TauLib · API Book IV

TauLib.BookIV.Electroweak.TauHiggs

TauLib.BookIV.Electroweak.TauHiggs

The τ-Higgs mechanism: crossing-point resolution, coherence functional, physical vacuum, and the surviving spin-0 excitation.

Registry Cross-References

  • [IV.D134] Higgs Mechanism as ω-Sector Resolution — HiggsMechanism

  • [IV.D135] Coherence Functional V_n — CoherenceFunctional

  • [IV.D136] Physical Vacuum — PhysicalVacuum

  • [IV.D137] Minimality Condition — MinimalityCondition

  • [IV.D138] EM-Nullity — EMNullity

  • [IV.D139] σ-Polarity of Higgs Excitation — SigmaPolarity

  • [IV.L06] Fiber Excitation Spin Decomposition — fiber_spin_decomposition

  • [IV.T63] Physical Vacuum Existence, Uniqueness, Stability — vacuum_existence_uniqueness_stability

  • [IV.T64] Surviving Excitation is Spin-0 — surviving_is_spin0

  • [IV.P72] Degenerate Vacuum Manifold — degenerate_vacuum_manifold

  • [IV.P73] V_n Minimum on S¹ — vn_minimum_on_circle

  • [IV.R34] Higgs Determines Sector Separation — structural remark

Mathematical Content

In the τ-framework, the Higgs mechanism is NOT an ad hoc scalar field added to the Lagrangian. It is the ω-sector resolution of the crossing singularity at L = S¹ ∨ S¹.

The coherence functional V_n on crossing excitations has a unique minimum (the physical vacuum) with vacuum expectation value v_EW ≈ 246 GeV. The minimum lies on a circle S¹ in field space (degenerate manifold), and the surviving excitation after eating Goldstone bosons is a spin-0 scalar with σ-polarity (σ = +1, unpolarized).

Key structural point: the Higgs field determines sector separation (which sectors get mass), NOT mass itself. Mass originates from the breathing operator on T².

Ground Truth Sources

  • Chapter 34 of Book IV (2nd Edition)

  • kappa_n_closing_identity_sprint.md §8

Tau.BookIV.Electroweak.HiggsMechanism

source structure Tau.BookIV.Electroweak.HiggsMechanism :Type

[IV.D134] The Higgs mechanism in the τ-framework: the ω-sector (crossing of B and C lobes) provides a smooth resolution of the crossing singularity at the lemniscate junction.

This is NOT a separate field — it is the structural content of the fifth generator ω acting at the B∩C intersection.

  • sector : BookIII.Sectors.Sector The mediating sector.

  • resolved_B : BookIII.Sectors.Sector The resolved sectors.

  • resolved_C : BookIII.Sectors.Sector

  • coupling_numer : ℕ The crossing coupling κ(B,C) governs the mechanism.

  • coupling_denom : ℕ
  • is_structural : Bool Not a separate field — structural resolution.

Instances For

Tau.BookIV.Electroweak.instReprHiggsMechanism.repr

source def Tau.BookIV.Electroweak.instReprHiggsMechanism.repr :HiggsMechanism → ℕ → Std.Format

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

Tau.BookIV.Electroweak.instReprHiggsMechanism

source instance Tau.BookIV.Electroweak.instReprHiggsMechanism :Repr HiggsMechanism

Equations

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

Tau.BookIV.Electroweak.higgs_mechanism

source def Tau.BookIV.Electroweak.higgs_mechanism :HiggsMechanism

Equations

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

Tau.BookIV.Electroweak.CoherenceFunctional

source structure Tau.BookIV.Electroweak.CoherenceFunctional :Type

[IV.D135] The coherence functional V_n on crossing excitations. V_n measures the coherence cost of displacing the ω-sector field from its equilibrium. The subscript n indicates evaluation at tower level n of the refinement tower.

V_n has the form of a Mexican hat potential, but this form is DERIVED from coherence constraints, not postulated.

  • tower_level : ℕ Tower level at which V_n is evaluated.

  • level_pos : self.tower_level > 0 Tower level is positive.

