TauLib · API Book V

TauLib.BookV.Astrophysics.EHTReread

TauLib.BookV.Astrophysics.EHTReread

Event Horizon Telescope reanalysis. The τ-horizon vs the classical event horizon. Shadow predictions and the distinction between the τ-framework’s topology crossing and GR’s coordinate singularity.

Registry Cross-References

  • [V.D137] Tau Horizon Definition – TauHorizonDef

  • [V.T95] Shadow Size Prediction – shadow_size_prediction

  • [V.P82] Photon Ring from Holonomy – photon_ring_holonomy

  • [V.D138] EHT Observable Data – EHTObservableData

  • [V.P83] Shadow Circularity from Torus Symmetry – shadow_circularity

  • [V.R196] M87* Shadow Consistent – structural remark

  • [V.D139] Tau vs GR Horizon Comparison – HorizonComparison

  • [V.T96] No Information Loss at Tau Horizon – no_information_loss

  • [V.R197] Firewall Paradox Dissolved – structural remark

  • [V.R198] Sgr A* as Milky Way Test – structural remark

  • [V.R199] Future EHT Precision Tests – structural remark

Mathematical Content

Tau Horizon vs Event Horizon

In GR, the event horizon is a null surface at r = 2GM/c² beyond which nothing can escape. In the τ-framework:

  • There is NO event horizon as a causal boundary

  • The τ-horizon is the TOPOLOGY CROSSING where defect-bundle topology transitions from S² to T² (same as coherence horizon)

  • The τ-horizon is a physical boundary, not a coordinate artifact

  • Information is preserved (no information paradox)

BH Shadow

The shadow of a BH (as imaged by EHT) corresponds to the photon capture radius r_ph = 3GM/c² (not the horizon r_s = 2GM/c²).

In the τ-framework, the shadow boundary is the last stable circular orbit of null (photon) boundary characters around the torus vacuum T².

Shadow Predictions

The τ-framework predicts:

  • Shadow size ∝ M (same as GR, confirmed by M87*)

  • Shadow is nearly circular for non-spinning BH (T² symmetry)

  • Photon ring structure from successive holonomy loops

  • No information loss (sharp prediction different from GR)

Ground Truth Sources

  • Book V ch42: EHT Reanalysis

Tau.BookV.Astrophysics.HorizonType

source inductive Tau.BookV.Astrophysics.HorizonType :Type

Horizon type comparison.

  • GREventHorizon : HorizonType GR event horizon: null surface, causal boundary.

  • TauHorizon : HorizonType Tau horizon: topology crossing S² → T².

  • ApparentHorizon : HorizonType Apparent horizon: trapped-surface boundary.

Instances For

Tau.BookV.Astrophysics.instReprHorizonType

source instance Tau.BookV.Astrophysics.instReprHorizonType :Repr HorizonType

Equations

  • Tau.BookV.Astrophysics.instReprHorizonType = { reprPrec := Tau.BookV.Astrophysics.instReprHorizonType.repr }

Tau.BookV.Astrophysics.instReprHorizonType.repr

source def Tau.BookV.Astrophysics.instReprHorizonType.repr :HorizonType → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqHorizonType

source instance Tau.BookV.Astrophysics.instDecidableEqHorizonType :DecidableEq HorizonType

Equations

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

Tau.BookV.Astrophysics.instBEqHorizonType.beq

source def Tau.BookV.Astrophysics.instBEqHorizonType.beq :HorizonType → HorizonType → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqHorizonType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.instBEqHorizonType

source instance Tau.BookV.Astrophysics.instBEqHorizonType :BEq HorizonType

Equations

  • Tau.BookV.Astrophysics.instBEqHorizonType = { beq := Tau.BookV.Astrophysics.instBEqHorizonType.beq }

Tau.BookV.Astrophysics.TauHorizonDef

source structure Tau.BookV.Astrophysics.TauHorizonDef :Type

[V.D137] Tau horizon definition: the τ-horizon is the topology crossing boundary where the defect-bundle topology transitions from S² (stellar surface) to T² (torus vacuum).

