TauLib · API Book V

TauLib.BookV.Cosmology.MergerNormalForm

TauLib.BookV.Cosmology.MergerNormalForm

BH merger normal form. Mass addition, spin alignment, GW emission. Ringdown damping, BH mass scale at depth n, primorial mass gap, Wilson loops, gravitational deconfinement, Aharonov-Bohm phase.

Registry Cross-References

  • [V.R228] Why overlap forces merger – structural remark

  • [V.T115] Merger Normal Form – merger_normal_form

  • [V.R229] What the Normal Form does not give – structural remark

  • [V.D175] Ringdown Mode – RingdownMode

  • [V.P97] Ringdown Damping is Structural – ringdown_damping_structural

  • [V.D176] BH Mass Scale at Depth n – BHMassScale

  • [V.P98] Mass Gap Between Adjacent Primorial Levels – mass_gap_primorial

  • [V.R230] The mass gap and the IMBH desert – structural remark

  • [V.R231] Scope note on mass spectrum predictions – structural remark

  • [V.D177] Base Wilson Loop – BaseWilsonLoop

  • [V.P99] Gravitational Deconfinement – gravitational_deconfinement

  • [V.R232] Contrast with the strong sector – structural remark

  • [V.P100] BH Gravitational Aharonov-Bohm Phase – bh_ab_phase

  • [V.P101] Radiated Energy Bound – radiated_energy_bound

  • [V.R233] The 1/√2 – structural remark

Mathematical Content

Merger Normal Form

When two BHs with masses M₁, M₂ and angular momenta J₁, J₂ satisfy the approach condition, the merger produces: M_final = M₁ + M₂ − ΔE/c² J_final = J₁ + J₂ − ΔJ

Ringdown

Post-merger, the excision oscillates as ringdown modes: r_n(t) = A_n · exp(−σ_n·t) · cos(ω_n·t + φ_n) All damping rates σ_n > 0 (ringdown terminates).

Primorial Mass Gap

BH mass scale at primorial depths: M_n = m_n · κ(D;1). Ratio between adjacent levels ~ p_{n+1} (next prime). This predicts a gap between supermassive and stellar BHs, possibly explaining IMBH scarcity.

Gravitational Deconfinement

Gravity (base τ¹) is deconfined: Wilson loops satisfy a perimeter law. Contrast: the strong force (fiber T²) is confined with area-law loops.

Radiated Energy Bound

ΔE/(M₁+M₂)c² ≤ 1 − 1/√2 ≈ 0.293 (Penrose extraction limit).

Ground Truth Sources

  • Book V ch52: Merger Normal Form

Tau.BookV.Cosmology.MergerNormalFormData

source structure Tau.BookV.Cosmology.MergerNormalFormData :Type

[V.T115] Merger normal form: when two single-excision BHs satisfy the approach condition, the merger produces a single excision with determined mass and angular momentum.

M_final ≤ M₁ + M₂ (energy radiated as GW). M_final ≥ max(M₁, M₂) (no-shrink for final BH).

  • mass_1 : ℕ Mass of BH 1 (scaled).

  • mass_2 : ℕ Mass of BH 2 (scaled).

  • mass_final : ℕ Mass of final BH (scaled).

  • radiated : ℕ Radiated energy (scaled, = M₁ + M₂ − M_final).

  • mass1_pos : self.mass_1 > 0 Both masses positive.

  • mass2_pos : self.mass_2 > 0

  • mass_balance : self.mass_final + self.radiated = self.mass_1 + self.mass_2 Final mass is sum minus radiated.

  • no_shrink : self.mass_final ≥ self.mass_1 ∨ self.mass_final ≥ self.mass_2 Final mass at least max of inputs.

Instances For

Tau.BookV.Cosmology.instReprMergerNormalFormData.repr

source def Tau.BookV.Cosmology.instReprMergerNormalFormData.repr :MergerNormalFormData → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprMergerNormalFormData

source instance Tau.BookV.Cosmology.instReprMergerNormalFormData :Repr MergerNormalFormData

Equations

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

Tau.BookV.Cosmology.merger_normal_form

source theorem Tau.BookV.Cosmology.merger_normal_form (m : MergerNormalFormData) :m.mass_final + m.radiated = m.mass_1 + m.mass_2

Merger normal form: mass is conserved (modulo radiation).

