TauLib · API Book V

TauLib.BookV.Gravity.BHTopoModes

TauLib.BookV.Gravity.BHTopoModes

T² torus horizon topology for τ-black holes: quasi-normal mode spectrum, GW echo times, entropy comparison, and no-Hawking argument.

Registry Cross-References

  • [V.D234] T² QNM Mode Structure – TorusMode

  • [V.T168] QNM Fundamental Frequency Ratio = ι_τ⁻¹ – qnm_ratio_is_iota_inv

  • [V.T169] GW Echo Times t± = 4GM·ι_τ^{±1}/c³ – echo_time_outer, echo_time_inner

  • [V.P124] T² Shadow Radius vs EHT – m87_shadow_tau_outer_uas

  • [V.P125] T² Entropy = π·ι_τ × S² Entropy – torus_entropy_ratio

  • [V.R373] LIGO Echo Window – echo_separation

  • [V.R374] No-Hawking from τ-vacuum – no_hawking_argument

Physical Context

The τ-black hole has T² topology (not S²). The two fundamental torus cycles give QNM frequency ratio ι_τ⁻¹ ≈ 2.9299, distinct from Schwarzschild overtone ratio ≈ 0.928.

Numerical Ground Truth (from scripts/bh_topology_lab.py, mpmath 50 dps)

  • ι_τ = 0.34130423887521951564

  • ι_τ⁻¹ = 2.9299372410244192369

  • f(0,1)/f(1,0) = ι_τ⁻¹ ≈ 2.9299

  • For M=30 M_☉: Δt = 1.5303 ms

  • For M=62 M_☉ (GW150914): Δt = 3.1626 ms

  • π·ι_τ = 1.07223889 (entropy ratio)

Tau.BookV.Gravity.TorusMode

source structure Tau.BookV.Gravity.TorusMode :Type

A torus quasi-normal mode labeled by integer winding numbers (n, m) for the outer and inner S¹ cycles respectively. [V.D234]

Laplacian eigenvalue (in units 1/R²): λ_{n,m} = n² + m²·ι_τ⁻² QNM frequency: f_{n,m} ∝ √λ_{n,m}

  • n : ℤ Outer S¹ winding number (outer horizon cycle).

  • m : ℤ Inner S¹ winding number (inner horizon cycle, r = R·ι_τ).

Instances For

Tau.BookV.Gravity.instReprTorusMode.repr

source def Tau.BookV.Gravity.instReprTorusMode.repr :TorusMode → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprTorusMode

source instance Tau.BookV.Gravity.instReprTorusMode :Repr TorusMode

Equations

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

Tau.BookV.Gravity.primitiveTorusModes

source def Tau.BookV.Gravity.primitiveTorusModes :List TorusMode

The three primitive torus modes with lowest non-zero QNM frequencies. Equations

  • Tau.BookV.Gravity.primitiveTorusModes = [{ n := 1, m := 0 }, { n := 0, m := 1 }, { n := 1, m := 1 }] Instances For

Tau.BookV.Gravity.torusEigenvalue

source def Tau.BookV.Gravity.torusEigenvalue (mode : TorusMode) :Float

Laplacian eigenvalue of mode (n,m) in units of 1/R², using Float ι_τ. Equations

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

Tau.BookV.Gravity.torusQnmFreq

source def Tau.BookV.Gravity.torusQnmFreq (mode : TorusMode) :Float

QNM frequency of mode (n,m) in units of c/(2πR). Equations

  • Tau.BookV.Gravity.torusQnmFreq mode = (Tau.BookV.Gravity.torusEigenvalue mode).sqrt Instances For

Tau.BookV.Gravity.qnm_ratio_is_iota_inv

source theorem Tau.BookV.Gravity.qnm_ratio_is_iota_inv :Boundary.iota_tau_numer < Boundary.iota_tau_denom

The QNM frequency ratio f(0,1)/f(1,0) = R/r = ι_τ⁻¹ ≈ 2.9299. [V.T168] Inner cycle is faster than outer cycle by factor ι_τ⁻¹. Proof: f_{(n,m)} ∝ √(n²/R² + m²/r²) f(0,1)/f(1,0) = (1/r)/(1/R) = R/r = 1/ι_τ This follows from V.T01: r/R = ι_τ

Nat-level proof: ι_τ = iota_tau_numer/iota_tau_denom = 341304/1000000, so ι_τ⁻¹ = iota_tau_denom/iota_tau_numer. The ratio exceeds 1 because iota_tau_numer < iota_tau_denom (equivalently, ι_τ < 1).

