TauLib · API Book IV

TauLib.BookIV.Sectors.SectorParameters

TauLib.BookIV.Sectors.SectorParameters

Sector Physics Template: the 4 defining parameters for each of the 5 sectors at the E₁ (physics) enrichment level.

Registry Cross-References

  • [IV.D01] Sector Physics Template — SectorPhysics, PolaritySign

  • [IV.D02] EM Sector at E₁ — em_sector

  • [IV.D03] Strong Sector at E₁ — strong_sector

  • [IV.D04] Higgs Sector at E₁ — higgs_sector

  • [IV.D05] Gravity Sector at E₁ — gravity_sector

  • [IV.D06] Weak Sector at E₁ — weak_sector

Mathematical Content

Book III established the abstract 4+1 sector template (III.D13) via character-lattice counting. Book IV instantiates each sector with its four defining parameters:

  • Self-coupling κ(S;d) — rational function of ι_τ at primorial depth d

  • Polarity signature — χ₊-dominant, balanced, χ₋-dominant, or crossing

  • Primorial depth — the level d ∈ {1, 2, 3} at which the coupling operates

  • Physical force — the E₁ incarnation of the sector

The five generators split into temporal (base τ¹) and spatial (fiber T²):

Sector Gen ABCD Carrier κ(S;d) Polarity Depth

B (EM) γ B Fiber T² ι_τ² χ₊-dom 2

C (Strong) η C Fiber T² ι_τ³/(1−ι_τ) χ₋-dom 3

ω (Higgs) γ∩η B∩C Fiber T² ι_τ³/(1+ι_τ) crossing 3

D (Gravity) α D Base τ¹ 1−ι_τ χ₊-dom 1

A (Weak) π A Base τ¹ ι_τ balanced 1

All couplings are determined by ι_τ = 2/(π+e) ≈ 0.341304 (No Knobs, III.T08).

Ground Truth Sources

  • physics_layer_sector_instantiation.md §4: 4+1 sector template at E₁

  • temporal_spatial_decomposition.md: generator-carrier correspondence

  • Book III editorial logbook Decision #31: canonical force mapping (LOCKED)

Tau.BookIV.Sectors.PolaritySign

source inductive Tau.BookIV.Sectors.PolaritySign :Type

[IV.D01] Spectral polarity signature of a sector. Determines which boundary characters dominate the sector’s physics.

  • ChiPlus : PolaritySign χ₊-dominant: multiplicative/spreading characters dominate.

  • Balanced : PolaritySign Balanced: equal χ₊ and χ₋ content (pol = 1).

  • ChiMinus : PolaritySign χ₋-dominant: additive/tightening characters dominate.

  • Crossing : PolaritySign Crossing: both lobes active simultaneously (ω-sector only).

Instances For

Tau.BookIV.Sectors.instReprPolaritySign

source instance Tau.BookIV.Sectors.instReprPolaritySign :Repr PolaritySign

Equations

  • Tau.BookIV.Sectors.instReprPolaritySign = { reprPrec := Tau.BookIV.Sectors.instReprPolaritySign.repr }

Tau.BookIV.Sectors.instReprPolaritySign.repr

source def Tau.BookIV.Sectors.instReprPolaritySign.repr :PolaritySign → ℕ → Std.Format

Equations

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

Tau.BookIV.Sectors.instDecidableEqPolaritySign

source instance Tau.BookIV.Sectors.instDecidableEqPolaritySign :DecidableEq PolaritySign

Equations

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

Tau.BookIV.Sectors.instBEqPolaritySign

source instance Tau.BookIV.Sectors.instBEqPolaritySign :BEq PolaritySign

Equations

  • Tau.BookIV.Sectors.instBEqPolaritySign = { beq := Tau.BookIV.Sectors.instBEqPolaritySign.beq }

Tau.BookIV.Sectors.instBEqPolaritySign.beq

source def Tau.BookIV.Sectors.instBEqPolaritySign.beq :PolaritySign → PolaritySign → Bool

Equations

  • Tau.BookIV.Sectors.instBEqPolaritySign.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIV.Sectors.SectorPhysics

source structure Tau.BookIV.Sectors.SectorPhysics :Type

[IV.D01] The four defining parameters of a sector at E₁. Every sector is completely characterized by these four values. All couplings are rational functions of ι_τ.

  • sector : BookIII.Sectors.Sector The abstract sector (from Book III Decomposition).

  • generator : Kernel.Generator The kernel generator seeding this sector.