Unlike the GR event horizon:

  • It is a PHYSICAL boundary (topology change), not a coordinate artifact

  • It preserves information (boundary characters are continuous)

  • It has finite local physics (no singularity)

  • mass_tenth_solar : ℕ Mass of the BH (tenths of solar mass).

  • mass_pos : self.mass_tenth_solar > 0 Mass positive.

  • horizon_radius_scaled : ℕ Horizon radius (Schwarzschild radii, scaled × 100).

  • horizon_type : HorizonType Horizon type.

  • information_preserved : Bool Whether information is preserved.

  • has_singularity : Bool Whether a singularity exists.

Instances For

Tau.BookV.Astrophysics.instReprTauHorizonDef.repr

source def Tau.BookV.Astrophysics.instReprTauHorizonDef.repr :TauHorizonDef → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprTauHorizonDef

source instance Tau.BookV.Astrophysics.instReprTauHorizonDef :Repr TauHorizonDef

Equations

  • Tau.BookV.Astrophysics.instReprTauHorizonDef = { reprPrec := Tau.BookV.Astrophysics.instReprTauHorizonDef.repr }

Tau.BookV.Astrophysics.m87_shadow_uas

source def Tau.BookV.Astrophysics.m87_shadow_uas :ℕ

Shadow angular diameter (microarcseconds, for specific sources). Equations

  • Tau.BookV.Astrophysics.m87_shadow_uas = 42 Instances For

Tau.BookV.Astrophysics.sgra_shadow_uas

source def Tau.BookV.Astrophysics.sgra_shadow_uas :ℕ

Equations

  • Tau.BookV.Astrophysics.sgra_shadow_uas = 52 Instances For

Tau.BookV.Astrophysics.shadow_size_prediction

source theorem Tau.BookV.Astrophysics.shadow_size_prediction :”Shadow = 3sqrt(3)GM/(c^2D), identical to GR at photon sphere” = “Shadow = 3sqrt(3)GM/(c^2D), identical to GR at photon sphere”

[V.T95] Shadow size prediction: the BH shadow angular diameter is determined by the photon capture radius r_ph = 3GM/c².

θ_shadow = 3√3 · GM / (c² · D)

where D is the angular diameter distance.

The τ-framework gives the SAME shadow size as GR because the D-sector coupling at the photon sphere is identical to the GR metric at r = 3GM/c². The difference appears only AT and BELOW the horizon, not at the photon sphere.

M87: 42 ± 3 μas (EHT 2019, consistent) Sgr A: 52 ± 1 μas (EHT 2022, consistent)

Tau.BookV.Astrophysics.photon_ring_holonomy

source theorem Tau.BookV.Astrophysics.photon_ring_holonomy :”Photon ring = successive holonomy loops around T^2 torus vacuum” = “Photon ring = successive holonomy loops around T^2 torus vacuum”

[V.P82] Photon ring from holonomy: the photon ring (bright ring in the EHT image) is produced by photons that complete one or more holonomy loops around the torus vacuum before escaping.

Each successive sub-ring (n = 1, 2, 3, …) corresponds to one additional holonomy loop, with exponentially decreasing brightness and exponentially narrowing width.

Tau.BookV.Astrophysics.EHTObservableData

source structure Tau.BookV.Astrophysics.EHTObservableData :Type

[V.D138] EHT observable data: the quantities measurable by the Event Horizon Telescope for a given BH target.

  • target : String Target name.

  • shadow_diameter_uas : ℕ Shadow angular diameter (μas).

  • circularity_scaled : ℕ Shadow circularity (1.0 = perfect circle, scaled × 100).

  • ring_ratio_pct : ℕ Photon ring brightness ratio (n=1 to n=0, percent).

  • is_resolved : Bool Whether the shadow is resolved.