  • mexican_hat : Bool The functional has a Mexican hat shape.

  • minimum_exists : Bool Minimum exists.

  • minimum_unique_mod_circle : Bool Minimum is unique (up to S¹ degeneracy).

Instances For

Tau.BookIV.Electroweak.instReprCoherenceFunctional

source instance Tau.BookIV.Electroweak.instReprCoherenceFunctional :Repr CoherenceFunctional

Equations

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

Tau.BookIV.Electroweak.instReprCoherenceFunctional.repr

source def Tau.BookIV.Electroweak.instReprCoherenceFunctional.repr :CoherenceFunctional → ℕ → Std.Format

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

Tau.BookIV.Electroweak.coherence_V1

source def Tau.BookIV.Electroweak.coherence_V1 :CoherenceFunctional

Coherence functional at level 1 (leading order). Equations

  • Tau.BookIV.Electroweak.coherence_V1 = { tower_level := 1, level_pos := Tau.BookIV.Electroweak.coherence_V1._proof_2 } Instances For

Tau.BookIV.Electroweak.PhysicalVacuum

source structure Tau.BookIV.Electroweak.PhysicalVacuum :Type

[IV.D136] The physical vacuum: the minimum of the coherence functional V_n. The vacuum expectation value (VEV) v_EW ≈ 246 GeV sets the electroweak scale.

In the τ-framework, v_EW is determined by ι_τ and the neutron mass anchor m_n, NOT as a free parameter.

  • vev_MeV : ℕ VEV in MeV (v_EW ≈ 246200 MeV).

  • unique : Bool VEV is unique (up to S¹ degeneracy).

  • stable : Bool Vacuum is stable (V_n has positive second derivative).

  • vev_nonzero : self.vev_MeV > 0 VEV is nonzero (spontaneous breaking occurs).

Instances For

Tau.BookIV.Electroweak.instReprPhysicalVacuum.repr

source def Tau.BookIV.Electroweak.instReprPhysicalVacuum.repr :PhysicalVacuum → ℕ → Std.Format

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

Tau.BookIV.Electroweak.instReprPhysicalVacuum

source instance Tau.BookIV.Electroweak.instReprPhysicalVacuum :Repr PhysicalVacuum

Equations

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

Tau.BookIV.Electroweak.physical_vacuum

source def Tau.BookIV.Electroweak.physical_vacuum :PhysicalVacuum

Equations

  • Tau.BookIV.Electroweak.physical_vacuum = { vev_nonzero := Tau.BookIV.Electroweak.physical_vacuum._proof_2 } Instances For

Tau.BookIV.Electroweak.MinimalityCondition

source structure Tau.BookIV.Electroweak.MinimalityCondition :Type

[IV.D137] Minimality condition: the first variation of V_n vanishes at the physical vacuum. Combined with the positive second variation (stability), this characterizes the VEV as a strict local minimum modulo the S¹ degeneracy.

  • first_variation_zero : Bool First variation vanishes.

  • second_variation_pos : Bool Second variation is positive (stability).

  • on_circle_orbit : Bool The minimum is on the S¹ orbit.

Instances For

Tau.BookIV.Electroweak.instReprMinimalityCondition.repr

source def Tau.BookIV.Electroweak.instReprMinimalityCondition.repr :MinimalityCondition → ℕ → Std.Format

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

Tau.BookIV.Electroweak.instReprMinimalityCondition

source instance Tau.BookIV.Electroweak.instReprMinimalityCondition :Repr MinimalityCondition

Equations

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

Tau.BookIV.Electroweak.minimality_condition

source def Tau.BookIV.Electroweak.minimality_condition :MinimalityCondition

Equations

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

Tau.BookIV.Electroweak.EMNullity

source structure Tau.BookIV.Electroweak.EMNullity :Type

[IV.D138] EM-nullity: the photon remains massless after electroweak symmetry breaking.

In the τ-framework, this is a THEOREM, not a tuning condition: the U(1)_EM generator is the unique combination of W³ and B that commutes with the ω-sector VEV. The VEV breaks SU(2)_L × U(1)_Y down to U(1)_EM, and the unbroken generator gives a massless photon.