Tau.BookV.Cosmology.RingdownMode

source structure Tau.BookV.Cosmology.RingdownMode :Type

[V.D175] Ringdown mode: the n-th quasi-normal mode of a merged excision, characterized by amplitude, damping rate, and frequency.

r_n(t) = A_n · exp(−σ_n·t) · cos(ω_n·t + φ_n)

Damping rate σ_n > 0 for all n ≥ 1.

  • mode_number : ℕ Mode number (≥ 1).

  • mode_pos : self.mode_number > 0 Mode number positive.

  • amplitude : ℕ Amplitude (scaled).

  • damping_rate : ℕ Damping rate (scaled, strictly positive).

  • damping_pos : self.damping_rate > 0 Damping positive.

  • frequency : ℕ Frequency (scaled).

Instances For

Tau.BookV.Cosmology.instReprRingdownMode.repr

source def Tau.BookV.Cosmology.instReprRingdownMode.repr :RingdownMode → ℕ → Std.Format

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Tau.BookV.Cosmology.instReprRingdownMode

source instance Tau.BookV.Cosmology.instReprRingdownMode :Repr RingdownMode

Equations

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

Tau.BookV.Cosmology.ringdown_damping_structural

source theorem Tau.BookV.Cosmology.ringdown_damping_structural (r : RingdownMode) :r.damping_rate > 0

[V.P97] Ringdown damping is structural: every mode has σ_n > 0. The ringdown terminates in finite time.

Tau.BookV.Cosmology.BHMassScale

source structure Tau.BookV.Cosmology.BHMassScale :Type

[V.D176] BH mass scale at refinement depth n: M_n = m_n · κ(D;1) = m_n · (1 − ι_τ).

κ(D;1) ≈ 0.6585. The mass scale decreases with depth because m_n decreases as the refinement goes deeper (smaller structures).

  • level : ℕ Primorial level index.

  • level_pos : self.level > 0 Level positive.

  • mass_numer : ℕ Mass scale numerator.

  • mass_denom : ℕ Mass scale denominator.

  • denom_pos : self.mass_denom > 0 Denominator positive.

Instances For

Tau.BookV.Cosmology.instReprBHMassScale

source instance Tau.BookV.Cosmology.instReprBHMassScale :Repr BHMassScale

Equations

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

Tau.BookV.Cosmology.instReprBHMassScale.repr

source def Tau.BookV.Cosmology.instReprBHMassScale.repr :BHMassScale → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.PrimorialMassGap

source structure Tau.BookV.Cosmology.PrimorialMassGap :Type

[V.P98] Mass gap between adjacent primorial levels: M_n / M_{n+1} ~ p_{n+1} (next prime).

The primorial mass hierarchy predicts a natural gap in the BH mass spectrum. This may explain the intermediate-mass black hole (IMBH) desert.

  • level_n : ℕ Level n.

  • level_np1 : ℕ Level n+1.

  • adjacent : self.level_np1 = self.level_n + 1 Adjacent levels.

  • gap_ratio : ℕ Gap ratio (approximate: next prime at that level).

  • gap_min : self.gap_ratio ≥ 2 Gap is at least 2 (smallest prime).

Instances For

Tau.BookV.Cosmology.instReprPrimorialMassGap

source instance Tau.BookV.Cosmology.instReprPrimorialMassGap :Repr PrimorialMassGap

Equations

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

Tau.BookV.Cosmology.instReprPrimorialMassGap.repr

source def Tau.BookV.Cosmology.instReprPrimorialMassGap.repr :PrimorialMassGap → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.mass_gap_primorial

source theorem Tau.BookV.Cosmology.mass_gap_primorial (g : PrimorialMassGap) :g.gap_ratio ≥ 2

Mass gap is at least factor of 2.

Tau.BookV.Cosmology.scope_note_mass_spectrum

source def Tau.BookV.Cosmology.scope_note_mass_spectrum :Prop

[V.R231] Scope note: the qualitative BH mass spectrum features (hierarchy, gap, IMBH scarcity) follow at the τ-effective level. Quantitative mass values require calibration against observation. Equations

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

Tau.BookV.Cosmology.scope_note_holds

source theorem Tau.BookV.Cosmology.scope_note_holds :scope_note_mass_spectrum

Tau.BookV.Cosmology.WilsonLawType

source inductive Tau.BookV.Cosmology.WilsonLawType :Type

[V.D177] Base Wilson loop W_n = Tr(Hol_∂(τ¹; n)): the trace of the boundary holonomy around τ¹ at refinement depth n.

Wilson loops diagnose confinement vs. deconfinement:

  • Area law ⟹ confinement (strong sector)

  • Perimeter law ⟹ deconfinement (gravitational sector)

  • PerimeterLaw : WilsonLawType Perimeter law: W ~ exp(−κ·L), deconfined.

