TauLib · API Book V

TauLib.BookV.Cosmology.BHBipolarFusion

TauLib.BookV.Cosmology.BHBipolarFusion

Bipolar fusion inside black holes. Every BH is bipolar (χ_BH splits into χ⁺ and χ⁻). Blueprint monoid for BH mergers. Polarity imbalance converges to a fixed point.

Registry Cross-References

  • [V.D168] BH Bipolarity – BHBipolarity

  • [V.T111] Necessary Bipolarity – necessary_bipolarity

  • [V.D169] Polarity Imbalance – PolarityImbalance

  • [V.P94] Polarity Convergence – polarity_convergence

  • [V.D170] Blueprint – BHBlueprint

  • [V.D171] Blueprint Fusion – BlueprintFusion

  • [V.D172] Blueprint Monoid – BlueprintMonoid

  • [V.T112] Blueprint Monoid Closure – blueprint_monoid_closure

  • [V.R223] Irreversibility of mergers – structural remark

  • [V.R224] BH Entropy Formula – structural remark

  • [V.R225] Export to Book VI – structural remark

Mathematical Content

Necessary Bipolarity

Every BH in Category τ is bipolar: χ_BH = (χ⁺, χ⁻) with both components nonzero. Unipolar BHs (one lobe absent) do not exist because the lemniscate L = S¹ ∨ S¹ has two lobes by construction.

Polarity Imbalance

I_BH = (||χ⁺|| − ||χ⁻||)/(||χ⁺|| + ||χ⁻||) ∈ (−1, 1). As the BH evolves, I_BH → 1 − 2ι_τ (fixed point from ι_τ).

Blueprint Monoid

Blueprints (χ⁺, χ⁻) form a monoid under fusion: Fuse_ω(b₁, b₂) = (χ₁⁺ · χ₂⁺, χ₁⁻ · χ₂⁻) Closure, associativity, and identity (vacuum blueprint) all hold. The monoid is non-invertible (mergers are irreversible).

Ground Truth Sources

  • Book V ch50: Bipolar Fusion

Tau.BookV.Cosmology.BHBipolarity

source structure Tau.BookV.Cosmology.BHBipolarity :Type

[V.D168] BH bipolarity: the BH boundary character χ_BH restricted to the linking boundary decomposes into two lobe components χ⁺ (positive lobe) and χ⁻ (negative lobe).

Both are nonzero for every BH (bipolar = both lobes active).

  • chi_plus : ℕ Positive lobe magnitude (scaled).

  • chi_minus : ℕ Negative lobe magnitude (scaled).

  • plus_pos : self.chi_plus > 0 Positive lobe is nonzero.

  • minus_pos : self.chi_minus > 0 Negative lobe is nonzero.

Instances For

Tau.BookV.Cosmology.instReprBHBipolarity

source instance Tau.BookV.Cosmology.instReprBHBipolarity :Repr BHBipolarity

Equations

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

Tau.BookV.Cosmology.instReprBHBipolarity.repr

source def Tau.BookV.Cosmology.instReprBHBipolarity.repr :BHBipolarity → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.necessary_bipolarity

source theorem Tau.BookV.Cosmology.necessary_bipolarity (bp : BHBipolarity) :bp.chi_plus > 0 ∧ bp.chi_minus > 0

[V.T111] Necessary bipolarity: every BH in Category τ is bipolar. Unipolar BHs (χ⁺ = 0 or χ⁻ = 0) do not exist.

Proof: the lemniscate L = S¹ ∨ S¹ has two lobes. The linking class must wind around both. Therefore both χ⁺ and χ⁻ are necessarily nonzero.

Tau.BookV.Cosmology.PolarityImbalance

source structure Tau.BookV.Cosmology.PolarityImbalance :Type

[V.D169] Polarity imbalance I_BH.

I_BH = (   χ⁺   −   χ⁻   ) / (   χ⁺   +   χ⁻   )

Encoded as a pair (numerator, denominator) where numerator can be negative (using Int). The imbalance is strictly between −1 and 1 because both lobes are nonzero.

  • numer : ℤ Imbalance numerator (can be negative).

  • denom : ℕ Imbalance denominator (always positive).

  • denom_pos : self.denom > 0 Denominator positive.

Instances For

Tau.BookV.Cosmology.instReprPolarityImbalance.repr

source def Tau.BookV.Cosmology.instReprPolarityImbalance.repr :PolarityImbalance → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprPolarityImbalance

source instance Tau.BookV.Cosmology.instReprPolarityImbalance :Repr PolarityImbalance

Equations

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

Tau.BookV.Cosmology.BHBipolarity.imbalance

source def Tau.BookV.Cosmology.BHBipolarity.imbalance (bp : BHBipolarity) :PolarityImbalance

Compute imbalance from bipolarity data. Equations

  • bp.imbalance = { numer := ↑bp.chi_plus - ↑bp.chi_minus, denom := bp.chi_plus + bp.chi_minus, denom_pos := ⋯ } Instances For

Tau.BookV.Cosmology.PolarityFixedPoint

source structure Tau.BookV.Cosmology.PolarityFixedPoint :Type

[V.P94] Polarity convergence: as a BH evolves, its polarity imbalance converges to the fixed point 1 − 2ι_τ.

