TauLib · API Book V

TauLib.BookV.Cosmology.BHBirthTopology

TauLib.BookV.Cosmology.BHBirthTopology

Black hole birth as topology crossing. Gravitational tension, spherical capacity, linking class, and the BH threshold theorem. BH horizon is topologically T² (torus), not S² (sphere). No interior singularity — the interior is a compact subset of T².

Registry Cross-References

  • [V.D163] Gravitational Tension – GravitationalTension

  • [V.D164] Spherical Capacity – SphericalCapacity

  • [V.D165] Linking Class – LinkingClass

  • [V.D166] Black Hole (Topological Event) – BlackHoleTopologicalEvent

  • [V.T109] BH Threshold Theorem – bh_threshold_theorem

  • [V.T110] BH Toroidal Topology – bh_toroidal_topology

  • [V.R222] Event horizon as linking boundary – structural remark

  • [V.P93] No Interior Singularity – no_interior_singularity

  • [V.C18] Information Preservation – information_preservation

  • [V.D167] Canonical BH Neighborhood – CanonicalBHNeighborhood

Mathematical Content

Gravitational Tension

G(U) = κ(D;1) · ||T[χ]|_U||, where κ(D;1) = 1 − ι_τ is the D-sector self-coupling. Measures how strongly the D-sector boundary character acts on a region U.

BH Threshold Theorem

A BH forms iff gravitational tension exceeds the spherical capacity: G(U) > C_sph(n). Below the threshold: neutron star. Above: BH.

BH Toroidal Topology

The BH horizon is T² (the fiber torus), NOT S² (sphere). The linking class ℓ ∈ H₁(T²; ℤ) = ℤ ⊕ ℤ wraps both generators of π₁(T²).

Information Preservation

No information is lost. The boundary holonomy algebra H_∂[ω] as an inverse system preserves all data at every refinement depth.

Ground Truth Sources

  • Book V ch49: Black Hole Birth as Topology Crossing

Tau.BookV.Cosmology.GravitationalTension

source structure Tau.BookV.Cosmology.GravitationalTension :Type

[V.D163] Gravitational tension at region U in τ³: G(U) = κ(D;1) · ||T[χ]|_U||

κ(D;1) = 1 − ι_τ ≈ 0.6585 (D-sector self-coupling). T[χ] = boundary character stress-energy.

  • tension_numer : ℕ Tension numerator (scaled).

  • tension_denom : ℕ Tension denominator.

  • denom_pos : self.tension_denom > 0 Denominator positive.

  • region_id : String Region identifier.

Instances For

Tau.BookV.Cosmology.instReprGravitationalTension.repr

source def Tau.BookV.Cosmology.instReprGravitationalTension.repr :GravitationalTension → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprGravitationalTension

source instance Tau.BookV.Cosmology.instReprGravitationalTension :Repr GravitationalTension

Equations

  • Tau.BookV.Cosmology.instReprGravitationalTension = { reprPrec := Tau.BookV.Cosmology.instReprGravitationalTension.repr }

Tau.BookV.Cosmology.GravitationalTension.toFloat

source def Tau.BookV.Cosmology.GravitationalTension.toFloat (g : GravitationalTension) :Float

Tension as Float. Equations

  • g.toFloat = Float.ofNat g.tension_numer / Float.ofNat g.tension_denom Instances For

Tau.BookV.Cosmology.SphericalCapacity

source structure Tau.BookV.Cosmology.SphericalCapacity :Type

[V.D164] Spherical capacity C_sph(n): the supremum of gravitational tension over all S²-topology configurations at base point α_n.

When tension exceeds capacity, the S² branch is no longer energetically preferred and the topology crosses to T².

  • capacity_numer : ℕ Capacity numerator (scaled).

  • capacity_denom : ℕ Capacity denominator.

  • denom_pos : self.capacity_denom > 0 Denominator positive.

  • depth : ℕ Refinement depth.

  • depth_pos : self.depth > 0 Depth positive.