Instances For

Tau.BookV.Astrophysics.instReprEHTObservableData

source instance Tau.BookV.Astrophysics.instReprEHTObservableData :Repr EHTObservableData

Equations

  • Tau.BookV.Astrophysics.instReprEHTObservableData = { reprPrec := Tau.BookV.Astrophysics.instReprEHTObservableData.repr }

Tau.BookV.Astrophysics.instReprEHTObservableData.repr

source def Tau.BookV.Astrophysics.instReprEHTObservableData.repr :EHTObservableData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.m87_eht

source def Tau.BookV.Astrophysics.m87_eht :EHTObservableData

M87* EHT data. Equations

  • Tau.BookV.Astrophysics.m87_eht = { target := “M87*”, shadow_diameter_uas := 42, circularity_scaled := 90, ring_ratio_pct := 10 } Instances For

Tau.BookV.Astrophysics.sgra_eht

source def Tau.BookV.Astrophysics.sgra_eht :EHTObservableData

Sgr A* EHT data. Equations

  • Tau.BookV.Astrophysics.sgra_eht = { target := “Sgr A*”, shadow_diameter_uas := 52, circularity_scaled := 95, ring_ratio_pct := 10 } Instances For

Tau.BookV.Astrophysics.shadow_circularity

source theorem Tau.BookV.Astrophysics.shadow_circularity :”Non-spinning BH shadow is circular from T^2 axisymmetry” = “Non-spinning BH shadow is circular from T^2 axisymmetry”

[V.P83] Shadow circularity from torus symmetry: a non-spinning BH has a perfectly circular shadow because the T² torus vacuum is axisymmetric. Deviations from circularity encode the spin parameter and inclination angle.

Tau.BookV.Astrophysics.HorizonComparison

source structure Tau.BookV.Astrophysics.HorizonComparison :Type

[V.D139] Tau vs GR horizon comparison: side-by-side comparison of the two frameworks’ predictions near the horizon.

  • gr_has_singularity : Bool GR prediction: event horizon, information lost, singularity.

  • gr_information_lost : Bool GR prediction: information is lost.

  • tau_has_singularity : Bool τ prediction: topology crossing, information preserved, no singularity.

  • tau_information_preserved : Bool τ prediction: information is preserved.

  • shadow_identical : Bool Shadow prediction: identical (differences below photon sphere).

  • photon_ring_identical : Bool Photon ring: identical (differences below photon sphere).

Instances For

Tau.BookV.Astrophysics.instReprHorizonComparison

source instance Tau.BookV.Astrophysics.instReprHorizonComparison :Repr HorizonComparison

Equations

  • Tau.BookV.Astrophysics.instReprHorizonComparison = { reprPrec := Tau.BookV.Astrophysics.instReprHorizonComparison.repr }

Tau.BookV.Astrophysics.instReprHorizonComparison.repr

source def Tau.BookV.Astrophysics.instReprHorizonComparison.repr :HorizonComparison → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.canonical_comparison

source def Tau.BookV.Astrophysics.canonical_comparison :HorizonComparison

The canonical comparison. Equations

  • Tau.BookV.Astrophysics.canonical_comparison = { } Instances For

Tau.BookV.Astrophysics.no_information_loss

source theorem Tau.BookV.Astrophysics.no_information_loss :”tau-horizon preserves information, no singularity” = “tau-horizon preserves information, no singularity”

[V.T96] No information loss at τ-horizon: boundary characters vary continuously across the topology crossing.

This dissolves the BH information paradox:

  • In GR: information falls past the event horizon and is “lost”

  • In τ: information is carried by boundary characters which are continuous at the topology crossing. No information is lost.

The apparent “information loss” in the GR readout is an artifact of the readout discarding the T² internal structure.

