TauLib · API Book V

TauLib.BookV.Gravity.Schwarzschild

TauLib.BookV.Gravity.Schwarzschild

Schwarzschild relation, black hole mass index, and the No-Shrink theorem.

Registry Cross-References

  • [V.D07] BH Mass Index — BHMassIndex

  • [V.D08] Schwarzschild Relation — SchwarzschildRelation

  • [V.T03] No-Shrink Theorem — NoShrinkProperty

  • [V.D09] BH Evolution Mode — BHEvolutionMode

  • [V.R02] Hawking evaporation forbidden — structural remark

Mathematical Content

Black Hole Mass Index

M_n(x) := MassIdx(NF_ω(x)) = α-Idx readout from normal-form stabilized torus vacuum.

Properties:

  • Not a primitive scalar but a resistance/scale index

  • Comes with minimal carrier that can host it

  • Monotone under admissible evolution (No-Shrink)

Tau-Schwarzschild Theorem

For all mature BH states x:

R_n(x) = 2 · G_τ · M_n(x)

Both R and M are readouts of a single surviving scale parameter on the stabilized torus vacuum. The linear coupling is the canonical invariance structure from the τ-kernel.

No-Shrink Theorem

For n ≥ n* (maturity horizon): no τ-admissible evolution step decreases M_n(x).

Three admissible BH evolution modes (all monotone in M):

  • Ringdown: Internal normalization (mass preserved)

  • Transport: Boundary-induced holomorphic transport (mass preserved)

  • Fusion: Merger/fission (mass strictly increases)

Consequences

  • Hawking evaporation is forbidden: The No-Shrink theorem directly contradicts BH mass loss. Orthodox Hawking radiation exists as a coarse-grain thermal READOUT but mass cannot decrease.

  • Bekenstein area-law entropy: Emerges as readout, not implication of mass loss.

  • Chandrasekhar limit = first major radius where ι_τ shape ratio can be refinement-invariantly realized = minimal maturity scale.

Ground Truth Sources

  • gravity-einstein.json: schwarzschild-relation, no-shrink-theorem

  • gravity-einstein.json: bh-mass-index, bh-evolution-modes

  • gravity-einstein.json: hawking-bekenstein-reinterpretation

Tau.BookV.Gravity.BHMassIndex

source structure Tau.BookV.Gravity.BHMassIndex :Type

[V.D07] Black hole mass index: the α-Idx readout from a normal-form stabilized torus vacuum.

M_n(x) := MassIdx(NF_ω(x))

Properties:

  • Resistance/scale index of stabilized torus (not primitive scalar)

  • Comes with minimal carrier that can host it

  • Monotone under admissible evolution (No-Shrink)

  • mass_numer : ℕ Mass index numerator (scaled).

  • mass_denom : ℕ Mass index denominator.

  • denom_pos : self.mass_denom > 0 Denominator positive.

  • mass_positive : self.mass_numer > 0 Mass is positive for any physical BH.

  • is_mature : Bool Whether this state is beyond the maturity horizon n*.

Instances For

Tau.BookV.Gravity.instReprBHMassIndex

source instance Tau.BookV.Gravity.instReprBHMassIndex :Repr BHMassIndex

Equations

  • Tau.BookV.Gravity.instReprBHMassIndex = { reprPrec := Tau.BookV.Gravity.instReprBHMassIndex.repr }

Tau.BookV.Gravity.instReprBHMassIndex.repr

source def Tau.BookV.Gravity.instReprBHMassIndex.repr :BHMassIndex → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.BHMassIndex.toFloat

source def Tau.BookV.Gravity.BHMassIndex.toFloat (m : BHMassIndex) :Float

Float display for BH mass. Equations

  • m.toFloat = Float.ofNat m.mass_numer / Float.ofNat m.mass_denom Instances For

Tau.BookV.Gravity.SchwarzschildRelation

source structure Tau.BookV.Gravity.SchwarzschildRelation :Type

[V.D08] Tau-Schwarzschild relation: R_n(x) = 2 · G_τ · M_n(x).

Linear coupling between major radius index and mass index, arising from the single surviving scale degree of freedom on the stabilized torus vacuum.

BH topology is T² (not S²) — only scale remains as free parameter.

Cross-multiplied form: radius_numer · mass_denom · g_denom = 2 · g_numer · mass_numer · radius_denom

  • radius_numer : ℕ Major radius index numerator R_n(x).

  • radius_denom : ℕ Major radius index denominator.

  • radius_denom_pos : self.radius_denom > 0 Radius denominator positive.

  • mass : BHMassIndex The BH mass index.

  • g_tau : GravConstant The gravitational constant.

