, chi_minus := 34, sum_eq := Tau.BookIV.Electroweak.pol_D._proof_2 } Instances For

Tau.BookIV.Electroweak.pol_A

source def Tau.BookIV.Electroweak.pol_A :PolarityIndex

Polarity index for A (Weak): balanced (50/50). Equations

  • Tau.BookIV.Electroweak.pol_A = { sector := Tau.BookIII.Sectors.Sector.A, chi_plus := 50, chi_minus := 50, sum_eq := Tau.BookIV.Electroweak.pol_A._proof_2 } Instances For

Tau.BookIV.Electroweak.pol_B

source def Tau.BookIV.Electroweak.pol_B :PolarityIndex

Polarity index for B (EM): chi_plus-dominant. Equations

  • Tau.BookIV.Electroweak.pol_B = { sector := Tau.BookIII.Sectors.Sector.B, chi_plus := 66, chi_minus := 34, sum_eq := Tau.BookIV.Electroweak.pol_D._proof_2 } Instances For

Tau.BookIV.Electroweak.pol_C

source def Tau.BookIV.Electroweak.pol_C :PolarityIndex

Polarity index for C (Strong): chi_minus-dominant. Equations

  • Tau.BookIV.Electroweak.pol_C = { sector := Tau.BookIII.Sectors.Sector.C, chi_plus := 34, chi_minus := 66, sum_eq := Tau.BookIV.Electroweak.pol_C._proof_2 } Instances For

Tau.BookIV.Electroweak.pol_Omega

source def Tau.BookIV.Electroweak.pol_Omega :PolarityIndex

Polarity index for omega (Higgs): crossing (both active). Equations

  • Tau.BookIV.Electroweak.pol_Omega = { sector := Tau.BookIII.Sectors.Sector.Omega, chi_plus := 50, chi_minus := 50, sum_eq := Tau.BookIV.Electroweak.pol_A._proof_2 } Instances For

Tau.BookIV.Electroweak.all_sector_polarities

source def Tau.BookIV.Electroweak.all_sector_polarities :List PolarityIndex

[IV.P52] Polarity signatures of all 5 sectors at E1. Equations

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

Tau.BookIV.Electroweak.all_sector_polarities_count

source theorem Tau.BookIV.Electroweak.all_sector_polarities_count :all_sector_polarities.length = 5

Tau.BookIV.Electroweak.ParityPreserving

source structure Tau.BookIV.Electroweak.ParityPreserving :Type

[IV.D108] An interaction is parity-preserving if it does not distinguish left from right chirality. EM, Strong, and Gravity are parity-preserving; Weak is not.

  • sector : BookIII.Sectors.Sector The sector.

  • preserves : Bool Whether the sector preserves parity.

Instances For

Tau.BookIV.Electroweak.instReprParityPreserving.repr

source def Tau.BookIV.Electroweak.instReprParityPreserving.repr :ParityPreserving → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprParityPreserving

source instance Tau.BookIV.Electroweak.instReprParityPreserving :Repr ParityPreserving

Equations

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

Tau.BookIV.Electroweak.parity_em

source def Tau.BookIV.Electroweak.parity_em :ParityPreserving

EM preserves parity. Equations

  • Tau.BookIV.Electroweak.parity_em = { sector := Tau.BookIII.Sectors.Sector.B, preserves := true } Instances For

Tau.BookIV.Electroweak.parity_strong

source def Tau.BookIV.Electroweak.parity_strong :ParityPreserving

Strong preserves parity. Equations

  • Tau.BookIV.Electroweak.parity_strong = { sector := Tau.BookIII.Sectors.Sector.C, preserves := true } Instances For

Tau.BookIV.Electroweak.parity_gravity

source def Tau.BookIV.Electroweak.parity_gravity :ParityPreserving

Gravity preserves parity. Equations

  • Tau.BookIV.Electroweak.parity_gravity = { sector := Tau.BookIII.Sectors.Sector.D, preserves := true } Instances For

Tau.BookIV.Electroweak.parity_weak

source def Tau.BookIV.Electroweak.parity_weak :ParityPreserving

Weak VIOLATES parity. Equations

  • Tau.BookIV.Electroweak.parity_weak = { sector := Tau.BookIII.Sectors.Sector.A, preserves := false } Instances For

Tau.BookIV.Electroweak.parity_higgs

source def Tau.BookIV.Electroweak.parity_higgs :ParityPreserving

Higgs preserves parity (scalar). Equations

  • Tau.BookIV.Electroweak.parity_higgs = { sector := Tau.BookIII.Sectors.Sector.Omega, preserves := true } Instances For

Tau.BookIV.Electroweak.ChiralityType

source inductive Tau.BookIV.Electroweak.ChiralityType :Type

[IV.D111] Chirality: left- or right-handedness of a fermion state. In the tau-framework, chirality is a boundary-character property: left-handed = sigma_A-admissible, right-handed = sigma_A-inadmissible.

  • Left : ChiralityType Left-handed: participates in weak interaction.

  • Right : ChiralityType Right-handed: invisible to weak force.