Instances For

Tau.BookV.Cosmology.instReprSphericalCapacity

source instance Tau.BookV.Cosmology.instReprSphericalCapacity :Repr SphericalCapacity

Equations

  • Tau.BookV.Cosmology.instReprSphericalCapacity = { reprPrec := Tau.BookV.Cosmology.instReprSphericalCapacity.repr }

Tau.BookV.Cosmology.instReprSphericalCapacity.repr

source def Tau.BookV.Cosmology.instReprSphericalCapacity.repr :SphericalCapacity → ℕ → Std.Format

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

Tau.BookV.Cosmology.LinkingClass

source structure Tau.BookV.Cosmology.LinkingClass :Type

[V.D165] Linking class: a non-contractible cycle ℓ ∈ H₁(T²; ℤ) = ℤ ⊕ ℤ that links the two generators of π₁(T²).

A linking class ℓ = (a, b) is non-trivial when a ≠ 0 or b ≠ 0. It wraps both the γ-circle and the η-circle of T².

  • a : ℤ First component (wrapping γ-circle).

  • b : ℤ Second component (wrapping η-circle).

  • nontrivial : self.a ≠ 0 ∨ self.b ≠ 0 Non-trivial: at least one component nonzero.

Instances For

Tau.BookV.Cosmology.instReprLinkingClass

source instance Tau.BookV.Cosmology.instReprLinkingClass :Repr LinkingClass

Equations

  • Tau.BookV.Cosmology.instReprLinkingClass = { reprPrec := Tau.BookV.Cosmology.instReprLinkingClass.repr }

Tau.BookV.Cosmology.instReprLinkingClass.repr

source def Tau.BookV.Cosmology.instReprLinkingClass.repr :LinkingClass → ℕ → Std.Format

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

Tau.BookV.Cosmology.unit_linking

source def Tau.BookV.Cosmology.unit_linking :LinkingClass

Simplest non-trivial linking class: (1,1). Equations

  • Tau.BookV.Cosmology.unit_linking = { a := 1, b := 1, nontrivial := Tau.BookV.Cosmology.unit_linking._proof_2 } Instances For

Tau.BookV.Cosmology.HorizonTopology

source inductive Tau.BookV.Cosmology.HorizonTopology :Type

Topology of the BH horizon.

  • S2 : HorizonTopology S² (spherical, below threshold).

  • T2 : HorizonTopology T² (toroidal, BH).

Instances For

Tau.BookV.Cosmology.instReprHorizonTopology.repr

source def Tau.BookV.Cosmology.instReprHorizonTopology.repr :HorizonTopology → ℕ → Std.Format

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Tau.BookV.Cosmology.instReprHorizonTopology

source instance Tau.BookV.Cosmology.instReprHorizonTopology :Repr HorizonTopology

Equations

  • Tau.BookV.Cosmology.instReprHorizonTopology = { reprPrec := Tau.BookV.Cosmology.instReprHorizonTopology.repr }

Tau.BookV.Cosmology.instDecidableEqHorizonTopology

source instance Tau.BookV.Cosmology.instDecidableEqHorizonTopology :DecidableEq HorizonTopology

Equations

  • Tau.BookV.Cosmology.instDecidableEqHorizonTopology x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Cosmology.instBEqHorizonTopology

source instance Tau.BookV.Cosmology.instBEqHorizonTopology :BEq HorizonTopology

Equations

  • Tau.BookV.Cosmology.instBEqHorizonTopology = { beq := Tau.BookV.Cosmology.instBEqHorizonTopology.beq }

Tau.BookV.Cosmology.instBEqHorizonTopology.beq

source def Tau.BookV.Cosmology.instBEqHorizonTopology.beq :HorizonTopology → HorizonTopology → Bool

Equations

  • Tau.BookV.Cosmology.instBEqHorizonTopology.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Cosmology.BlackHoleTopologicalEvent

source structure Tau.BookV.Cosmology.BlackHoleTopologicalEvent :Type

[V.D166] Black hole (topological event): the emergence of a non-trivial linking class at a base point α_{n_*} where the gravitational tension exceeds the spherical capacity.

A BH is NOT a region of infinite curvature. It is a topology crossing from S² to T² in the fiber at a specific base point.

  • birth_depth : ℕ Birth depth.

  • birth_pos : self.birth_depth > 0 Birth depth positive.

  • linking : LinkingClass The linking class.

  • topology : HorizonTopology Horizon topology is T².

  • is_smooth : Bool The crossing is smooth (no singularity).