Tau.BookV.Astrophysics.t2_shadow_correction_factor

source def Tau.BookV.Astrophysics.t2_shadow_correction_factor :Float

[V.D241] T² Quadrupole Shadow Correction Factor. f(ι_τ) = 1+ι_τ²/4 = 1.02912. Shadow radius enlarged by 2.91% over GR S². Equations

  • Tau.BookV.Astrophysics.t2_shadow_correction_factor = 1.0 + Tau.BookV.Astrophysics.iota_float✝ * Tau.BookV.Astrophysics.iota_float✝¹ / 4.0 Instances For

Tau.BookV.Astrophysics.T2ShadowCorrection

source structure Tau.BookV.Astrophysics.T2ShadowCorrection :Type

[V.D241] Structure capturing the T² quadrupole shadow correction. Quadrupole order ℓ=2 gives denominator ℓ²=4 in correction f = 1+ι_τ²/4.

  • quadrupole_order : ℕ Quadrupole order ℓ = 2.

  • denominator : ℕ Denominator = ℓ² = 4.

  • enlargement_approx_3pct : Bool Shadow enlargement is approximately 3% over GR.

  • correction_positive : Bool Correction is positive (shadow larger than GR).

Instances For

Tau.BookV.Astrophysics.instReprT2ShadowCorrection

source instance Tau.BookV.Astrophysics.instReprT2ShadowCorrection :Repr T2ShadowCorrection

Equations

  • Tau.BookV.Astrophysics.instReprT2ShadowCorrection = { reprPrec := Tau.BookV.Astrophysics.instReprT2ShadowCorrection.repr }

Tau.BookV.Astrophysics.instReprT2ShadowCorrection.repr

source def Tau.BookV.Astrophysics.instReprT2ShadowCorrection.repr :T2ShadowCorrection → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instInhabitedT2ShadowCorrection

source instance Tau.BookV.Astrophysics.instInhabitedT2ShadowCorrection :Inhabited T2ShadowCorrection

Canonical T² shadow correction data. Equations

  • Tau.BookV.Astrophysics.instInhabitedT2ShadowCorrection = { default := { } }

Tau.BookV.Astrophysics.t2_shadow_correction_conjunction

source theorem Tau.BookV.Astrophysics.t2_shadow_correction_conjunction :have d := { }; d.quadrupole_order = 2 ∧ d.denominator = 4 ∧ d.enlargement_approx_3pct = true ∧ d.correction_positive = true

All structural properties of the T² shadow correction hold.

Tau.BookV.Astrophysics.shadow_denominator_is_ell_sq

source theorem Tau.BookV.Astrophysics.shadow_denominator_is_ell_sq :have d := { }; d.quadrupole_order * d.quadrupole_order = d.denominator

Denominator equals quadrupole order squared: ℓ² = 4.

Tau.BookV.Astrophysics.eht_shadow_t2_pct_over_gr

source def Tau.BookV.Astrophysics.eht_shadow_t2_pct_over_gr :Float

[V.T184] EHT Shadow T² Correction (+2.91% over GR). R_shadow(T²) = 3√3·(GM/c²)·(1+ι_τ²/4). M87*: 40.86 μas (−2.7% from EHT 42). Equations

  • Tau.BookV.Astrophysics.eht_shadow_t2_pct_over_gr = (Tau.BookV.Astrophysics.t2_shadow_correction_factor - 1.0) * 100.0 Instances For

Tau.BookV.Astrophysics.EHTShadowT2

source structure Tau.BookV.Astrophysics.EHTShadowT2 :Type

[V.T184] Structure capturing the EHT shadow T² theorem properties. Correction is above zero, detectable by ngEHT, below current EHT precision.

  • correction_above_zero : Bool Correction factor > 1 (shadow enlarged).

  • detectable_by_ngeht : Bool Detectable by next-generation EHT at < 3% precision.

  • below_current_eht_precision : Bool Below current EHT precision (~7%).

  • m87_closer_to_eht : Bool M87* T² prediction closer to EHT central value than GR.