  • depth : Denotation.TauIdx Primorial depth at which the coupling operates.

  • polarity : PolaritySign Spectral polarity signature.

  • coupling_numer : ℕ Self-coupling numerator (scaled by coupling_denom).

  • coupling_denom : ℕ Self-coupling denominator (common scale).

  • denom_pos : self.coupling_denom > 0 Denominator is positive.

Instances For

Tau.BookIV.Sectors.instReprSectorPhysics

source instance Tau.BookIV.Sectors.instReprSectorPhysics :Repr SectorPhysics

Equations

  • Tau.BookIV.Sectors.instReprSectorPhysics = { reprPrec := Tau.BookIV.Sectors.instReprSectorPhysics.repr }

Tau.BookIV.Sectors.instReprSectorPhysics.repr

source def Tau.BookIV.Sectors.instReprSectorPhysics.repr :SectorPhysics → ℕ → Std.Format

Equations

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

Tau.BookIV.Sectors.CouplingScale

source@[reducible, inline]

abbrev Tau.BookIV.Sectors.CouplingScale :ℕ

The common denominator for coupling computations. We use 10¹² to preserve precision through multiplication. Equations

  • Tau.BookIV.Sectors.CouplingScale = 1000000000000 Instances For

Tau.BookIV.Sectors.iota

source@[reducible, inline]

abbrev Tau.BookIV.Sectors.iota :ℕ

ι_τ numerator at scale 10⁶ (from Iota.lean). Equations

  • Tau.BookIV.Sectors.iota = Tau.Boundary.iota_tau_numer Instances For

Tau.BookIV.Sectors.iotaD

source@[reducible, inline]

abbrev Tau.BookIV.Sectors.iotaD :ℕ

ι_τ denominator at scale 10⁶ (from Iota.lean). Equations

  • Tau.BookIV.Sectors.iotaD = Tau.Boundary.iota_tau_denom Instances For

Tau.BookIV.Sectors.iota_sq_numer

source def Tau.BookIV.Sectors.iota_sq_numer :ℕ

ι_τ² numerator: iota² = 341304² = 116,594,274,681. Equations

  • Tau.BookIV.Sectors.iota_sq_numer = Tau.BookIV.Sectors.iota * Tau.BookIV.Sectors.iota Instances For

Tau.BookIV.Sectors.iota_sq_denom

source def Tau.BookIV.Sectors.iota_sq_denom :ℕ

ι_τ² denominator: 10¹². Equations

  • Tau.BookIV.Sectors.iota_sq_denom = Tau.BookIV.Sectors.iotaD * Tau.BookIV.Sectors.iotaD Instances For

Tau.BookIV.Sectors.iota_cu_numer

source def Tau.BookIV.Sectors.iota_cu_numer :ℕ

ι_τ³ numerator: iota³. Equations

  • Tau.BookIV.Sectors.iota_cu_numer = Tau.BookIV.Sectors.iota * Tau.BookIV.Sectors.iota * Tau.BookIV.Sectors.iota Instances For

Tau.BookIV.Sectors.iota_cu_denom

source def Tau.BookIV.Sectors.iota_cu_denom :ℕ

ι_τ³ denominator: 10¹⁸. Equations

  • Tau.BookIV.Sectors.iota_cu_denom = Tau.BookIV.Sectors.iotaD * Tau.BookIV.Sectors.iotaD * Tau.BookIV.Sectors.iotaD Instances For

Tau.BookIV.Sectors.em_sector

source def Tau.BookIV.Sectors.em_sector :SectorPhysics

[IV.D02] EM Sector (B): γ-generator, electromagnetic force. Self-coupling κ(B;2) = ι_τ². Polarity: χ₊-dominant (spreading/multiplicative). Depth: 2 (second primorial level). Physical: photon transport, Maxwell equations, fine structure. Equations

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

Tau.BookIV.Sectors.strong_sector

source def Tau.BookIV.Sectors.strong_sector :SectorPhysics

[IV.D03] Strong Sector (C): η-generator, strong force. Self-coupling κ(C;3) = ι_τ³/(1−ι_τ). Polarity: χ₋-dominant (tightening/additive). Depth: 3 (third primorial level). Physical: color holonomy, confinement, mass gap. The (1−ι_τ) denominator is the structural signature of confinement. Equations