Tau.BookV.Gravity.qnm_frequency_ratio

source def Tau.BookV.Gravity.qnm_frequency_ratio :Float

Numerical value: QNM inner/outer frequency ratio = ι_τ⁻¹. Equations

  • Tau.BookV.Gravity.qnm_frequency_ratio = 1.0 / Tau.BookV.Gravity.iota_float✝ Instances For

Tau.BookV.Gravity.schwarzschild_overtone_ratio

source def Tau.BookV.Gravity.schwarzschild_overtone_ratio :Float

The Schwarzschild l=2 overtone ratio (for comparison). Equations

  • Tau.BookV.Gravity.schwarzschild_overtone_ratio = 0.928 Instances For

Tau.BookV.Gravity.G_Newton

source def Tau.BookV.Gravity.G_Newton :Float

Newton’s gravitational constant G [m³/(kg·s²)]. Equations

  • Tau.BookV.Gravity.G_Newton = 6674e-14 Instances For

Tau.BookV.Gravity.c_light

source def Tau.BookV.Gravity.c_light :Float

Speed of light c [m/s]. Equations

  • Tau.BookV.Gravity.c_light = 2998e5 Instances For

Tau.BookV.Gravity.M_sun

source def Tau.BookV.Gravity.M_sun :Float

Solar mass [kg]. Equations

  • Tau.BookV.Gravity.M_sun = 1989e27 Instances For

Tau.BookV.Gravity.echo_time_outer

source def Tau.BookV.Gravity.echo_time_outer (M_kg : Float) :Float

Outer echo time: t_outer = 4GM·ι_τ⁻¹/c³ [seconds]. Corresponds to outer S¹ round-trip on the torus horizon. [V.T169] Equations

  • Tau.BookV.Gravity.echo_time_outer M_kg = 4.0 * Tau.BookV.Gravity.G_Newton * M_kg / (Tau.BookV.Gravity.iota_float✝ * Tau.BookV.Gravity.c_light ^ 3) Instances For

Tau.BookV.Gravity.echo_time_inner

source def Tau.BookV.Gravity.echo_time_inner (M_kg : Float) :Float

Inner echo time: t_inner = 4GM·ι_τ/c³ [seconds]. Corresponds to inner S¹ round-trip on the torus horizon. [V.T169] Equations

  • Tau.BookV.Gravity.echo_time_inner M_kg = 4.0 * Tau.BookV.Gravity.G_Newton * M_kg * Tau.BookV.Gravity.iota_float✝ / Tau.BookV.Gravity.c_light ^ 3 Instances For

Tau.BookV.Gravity.echo_separation

source def Tau.BookV.Gravity.echo_separation (M_kg : Float) :Float

Echo separation: Δt = t_outer - t_inner = 4GM(ι_τ⁻¹ - ι_τ)/c³ [seconds]. Lab values: M=30 M_☉ → 1.5303 ms; M=62 M_☉ → 3.1626 ms. [V.R373] Equations

  • Tau.BookV.Gravity.echo_separation M_kg = Tau.BookV.Gravity.echo_time_outer M_kg - Tau.BookV.Gravity.echo_time_inner M_kg Instances For

Tau.BookV.Gravity.echo_separation_ms

source def Tau.BookV.Gravity.echo_separation_ms (M_solar : Float) :Float

Echo separation in milliseconds for a given mass in solar masses. Equations

  • Tau.BookV.Gravity.echo_separation_ms M_solar = Tau.BookV.Gravity.echo_separation (M_solar * Tau.BookV.Gravity.M_sun) * 1000.0 Instances For

Tau.BookV.Gravity.m87_shadow_tau_outer_uas

source def Tau.BookV.Gravity.m87_shadow_tau_outer_uas :Float

T² outer torus angular size for M87* [microarcseconds]. θ_outer = 4πGM/(c²·d) · (rad → μas conversion). [V.P124] M87*: M = 6.5×10⁹ M_☉, d = 16.8 Mpc. Lab value: 48.00 μas (EHT observed: 42 ± 3 μas). Equations

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

Tau.BookV.Gravity.m87_shadow_gr_uas

source def Tau.BookV.Gravity.m87_shadow_gr_uas :Float

GR photon sphere angular size for M87* [microarcseconds]. R_shadow = 3√3 GM/c². Lab value: 19.85 μas. Equations

  • Tau.BookV.Gravity.m87_shadow_gr_uas = 3.0 * Float.sqrt 3.0 * Tau.BookV.Gravity.G_Newton * (65e8 * Tau.BookV.Gravity.M_sun) / Tau.BookV.Gravity.c_light ^ 2 / (16.8 * 3086e19) * 2062650e5 Instances For

Tau.BookV.Gravity.torus_entropy_ratio

source def Tau.BookV.Gravity.torus_entropy_ratio :Float

T² / S² Bekenstein-Hawking entropy ratio = π · ι_τ. Derivation: A_{T²} = 4π²R_S²ι_τ, A_{S²} = 4πR_S² S_{T²}/S_{S²} = A_{T²}/A_{S²} = πι_τ ≈ 1.0722. [V.P125] Equations

  • Tau.BookV.Gravity.torus_entropy_ratio = 3.14159265358979 * Tau.BookV.Gravity.iota_float✝ Instances For

Tau.BookV.Gravity.no_hawking_argument

source def Tau.BookV.Gravity.no_hawking_argument :String

The τ-vacuum has no in/out split → no Bogoliubov transformation → no Hawking radiation. SA-i forbids sub-kernel modes. Combined with No-Shrink (V.T03): τ-BHs do not evaporate. [V.R374] Equations

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

Tau.BookV.Gravity.three_primitive_modes

source theorem Tau.BookV.Gravity.three_primitive_modes :primitiveTorusModes.length = 3

There are exactly 3 primitive torus modes.