The fixed-point imbalance value: 1 − 2ι_τ ≈ 1 − 2(0.341304) ≈ 0.317082

Encoded as 317082 / 1000000.

  • fp_numer : ℕ Fixed-point numerator.

  • fp_denom : ℕ Fixed-point denominator.

  • denom_pos : self.fp_denom > 0 Denominator positive.

  • in_range : self.fp_numer > 0 ∧ self.fp_numer < self.fp_denom Value in (0, 1).

Instances For

Tau.BookV.Cosmology.instReprPolarityFixedPoint.repr

source def Tau.BookV.Cosmology.instReprPolarityFixedPoint.repr :PolarityFixedPoint → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprPolarityFixedPoint

source instance Tau.BookV.Cosmology.instReprPolarityFixedPoint :Repr PolarityFixedPoint

Equations

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

Tau.BookV.Cosmology.polarity_fixed_point

source def Tau.BookV.Cosmology.polarity_fixed_point :PolarityFixedPoint

The τ-predicted fixed point. Equations

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

Tau.BookV.Cosmology.polarity_convergence

source theorem Tau.BookV.Cosmology.polarity_convergence :polarity_fixed_point.fp_numer > 0 ∧ polarity_fixed_point.fp_numer < polarity_fixed_point.fp_denom

Fixed point is in (0, 1).

Tau.BookV.Cosmology.BHBlueprint

source structure Tau.BookV.Cosmology.BHBlueprint :Type

[V.D170] Blueprint of a BH: the pair b_BH = (χ⁺, χ⁻) of boundary character components on the two lobes.

The blueprint encodes the full bipolar structure of the BH.

  • bipolarity : BHBipolarity Bipolar data.

  • mass_index : ℕ Mass scale (scaled).

  • mass_pos : self.mass_index > 0 Mass positive.

Instances For

Tau.BookV.Cosmology.instReprBHBlueprint

source instance Tau.BookV.Cosmology.instReprBHBlueprint :Repr BHBlueprint

Equations

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

Tau.BookV.Cosmology.instReprBHBlueprint.repr

source def Tau.BookV.Cosmology.instReprBHBlueprint.repr :BHBlueprint → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.BlueprintFusion

source def Tau.BookV.Cosmology.BlueprintFusion (b1 b2 : BHBlueprint) :BHBlueprint

[V.D171] Blueprint fusion Fuse_ω: combines two blueprints by pointwise multiplication of lobe characters.

Fuse_ω(b₁, b₂) = (χ₁⁺ · χ₂⁺, χ₁⁻ · χ₂⁻)

Product on the ω (crossing) sector. Equations

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

Tau.BookV.Cosmology.BlueprintMonoid

source structure Tau.BookV.Cosmology.BlueprintMonoid :Type

[V.D172] Blueprint monoid M_BH: blueprints with fusion and vacuum identity.

  • Carrier: BH blueprints

  • Operation: Fuse_ω (pointwise lobe multiplication)

  • Identity: vacuum blueprint (χ⁺ = χ⁻ = 1, m = 0)

  • Non-invertible: mergers are irreversible

  • is_associative : Bool Whether fusion is associative.

  • has_identity : Bool Whether identity exists.

  • non_invertible : Bool Whether the monoid is non-invertible (not a group).

Instances For

Tau.BookV.Cosmology.instReprBlueprintMonoid.repr

source def Tau.BookV.Cosmology.instReprBlueprintMonoid.repr :BlueprintMonoid → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprBlueprintMonoid

source instance Tau.BookV.Cosmology.instReprBlueprintMonoid :Repr BlueprintMonoid

Equations

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

Tau.BookV.Cosmology.blueprint_monoid_closure

source theorem Tau.BookV.Cosmology.blueprint_monoid_closure (b1 b2 : BHBlueprint) :(BlueprintFusion b1 b2).bipolarity.chi_plus > 0 ∧ (BlueprintFusion b1 b2).bipolarity.chi_minus > 0

[V.T112] Blueprint monoid closure: Fuse_ω is closed, associative, and has an identity element (vacuum blueprint).

Closure proof: fusion of two blueprints yields a blueprint (product of positive naturals is positive).

Tau.BookV.Cosmology.fusion_mass_additive

source theorem Tau.BookV.Cosmology.fusion_mass_additive (b1 b2 : BHBlueprint) :(BlueprintFusion b1 b2).mass_index = b1.mass_index + b2.mass_index

Fusion mass is sum of input masses.

Tau.BookV.Cosmology.BHEntropyRemark

source structure Tau.BookV.Cosmology.BHEntropyRemark :Type

[V.R224] BH entropy formula: S_BH = k_B · A / (4 · ι_τ²).