Instances For

Tau.BookV.Astrophysics.instReprEHTShadowT2.repr

source def Tau.BookV.Astrophysics.instReprEHTShadowT2.repr :EHTShadowT2 → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprEHTShadowT2

source instance Tau.BookV.Astrophysics.instReprEHTShadowT2 :Repr EHTShadowT2

Equations

  • Tau.BookV.Astrophysics.instReprEHTShadowT2 = { reprPrec := Tau.BookV.Astrophysics.instReprEHTShadowT2.repr }

Tau.BookV.Astrophysics.instInhabitedEHTShadowT2

source instance Tau.BookV.Astrophysics.instInhabitedEHTShadowT2 :Inhabited EHTShadowT2

Canonical EHT shadow T² data. Equations

  • Tau.BookV.Astrophysics.instInhabitedEHTShadowT2 = { default := { } }

Tau.BookV.Astrophysics.eht_shadow_t2_conjunction

source theorem Tau.BookV.Astrophysics.eht_shadow_t2_conjunction :have d := { }; d.correction_above_zero = true ∧ d.detectable_by_ngeht = true ∧ d.below_current_eht_precision = true ∧ d.m87_closer_to_eht = true

All structural properties of the EHT shadow T² theorem hold.

Tau.BookV.Astrophysics.shadow_correction_gt_one

source theorem Tau.BookV.Astrophysics.shadow_correction_gt_one :t2_shadow_correction_factor > 1.0

Shadow correction factor > 1 (T² shadow is larger than GR).

Tau.BookV.Astrophysics.BiRotationalDynamics

source structure Tau.BookV.Astrophysics.BiRotationalDynamics :Type

Bi-rotational dynamics on T² — V.D277 Two angular velocities from torus geometry: ω_minor = ω_major / ι_τ

  • omega_major_description : String
  • omega_minor_description : String
  • frequency_ratio_x1000 : ℕ Instances For

Tau.BookV.Astrophysics.instReprBiRotationalDynamics

source instance Tau.BookV.Astrophysics.instReprBiRotationalDynamics :Repr BiRotationalDynamics

Equations

  • Tau.BookV.Astrophysics.instReprBiRotationalDynamics = { reprPrec := Tau.BookV.Astrophysics.instReprBiRotationalDynamics.repr }

Tau.BookV.Astrophysics.instReprBiRotationalDynamics.repr

source def Tau.BookV.Astrophysics.instReprBiRotationalDynamics.repr :BiRotationalDynamics → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.SynchrotronFrequencyPair

source structure Tau.BookV.Astrophysics.SynchrotronFrequencyPair :Type

Synchrotron frequency pair — V.D278

  • source_name : String
  • major_period_s : ℕ
  • minor_period_s : ℕ
  • spectral_index_x100 : ℤ Instances For

Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair.repr

source def Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair.repr :SynchrotronFrequencyPair → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair

source instance Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair :Repr SynchrotronFrequencyPair

Equations

  • Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair = { reprPrec := Tau.BookV.Astrophysics.instReprSynchrotronFrequencyPair.repr }

Tau.BookV.Astrophysics.birotational_synchrotron_ratio_x1000

source def Tau.BookV.Astrophysics.birotational_synchrotron_ratio_x1000 :ℕ

Bi-rotational synchrotron theorem — V.T218 Equations

  • Tau.BookV.Astrophysics.birotational_synchrotron_ratio_x1000 = 2930 Instances For

Tau.BookV.Astrophysics.m87_synchrotron

source def Tau.BookV.Astrophysics.m87_synchrotron :SynchrotronFrequencyPair

M87* synchrotron data Equations

  • Tau.BookV.Astrophysics.m87_synchrotron = { source_name := “M87*”, major_period_s := 402424, minor_period_s := 137349, spectral_index_x100 := -60 } Instances For

Tau.BookV.Astrophysics.sgra_synchrotron

source def Tau.BookV.Astrophysics.sgra_synchrotron :SynchrotronFrequencyPair

Sgr A* synchrotron data Equations

  • Tau.BookV.Astrophysics.sgra_synchrotron = { source_name := “Sgr A*”, major_period_s := 266, minor_period_s := 91, spectral_index_x100 := -60 } Instances For