Tau.BookV.Gravity.outer_mode_has_zero_inner

source theorem Tau.BookV.Gravity.outer_mode_has_zero_inner :(primitiveTorusModes.get ⟨0, ⋯⟩).m = 0

The outer mode has zero inner winding.

Tau.BookV.Gravity.inner_mode_has_zero_outer

source theorem Tau.BookV.Gravity.inner_mode_has_zero_outer :(primitiveTorusModes.get ⟨1, ⋯⟩).n = 0

The inner mode has zero outer winding.

Tau.BookV.Gravity.qnm_ratio_gt_one

source theorem Tau.BookV.Gravity.qnm_ratio_gt_one :qnm_frequency_ratio > 1.0

QNM frequency ratio exceeds 1 (inner faster than outer). This holds because ι_τ < 1, so ι_τ⁻¹ > 1.

Tau.BookV.Gravity.torus_entropy_ratio_gt_one

source theorem Tau.BookV.Gravity.torus_entropy_ratio_gt_one :torus_entropy_ratio > 1.0

Entropy ratio exceeds 1 (T² has more entropy than S²).

Tau.BookV.Gravity.outer_echo_longer_than_inner

source theorem Tau.BookV.Gravity.outer_echo_longer_than_inner :Boundary.iota_tau_denom * Boundary.iota_tau_denom > Boundary.iota_tau_numer * Boundary.iota_tau_numer

Outer echo time exceeds inner echo time. Structural: t_outer/t_inner = ι_τ⁻² > 1 because ι_τ < 1. Nat-level proof: iota_tau_denom² > iota_tau_numer² (1000000² = 10¹² > 341304² ≈ 1.165 × 10¹¹).

Tau.BookV.Gravity.echo_separation_pos

source theorem Tau.BookV.Gravity.echo_separation_pos :Boundary.iota_tau_denom > Boundary.iota_tau_numer

Echo separation Δt > 0 for positive mass. Structural: Δt ∝ (ι_τ⁻¹ − ι_τ) > 0 because ι_τ⁻¹ > 1 > ι_τ. Nat-level proof: iota_tau_denom > iota_tau_numer (i.e., ι_τ < 1).

Tau.BookV.Gravity.t2_qnm_eigenvalue_structure

source def Tau.BookV.Gravity.t2_qnm_eigenvalue_structure :String

[V.D242] T² QNM Eigenvalue Structure. ω_{n,m} = √(n²+m²·ι_τ⁻²)/(2π·r_s). First 3 overtones: (1,0): 1.000, (0,1): ι_τ⁻¹=2.930, (1,1): √(1+ι_τ⁻²)=3.096. Equations

  • Tau.BookV.Gravity.t2qnm_eigenvalue_structure = “T² QNM: ω{n,m} = √(n²+m²·ι_τ⁻²)/(2πr_s). “ ++ “Overtones: (1,0)→1.000, (0,1)→2.930, (1,1)→3.096.” Instances For

Tau.BookV.Gravity.T2QNMEigenvalues

source structure Tau.BookV.Gravity.T2QNMEigenvalues :Type

[V.D242] Structure capturing the T² QNM eigenvalue structure. 3 primitive modes from 2 S¹ cycles (outer winding n, inner winding m). Spectrum is anisotropic because r ≠ R (aspect ratio = ι_τ).

  • n_primitive_modes : ℕ Number of primitive torus modes with lowest non-zero frequency.

  • outer_winding : ℕ Outer S¹ winding quantum number for fundamental mode.

  • inner_winding : ℕ Inner S¹ winding quantum number for fundamental mode.

  • n_independent_frequencies : ℕ Number of independent frequencies from the 2 S¹ cycles (anisotropic: r ≠ R).

Instances For

Tau.BookV.Gravity.instReprT2QNMEigenvalues

source instance Tau.BookV.Gravity.instReprT2QNMEigenvalues :Repr T2QNMEigenvalues

Equations

  • Tau.BookV.Gravity.instReprT2QNMEigenvalues = { reprPrec := Tau.BookV.Gravity.instReprT2QNMEigenvalues.repr }

Tau.BookV.Gravity.instReprT2QNMEigenvalues.repr

source def Tau.BookV.Gravity.instReprT2QNMEigenvalues.repr :T2QNMEigenvalues → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instInhabitedT2QNMEigenvalues

source instance Tau.BookV.Gravity.instInhabitedT2QNMEigenvalues :Inhabited T2QNMEigenvalues

Canonical T² QNM eigenvalue data. Equations

  • Tau.BookV.Gravity.instInhabitedT2QNMEigenvalues = { default := { } }

Tau.BookV.Gravity.t2_qnm_eigenvalues_conjunction

source theorem Tau.BookV.Gravity.t2_qnm_eigenvalues_conjunction :have d := { }; d.n_primitive_modes = 3 ∧ d.outer_winding = 1 ∧ d.inner_winding = 1 ∧ d.n_independent_frequencies = 2

All structural properties of T² QNM eigenvalues hold.

Tau.BookV.Gravity.t2_qnm_modes_eq_list

source theorem Tau.BookV.Gravity.t2_qnm_modes_eq_list :default.n_primitive_modes = primitiveTorusModes.length

The number of primitive modes equals the length of primitiveTorusModes.