Instances For

Tau.BookIV.Electroweak.instReprChiralityType

source instance Tau.BookIV.Electroweak.instReprChiralityType :Repr ChiralityType

Equations

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

Tau.BookIV.Electroweak.instReprChiralityType.repr

source def Tau.BookIV.Electroweak.instReprChiralityType.repr :ChiralityType → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instDecidableEqChiralityType

source instance Tau.BookIV.Electroweak.instDecidableEqChiralityType :DecidableEq ChiralityType

Equations

  • Tau.BookIV.Electroweak.instDecidableEqChiralityType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIV.Electroweak.instBEqChiralityType.beq

source def Tau.BookIV.Electroweak.instBEqChiralityType.beq :ChiralityType → ChiralityType → Bool

Equations

  • Tau.BookIV.Electroweak.instBEqChiralityType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIV.Electroweak.instBEqChiralityType

source instance Tau.BookIV.Electroweak.instBEqChiralityType :BEq ChiralityType

Equations

  • Tau.BookIV.Electroweak.instBEqChiralityType = { beq := Tau.BookIV.Electroweak.instBEqChiralityType.beq }

Tau.BookIV.Electroweak.sigma_a_admissible

source def Tau.BookIV.Electroweak.sigma_a_admissible (c : ChiralityType) :Bool

[IV.D112] sigma_A-admissibility: a fermion state is sigma_A-admissible iff it is left-handed. The weak sector involution sigma_A acts only on the left-handed projection. Equations

  • Tau.BookIV.Electroweak.sigma_a_admissible Tau.BookIV.Electroweak.ChiralityType.Left = true
  • Tau.BookIV.Electroweak.sigma_a_admissible Tau.BookIV.Electroweak.ChiralityType.Right = false Instances For

Tau.BookIV.Electroweak.sigma_a_iff_left

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

[IV.L05] sigma_A-admissible if and only if left-handed.

Tau.BookIV.Electroweak.WBosonMode

source structure Tau.BookIV.Electroweak.WBosonMode :Type

[IV.D109] W boson: the polarity-switching transport mode in the A-sector. Carries charge and mediates charged-current weak interactions. W is massive because omega (lemniscate crossing) provides a conical singularity that fixes the coherence scale.

  • name : String Name identifier.

  • charge : ℤ Electric charge (in units of e): +1, -1, or 0.

  • massive : Bool Massive (true) or massless (false).

  • left_only : Bool Chirality coupling: left-only.

Instances For

Tau.BookIV.Electroweak.instReprWBosonMode.repr

source def Tau.BookIV.Electroweak.instReprWBosonMode.repr :WBosonMode → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprWBosonMode

source instance Tau.BookIV.Electroweak.instReprWBosonMode :Repr WBosonMode

Equations

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

Tau.BookIV.Electroweak.w_plus

source def Tau.BookIV.Electroweak.w_plus :WBosonMode

W-plus boson. Equations

  • Tau.BookIV.Electroweak.w_plus = { name := “W+”, charge := 1, massive := true, left_only := true } Instances For

Tau.BookIV.Electroweak.w_minus

source def Tau.BookIV.Electroweak.w_minus :WBosonMode

W-minus boson. Equations

  • Tau.BookIV.Electroweak.w_minus = { name := “W-“, charge := -1, massive := true, left_only := true } Instances For

Tau.BookIV.Electroweak.ZBosonMode

source structure Tau.BookIV.Electroweak.ZBosonMode :Type

[IV.D110] Z boson: the neutral weak transport mode. Mediates neutral-current weak interactions. Also massive via omega-coupling.

  • name : String Name identifier.

  • charge : ℤ Electric charge: always 0.

  • charge_zero : self.charge = 0

  • massive : Bool Massive.

  • massive_true : self.massive = true Instances For

Tau.BookIV.Electroweak.instReprZBosonMode

source instance Tau.BookIV.Electroweak.instReprZBosonMode :Repr ZBosonMode

Equations

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

Tau.BookIV.Electroweak.instReprZBosonMode.repr

source def Tau.BookIV.Electroweak.instReprZBosonMode.repr :ZBosonMode → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.z_zero

source def Tau.BookIV.Electroweak.z_zero :ZBosonMode

Z-zero boson. Equations

  • Tau.BookIV.Electroweak.z_zero = { name := “Z0”, charge := 0, charge_zero := Tau.BookIV.Electroweak.z_zero._proof_1, massive := true, massive_true := ⋯ } Instances For

Tau.BookIV.Electroweak.ParityTransformation

source structure Tau.BookIV.Electroweak.ParityTransformation :Type

[IV.D113] Parity transformation P: spatial inversion x to -x. In the tau-framework, P is the sigma-involution restricted to spatial coordinates (fiber T-squared). It swaps chi_plus and chi_minus on the fiber.

  • spatial_dims : ℕ Spatial dimension count affected.

  • spatial_eq : self.spatial_dims = 3

  • det_sign : ℤ Determinant of P: det(P) = (-1)^d = -1 for d = 3.

