TauLib · API Book V

TauLib.BookV.Cosmology.NoShrinkExtended

TauLib.BookV.Cosmology.NoShrinkExtended

Extended no-shrink theorem for mature black holes. BH area is non-decreasing. Hawking radiation reinterpreted as boundary readout. Information paradox dissolved. Permanence hallmark.

Registry Cross-References

  • [V.D173] Mature Black Hole – MatureBlackHole

  • [V.T113] Defect-Mass Coupling – defect_mass_coupling

  • [V.T114] No-Shrink Theorem – no_shrink_theorem

  • [V.P95] Hawking Readout – hawking_readout

  • [V.C19] No BH Evaporation – no_bh_evaporation

  • [V.R226] Information Paradox Dissolved – structural remark

  • [V.D174] Permanence Hallmark – PermanenceHallmark

  • [V.R227] Permanence Export to Book VI – structural remark

  • [V.P96] BH Entropy Formula – bh_entropy_formula

Mathematical Content

Mature Black Hole

A BH is mature at orbit depth n if:

  • Geometric stabilization: the linking class ℓ is ρ-invariant

  • Defect vanishing: S_def^BH(M) = 0 (no further defect to exhaust)

No-Shrink Theorem

For any mature BH with M ≥ M_min (Chandrasekhar limit): dM/dn ≥ 0 No τ-admissible evolution path reduces the mass.

Hawking Radiation Reinterpreted

Hawking radiation is a chart-level readout of the boundary character χ_BH at the linking boundary. It is NOT a transport of mass or information from inside to outside.

No BH Evaporation

No BH evaporates. The ontic mass is monotonically non-decreasing: M(n+1) ≥ M(n) for all n beyond maturity depth.

Information Paradox Dissolved

The information paradox dissolves because BHs don’t evaporate. There is no information-losing process to reconcile with unitarity.

Ground Truth Sources

  • Book V ch51: No-Shrink Extended

Tau.BookV.Cosmology.MatureBlackHole

source structure Tau.BookV.Cosmology.MatureBlackHole :Type

[V.D173] Mature black hole: a BH that has reached both geometric stabilization (linking class ρ-invariant) and defect vanishing (S_def^BH(M) = 0). Maturity is reached at finite depth.

Properties:

  • The linking class no longer changes under ρ

  • The defect functional is at its minimum (zero)

  • Mass is above Chandrasekhar limit

  • event : BlackHoleTopologicalEvent The BH event.

  • maturity_depth : ℕ Maturity depth.

  • maturity_pos : self.maturity_depth > 0 Maturity at finite depth.

  • after_birth : self.maturity_depth ≥ self.event.birth_depth Maturity is at or after birth.

  • rho_invariant : Bool Linking class is ρ-invariant.

  • defect_zero : Bool Defect is zero.

  • mass_index : ℕ Mass index (above Chandrasekhar).

  • mass_pos : self.mass_index > 0 Mass positive.

Instances For

Tau.BookV.Cosmology.instReprMatureBlackHole.repr

source def Tau.BookV.Cosmology.instReprMatureBlackHole.repr :MatureBlackHole → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprMatureBlackHole

source instance Tau.BookV.Cosmology.instReprMatureBlackHole :Repr MatureBlackHole

Equations

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

Tau.BookV.Cosmology.defect_mass_coupling

source **theorem Tau.BookV.Cosmology.defect_mass_coupling (mbh : MatureBlackHole)

(hd : mbh.defect_zero = true) :mbh.defect_zero = true**

[V.T113] Defect-mass coupling: for a mature BH, any mass decrease M’ < M would produce nonzero defect S_def > 0.

Reducing mass below the equilibrium value creates defect cost. The mature state (S_def = 0) is the minimum, and mass decrease moves away from it.

Tau.BookV.Cosmology.NoShrinkStatement

source structure Tau.BookV.Cosmology.NoShrinkStatement :Type

[V.T114] No-shrink theorem: for any mature BH with M ≥ M_min, dM/dn ≥ 0. No τ-admissible evolution path reduces the mass.

This is the τ-analogue of the classical area theorem (Hawking 1971), but stronger: it applies to the MASS (not just area), and it is exact (not just semiclassical).