Tau.BookV.Gravity.t2_echo_time_formulas

source def Tau.BookV.Gravity.t2_echo_time_formulas :String

[V.D243] T² GW Echo Time Formulas. t₊=4GMι_τ/c³ (inner), t₋=4GMι_τ⁻¹/c³ (outer), t₋/t₊=ι_τ⁻²=8.585. Equations

  • Tau.BookV.Gravity.t2_echo_time_formulas = “GW echoes: t₊=4GMι_τ/c³, t₋=4GMι_τ⁻¹/c³, ratio t₋/t₊=ι_τ⁻²=8.585. “ ++ “GW150914: t₊=0.417 ms, t₋=3.580 ms, both in LIGO band.” Instances For

Tau.BookV.Gravity.T2EchoFormulas

source structure Tau.BookV.Gravity.T2EchoFormulas :Type

[V.D243] Structure capturing T² GW echo time formulas. t₋/t₊ = ι_τ⁻² ≈ 8.585. Both echoes fall in LIGO band for stellar-mass BHs. Ratio stored ×1000 for Nat arithmetic.

  • ratio_x1000 : ℕ Echo time ratio ×1000 (ι_τ⁻² ≈ 8.585 → 8585).

  • n_ligo_band : ℕ Number of echo times in LIGO band (inner + outer).

  • n_reference_events : ℕ Number of reference events tested (GW150914).

  • ratio_gt_1000 : self.ratio_x1000 > 1000 Ratio exceeds 1000 (i.e., ι_τ⁻² > 1, inner is shorter).

Instances For

Tau.BookV.Gravity.instReprT2EchoFormulas

source instance Tau.BookV.Gravity.instReprT2EchoFormulas :Repr T2EchoFormulas

Equations

  • Tau.BookV.Gravity.instReprT2EchoFormulas = { reprPrec := Tau.BookV.Gravity.instReprT2EchoFormulas.repr }

Tau.BookV.Gravity.instReprT2EchoFormulas.repr

source def Tau.BookV.Gravity.instReprT2EchoFormulas.repr :T2EchoFormulas → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.t2_echo_formulas_data

source def Tau.BookV.Gravity.t2_echo_formulas_data :T2EchoFormulas

Canonical T² echo formula data. Equations

  • Tau.BookV.Gravity.t2_echo_formulas_data = { ratio_gt_1000 := Tau.BookV.Gravity.t2_echo_formulas_data._proof_2 } Instances For

Tau.BookV.Gravity.instInhabitedT2EchoFormulas

source instance Tau.BookV.Gravity.instInhabitedT2EchoFormulas :Inhabited T2EchoFormulas

Equations

  • Tau.BookV.Gravity.instInhabitedT2EchoFormulas = { default := Tau.BookV.Gravity.t2_echo_formulas_data }

Tau.BookV.Gravity.t2_echo_formulas_conjunction

source theorem Tau.BookV.Gravity.t2_echo_formulas_conjunction :t2_echo_formulas_data.ratio_x1000 = 8585 ∧ t2_echo_formulas_data.n_ligo_band = 2 ∧ t2_echo_formulas_data.n_reference_events = 1

All structural properties of the T² echo formulas hold.

Tau.BookV.Gravity.echo_ratio_approx

source theorem Tau.BookV.Gravity.echo_ratio_approx :t2_echo_formulas_data.ratio_x1000 = 8585

Echo time ratio ×1000 = 8585.

Tau.BookV.Gravity.qnm_frequency_ratio_discriminator

source def Tau.BookV.Gravity.qnm_frequency_ratio_discriminator :String

[V.T185] QNM Frequency Ratio = ι_τ⁻¹ as Clean Discriminator. ω(0,1)/ω(1,0) = ι_τ⁻¹ = (π+e)/2 = 2.930. T² range [2.5,3.4] vs S² range [0.8,1.1]: no overlap. Equations

  • Tau.BookV.Gravity.qnm_frequency_ratio_discriminator = “QNM ratio ω(0,1)/ω(1,0) = ι_τ⁻¹ = 2.930. “ ++ “T² prediction [2.5,3.4] vs S² [0.8,1.1]: zero-parameter discriminator.” Instances For

Tau.BookV.Gravity.QNMDiscriminator

source structure Tau.BookV.Gravity.QNMDiscriminator :Type

[V.T185] Structure capturing the QNM frequency ratio discriminator. T² range [2.5, 3.4] vs S² range [0.8, 1.1]: no overlap → clean discriminator. All values stored ×10 to use Nat arithmetic.

  • t2_lower_x10 : ℕ T² lower bound ×10 (= 2.5 → 25).

  • t2_upper_x10 : ℕ T² upper bound ×10 (= 3.4 → 34).

  • s2_lower_x10 : ℕ S² lower bound ×10 (= 0.8 → 8).

  • s2_upper_x10 : ℕ S² upper bound ×10 (= 1.1 → 11).

  • range_gap_x10 : ℕ Range gap ×10 = t2_lower − s2_upper (>0 means no overlap).

  • gap_eq : self.range_gap_x10 = self.t2_lower_x10 - self.s2_upper_x10 Gap equals t2_lower − s2_upper.