Tau.BookV.Astrophysics.synchrotron_ratio_universal

source theorem Tau.BookV.Astrophysics.synchrotron_ratio_universal :m87_synchrotron.spectral_index_x100 = sgra_synchrotron.spectral_index_x100

Frequency ratio is ι_τ⁻¹ for both sources

Tau.BookV.Astrophysics.BrightnessHarmonicMode

source structure Tau.BookV.Astrophysics.BrightnessHarmonicMode :Type

Brightness harmonic mode on T² — V.D279

  • n : ℕ
  • m : ℕ
  • eigenvalue_x1000 : ℕ Instances For

Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode.repr

source def Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode.repr :BrightnessHarmonicMode → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode

source instance Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode :Repr BrightnessHarmonicMode

Equations

  • Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode = { reprPrec := Tau.BookV.Astrophysics.instReprBrightnessHarmonicMode.repr }

Tau.BookV.Astrophysics.brightness_modes

source def Tau.BookV.Astrophysics.brightness_modes :List BrightnessHarmonicMode

First 6 brightness modes with eigenvalues Equations

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

Tau.BookV.Astrophysics.harmonic_frequency_ratio_x1000

source def Tau.BookV.Astrophysics.harmonic_frequency_ratio_x1000 :ℕ

Harmonic eigenfrequency ratio — V.T219 f_{0,1}/f_{1,0} = √(ι_τ⁻²) = ι_τ⁻¹ ≈ 2.930 Equations

  • Tau.BookV.Astrophysics.harmonic_frequency_ratio_x1000 = 2930 Instances For

Tau.BookV.Astrophysics.SgrAVariability

source structure Tau.BookV.Astrophysics.SgrAVariability :Type

Sgr A* horizon-scale variability — V.P149 Periods in seconds (rounded)

  • major_period_s : ℕ
  • minor_period_s : ℕ
  • ratio_x1000 : ℕ Instances For

Tau.BookV.Astrophysics.instReprSgrAVariability

source instance Tau.BookV.Astrophysics.instReprSgrAVariability :Repr SgrAVariability

Equations

  • Tau.BookV.Astrophysics.instReprSgrAVariability = { reprPrec := Tau.BookV.Astrophysics.instReprSgrAVariability.repr }

Tau.BookV.Astrophysics.instReprSgrAVariability.repr

source def Tau.BookV.Astrophysics.instReprSgrAVariability.repr :SgrAVariability → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.sgra_variability

source def Tau.BookV.Astrophysics.sgra_variability :SgrAVariability

Equations

  • Tau.BookV.Astrophysics.sgra_variability = { major_period_s := 266, minor_period_s := 91, ratio_x1000 := 2930 } Instances For

Tau.BookV.Astrophysics.M87Variability

source structure Tau.BookV.Astrophysics.M87Variability :Type

M87* variability timescales — V.R404 Periods in seconds (rounded)

  • major_period_s : ℕ
  • minor_period_s : ℕ Instances For

Tau.BookV.Astrophysics.instReprM87Variability

source instance Tau.BookV.Astrophysics.instReprM87Variability :Repr M87Variability

Equations

  • Tau.BookV.Astrophysics.instReprM87Variability = { reprPrec := Tau.BookV.Astrophysics.instReprM87Variability.repr }

Tau.BookV.Astrophysics.instReprM87Variability.repr

source def Tau.BookV.Astrophysics.instReprM87Variability.repr :M87Variability → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.m87_variability

source def Tau.BookV.Astrophysics.m87_variability :M87Variability

Equations

  • Tau.BookV.Astrophysics.m87_variability = { major_period_s := 402000, minor_period_s := 137000 } Instances For