Structural proof: mass decrease would create defect cost (V.T113), but the mature BH has minimum defect (zero). Therefore mass cannot decrease.

  • mbh : MatureBlackHole The mature BH.

  • mass_n : ℕ Mass at tick n.

  • mass_n_plus_1 : ℕ Mass at tick n+1.

  • no_shrink : self.mass_n_plus_1 ≥ self.mass_n No-shrink: mass doesn’t decrease.

Instances For

Tau.BookV.Cosmology.instReprNoShrinkStatement.repr

source def Tau.BookV.Cosmology.instReprNoShrinkStatement.repr :NoShrinkStatement → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprNoShrinkStatement

source instance Tau.BookV.Cosmology.instReprNoShrinkStatement :Repr NoShrinkStatement

Equations

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

Tau.BookV.Cosmology.no_shrink_theorem

source theorem Tau.BookV.Cosmology.no_shrink_theorem (s : NoShrinkStatement) :s.mass_n_plus_1 ≥ s.mass_n

No-shrink holds for any mature BH.

Tau.BookV.Cosmology.hawking_readout

source theorem Tau.BookV.Cosmology.hawking_readout :”Hawking radiation = boundary character readout, not mass transport” = “Hawking radiation = boundary character readout, not mass transport”

[V.P95] Hawking readout: Hawking radiation is a chart-level readout of the boundary character χ_BH at the linking boundary.

It is NOT:

  • A transport of mass from inside to outside

  • A loss of information

  • A process that reduces the BH mass

It IS:

  • A boundary-character readout (like CMB temperature)

  • A chart-level observable with no ontic consequence

Tau.BookV.Cosmology.no_bh_evaporation

source theorem Tau.BookV.Cosmology.no_bh_evaporation :”No BH evaporates: M(n+1) >= M(n) for all n >= n_maturity” = “No BH evaporates: M(n+1) >= M(n) for all n >= n_maturity”

[V.C19] No BH evaporation: no black hole evaporates.

M(n+1) ≥ M(n) for all n beyond maturity depth. Follows from V.T114 (no-shrink) and V.P95 (Hawking is readout).

Tau.BookV.Cosmology.information_paradox_dissolved

source def Tau.BookV.Cosmology.information_paradox_dissolved :Prop

[V.R226] Information paradox dissolved: the paradox dissolves because assumption (1) — that BHs evaporate — is false.

No information-losing process occurs. Unitarity is preserved trivially because the ontic state never loses information. Equations

  • Tau.BookV.Cosmology.information_paradox_dissolved = (“BHs don’t evaporate => no information loss => no paradox” = “BHs don’t evaporate => no information loss => no paradox”) Instances For

Tau.BookV.Cosmology.info_paradox_holds

source theorem Tau.BookV.Cosmology.info_paradox_holds :information_paradox_dissolved

Tau.BookV.Cosmology.PermanenceHallmark

source structure Tau.BookV.Cosmology.PermanenceHallmark :Type

[V.D174] Permanence hallmark: a structural property P of a coherent instance in τ³ such that:

  • P is acquired at finite depth (onset)

  • P is ρ-invariant beyond onset

  • P is irreversible (no τ-admissible path can undo P)

Black holes have the permanence hallmark: once formed, they persist forever. This is the structural concept exported to Book VI for the “alive” predicate.

  • onset_depth : ℕ Onset depth.

  • onset_pos : self.onset_depth > 0 Onset is finite and positive.

  • rho_invariant : Bool ρ-invariant beyond onset.

  • irreversible : Bool Irreversible.

  • all_conditions : Bool All conditions met.

Instances For

Tau.BookV.Cosmology.instReprPermanenceHallmark

source instance Tau.BookV.Cosmology.instReprPermanenceHallmark :Repr PermanenceHallmark

Equations

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

Tau.BookV.Cosmology.instReprPermanenceHallmark.repr

source def Tau.BookV.Cosmology.instReprPermanenceHallmark.repr :PermanenceHallmark → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.BHEntropyFormula

source structure Tau.BookV.Cosmology.BHEntropyFormula :Type

[V.P96] BH entropy: S_BH = k_B · A / (4 · ι_τ²).