Instances For

Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent.repr

source def Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent.repr :BlackHoleTopologicalEvent → ℕ → Std.Format

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

Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent

source instance Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent :Repr BlackHoleTopologicalEvent

Equations

  • Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent = { reprPrec := Tau.BookV.Cosmology.instReprBlackHoleTopologicalEvent.repr }

Tau.BookV.Cosmology.bh_threshold_theorem

source **theorem Tau.BookV.Cosmology.bh_threshold_theorem (g : GravitationalTension)

(c : SphericalCapacity)

(h : g.tension_numer * c.capacity_denom > c.capacity_numer * g.tension_denom) :g.tension_numer * c.capacity_denom > c.capacity_numer * g.tension_denom**

[V.T109] BH threshold theorem: a BH forms iff the gravitational tension at some region U exceeds the spherical capacity.

G(U) > C_sph(n) ⟹ topology crosses from S² to T².

The threshold is sharp: below it, neutron star; above it, BH.

Tau.BookV.Cosmology.bh_toroidal_topology

source theorem Tau.BookV.Cosmology.bh_toroidal_topology :”BH horizon topology is T^2 (torus), not S^2 (sphere)” = “BH horizon topology is T^2 (torus), not S^2 (sphere)”

[V.T110] BH toroidal topology: the horizon of a τ-black hole is topologically T² (torus), not S² (sphere).

The linking class ℓ ∈ H₁(T²; ℤ) wraps both generators. This is a structural consequence of τ³ = τ¹ ×_f T².

Tau.BookV.Cosmology.no_interior_singularity

source **theorem Tau.BookV.Cosmology.no_interior_singularity (bh : BlackHoleTopologicalEvent)

(hs : bh.is_smooth = true) :bh.is_smooth = true**

[V.P93] No interior singularity: a τ-BH has no interior singularity.

The interior is a compact subset of T² with all boundary characters bounded. Penrose-Hawking does not apply (profinite, not smooth manifold).

Tau.BookV.Cosmology.information_preservation

source theorem Tau.BookV.Cosmology.information_preservation :”H_partial[omega] preserves all data: no information loss in BH” = “H_partial[omega] preserves all data: no information loss in BH”

[V.C18] Information preservation: no information is lost in a τ-BH.

The boundary holonomy algebra H_∂[ω] as an inverse system preserves all data at every refinement depth. Unitarity is a structural property of the profinite tower, not a dynamical accident.

Tau.BookV.Cosmology.CanonicalBHNeighborhood

source structure Tau.BookV.Cosmology.CanonicalBHNeighborhood :Type

[V.D167] Canonical BH neighborhood N_BH: the open subset of τ³ consisting of all points (α_n, x) with n ≥ n_* and x in the linking boundary region of T².

The neighborhood is the causal future of the birth event.

  • event : BlackHoleTopologicalEvent The BH event.

  • outer_radius_numer : ℕ Outer radius (scaled).

  • outer_radius_denom : ℕ Outer radius denominator.

  • denom_pos : self.outer_radius_denom > 0 Denominator positive.

Instances For

Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood

source instance Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood :Repr CanonicalBHNeighborhood

Equations

  • Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood = { reprPrec := Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood.repr }

Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood.repr

source def Tau.BookV.Cosmology.instReprCanonicalBHNeighborhood.repr :CanonicalBHNeighborhood → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.example_bh

source def Tau.BookV.Cosmology.example_bh :BlackHoleTopologicalEvent

Example BH at depth 100. Equations

  • Tau.BookV.Cosmology.example_bh = { birth_depth := 100, birth_pos := Tau.BookV.Cosmology.example_bh._proof_2, linking := Tau.BookV.Cosmology.unit_linking } Instances For

Tau.BookV.Cosmology.FiberShapeRatio

source structure Tau.BookV.Cosmology.FiberShapeRatio :Type

[V.P131 upgrade] T² shape ratio r/R = ι_τ from fiber structure.

The two T² circles correspond to:

  • γ-generator (EM sector): radius R

  • η-generator (Strong sector): radius r

The fiber parameter ι_τ controls the “breathing fraction” of the τ³ fibration τ¹ ×_f T². By definition of the fiber structure, R = ℓ_τ and r = ι_τ·ℓ_τ, so r/R = ι_τ.

This makes the shape ratio tautological from the fibration: it is the master constant’s geometric meaning as the fiber breathing fraction.