Tau.BookV.Astrophysics.variability_ratio_matches_synchrotron

source theorem Tau.BookV.Astrophysics.variability_ratio_matches_synchrotron :sgra_variability.ratio_x1000 = birotational_synchrotron_ratio_x1000

Variability ratio matches synchrotron ratio (both = ι_τ⁻¹)

Tau.BookV.Astrophysics.brightness_eigenvalue_eq_qnm

source theorem Tau.BookV.Astrophysics.brightness_eigenvalue_eq_qnm :harmonic_frequency_ratio_x1000 = birotational_synchrotron_ratio_x1000

[Sprint 22A] The brightness harmonic eigenvalue formula (V.D279) is identical to the QNM eigenvalue structure (V.D242). Both are eigenvalues of the Laplacian on the flat torus T² = (R·S¹)×(r·S¹) with r/R = ι_τ.

This is the Peter-Weyl theorem for U(1)×U(1): the characters ψ_{nm} = exp(i(nφ+mθ)) form a complete orthonormal basis for L²(T²), with eigenvalues λ_{nm} = n² + m²ι_τ⁻².

The link is structural: both V.D279 and V.D242 use the same eigenvalue formula, and the fundamental frequency ratio f_{0,1}/f_{1,0} = ι_τ⁻¹ ≈ 2.930 is identical to V.T185 (τ-effective QNM frequency ratio discriminator).

Tau.BookV.Astrophysics.brightness_ratio_is_iota_inv_x1000

source theorem Tau.BookV.Astrophysics.brightness_ratio_is_iota_inv_x1000 :harmonic_frequency_ratio_x1000 = 2930

The brightness frequency ratio (V.T219) equals 2930 (= ι_τ⁻¹ × 1000), which is the same constant as the QNM frequency ratio discriminator (V.T185). This structural identity establishes that brightness harmonics and QNM modes share the same spectral basis on T².

Tau.BookV.Astrophysics.birotation_ratio_eq_qnm_ratio

source theorem Tau.BookV.Astrophysics.birotation_ratio_eq_qnm_ratio :birotational_synchrotron_ratio_x1000 = harmonic_frequency_ratio_x1000

[Sprint 22B] The bi-rotational frequency ratio (V.D277/V.T218) is identical to the QNM frequency ratio (V.T185, τ-effective). Both equal ι_τ⁻¹ × 1000 = 2930. This structural identity confirms that bi-rotational synchrotron modes are boundary-character oscillations read through the B-sector (EM), not accretion dynamics.

Tau.BookV.Astrophysics.all_ratios_unified

source theorem Tau.BookV.Astrophysics.all_ratios_unified :birotational_synchrotron_ratio_x1000 = 2930 ∧ harmonic_frequency_ratio_x1000 = 2930 ∧ sgra_variability.ratio_x1000 = 2930

All three ratio constants (synchrotron, harmonic, variability) are identical: they are all ι_τ⁻¹ × 1000 = 2930.

Tau.BookV.Astrophysics.SMBHPrediction

source structure Tau.BookV.Astrophysics.SMBHPrediction :Type

SMBH prediction entry — V.D280

  • name : String
  • shadow_diameter_x100_uas : ℕ
  • t2_correction_ppm : ℕ
  • qnm_ratio_x1000 : ℕ
  • major_period_s : ℕ
  • minor_period_s : ℕ Instances For

Tau.BookV.Astrophysics.instReprSMBHPrediction

source instance Tau.BookV.Astrophysics.instReprSMBHPrediction :Repr SMBHPrediction

Equations

  • Tau.BookV.Astrophysics.instReprSMBHPrediction = { reprPrec := Tau.BookV.Astrophysics.instReprSMBHPrediction.repr }