  • AreaLaw : WilsonLawType Area law: W ~ exp(−σ·A), confined.

Instances For

Tau.BookV.Cosmology.instReprWilsonLawType.repr

source def Tau.BookV.Cosmology.instReprWilsonLawType.repr :WilsonLawType → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprWilsonLawType

source instance Tau.BookV.Cosmology.instReprWilsonLawType :Repr WilsonLawType

Equations

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

Tau.BookV.Cosmology.instDecidableEqWilsonLawType

source instance Tau.BookV.Cosmology.instDecidableEqWilsonLawType :DecidableEq WilsonLawType

Equations

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

Tau.BookV.Cosmology.instBEqWilsonLawType

source instance Tau.BookV.Cosmology.instBEqWilsonLawType :BEq WilsonLawType

Equations

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

Tau.BookV.Cosmology.instBEqWilsonLawType.beq

source def Tau.BookV.Cosmology.instBEqWilsonLawType.beq :WilsonLawType → WilsonLawType → Bool

Equations

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

Tau.BookV.Cosmology.BaseWilsonLoop

source structure Tau.BookV.Cosmology.BaseWilsonLoop :Type

Wilson loop on the base circle.

  • depth : ℕ Refinement depth.

  • depth_pos : self.depth > 0 Depth positive.

  • law : WilsonLawType Law type.

  • coupling_numer : ℕ Coupling (scaled).

Instances For

Tau.BookV.Cosmology.instReprBaseWilsonLoop.repr

source def Tau.BookV.Cosmology.instReprBaseWilsonLoop.repr :BaseWilsonLoop → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprBaseWilsonLoop

source instance Tau.BookV.Cosmology.instReprBaseWilsonLoop :Repr BaseWilsonLoop

Equations

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

Tau.BookV.Cosmology.gravitational_deconfinement

source theorem Tau.BookV.Cosmology.gravitational_deconfinement :”D-sector deconfined: perimeter law for base Wilson loops” = “D-sector deconfined: perimeter law for base Wilson loops”

[V.P99] Gravitational deconfinement: the D-sector is deconfined. Base Wilson loops satisfy a perimeter law: W_n ~ exp(−κ_τ · L(τ¹; n))

Gravity propagates freely at all distances (no confinement).

Tau.BookV.Cosmology.BHABPhase

source structure Tau.BookV.Cosmology.BHABPhase :Type

[V.P100] BH gravitational Aharonov-Bohm phase: Φ_BH = G·M / ι_τ = (c³/ℏ) · ι_τ · M.

The phase is proportional to M and inversely proportional to ι_τ. It is detectable (in principle) via gravitational interference.

  • mass_index : ℕ Mass index (scaled).

  • mass_pos : self.mass_index > 0 Mass positive.

  • proportional_to_mass : Bool Phase is proportional to M.

Instances For

Tau.BookV.Cosmology.instReprBHABPhase

source instance Tau.BookV.Cosmology.instReprBHABPhase :Repr BHABPhase

Equations

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

Tau.BookV.Cosmology.instReprBHABPhase.repr

source def Tau.BookV.Cosmology.instReprBHABPhase.repr :BHABPhase → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.bh_ab_phase

source **theorem Tau.BookV.Cosmology.bh_ab_phase (p : BHABPhase)

(h : p.proportional_to_mass = true) :p.proportional_to_mass = true**

AB phase is proportional to mass.

Tau.BookV.Cosmology.RadiatedEnergyBound

source structure Tau.BookV.Cosmology.RadiatedEnergyBound :Type

[V.P101] Radiated energy bound: ΔE / ((M₁+M₂)c²) ≤ 1 − 1/√2.

1 − 1/√2 ≈ 0.2929. Encoded as 2929/10000. This is the Penrose extraction limit for Kerr BHs. In τ, the same bound arises from the blueprint monoid structure.

  • bound_numer : ℕ Bound numerator.

  • bound_denom : ℕ Bound denominator.

  • denom_pos : self.bound_denom > 0 Denominator positive.

  • approx : self.bound_numer > 2900 ∧ self.bound_numer < 3000 Bound is approximately 0.293: 2929/10000.