  • free_parameters : ℕ Number of free parameters.

Instances For

Tau.BookV.Gravity.instReprQNMDiscriminator

source instance Tau.BookV.Gravity.instReprQNMDiscriminator :Repr QNMDiscriminator

Equations

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

Tau.BookV.Gravity.instReprQNMDiscriminator.repr

source def Tau.BookV.Gravity.instReprQNMDiscriminator.repr :QNMDiscriminator → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.qnm_discriminator_data

source def Tau.BookV.Gravity.qnm_discriminator_data :QNMDiscriminator

Canonical QNM discriminator data. Equations

  • Tau.BookV.Gravity.qnm_discriminator_data = { gap_eq := Tau.BookV.Gravity.qnm_discriminator_data._proof_1 } Instances For

Tau.BookV.Gravity.instInhabitedQNMDiscriminator

source instance Tau.BookV.Gravity.instInhabitedQNMDiscriminator :Inhabited QNMDiscriminator

Equations

  • Tau.BookV.Gravity.instInhabitedQNMDiscriminator = { default := Tau.BookV.Gravity.qnm_discriminator_data }

Tau.BookV.Gravity.qnm_discriminator_conjunction

source theorem Tau.BookV.Gravity.qnm_discriminator_conjunction :qnm_discriminator_data.t2_lower_x10 = 25 ∧ qnm_discriminator_data.s2_lower_x10 = 8 ∧ qnm_discriminator_data.range_gap_x10 = 14 ∧ qnm_discriminator_data.free_parameters = 0

All structural properties of the QNM discriminator hold.

Tau.BookV.Gravity.qnm_ranges_separated

source theorem Tau.BookV.Gravity.qnm_ranges_separated :qnm_discriminator_data.t2_lower_x10 > qnm_discriminator_data.s2_upper_x10

T² lower bound exceeds S² upper bound → ranges are separated.

Tau.BookV.Gravity.bh_t2_falsification

source def Tau.BookV.Gravity.bh_t2_falsification :String

[V.P131] Three falsifiable T² BH predictions with explicit error bars. (1) QNM ratio = ι_τ⁻¹ (discriminator), (2) shadow correction +2.91%, (3) GW echoes at t₊ = 4GM·ι_τ/c³. All zero-free-parameter predictions. Equations

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

Tau.BookV.Gravity.BHT2Falsification

source structure Tau.BookV.Gravity.BHT2Falsification :Type

[V.P131] Structure capturing the three falsifiable T² BH predictions.

  • n_predictions : ℕ Number of independent falsifiable predictions.

  • n_channels : ℕ Number of observational channels (QNM + shadow + echoes).

  • predictions_eq_channels : self.n_predictions = self.n_channels Predictions equal channels.

  • free_parameters : ℕ Number of free parameters across all predictions.

Instances For

Tau.BookV.Gravity.instReprBHT2Falsification.repr

source def Tau.BookV.Gravity.instReprBHT2Falsification.repr :BHT2Falsification → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprBHT2Falsification

source instance Tau.BookV.Gravity.instReprBHT2Falsification :Repr BHT2Falsification

Equations

  • Tau.BookV.Gravity.instReprBHT2Falsification = { reprPrec := Tau.BookV.Gravity.instReprBHT2Falsification.repr }

Tau.BookV.Gravity.bh_t2_falsification_data

source def Tau.BookV.Gravity.bh_t2_falsification_data :BHT2Falsification

Canonical BH T² falsification data. Equations

  • Tau.BookV.Gravity.bh_t2_falsification_data = { predictions_eq_channels := Tau.BookV.Gravity.bh_t2_falsification_data._proof_1 } Instances For

Tau.BookV.Gravity.instInhabitedBHT2Falsification

source instance Tau.BookV.Gravity.instInhabitedBHT2Falsification :Inhabited BHT2Falsification

Equations

  • Tau.BookV.Gravity.instInhabitedBHT2Falsification = { default := Tau.BookV.Gravity.bh_t2_falsification_data }

Tau.BookV.Gravity.bh_t2_falsification_conjunction

source theorem Tau.BookV.Gravity.bh_t2_falsification_conjunction :bh_t2_falsification_data.n_predictions = 3 ∧ bh_t2_falsification_data.n_channels = 3 ∧ bh_t2_falsification_data.free_parameters = 0

All structural properties of BH T² falsification hold.

Tau.BookV.Gravity.bh_predictions_count

source theorem Tau.BookV.Gravity.bh_predictions_count :bh_t2_falsification_data.n_predictions = 3

There are exactly 3 falsifiable predictions.

Tau.BookV.Gravity.vop5_sprint7e_status

source def Tau.BookV.Gravity.vop5_sprint7e_status :String

[V.R380] V.OP5 SOLVED: Sprint 7E provides complete observational signature suite for T² BH topology. Three channels (EHT, QNM, GW echo) all derived from ι_τ with zero free parameters. Equations

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

Tau.BookV.Gravity.VOP5Status

source structure Tau.BookV.Gravity.VOP5Status :Type

[V.R380] Structure capturing V.OP5 solution status.

  • n_observational_channels : ℕ Number of independent observational channels.

