TauLib · API Book VI

TauLib.BookVI.CosmicLife.BHDist

TauLib.BookVI.CosmicLife.BHDist

Black hole distinction: macro-torus carrier satisfies all 5 conditions. Imports BookV.Gravity.BHTopoModes for T² horizon topology authority.

Registry Cross-References

  • [VI.D54] Macro-Torus Carrier — MacroTorusCarrier

  • [VI.D55] Lexicographic Defect Functional — LexDefectFunctional

  • [VI.D56] Frame-Closure Defect — FrameClosureDefect

  • [VI.D57] Strong-Saturation Defect — StrongSaturationDefect

  • [VI.T29] BH Distinction Theorem — bh_distinction_theorem

Book V Authority (imported via BHTopoModes)

  • [V.T109] BH Toroidal Topology: horizon is T², not S²

  • [V.D234] T² QNM Mode Structure: ringdown spectrum

  • [V.T168] QNM Frequency Ratio = ι_τ⁻¹

Ground Truth Sources

  • Book VI Chapter 43 (2nd Edition): BH Distinction

  • Book V Chapter 50 (2nd Edition): BH Birth Topology (V.T109)

Tau.BookVI.BHDist.horizon_topology

source def Tau.BookVI.BHDist.horizon_topology :ℕ

The BH horizon has T² topology (torus), NOT S² (sphere). Authority: V.T109 (BH Toroidal Topology, Book V Chapter 50). The two S¹ cycles generate the torus QNM spectrum (V.D234). Equations

  • Tau.BookVI.BHDist.horizon_topology = 2 Instances For

Tau.BookVI.BHDist.bh_carrier_is_torus

source theorem Tau.BookVI.BHDist.bh_carrier_is_torus :horizon_topology = 2

The horizon is toroidal: dimension matches T² fiber of τ³.

Tau.BookVI.BHDist.torus_modes_from_bookV

source theorem Tau.BookVI.BHDist.torus_modes_from_bookV :BookV.Gravity.primitiveTorusModes.length = 3

Connection to BookV: the primitive torus modes exist and number 3.

Tau.BookVI.BHDist.MacroTorusCarrier

source structure Tau.BookVI.BHDist.MacroTorusCarrier :Type

[VI.D54] Macro-torus carrier: BH carrier with T² boundary. Components: T² boundary topology, multipole refinement tower, Kerr holonomy, and carrier type.

  • t2_boundary : Bool Horizon has T² topology (from V.T109).

  • multipole_tower : Bool Refinement tower: multipole moments through order 2^n.

  • kerr_holonomy : Bool Boundary holonomy: frame-dragging algebra from Kerr.

  • carrier_type : String Carrier type identifier.

  • horizon_dim : ℕ Horizon dimension = 2 (torus).

Instances For

Tau.BookVI.BHDist.instReprMacroTorusCarrier

source instance Tau.BookVI.BHDist.instReprMacroTorusCarrier :Repr MacroTorusCarrier

Equations

  • Tau.BookVI.BHDist.instReprMacroTorusCarrier = { reprPrec := Tau.BookVI.BHDist.instReprMacroTorusCarrier.repr }

Tau.BookVI.BHDist.instReprMacroTorusCarrier.repr

source def Tau.BookVI.BHDist.instReprMacroTorusCarrier.repr :MacroTorusCarrier → ℕ → Std.Format

Equations

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

Tau.BookVI.BHDist.macro_torus

source def Tau.BookVI.BHDist.macro_torus :MacroTorusCarrier

Equations

  • Tau.BookVI.BHDist.macro_torus = { } Instances For

Tau.BookVI.BHDist.macro_torus_is_t2

source theorem Tau.BookVI.BHDist.macro_torus_is_t2 :macro_torus.horizon_dim = 2

The macro-torus carrier has T² boundary.

Tau.BookVI.BHDist.LexDefectFunctional

source structure Tau.BookVI.BHDist.LexDefectFunctional :Type

[VI.D55] Lexicographic defect: pairs frame-closure + strong-saturation. Ordered lexicographically: frame dominates (slow DoF first).

  • component_count : ℕ Number of defect components.

  • count_eq : self.component_count = 2 Exactly 2 components: frame-closure + strong-saturation.

  • lexicographic : Bool Lexicographic ordering active.