  • schwarzschild_identity : self.radius_numer * self.mass.mass_denom * self.g_tau.g_denom = 2 * self.g_tau.g_numer * self.mass.mass_numer * self.radius_denom The Schwarzschild identity: R = 2 G_τ M (cross-multiplied).

Instances For

Tau.BookV.Gravity.instReprSchwarzschildRelation.repr

source def Tau.BookV.Gravity.instReprSchwarzschildRelation.repr :SchwarzschildRelation → ℕ → Std.Format

Equations

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Tau.BookV.Gravity.instReprSchwarzschildRelation

source instance Tau.BookV.Gravity.instReprSchwarzschildRelation :Repr SchwarzschildRelation

Equations

  • Tau.BookV.Gravity.instReprSchwarzschildRelation = { reprPrec := Tau.BookV.Gravity.instReprSchwarzschildRelation.repr }

Tau.BookV.Gravity.SchwarzschildRelation.radiusFloat

source def Tau.BookV.Gravity.SchwarzschildRelation.radiusFloat (s : SchwarzschildRelation) :Float

Radius as Float. Equations

  • s.radiusFloat = Float.ofNat s.radius_numer / Float.ofNat s.radius_denom Instances For

Tau.BookV.Gravity.NoShrinkProperty

source structure Tau.BookV.Gravity.NoShrinkProperty :Type

[V.T03] No-Shrink Theorem: beyond the maturity horizon n*, no τ-admissible evolution step can decrease the BH mass index.

This is the τ-native mass monotonicity principle.

Consequences:

  • Hawking evaporation is forbidden (mass cannot decrease)

  • Bekenstein area-law entropy = readout, not mass loss implication

  • BH is permanent ontic object (no information loss)

  • mass : BHMassIndex The BH mass that cannot shrink.

  • mature_proof : self.mass.is_mature = true The BH must be mature (beyond maturity horizon).

Instances For

Tau.BookV.Gravity.instReprNoShrinkProperty.repr

source def Tau.BookV.Gravity.instReprNoShrinkProperty.repr :NoShrinkProperty → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprNoShrinkProperty

source instance Tau.BookV.Gravity.instReprNoShrinkProperty :Repr NoShrinkProperty

Equations

  • Tau.BookV.Gravity.instReprNoShrinkProperty = { reprPrec := Tau.BookV.Gravity.instReprNoShrinkProperty.repr }

Tau.BookV.Gravity.BHEvolutionMode

source inductive Tau.BookV.Gravity.BHEvolutionMode :Type

[V.D09] The three admissible BH evolution modes.

All three are monotone in the mass index M_n(x):

  • Ringdown preserves M

  • Transport preserves M

  • Fusion strictly increases M

No other τ-admissible evolution exists for mature BH states.

  • Ringdown : BHEvolutionMode Internal ringdown normalization. Mass preserved; internal degrees of freedom settle.

  • Transport : BHEvolutionMode Boundary-induced holomorphic transport. Mass preserved; BH moves or deforms within carrier bounds.

  • Fusion : BHEvolutionMode Merger/fusion of two BH states. Mass strictly increases: M(result) > max(M₁, M₂). Gen_ω(g₁, g₂) := Norm_ω(Fuse_ω(g₁, g₂)).

Instances For

Tau.BookV.Gravity.instReprBHEvolutionMode

source instance Tau.BookV.Gravity.instReprBHEvolutionMode :Repr BHEvolutionMode

Equations

  • Tau.BookV.Gravity.instReprBHEvolutionMode = { reprPrec := Tau.BookV.Gravity.instReprBHEvolutionMode.repr }

Tau.BookV.Gravity.instReprBHEvolutionMode.repr

source def Tau.BookV.Gravity.instReprBHEvolutionMode.repr :BHEvolutionMode → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instDecidableEqBHEvolutionMode

source instance Tau.BookV.Gravity.instDecidableEqBHEvolutionMode :DecidableEq BHEvolutionMode

Equations

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

Tau.BookV.Gravity.instBEqBHEvolutionMode

source instance Tau.BookV.Gravity.instBEqBHEvolutionMode :BEq BHEvolutionMode

Equations

  • Tau.BookV.Gravity.instBEqBHEvolutionMode = { beq := Tau.BookV.Gravity.instBEqBHEvolutionMode.beq }

Tau.BookV.Gravity.instBEqBHEvolutionMode.beq

source def Tau.BookV.Gravity.instBEqBHEvolutionMode.beq :BHEvolutionMode → BHEvolutionMode → Bool

Equations

  • Tau.BookV.Gravity.instBEqBHEvolutionMode.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Gravity.instInhabitedBHEvolutionMode.default

source def Tau.BookV.Gravity.instInhabitedBHEvolutionMode.default :BHEvolutionMode