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

Tau.BookIV.Sectors.higgs_sector

source def Tau.BookIV.Sectors.higgs_sector :SectorPhysics

[IV.D04] Higgs Sector (ω): γ∩η crossing, Higgs/mass mechanism. Self-coupling κ(B,C) = ι_τ³/(1+ι_τ). Polarity: crossing (both lobes active simultaneously). Depth: 3 (third primorial level). Physical: mass generation, dense spatial occupancy. The unique +1 derived sector from the lemniscate crossing. Equations

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

Tau.BookIV.Sectors.gravity_sector

source def Tau.BookIV.Sectors.gravity_sector :SectorPhysics

[IV.D05] Gravity Sector (D): α-generator, gravitational force. Self-coupling κ(D;1) = 1−ι_τ. Polarity: χ₊-dominant. Depth: 1 (first primorial level). Physical: frame holonomy, temporal flow, G = (c³/ℏ)·ι_τ². Equations

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

Tau.BookIV.Sectors.weak_sector

source def Tau.BookIV.Sectors.weak_sector :SectorPhysics

[IV.D06] Weak Sector (A): π-generator, weak force. Self-coupling κ(A;1) = ι_τ. Polarity: balanced (unique sector with pol = 1). Depth: 1 (first primorial level). Physical: temporal arrow, parity violation, beta decay. The master constant ι_τ itself IS the weak self-coupling. Equations

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

Tau.BookIV.Sectors.all_sectors

source def Tau.BookIV.Sectors.all_sectors :List SectorPhysics

All five sector instantiations as a list. Equations

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

Tau.BookIV.Sectors.sector_physics

source def Tau.BookIV.Sectors.sector_physics (s : BookIII.Sectors.Sector) :SectorPhysics

Lookup sector physics by abstract sector. Equations

  • Tau.BookIV.Sectors.sector_physics Tau.BookIII.Sectors.Sector.D = Tau.BookIV.Sectors.gravity_sector
  • Tau.BookIV.Sectors.sector_physics Tau.BookIII.Sectors.Sector.A = Tau.BookIV.Sectors.weak_sector
  • Tau.BookIV.Sectors.sector_physics Tau.BookIII.Sectors.Sector.B = Tau.BookIV.Sectors.em_sector
  • Tau.BookIV.Sectors.sector_physics Tau.BookIII.Sectors.Sector.C = Tau.BookIV.Sectors.strong_sector
  • Tau.BookIV.Sectors.sector_physics Tau.BookIII.Sectors.Sector.Omega = Tau.BookIV.Sectors.higgs_sector Instances For

Tau.BookIV.Sectors.gen_sector_injective

source theorem Tau.BookIV.Sectors.gen_sector_injective :gravity_sector.generator ≠ weak_sector.generator ∧ weak_sector.generator ≠ em_sector.generator ∧ em_sector.generator ≠ strong_sector.generator

Generator-sector correspondence is injective on primitive sectors.

Tau.BookIV.Sectors.all_couplings_pos

source theorem Tau.BookIV.Sectors.all_couplings_pos :gravity_sector.coupling_numer > 0 ∧ weak_sector.coupling_numer > 0 ∧ em_sector.coupling_numer > 0 ∧ strong_sector.coupling_numer > 0 ∧ higgs_sector.coupling_numer > 0

Every sector has a positive coupling.

Tau.BookIV.Sectors.temporal_depth_one

source theorem Tau.BookIV.Sectors.temporal_depth_one :gravity_sector.depth = 1 ∧ weak_sector.depth = 1

Temporal sectors have depth 1; spatial sectors have depth ≥ 2.

Tau.BookIV.Sectors.spatial_depth_ge_two

source theorem Tau.BookIV.Sectors.spatial_depth_ge_two :em_sector.depth ≥ 2 ∧ strong_sector.depth ≥ 2 ∧ higgs_sector.depth ≥ 2

Tau.BookIV.Sectors.weak_unique_balanced

source theorem Tau.BookIV.Sectors.weak_unique_balanced :weak_sector.polarity = PolaritySign.Balanced ∧ gravity_sector.polarity ≠ PolaritySign.Balanced ∧ em_sector.polarity ≠ PolaritySign.Balanced ∧ strong_sector.polarity ≠ PolaritySign.Balanced ∧ higgs_sector.polarity ≠ PolaritySign.Balanced

The weak sector is the unique balanced sector.

Tau.BookIV.Sectors.SectorPhysics.coupling_float

source def Tau.BookIV.Sectors.SectorPhysics.coupling_float (s : SectorPhysics) :Float

Coupling as a Float (for display purposes). Equations

  • s.coupling_float = Float.ofNat s.coupling_numer / Float.ofNat s.coupling_denom Instances For