  • det_eq : self.det_sign = -1 Instances For

Tau.BookIV.Electroweak.instReprParityTransformation.repr

source def Tau.BookIV.Electroweak.instReprParityTransformation.repr :ParityTransformation → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprParityTransformation

source instance Tau.BookIV.Electroweak.instReprParityTransformation :Repr ParityTransformation

Equations

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

Tau.BookIV.Electroweak.parity_3d

source def Tau.BookIV.Electroweak.parity_3d :ParityTransformation

Canonical parity transformation in 3+1 dimensions. Equations

  • Tau.BookIV.Electroweak.parity_3d = { spatial_dims := 3, spatial_eq := Tau.BookIV.Electroweak.parity_3d._proof_1, det_sign := -1, det_eq := Tau.BookIV.Electroweak.parity_3d._proof_2 } Instances For

Tau.BookIV.Electroweak.ParityViolation

source structure Tau.BookIV.Electroweak.ParityViolation :Type

[IV.D114] Parity violation measure V(S) for sector S: V = 0 for parity-preserving, V = 100 for maximal violation. The weak sector has V = 100 (couples only to left-handed fermions).

  • sector : BookIII.Sectors.Sector Sector.

  • violation : ℕ Violation measure: 0 (preserving) or 100 (maximal), scaled.

  • bounded : self.violation ≤ 100 Bounded by 100.

Instances For

Tau.BookIV.Electroweak.instReprParityViolation.repr

source def Tau.BookIV.Electroweak.instReprParityViolation.repr :ParityViolation → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprParityViolation

source instance Tau.BookIV.Electroweak.instReprParityViolation :Repr ParityViolation

Equations

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

Tau.BookIV.Electroweak.pv_gravity

source def Tau.BookIV.Electroweak.pv_gravity :ParityViolation

Parity violation for each sector. Equations

  • Tau.BookIV.Electroweak.pv_gravity = { sector := Tau.BookIII.Sectors.Sector.D, violation := 0, bounded := Tau.BookIV.Electroweak.pv_gravity._proof_2 } Instances For

Tau.BookIV.Electroweak.pv_weak

source def Tau.BookIV.Electroweak.pv_weak :ParityViolation

Equations

  • Tau.BookIV.Electroweak.pv_weak = { sector := Tau.BookIII.Sectors.Sector.A, violation := 100, bounded := Tau.BookIV.Electroweak.pv_weak._proof_2 } Instances For

Tau.BookIV.Electroweak.pv_em

source def Tau.BookIV.Electroweak.pv_em :ParityViolation

Equations

  • Tau.BookIV.Electroweak.pv_em = { sector := Tau.BookIII.Sectors.Sector.B, violation := 0, bounded := Tau.BookIV.Electroweak.pv_gravity._proof_2 } Instances For

Tau.BookIV.Electroweak.pv_strong

source def Tau.BookIV.Electroweak.pv_strong :ParityViolation

Equations

  • Tau.BookIV.Electroweak.pv_strong = { sector := Tau.BookIII.Sectors.Sector.C, violation := 0, bounded := Tau.BookIV.Electroweak.pv_gravity._proof_2 } Instances For

Tau.BookIV.Electroweak.pv_higgs

source def Tau.BookIV.Electroweak.pv_higgs :ParityViolation

Equations

  • Tau.BookIV.Electroweak.pv_higgs = { sector := Tau.BookIII.Sectors.Sector.Omega, violation := 0, bounded := Tau.BookIV.Electroweak.pv_gravity._proof_2 } Instances For

Tau.BookIV.Electroweak.weak_max_parity_violation

source theorem Tau.BookIV.Electroweak.weak_max_parity_violation :pv_weak.violation = 100 ∧ pv_gravity.violation = 0 ∧ pv_em.violation = 0 ∧ pv_strong.violation = 0 ∧ pv_higgs.violation = 0

[IV.T51] The weak interaction maximally violates parity: V(A) = 100 (maximal) while V(D) = V(B) = V(C) = V(omega) = 0. This is structural: the A-sector balanced polarity (pol = 1) means sigma_A acts non-trivially, selecting one chirality.

Tau.BookIV.Electroweak.weak_unique_violator

source theorem Tau.BookIV.Electroweak.weak_unique_violator :pv_weak.violation > 0 ∧ pv_gravity.violation = 0 ∧ pv_em.violation = 0 ∧ pv_strong.violation = 0 ∧ pv_higgs.violation = 0

Only the weak sector has nonzero parity violation.

Tau.BookIV.Electroweak.wz_massive_from_omega

source theorem Tau.BookIV.Electroweak.wz_massive_from_omega :w_plus.massive = true ∧ w_minus.massive = true ∧ z_zero.massive = true

[IV.P53] W and Z bosons are massive because omega is the conical singularity of L (lemniscate crossing point). The crossing fixes a coherence scale, breaking the would-be massless gauge symmetry. Structural: massive = true for both W and Z.

Tau.BookIV.Electroweak.w_left_only

source theorem Tau.BookIV.Electroweak.w_left_only :w_plus.left_only = true ∧ w_minus.left_only = true

W bosons couple only to left-handed fermions.