Equations

  • Tau.BookV.Gravity.instInhabitedBHEvolutionMode.default = Tau.BookV.Gravity.BHEvolutionMode.Ringdown Instances For

Tau.BookV.Gravity.instInhabitedBHEvolutionMode

source instance Tau.BookV.Gravity.instInhabitedBHEvolutionMode :Inhabited BHEvolutionMode

Equations

  • Tau.BookV.Gravity.instInhabitedBHEvolutionMode = { default := Tau.BookV.Gravity.instInhabitedBHEvolutionMode.default }

Tau.BookV.Gravity.BHEvolutionMode.preserves_mass

source def Tau.BookV.Gravity.BHEvolutionMode.preserves_mass :BHEvolutionMode → Bool

Whether an evolution mode preserves mass (vs. increases). Equations

  • Tau.BookV.Gravity.BHEvolutionMode.Ringdown.preserves_mass = true
  • Tau.BookV.Gravity.BHEvolutionMode.Transport.preserves_mass = true
  • Tau.BookV.Gravity.BHEvolutionMode.Fusion.preserves_mass = false Instances For

Tau.BookV.Gravity.BHEvolutionMode.is_internal

source def Tau.BookV.Gravity.BHEvolutionMode.is_internal :BHEvolutionMode → Bool

Whether an evolution mode is internal (vs. requires external input). Equations

  • Tau.BookV.Gravity.BHEvolutionMode.Ringdown.is_internal = true
  • Tau.BookV.Gravity.BHEvolutionMode.Transport.is_internal = false
  • Tau.BookV.Gravity.BHEvolutionMode.Fusion.is_internal = false Instances For

Tau.BookV.Gravity.ChandrasekharLimit

source structure Tau.BookV.Gravity.ChandrasekharLimit :Type

[V.R02] The Chandrasekhar limit reinterpreted in the τ-framework.

M_Chandrasekhar = first major radius where the ι_τ shape ratio can be refinement-invariantly realized = minimal maturity scale.

This is NOT a PDE equilibrium (TOV solution) but a threshold where the torus vacuum first achieves ontic stability.

The Hawking-Bekenstein radiation exists as coarse-grain thermal readout on the empirical layer, but evaporation is forbidden by the No-Shrink theorem (mass monotonicity).

  • minimal_mass : BHMassIndex Minimal mature mass index.

  • is_mature : self.minimal_mass.is_mature = true Must be mature by definition.

  • is_minimal : Bool No smaller mature BH exists (minimality).

Instances For

Tau.BookV.Gravity.instReprChandrasekharLimit.repr

source def Tau.BookV.Gravity.instReprChandrasekharLimit.repr :ChandrasekharLimit → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprChandrasekharLimit

source instance Tau.BookV.Gravity.instReprChandrasekharLimit :Repr ChandrasekharLimit

Equations

  • Tau.BookV.Gravity.instReprChandrasekharLimit = { reprPrec := Tau.BookV.Gravity.instReprChandrasekharLimit.repr }

Tau.BookV.Gravity.three_evolution_modes

source theorem Tau.BookV.Gravity.three_evolution_modes (m : BHEvolutionMode) :m = BHEvolutionMode.Ringdown ∨ m = BHEvolutionMode.Transport ∨ m = BHEvolutionMode.Fusion

Exactly 3 BH evolution modes.

Tau.BookV.Gravity.fusion_increases_mass

source theorem Tau.BookV.Gravity.fusion_increases_mass :BHEvolutionMode.Fusion.preserves_mass = false

Fusion increases mass (does not preserve).

Tau.BookV.Gravity.ringdown_preserves_mass

source theorem Tau.BookV.Gravity.ringdown_preserves_mass :BHEvolutionMode.Ringdown.preserves_mass = true

Ringdown preserves mass.

Tau.BookV.Gravity.transport_preserves_mass

source theorem Tau.BookV.Gravity.transport_preserves_mass :BHEvolutionMode.Transport.preserves_mass = true

Transport preserves mass.

Tau.BookV.Gravity.ringdown_internal

source theorem Tau.BookV.Gravity.ringdown_internal :BHEvolutionMode.Ringdown.is_internal = true

Ringdown is internal.

Tau.BookV.Gravity.no_shrink_requires_maturity

source theorem Tau.BookV.Gravity.no_shrink_requires_maturity (p : NoShrinkProperty) :p.mass.is_mature = true

No-Shrink requires maturity.

Tau.BookV.Gravity.schwarzschild_linear

source theorem Tau.BookV.Gravity.schwarzschild_linear (s : SchwarzschildRelation) :s.radius_numer * s.mass.mass_denom * s.g_tau.g_denom = 2 * s.g_tau.g_numer * s.mass.mass_numer * s.radius_denom

Schwarzschild is linear: R is proportional to M (the proportionality constant is 2G_τ).