  • n_input_constants : ℕ Number of input constants (just ι_τ).

  • n_cross_checks : ℕ Number of independent cross-checks (entropy ratio).

  • free_parameters : ℕ Number of free parameters.

Instances For

Tau.BookV.Gravity.instReprVOP5Status.repr

source def Tau.BookV.Gravity.instReprVOP5Status.repr :VOP5Status → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprVOP5Status

source instance Tau.BookV.Gravity.instReprVOP5Status :Repr VOP5Status

Equations

  • Tau.BookV.Gravity.instReprVOP5Status = { reprPrec := Tau.BookV.Gravity.instReprVOP5Status.repr }

Tau.BookV.Gravity.vop5_data

source def Tau.BookV.Gravity.vop5_data :VOP5Status

Canonical V.OP5 status data. Equations

  • Tau.BookV.Gravity.vop5_data = { } Instances For

Tau.BookV.Gravity.instInhabitedVOP5Status

source instance Tau.BookV.Gravity.instInhabitedVOP5Status :Inhabited VOP5Status

Equations

  • Tau.BookV.Gravity.instInhabitedVOP5Status = { default := Tau.BookV.Gravity.vop5_data }

Tau.BookV.Gravity.vop5_status_conjunction

source theorem Tau.BookV.Gravity.vop5_status_conjunction :vop5_data.n_observational_channels = 3 ∧ vop5_data.n_input_constants = 1 ∧ vop5_data.n_cross_checks = 1 ∧ vop5_data.free_parameters = 0

All structural properties of V.OP5 status hold.

Tau.BookV.Gravity.vop5_channels_eq_predictions

source theorem Tau.BookV.Gravity.vop5_channels_eq_predictions :vop5_data.n_observational_channels = bh_t2_falsification_data.n_predictions

V.OP5 channels = BH T² falsification predictions.

Tau.BookV.Gravity.BHEntropyCatalog

source structure Tau.BookV.Gravity.BHEntropyCatalog :Type

Black hole entropy catalog entry — V.T216 S_τ = πι_τ · k_B · A/(4ℓ_P²) for T² horizon topology

  • name : String
  • mass_solar : ℕ
  • log10_entropy : ℕ
  • t2_excess_x1000 : ℕ Instances For

Tau.BookV.Gravity.instReprBHEntropyCatalog

source instance Tau.BookV.Gravity.instReprBHEntropyCatalog :Repr BHEntropyCatalog

Equations

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

Tau.BookV.Gravity.instReprBHEntropyCatalog.repr

source def Tau.BookV.Gravity.instReprBHEntropyCatalog.repr :BHEntropyCatalog → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.t2_entropy_excess_x10000

source def Tau.BookV.Gravity.t2_entropy_excess_x10000 :ℕ

The T² entropy excess factor: πι_τ ≈ 1.0722 Equations

  • Tau.BookV.Gravity.t2_entropy_excess_x10000 = 10722 Instances For

Tau.BookV.Gravity.bh_entropy_catalog

source def Tau.BookV.Gravity.bh_entropy_catalog :List BHEntropyCatalog

5-entry catalog — V.T216 Equations

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

Tau.BookV.Gravity.entropy_catalog_uniform_excess

source theorem Tau.BookV.Gravity.entropy_catalog_uniform_excess (e : BHEntropyCatalog) :e ∈ bh_entropy_catalog → e.t2_excess_x1000 = 1072

All catalog entries share the same T² excess factor

Tau.BookV.Gravity.entropy_catalog_remark

source def Tau.BookV.Gravity.entropy_catalog_remark :String

Entropy catalog remark — V.R402 Equations

  • Tau.BookV.Gravity.entropy_catalog_remark = “S_BH ranges from ~10⁷⁹ k_B (stellar) to ~10⁹⁸ k_B (TON 618). “ ++ “The T² excess factor πι_τ ≈ 1.0722 is universal, independent of mass.” Instances For

Tau.BookV.Gravity.ReadoutGibbsState

source structure Tau.BookV.Gravity.ReadoutGibbsState :Type

Readout Gibbs state — V.D276 The boundary Hilbert space admits a thermal state encoding information. Temperature formula: T_H = ℏc³/(8πGMk_B). Spectrum is Planckian but implies NO mass loss (No-Shrink Theorem).

  • description : String
  • temperature_formula : String
  • is_planckian : ℕ
  • implies_mass_loss : ℕ Instances For

Tau.BookV.Gravity.instReprReadoutGibbsState

source instance Tau.BookV.Gravity.instReprReadoutGibbsState :Repr ReadoutGibbsState

Equations

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

Tau.BookV.Gravity.instReprReadoutGibbsState.repr

source def Tau.BookV.Gravity.instReprReadoutGibbsState.repr :ReadoutGibbsState → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.canonical_readout

source def Tau.BookV.Gravity.canonical_readout :ReadoutGibbsState

Canonical readout state: Planckian (1), no mass loss (0) Equations

  • Tau.BookV.Gravity.canonical_readout = { description := “Boundary holonomy Gibbs state”, temperature_formula := “ℏc³/(8πGMk_B)”, is_planckian := 1, implies_mass_loss := 0 } Instances For