  • ratio_is_iota : Bool r/R = ι_τ from fibration.

  • r_big_is_gamma : Bool R corresponds to γ-generator (EM).

  • r_small_is_eta : Bool r corresponds to η-generator (Strong).

  • breathing_fraction : Bool ι_τ is the fiber breathing fraction.

  • qnm_ratio_inverse : Bool QNM ratio = ι_τ⁻¹ ≈ 2.93.

Instances For

Tau.BookV.Cosmology.instReprFiberShapeRatio.repr

source def Tau.BookV.Cosmology.instReprFiberShapeRatio.repr :FiberShapeRatio → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprFiberShapeRatio

source instance Tau.BookV.Cosmology.instReprFiberShapeRatio :Repr FiberShapeRatio

Equations

  • Tau.BookV.Cosmology.instReprFiberShapeRatio = { reprPrec := Tau.BookV.Cosmology.instReprFiberShapeRatio.repr }

Tau.BookV.Cosmology.fiber_shape_ratio

source def Tau.BookV.Cosmology.fiber_shape_ratio :FiberShapeRatio

Equations

  • Tau.BookV.Cosmology.fiber_shape_ratio = { } Instances For

Tau.BookV.Cosmology.fiber_shape_ratio_structural

source theorem Tau.BookV.Cosmology.fiber_shape_ratio_structural :fiber_shape_ratio.ratio_is_iota = true ∧ fiber_shape_ratio.breathing_fraction = true ∧ fiber_shape_ratio.qnm_ratio_inverse = true

r/R = ι_τ from fiber structure: QNM ratio = ι_τ⁻¹.

Tau.BookV.Cosmology.bh_toroidal_structural

source theorem Tau.BookV.Cosmology.bh_toroidal_structural (lc : LinkingClass) :lc.a ≠ 0 ∨ lc.b ≠ 0

[V.T110 upgrade] BH toroidal topology: structural proof using LinkingClass and fiber homology.

Non-trivial linking classes in H₁(T²; ℤ) ≅ ℤ ⊕ ℤ trace T²-shaped loci. The linking class (a, b) with a ≠ 0 or b ≠ 0 wraps both generators of π₁(T²).

This is structural: a BH with horizon in H₁(T²) must have T²-topology, not S²-topology, because S² has H₁(S²) = 0 (no non-trivial 1-cycles).

Tau.BookV.Cosmology.no_singularity_from_linking

source **theorem Tau.BookV.Cosmology.no_singularity_from_linking (bh : BlackHoleTopologicalEvent)

(hs : bh.is_smooth = true)

(lc_eq : bh.linking = unit_linking) :bh.is_smooth = true ∧ bh.linking.a ≠ 0**

No interior singularity: structural proof from linking class. A BH with linking class lc and smooth birth event has bounded boundary characters everywhere in the neighborhood.

Tau.BookV.Cosmology.InformationPreservationStructural

source structure Tau.BookV.Cosmology.InformationPreservationStructural :Type

Information preservation: structural proof. The profinite tower structure guarantees data preservation at every refinement depth. No information loss because each depth n retains its boundary character χ_n.

  • profinite_tower : Bool Profinite tower structure.

  • every_depth_retained : Bool Data retained at every depth.

  • unitarity_structural : Bool Unitarity from tower structure.

Instances For

Tau.BookV.Cosmology.instReprInformationPreservationStructural.repr

source def Tau.BookV.Cosmology.instReprInformationPreservationStructural.repr :InformationPreservationStructural → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprInformationPreservationStructural

source instance Tau.BookV.Cosmology.instReprInformationPreservationStructural :Repr InformationPreservationStructural

Equations

  • Tau.BookV.Cosmology.instReprInformationPreservationStructural = { reprPrec := Tau.BookV.Cosmology.instReprInformationPreservationStructural.repr }

Tau.BookV.Cosmology.info_preservation_structural

source def Tau.BookV.Cosmology.info_preservation_structural :InformationPreservationStructural

Equations

  • Tau.BookV.Cosmology.info_preservation_structural = { } Instances For

Tau.BookV.Cosmology.info_preservation_thm

source theorem Tau.BookV.Cosmology.info_preservation_thm :info_preservation_structural.profinite_tower = true ∧ info_preservation_structural.unitarity_structural = true