  • photon_massless : Bool The photon is massless.

  • unbroken_symmetry : String The unbroken symmetry is U(1)_EM.

  • forced_by_vev : Bool This is forced by the VEV structure.

Instances For

Tau.BookIV.Electroweak.instReprEMNullity

source instance Tau.BookIV.Electroweak.instReprEMNullity :Repr EMNullity

Equations

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

Tau.BookIV.Electroweak.instReprEMNullity.repr

source def Tau.BookIV.Electroweak.instReprEMNullity.repr :EMNullity → ℕ → Std.Format

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

Tau.BookIV.Electroweak.em_nullity

source def Tau.BookIV.Electroweak.em_nullity :EMNullity

Equations

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

Tau.BookIV.Electroweak.SigmaPolarity

source structure Tau.BookIV.Electroweak.SigmaPolarity :Type

[IV.D139] The surviving Higgs excitation has σ-polarity σ = +1, meaning it is unpolarized (neither χ₊ nor χ₋ dominant). This reflects its origin at the CROSSING point of the lemniscate, where both lobes meet.

  • sigma : ℤ Polarity value: +1 = unpolarized.

  • at_crossing : Bool At crossing point.

  • unpolarized : Bool Neither lobe dominates.

Instances For

Tau.BookIV.Electroweak.instReprSigmaPolarity.repr

source def Tau.BookIV.Electroweak.instReprSigmaPolarity.repr :SigmaPolarity → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprSigmaPolarity

source instance Tau.BookIV.Electroweak.instReprSigmaPolarity :Repr SigmaPolarity

Equations

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

Tau.BookIV.Electroweak.sigma_polarity

source def Tau.BookIV.Electroweak.sigma_polarity :SigmaPolarity

Equations

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

Tau.BookIV.Electroweak.FiberSpinDecomposition

source structure Tau.BookIV.Electroweak.FiberSpinDecomposition :Type

[IV.L06] Fiber excitations on T² decompose by spin at the crossing point:

  • Spin 0: scalar (survives as Higgs)

  • Spin 1: vector (eaten as longitudinal W/Z modes = Goldstones)

This decomposition is forced by the SO(2) symmetry of the crossing-point tangent space.

  • spin0_count : ℕ Number of spin-0 modes at crossing.

  • spin1_count : ℕ Number of spin-1 modes at crossing.

  • total : ℕ Total excitation count.

  • total_check : self.total = self.spin0_count + self.spin1_count Total equals spin-0 + spin-1.

Instances For

Tau.BookIV.Electroweak.instReprFiberSpinDecomposition

source instance Tau.BookIV.Electroweak.instReprFiberSpinDecomposition :Repr FiberSpinDecomposition

Equations

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

Tau.BookIV.Electroweak.instReprFiberSpinDecomposition.repr

source def Tau.BookIV.Electroweak.instReprFiberSpinDecomposition.repr :FiberSpinDecomposition → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.fiber_spin_decomposition

source def Tau.BookIV.Electroweak.fiber_spin_decomposition :FiberSpinDecomposition

Equations

  • Tau.BookIV.Electroweak.fiber_spin_decomposition = { total_check := Tau.BookIV.Electroweak.fiber_spin_decomposition._proof_2 } Instances For

Tau.BookIV.Electroweak.vacuum_existence_uniqueness_stability

source theorem Tau.BookIV.Electroweak.vacuum_existence_uniqueness_stability :physical_vacuum.unique = true ∧ physical_vacuum.stable = true ∧ physical_vacuum.vev_MeV > 0

[IV.T63] The physical vacuum exists, is unique (mod S¹), and is stable.

Existence: V_n is bounded below and continuous on a compact domain. Uniqueness: The Mexican hat potential has a unique radial minimum. Stability: The Hessian at the minimum has all positive eigenvalues in the radial direction (the angular direction is flat = Goldstone).