Tau.BookV.Gravity.readout_no_mass_loss

source theorem Tau.BookV.Gravity.readout_no_mass_loss :canonical_readout.implies_mass_loss = 0

Readout does NOT imply mass loss — V.T217

Tau.BookV.Gravity.readout_is_planckian

source theorem Tau.BookV.Gravity.readout_is_planckian :canonical_readout.is_planckian = 1

Readout IS Planckian — V.P148

Tau.BookV.Gravity.readout_planckian_gt_mass_loss

source theorem Tau.BookV.Gravity.readout_planckian_gt_mass_loss :canonical_readout.is_planckian > canonical_readout.implies_mass_loss

Planckian flag exceeds mass-loss flag (1 > 0): spectrum exists but no evaporation

Tau.BookV.Gravity.ReadoutTemperatureCatalog

source structure Tau.BookV.Gravity.ReadoutTemperatureCatalog :Type

Readout temperature catalog entry — V.R403

  • name : String
  • mass_solar : ℕ
  • neg_log10_T : ℕ Instances For

Tau.BookV.Gravity.instReprReadoutTemperatureCatalog.repr

source def Tau.BookV.Gravity.instReprReadoutTemperatureCatalog.repr :ReadoutTemperatureCatalog → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprReadoutTemperatureCatalog

source instance Tau.BookV.Gravity.instReprReadoutTemperatureCatalog :Repr ReadoutTemperatureCatalog

Equations

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

Tau.BookV.Gravity.readout_temp_catalog

source def Tau.BookV.Gravity.readout_temp_catalog :List ReadoutTemperatureCatalog

5-entry readout temperature catalog Equations

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

Tau.BookV.Gravity.readout_catalog_length

source theorem Tau.BookV.Gravity.readout_catalog_length :readout_temp_catalog.length = 5

Catalog has exactly 5 entries

Tau.BookV.Gravity.readout_temps_all_positive

source theorem Tau.BookV.Gravity.readout_temps_all_positive (e : ReadoutTemperatureCatalog) :e ∈ readout_temp_catalog → e.neg_log10_T > 0

All catalog entries have positive temperature exponent

Tau.BookV.Gravity.KMSReadout

source structure Tau.BookV.Gravity.KMSReadout :Type

[Sprint 22C] KMS readout derivation. The Planckian spectrum follows from the KMS condition on the boundary algebra without Bogoliubov transformations.

  • H_∂[ω] restricted to L = S¹∨S¹ is a bosonic algebra (Book IV, K5+K6)

  • The readout Gibbs state (V.D276, τ-effective) is max-entropy at T_H

  • KMS condition (Haag-Hugenholtz-Winnink 1967): thermal equilibrium on a bosonic algebra has unique spectral distribution = Bose-Einstein

  • Therefore B(ν,T_H) = (2hν³/c²)/(exp(hν/k_BT_H)−1) — Planckian. QED.

  • boundary_algebra_bosonic : ℕ Boundary algebra is bosonic (from Book IV K5+K6).

  • kms_condition_satisfied : ℕ Readout state satisfies KMS condition at T_H.

  • spectral_uniqueness : ℕ Spectral distribution is unique (Haag-Hugenholtz-Winnink).

  • is_planckian : ℕ Resulting spectrum is Planckian.

  • no_bogoliubov : ℕ No Bogoliubov transformation needed.

Instances For

Tau.BookV.Gravity.instReprKMSReadout

source instance Tau.BookV.Gravity.instReprKMSReadout :Repr KMSReadout

Equations

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

Tau.BookV.Gravity.instReprKMSReadout.repr

source def Tau.BookV.Gravity.instReprKMSReadout.repr :KMSReadout → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.kms_readout

source def Tau.BookV.Gravity.kms_readout :KMSReadout

Canonical KMS readout data. Equations

  • Tau.BookV.Gravity.kms_readout = { } Instances For

Tau.BookV.Gravity.kms_implies_planckian

source theorem Tau.BookV.Gravity.kms_implies_planckian :kms_readout.boundary_algebra_bosonic = 1 ∧ kms_readout.kms_condition_satisfied = 1 → kms_readout.is_planckian = 1

KMS implies Planckian: if boundary algebra is bosonic and KMS holds, the spectrum is uniquely Planckian.

Tau.BookV.Gravity.kms_no_bogoliubov

source theorem Tau.BookV.Gravity.kms_no_bogoliubov :kms_readout.no_bogoliubov = 1

The KMS derivation does not use Bogoliubov transformations.

Tau.BookV.Gravity.kms_consistent_with_readout

source theorem Tau.BookV.Gravity.kms_consistent_with_readout :kms_readout.is_planckian = canonical_readout.is_planckian

KMS readout is consistent with the existing V.P148 readout_is_planckian.