Instances For

Tau.BookV.Cosmology.instReprRadiatedEnergyBound.repr

source def Tau.BookV.Cosmology.instReprRadiatedEnergyBound.repr :RadiatedEnergyBound → ℕ → Std.Format

Equations

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Tau.BookV.Cosmology.instReprRadiatedEnergyBound

source instance Tau.BookV.Cosmology.instReprRadiatedEnergyBound :Repr RadiatedEnergyBound

Equations

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

Tau.BookV.Cosmology.canonical_energy_bound

source def Tau.BookV.Cosmology.canonical_energy_bound :RadiatedEnergyBound

The canonical radiated energy bound. Equations

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

Tau.BookV.Cosmology.radiated_energy_bound

source theorem Tau.BookV.Cosmology.radiated_energy_bound :canonical_energy_bound.bound_numer > 2900 ∧ canonical_energy_bound.bound_numer < 3000

The bound is approximately 0.293.

Tau.BookV.Cosmology.the_sqrt2_remark

source def Tau.BookV.Cosmology.the_sqrt2_remark :Prop

[V.R233] The 1/√2 radiated energy bound is a classical GR result (Penrose extraction limit for Kerr BHs). In τ, the same bound arises structurally from the blueprint fusion operation: energy extraction cannot exceed the bipolar imbalance limit. Equations

  • Tau.BookV.Cosmology.the_sqrt2_remark = (“1/sqrt(2) bound from GR = blueprint fusion limit in tau” = “1/sqrt(2) bound from GR = blueprint fusion limit in tau”) Instances For

Tau.BookV.Cosmology.sqrt2_remark_holds

source theorem Tau.BookV.Cosmology.sqrt2_remark_holds :the_sqrt2_remark

Tau.BookV.Cosmology.example_merger

source def Tau.BookV.Cosmology.example_merger :MergerNormalFormData

Example merger. Equations

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

Tau.BookV.Cosmology.mode1

source def Tau.BookV.Cosmology.mode1 :RingdownMode

Example ringdown mode 1. Equations

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

Tau.BookV.Cosmology.BlueprintFusionEnergy

source structure Tau.BookV.Cosmology.BlueprintFusionEnergy :Type

[V.D282] Blueprint fusion energy: radiated fraction η = ι_τ² · ν where ν = q/(1+q)² is the symmetric mass ratio.

Derived from linking-class reduction during blueprint fusion at the lemniscate crossing point: pre-merger H₁(L₁)⊕H₁(L₂) ≅ ℤ⁴ reduces to post-merger H₁(L_final) ≅ ℤ². The two lost classes release energy proportional to ι_τ² (D-sector holonomy constraint).

  • description : String
  • formula : String
  • iota_sq_x10000 : ℕ Instances For

Tau.BookV.Cosmology.instReprBlueprintFusionEnergy.repr

source def Tau.BookV.Cosmology.instReprBlueprintFusionEnergy.repr :BlueprintFusionEnergy → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprBlueprintFusionEnergy

source instance Tau.BookV.Cosmology.instReprBlueprintFusionEnergy :Repr BlueprintFusionEnergy

Equations

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

Tau.BookV.Cosmology.merger_energy_formula

source def Tau.BookV.Cosmology.merger_energy_formula :BlueprintFusionEnergy

[V.T224] Merger energy theorem: non-spinning radiated fraction from blueprint fusion. Equations

  • Tau.BookV.Cosmology.merger_energy_formula = { description := “Non-spinning radiated fraction from blueprint fusion”, formula := “η = ι_τ² · ν, ν = q/(1+q)²”, iota_sq_x10000 := 1165 } Instances For

Tau.BookV.Cosmology.equal_mass_eta_ppm

source def Tau.BookV.Cosmology.equal_mass_eta_ppm :ℕ

[V.P150] Equal-mass energy fraction: η(q=1) = ι_τ²/4 ≈ 0.02912, stored as parts per million. Equations

  • Tau.BookV.Cosmology.equal_mass_eta_ppm = 29122 Instances For

Tau.BookV.Cosmology.equal_mass_eta_positive

source theorem Tau.BookV.Cosmology.equal_mass_eta_positive :equal_mass_eta_ppm > 0

Equal-mass fraction is positive.

Tau.BookV.Cosmology.equal_mass_eta_below_bound

source theorem Tau.BookV.Cosmology.equal_mass_eta_below_bound :equal_mass_eta_ppm < 293000

Equal-mass fraction is below upper bound (1−1/√2 ≈ 0.293).

Tau.BookV.Cosmology.iota_sq_canonical

source theorem Tau.BookV.Cosmology.iota_sq_canonical :merger_energy_formula.iota_sq_x10000 = 1165

iota_tau^2 × 10000 matches the canonical value.