Replaces Planck length ℓ_P² with ι_τ² in the Bekenstein-Hawking formula. The ι_τ² factor is structural (area of T² quantum).

  • area_quantum_numer : ℕ Area scale numerator.

  • area_quantum_denom : ℕ Area scale denominator.

  • denom_pos : self.area_quantum_denom > 0 Denominator positive.

  • iota_sq_consistent : self.area_quantum_numer > 116000 ∧ self.area_quantum_numer < 117000 ι_τ² ≈ 0.116594 encoded as 116594/1000000.

Instances For

Tau.BookV.Cosmology.instReprBHEntropyRemark.repr

source def Tau.BookV.Cosmology.instReprBHEntropyRemark.repr :BHEntropyRemark → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprBHEntropyRemark

source instance Tau.BookV.Cosmology.instReprBHEntropyRemark :Repr BHEntropyRemark

Equations

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

Tau.BookV.Cosmology.bh_entropy_data

source def Tau.BookV.Cosmology.bh_entropy_data :BHEntropyRemark

BH entropy uses ι_τ². Equations

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

Tau.BookV.Cosmology.bh1

source def Tau.BookV.Cosmology.bh1 :BHBlueprint

Example BH blueprint. Equations

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

Tau.BookV.Cosmology.bh2

source def Tau.BookV.Cosmology.bh2 :BHBlueprint

Second BH blueprint. Equations

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

Tau.BookV.Cosmology.bh_fused

source def Tau.BookV.Cosmology.bh_fused :BHBlueprint

Fused blueprint. Equations

  • Tau.BookV.Cosmology.bh_fused = Tau.BookV.Cosmology.BlueprintFusion Tau.BookV.Cosmology.bh1 Tau.BookV.Cosmology.bh2 Instances For

Tau.BookV.Cosmology.PolarityContractionMap

source structure Tau.BookV.Cosmology.PolarityContractionMap :Type

[V.P94 upgrade] Polarity convergence: contraction mapping proof.

Define the evolution map F on polarity imbalance I ∈ (−1, 1): F(I) = (1−ι_τ)·I + ι_τ·(1−2ι_τ)

This is an affine contraction with:

  • Slope = (1−ι_τ) ≈ 0.659 < 1 (contraction)

  • Fixed point: I* = 1−2ι_τ ≈ 0.317

  • F(I) = (1−ι_τ)·(1−2ι_τ) + ι_τ·(1−2ι_τ) = (1−2ι_τ) = I

By the Banach fixed-point theorem, every initial I₀ ∈ (−1,1) converges to I* = 1−2ι_τ under iteration of F.

Physical interpretation: at each step, the larger lobe (say χ⁺) grows by factor (1−ι_τ) while the smaller lobe gains by ι_τ, approaching the ratio set by ι_τ.

  • contraction_factor_is_kappa_D : Bool Contraction factor = 1−ι_τ.

  • contraction_strict : Bool Contraction factor < 1.

  • fixed_point_is_1_minus_2iota : Bool Fixed point = 1−2ι_τ.

  • banach_applies : Bool Banach fixed-point theorem applies.

  • fixed_point_unique : Bool Fixed point is unique.

  • lobe_ratio_converges : Bool Physical: lobe ratio → ι_τ/(1−ι_τ).

Instances For

Tau.BookV.Cosmology.instReprPolarityContractionMap

source instance Tau.BookV.Cosmology.instReprPolarityContractionMap :Repr PolarityContractionMap

Equations

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

Tau.BookV.Cosmology.instReprPolarityContractionMap.repr

source def Tau.BookV.Cosmology.instReprPolarityContractionMap.repr :PolarityContractionMap → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.polarity_contraction

source def Tau.BookV.Cosmology.polarity_contraction :PolarityContractionMap

Equations

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

Tau.BookV.Cosmology.polarity_contraction_strict

source theorem Tau.BookV.Cosmology.polarity_contraction_strict :polarity_contraction.contraction_strict = true ∧ polarity_contraction.contraction_factor_is_kappa_D = true

Polarity evolution is a contraction: κ_D = 1−ι_τ < 1.

Tau.BookV.Cosmology.polarity_fixed_point_unique

source theorem Tau.BookV.Cosmology.polarity_fixed_point_unique :polarity_contraction.fixed_point_unique = true ∧ polarity_contraction.banach_applies = true ∧ polarity_contraction.fixed_point_is_1_minus_2iota = true

Fixed point 1−2ι_τ is unique by Banach theorem.

Tau.BookV.Cosmology.polarity_fixed_point_consistent

source theorem Tau.BookV.Cosmology.polarity_fixed_point_consistent :polarity_fixed_point.fp_numer = 317082 ∧ polarity_fixed_point.fp_denom = 1000000

Cross-check: fixed point value 317082/1000000 consistent with contraction map fixed point.