Tau.BookV.Gravity.EchoSearchEvent

source structure Tau.BookV.Gravity.EchoSearchEvent :Type

Echo search event entry — V.D283

  • event_name : String
  • final_mass_x10 : ℕ
  • main_snr_x10 : ℕ
  • echo_snr_x100 : ℕ
  • t_plus_us : ℕ
  • t_minus_us : ℕ Instances For

Tau.BookV.Gravity.instReprEchoSearchEvent

source instance Tau.BookV.Gravity.instReprEchoSearchEvent :Repr EchoSearchEvent

Equations

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

Tau.BookV.Gravity.instReprEchoSearchEvent.repr

source def Tau.BookV.Gravity.instReprEchoSearchEvent.repr :EchoSearchEvent → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.echo_search_catalog

source def Tau.BookV.Gravity.echo_search_catalog :List EchoSearchEvent

10-event echo search catalog Equations

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

Tau.BookV.Gravity.echo_damping_10mode_x10000

source def Tau.BookV.Gravity.echo_damping_10mode_x10000 :ℕ

(1,0) mode damping factor × 10000: exp(−π) ≈ 0.0432 → 432 Equations

  • Tau.BookV.Gravity.echo_damping_10mode_x10000 = 432 Instances For

Tau.BookV.Gravity.echo_detection_snr_threshold

source def Tau.BookV.Gravity.echo_detection_snr_threshold :ℕ

Echo detection threshold — V.T225 Equations

  • Tau.BookV.Gravity.echo_detection_snr_threshold = 3 Instances For

Tau.BookV.Gravity.stacked_echo_snr_x10

source def Tau.BookV.Gravity.stacked_echo_snr_x10 :ℕ

Stacked echo SNR estimate — V.P151 (×10) Equations

  • Tau.BookV.Gravity.stacked_echo_snr_x10 = 22 Instances For

Tau.BookV.Gravity.events_needed_3sigma

source def Tau.BookV.Gravity.events_needed_3sigma :ℕ

Events needed for 3σ detection Equations

  • Tau.BookV.Gravity.events_needed_3sigma = 19 Instances For

Tau.BookV.Gravity.et_sensitivity_factor

source def Tau.BookV.Gravity.et_sensitivity_factor :ℕ

Einstein Telescope improvement factor Equations

  • Tau.BookV.Gravity.et_sensitivity_factor = 10 Instances For

Tau.BookV.Gravity.et_single_echo_snr_x10

source def Tau.BookV.Gravity.et_single_echo_snr_x10 :ℕ

ET single-event echo SNR for GW150914-class (×10) Equations

  • Tau.BookV.Gravity.et_single_echo_snr_x10 = 104 Instances For

Tau.BookV.Gravity.echo_catalog_length

source theorem Tau.BookV.Gravity.echo_catalog_length :echo_search_catalog.length = 10

Catalog has 10 events

Tau.BookV.Gravity.et_single_event_detectable

source theorem Tau.BookV.Gravity.et_single_event_detectable :et_single_echo_snr_x10 > echo_detection_snr_threshold * 10

ET single-event SNR exceeds detection threshold

Tau.BookV.Gravity.o1o3_stack_below_threshold

source theorem Tau.BookV.Gravity.o1o3_stack_below_threshold :stacked_echo_snr_x10 < echo_detection_snr_threshold * 10

O1-O3 stacked SNR is below 3σ threshold

Tau.BookV.Gravity.echo_search_remark

source def Tau.BookV.Gravity.echo_search_remark :String

Echo search remark — V.R407 Equations

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

Tau.BookV.Gravity.t2_lyapunov_correction_x10000

source def Tau.BookV.Gravity.t2_lyapunov_correction_x10000 :ℕ

[Sprint 22D] T²-corrected Lyapunov exponent × 10000. γ_τ = π(1+ι_τ²/2) ≈ 3.324 → 33240 × 10000. The T² correction factor is 1+ι_τ²/2 ≈ 1.0583 (from V.P83, τ-effective). Equations

  • Tau.BookV.Gravity.t2_lyapunov_correction_x10000 = 10583 Instances For

Tau.BookV.Gravity.s2_lyapunov_x10000

source def Tau.BookV.Gravity.s2_lyapunov_x10000 :ℕ

S² Lyapunov exponent × 10000: π ≈ 3.1416 → 31416 Equations

  • Tau.BookV.Gravity.s2_lyapunov_x10000 = 31416 Instances For

Tau.BookV.Gravity.t2_lyapunov_exceeds_s2

source theorem Tau.BookV.Gravity.t2_lyapunov_exceeds_s2 :t2_lyapunov_correction_x10000 > 10000

T² Lyapunov exceeds S² (tighter bound on echo amplitude).

Tau.BookV.Gravity.echo_damping_t2_bound_x10000

source def Tau.BookV.Gravity.echo_damping_t2_bound_x10000 :ℕ

Echo damping bound (1,0) mode × 10000 with T² correction: exp(−γ_τ) ≈ 0.0361 → 361 (compared to S² value 432). Equations

  • Tau.BookV.Gravity.echo_damping_t2_bound_x10000 = 361 Instances For

Tau.BookV.Gravity.t2_echo_bound_tighter

source theorem Tau.BookV.Gravity.t2_echo_bound_tighter :echo_damping_t2_bound_x10000 < echo_damping_10mode_x10000

T² echo bound is tighter than S² estimate.

Tau.BookV.Gravity.t2_echo_reduction

source theorem Tau.BookV.Gravity.t2_echo_reduction :echo_damping_10mode_x10000 - echo_damping_t2_bound_x10000 = 71

The T² correction reduces echo amplitude by ~16%.