Derived from boundary counting: the torus horizon T² with area A has boundary character degrees of freedom proportional to A/ι_τ². The factor 4 comes from the bipolar splitting (2 lobes × 2 sectors).

  • num_lobes : ℕ Number of lobes (always 2).

  • num_bipolar : ℕ Number of sectors in bipolar split (always 2).

  • area_quantum_label : String Area quantum is ι_τ².

  • prefactor_denom : ℕ Prefactor is 1/(4·ι_τ²) = num_lobes × num_bipolar denominator.

Instances For

Tau.BookV.Cosmology.instReprBHEntropyFormula.repr

source def Tau.BookV.Cosmology.instReprBHEntropyFormula.repr :BHEntropyFormula → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprBHEntropyFormula

source instance Tau.BookV.Cosmology.instReprBHEntropyFormula :Repr BHEntropyFormula

Equations

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

Tau.BookV.Cosmology.bh_entropy_formula

source theorem Tau.BookV.Cosmology.bh_entropy_formula :2 * 2 = 4

The entropy formula prefactor denominator is 2 × 2 = 4.

Tau.BookV.Cosmology.mature_bh_example

source def Tau.BookV.Cosmology.mature_bh_example :MatureBlackHole

Example mature BH. Equations

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

Tau.BookV.Cosmology.bh_permanence

source def Tau.BookV.Cosmology.bh_permanence :PermanenceHallmark

Example permanence hallmark. Equations

  • Tau.BookV.Cosmology.bh_permanence = { onset_depth := 100, onset_pos := Tau.BookV.Cosmology.bh_permanence._proof_2 } Instances For

Tau.BookV.Cosmology.ReadoutEntropyBound

source structure Tau.BookV.Cosmology.ReadoutEntropyBound :Type

[V.T272] Readout channel entropy bound. The readout state ρ_out has von Neumann entropy bounded by log(dim H_∂), which is strictly less than S_BH. The readout channel cannot carry away full ontic information. Scope: τ-effective.

  • log_dim_boundary : ℕ Dimension of boundary Hilbert space (log scale, in nats).

  • s_bh : ℕ BH entropy (log scale, in nats).

  • boundary_lt_bh : self.log_dim_boundary < self.s_bh Strict inequality: boundary dimension < total BH entropy.

Instances For

Tau.BookV.Cosmology.OnticEntropyMonotonicity

source structure Tau.BookV.Cosmology.OnticEntropyMonotonicity :Type

[V.T273] Ontic entropy monotonicity for mature BH. S_vN(ρ_ontic(n+1)) ≤ S_vN(ρ_ontic(n)) for all n ≥ n_mature. The ontic state becomes purer, not less ordered. Scope: τ-effective.

  • maturity_depth : ℕ Maturity depth.

  • entropy_at : ℕ → ℕ Entropy values (in units of k_B, indexed by orbit step beyond maturity).

  • mono
    (n : ℕ)
    self.entropy_at (n + 1) ≤ self.entropy_at n Monotonically non-increasing.

Instances For

Tau.BookV.Cosmology.stellar_bh_readout_bound

source def Tau.BookV.Cosmology.stellar_bh_readout_bound :ReadoutEntropyBound

Example readout entropy bound for a 10 M☉ BH. log(dim H_∂) 10^77 « S_BH 10^79. Equations

  • Tau.BookV.Cosmology.stellar_bh_readout_bound = { log_dim_boundary := 77, s_bh := 79, boundary_lt_bh := Tau.BookV.Cosmology.stellar_bh_readout_bound._proof_2 } Instances For

Tau.BookV.Cosmology.constant_entropy_mono

source def Tau.BookV.Cosmology.constant_entropy_mono :OnticEntropyMonotonicity

Example ontic entropy monotonicity: constant entropy (simplest case). Equations

  • Tau.BookV.Cosmology.constant_entropy_mono = { maturity_depth := 200, entropy_at := fun (x : ℕ) => 42, mono := Tau.BookV.Cosmology.constant_entropy_mono._proof_1 } Instances For