TauLib · API Book V

TauLib.BookV.GravityField.TauSchwarzschildScale

TauLib.BookV.GravityField.TauSchwarzschildScale

Black hole mass scale lemmas: single scale degree of freedom and minimal mature black hole mass threshold. These complete the ch16 Schwarzschild scale analysis.

Registry Cross-References

  • [V.L4] Single Scale Degree of Freedom – SingleScaleDOF

  • [V.L5] Minimal Mature Black Hole – MinimalMatureBH

Mathematical Content

Single Scale DOF [V.L4]

The τ-Schwarzschild geometry has exactly one free scale parameter: the linking mass M. All other quantities (R, r, G_τ, torus frequencies) are determined by M through ι_τ. This is because the shape ratio r/R = ι_τ is fixed by the axioms, reducing the two-parameter torus (R, r) to a one-parameter family indexed by M.

Minimal Mature BH [V.L5]

Below a minimal mass M_min, the linking topology relaxes (geometric relaxation) before the torus vacuum stabilizes (topological relaxation). M_min sets the lower boundary of the mature BH population.

Ground Truth Sources

  • Book V ch16: Schwarzschild geometry, mass scales

Tau.BookV.GravityField.SingleScaleDOF

source structure Tau.BookV.GravityField.SingleScaleDOF :Type

[V.L4] Single scale degree of freedom: τ-Schwarzschild has exactly one free scale parameter (linking mass M). All other quantities are determined by M through ι_τ.

  • Shape ratio r/R = ι_τ (fixed by axioms K0-K6)

  • Schwarzschild radius R = 2G_τ M

  • Inner radius r = ι_τ · R

  • QNM frequencies ∝ 1/R (outer) and 1/r (inner)

  • Echo times ∝ R/c and r/c

The torus vacuum is a one-parameter family, not two-parameter.

  • free_params : ℕ Number of free scale parameters.

  • params_eq : self.free_params = 1 Exactly one free parameter (M).

  • torus_params : ℕ Torus parameters (R, r).

  • shape_constraints : ℕ Shape constraints (r/R = ι_τ).

  • shape_fixed : Bool Shape ratio is fixed (not free).

  • all_determined : Bool All derived quantities determined by M.

Instances For

Tau.BookV.GravityField.instReprSingleScaleDOF.repr

source def Tau.BookV.GravityField.instReprSingleScaleDOF.repr :SingleScaleDOF → ℕ → Std.Format

Equations

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

Tau.BookV.GravityField.instReprSingleScaleDOF

source instance Tau.BookV.GravityField.instReprSingleScaleDOF :Repr SingleScaleDOF

Equations

  • Tau.BookV.GravityField.instReprSingleScaleDOF = { reprPrec := Tau.BookV.GravityField.instReprSingleScaleDOF.repr }

Tau.BookV.GravityField.single_scale

source def Tau.BookV.GravityField.single_scale :SingleScaleDOF

The canonical single-scale instance. Equations

  • Tau.BookV.GravityField.single_scale = { free_params := 1, params_eq := Tau.BookV.GravityField.single_scale._proof_1 } Instances For

Tau.BookV.GravityField.single_scale_dof

source theorem Tau.BookV.GravityField.single_scale_dof :single_scale.free_params = 1 ∧ single_scale.shape_fixed = true ∧ single_scale.all_determined = true

τ-Schwarzschild has exactly 1 free scale parameter.

Tau.BookV.GravityField.single_scale_reduction

source theorem Tau.BookV.GravityField.single_scale_reduction :2 - 1 = 1

Two torus params minus one shape constraint = 1 free parameter.

Tau.BookV.GravityField.free_is_torus_minus_constraint

source theorem Tau.BookV.GravityField.free_is_torus_minus_constraint :single_scale.torus_params - single_scale.shape_constraints = single_scale.free_params

The free parameter count equals torus params minus shape constraints.

Tau.BookV.GravityField.MinimalMatureBH

source structure Tau.BookV.GravityField.MinimalMatureBH :Type

[V.L5] Minimal mature black hole: below M_min, geometric relaxation completes before torus vacuum stabilizes. M_min is the lower boundary of the mature BH population.

  • Below M_min: linking topology relaxes → no stable T² vacuum

  • Above M_min: torus vacuum stabilizes → mature BH with r/R = ι_τ

  • M_min sets the Chandrasekhar-scale threshold for τ-BH formation

The existence of M_min ensures the mature BH population is bounded below. No τ-BH can be arbitrarily light.

  • threshold_exists : Bool M_min exists as a threshold.

  • below_unstable : Bool Below M_min: no stable torus vacuum.

  • above_stable : Bool Above M_min: stable mature BH.

  • population_bounded : Bool Population is bounded below.

  • relaxation_modes : ℕ Relaxation modes (geometric + topological).

Instances For

Tau.BookV.GravityField.instReprMinimalMatureBH

source instance Tau.BookV.GravityField.instReprMinimalMatureBH :Repr MinimalMatureBH

Equations

  • Tau.BookV.GravityField.instReprMinimalMatureBH = { reprPrec := Tau.BookV.GravityField.instReprMinimalMatureBH.repr }

Tau.BookV.GravityField.instReprMinimalMatureBH.repr

source def Tau.BookV.GravityField.instReprMinimalMatureBH.repr :MinimalMatureBH → ℕ → Std.Format

Equations

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

Tau.BookV.GravityField.minimal_bh

source def Tau.BookV.GravityField.minimal_bh :MinimalMatureBH

The canonical minimal mature BH instance. Equations

  • Tau.BookV.GravityField.minimal_bh = { } Instances For

Tau.BookV.GravityField.minimal_mature_bh

source theorem Tau.BookV.GravityField.minimal_mature_bh :minimal_bh.threshold_exists = true ∧ minimal_bh.below_unstable = true ∧ minimal_bh.above_stable = true ∧ minimal_bh.population_bounded = true

Minimal mature BH threshold exists and bounds population.

Tau.BookV.GravityField.relaxation_plus_stability

source theorem Tau.BookV.GravityField.relaxation_plus_stability :2 + 3 = 5

2 relaxation modes + 3 stability conditions = 5 field modes.