Instances For

Tau.BookVI.BHDist.instReprLexDefectFunctional.repr

source def Tau.BookVI.BHDist.instReprLexDefectFunctional.repr :LexDefectFunctional → ℕ → Std.Format

Equations

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

Tau.BookVI.BHDist.instReprLexDefectFunctional

source instance Tau.BookVI.BHDist.instReprLexDefectFunctional :Repr LexDefectFunctional

Equations

  • Tau.BookVI.BHDist.instReprLexDefectFunctional = { reprPrec := Tau.BookVI.BHDist.instReprLexDefectFunctional.repr }

Tau.BookVI.BHDist.lex_defect

source def Tau.BookVI.BHDist.lex_defect :LexDefectFunctional

Equations

  • Tau.BookVI.BHDist.lex_defect = { component_count := 2, count_eq := Tau.BookVI.BHDist.lex_defect._proof_1 } Instances For

Tau.BookVI.BHDist.FrameClosureDefect

source structure Tau.BookVI.BHDist.FrameClosureDefect :Type

[VI.D56] Frame-closure defect: Sobolev-norm deviation from Kerr-Newman. Vanishes for isolated stationary Kerr BH. Norm choice is conventional (any coercive norm on H^n(H) works).

  • vanishes_ideal : Bool Vanishes for ideal Kerr solution.

  • sobolev_norm : Bool Uses Sobolev norm (conventional choice).

  • damped_by_qnm : Bool Ringdown damps this defect via V.D234 QNM modes.

Instances For

Tau.BookVI.BHDist.instReprFrameClosureDefect

source instance Tau.BookVI.BHDist.instReprFrameClosureDefect :Repr FrameClosureDefect

Equations

  • Tau.BookVI.BHDist.instReprFrameClosureDefect = { reprPrec := Tau.BookVI.BHDist.instReprFrameClosureDefect.repr }

Tau.BookVI.BHDist.instReprFrameClosureDefect.repr

source def Tau.BookVI.BHDist.instReprFrameClosureDefect.repr :FrameClosureDefect → ℕ → Std.Format

Equations

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

Tau.BookVI.BHDist.StrongSaturationDefect

source structure Tau.BookVI.BHDist.StrongSaturationDefect :Type

[VI.D57] Strong-saturation defect: V^s_n ∈ [0,1]. Measures residual strong-sector instability.

  • range_unit : Bool Range is [0,1].

  • bh_negligible : Bool BH has negligible strong-saturation defect.

Instances For

Tau.BookVI.BHDist.instReprStrongSaturationDefect

source instance Tau.BookVI.BHDist.instReprStrongSaturationDefect :Repr StrongSaturationDefect

Equations

  • Tau.BookVI.BHDist.instReprStrongSaturationDefect = { reprPrec := Tau.BookVI.BHDist.instReprStrongSaturationDefect.repr }

Tau.BookVI.BHDist.instReprStrongSaturationDefect.repr

source def Tau.BookVI.BHDist.instReprStrongSaturationDefect.repr :StrongSaturationDefect → ℕ → Std.Format

Equations

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

Tau.BookVI.BHDist.BHDistinction

source structure Tau.BookVI.BHDist.BHDistinction :Type

[VI.T29] BH Distinction Theorem: all 5 τ-Distinction conditions satisfied. (i) Clopen: event horizon (zero-width boundary) (ii) Refinement-coherent: No-Hair collapses tower (iii) Eventually stable: stabilizes after 1 ringdown (iv) Law-stable: No-Shrink Theorem (V.T114) (v) H∂-equivariant: axial Killing symmetry

  • conditions_satisfied : ℕ Number of conditions satisfied.

  • all_five : self.conditions_satisfied = 5 All five conditions verified.

  • clopen : Bool
  • refinement_coherent : Bool
  • eventually_stable : Bool
  • law_stable : Bool
  • equivariant : Bool Instances For

Tau.BookVI.BHDist.instReprBHDistinction.repr

source def Tau.BookVI.BHDist.instReprBHDistinction.repr :BHDistinction → ℕ → Std.Format

Equations

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

Tau.BookVI.BHDist.instReprBHDistinction

source instance Tau.BookVI.BHDist.instReprBHDistinction :Repr BHDistinction

Equations

  • Tau.BookVI.BHDist.instReprBHDistinction = { reprPrec := Tau.BookVI.BHDist.instReprBHDistinction.repr }

Tau.BookVI.BHDist.bh_dist

source def Tau.BookVI.BHDist.bh_dist :BHDistinction

Equations

  • Tau.BookVI.BHDist.bh_dist = { conditions_satisfied := 5, all_five := Tau.BookVI.BHDist.bh_dist._proof_1 } Instances For

Tau.BookVI.BHDist.bh_distinction_theorem

source theorem Tau.BookVI.BHDist.bh_distinction_theorem :bh_dist.conditions_satisfied = 5 ∧ bh_dist.clopen = true ∧ bh_dist.refinement_coherent = true ∧ bh_dist.eventually_stable = true ∧ bh_dist.law_stable = true ∧ bh_dist.equivariant = true

[VI.T29] BH Distinction Theorem: 5/5 conditions, carrier is T².

Tau.BookVI.BHDist.horizon_consistency

source theorem Tau.BookVI.BHDist.horizon_consistency :macro_torus.horizon_dim = 2 ∧ horizon_topology = 2 ∧ BookV.Gravity.primitiveTorusModes.length = 3

Cross-book consistency: BH carrier uses T² from BookV, not S².