Tau.BookIV.Electroweak.surviving_is_spin0

source theorem Tau.BookIV.Electroweak.surviving_is_spin0 :fiber_spin_decomposition.spin0_count = 1 ∧ fiber_spin_decomposition.spin1_count = 3 ∧ fiber_spin_decomposition.total = 4

[IV.T64] After Goldstone bosons are eaten by W± and Z, the single surviving excitation is spin-0 (the Higgs boson).

Counting: 4 real components − 3 Goldstones = 1 physical scalar.

Tau.BookIV.Electroweak.surviving_at_crossing

source theorem Tau.BookIV.Electroweak.surviving_at_crossing :sigma_polarity.at_crossing = true ∧ sigma_polarity.sigma = 1

The surviving mode is at the crossing point with σ = +1.

Tau.BookIV.Electroweak.DegenerateVacuumManifold

source structure Tau.BookIV.Electroweak.DegenerateVacuumManifold :Type

[IV.P72] The vacuum manifold is S¹: the set of minima of V_n forms a circle in field space. This degeneracy is the geometric origin of Goldstone bosons.

In the τ-framework, S¹ is one lobe of the lemniscate L = S¹ ∨ S¹. The vacuum selects a point on one lobe, breaking the continuous S¹ symmetry spontaneously.

  • topology : String Vacuum manifold topology.

  • dim : ℕ Dimension of manifold.

  • goldstone_count : ℕ Number of Goldstone bosons = dim of manifold.

  • from_su2_breaking : Bool The 3 Goldstones come from SU(2)_L breaking, not just S¹.

Instances For

Tau.BookIV.Electroweak.instReprDegenerateVacuumManifold

source instance Tau.BookIV.Electroweak.instReprDegenerateVacuumManifold :Repr DegenerateVacuumManifold

Equations

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

Tau.BookIV.Electroweak.instReprDegenerateVacuumManifold.repr

source def Tau.BookIV.Electroweak.instReprDegenerateVacuumManifold.repr :DegenerateVacuumManifold → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.degenerate_vacuum_manifold

source def Tau.BookIV.Electroweak.degenerate_vacuum_manifold :DegenerateVacuumManifold

Equations

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

Tau.BookIV.Electroweak.VnMinimumCircle

source structure Tau.BookIV.Electroweak.VnMinimumCircle :Type

[IV.P73] The minimum of V_n lies on a circle S¹ of radius v_EW in the (Re φ, Im φ) plane. All points on this circle are physically equivalent (related by a U(1) gauge transformation).

  • radius_MeV : ℕ Radius of the minimum circle in MeV.

  • is_gauge_orbit : Bool The circle is a gauge orbit.

  • all_equivalent : Bool All points physically equivalent.

Instances For

Tau.BookIV.Electroweak.instReprVnMinimumCircle.repr

source def Tau.BookIV.Electroweak.instReprVnMinimumCircle.repr :VnMinimumCircle → ℕ → Std.Format

Equations

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

Tau.BookIV.Electroweak.instReprVnMinimumCircle

source instance Tau.BookIV.Electroweak.instReprVnMinimumCircle :Repr VnMinimumCircle

Equations

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

Tau.BookIV.Electroweak.vn_minimum_on_circle

source def Tau.BookIV.Electroweak.vn_minimum_on_circle :VnMinimumCircle

Equations

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

Tau.BookIV.Electroweak.remark_sector_separation

source def Tau.BookIV.Electroweak.remark_sector_separation :String

[IV.R34] The Higgs mechanism determines SECTOR SEPARATION (which sectors acquire mass via coupling to the VEV), NOT mass origin itself. Mass originates from the breathing operator on T² (Book IV Part III). The Higgs mechanism tells us which particles couple to the VEV and therefore which particles get mass from the EW sector.

This distinction resolves the conceptual confusion in the SM where the Higgs “gives mass” — in τ, it mediates the assignment of mass to sectors, while mass itself is a fiber-geometric quantity. Equations

  • Tau.BookIV.Electroweak.remark_sector_separation = “Higgs determines sector separation (mass assignment), not mass origin (breathing operator on T2)” Instances For