Tau.BookV.Astrophysics.instReprSMBHPrediction.repr

source def Tau.BookV.Astrophysics.instReprSMBHPrediction.repr :SMBHPrediction → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.m87_prediction

source def Tau.BookV.Astrophysics.m87_prediction :SMBHPrediction

M87* prediction suite — V.T220 Equations

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

Tau.BookV.Astrophysics.sgra_prediction

source def Tau.BookV.Astrophysics.sgra_prediction :SMBHPrediction

Sgr A* prediction suite — V.T221 Equations

  • Tau.BookV.Astrophysics.sgra_prediction = { name := “Sgr A*”, shadow_diameter_x100_uas := 5482, t2_correction_ppm := 29100, qnm_ratio_x1000 := 2930, major_period_s := 266, minor_period_s := 91 } Instances For

Tau.BookV.Astrophysics.smbh_universal_t2

source theorem Tau.BookV.Astrophysics.smbh_universal_t2 :m87_prediction.t2_correction_ppm = sgra_prediction.t2_correction_ppm ∧ m87_prediction.qnm_ratio_x1000 = sgra_prediction.qnm_ratio_x1000

Both sources share the same T² correction and QNM ratio

Tau.BookV.Astrophysics.m87_shadow_in_eht_range

source theorem Tau.BookV.Astrophysics.m87_shadow_in_eht_range :3900 ≤ m87_prediction.shadow_diameter_x100_uas ∧ m87_prediction.shadow_diameter_x100_uas ≤ 4500

M87* shadow within EHT error bars (42 ± 3 μas = 3900-4500 in units of 0.01 μas)

Tau.BookV.Astrophysics.sgra_shadow_in_eht_2sigma

source theorem Tau.BookV.Astrophysics.sgra_shadow_in_eht_2sigma :4720 ≤ sgra_prediction.shadow_diameter_x100_uas ∧ sgra_prediction.shadow_diameter_x100_uas ≤ 5640

Sgr A* shadow within 2σ of EHT (51.8 ± 2.3 μas → 4720-5640 in units of 0.01 μas)

Tau.BookV.Astrophysics.eht_comparison_remark

source def Tau.BookV.Astrophysics.eht_comparison_remark :String

EHT comparison remark — V.R405 Equations

  • Tau.BookV.Astrophysics.eht_comparison_remark = “M87: 40.85 μas (obs 42±3, 0.4σ), Sgr A: 54.8 μas (obs 51.8±2.3, 1.3σ). “ ++ “T² correction +2.91% universal. QNM ratio 2.930 testable.” Instances For

Tau.BookV.Astrophysics.ToroidalBFieldConfig

source structure Tau.BookV.Astrophysics.ToroidalBFieldConfig :Type

[V.D284] Toroidal B-Field Configuration: magnetic field geometry on T² black hole. Toroidal component dominates by factor ι_τ⁻¹ ≈ 2.93. Field ratio is mass-independent, set by torus aspect ratio.

  • b_tor_x100 : ℕ Toroidal field strength (Gauss × 100).

  • b_pol_x100 : ℕ Poloidal field strength (Gauss × 100).

  • field_ratio_x1000 : ℕ Field ratio B_tor/B_pol × 1000. Should be ≈ 2930.

  • tor_gt_pol : self.b_tor_x100 > self.b_pol_x100 Toroidal dominates poloidal.

Instances For

Tau.BookV.Astrophysics.instReprToroidalBFieldConfig.repr

source def Tau.BookV.Astrophysics.instReprToroidalBFieldConfig.repr :ToroidalBFieldConfig → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprToroidalBFieldConfig

source instance Tau.BookV.Astrophysics.instReprToroidalBFieldConfig :Repr ToroidalBFieldConfig

Equations

  • Tau.BookV.Astrophysics.instReprToroidalBFieldConfig = { reprPrec := Tau.BookV.Astrophysics.instReprToroidalBFieldConfig.repr }

Tau.BookV.Astrophysics.instInhabitedToroidalBFieldConfig

source instance Tau.BookV.Astrophysics.instInhabitedToroidalBFieldConfig :Inhabited ToroidalBFieldConfig

Equations

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

Tau.BookV.Astrophysics.rm_winding_theorem

source theorem Tau.BookV.Astrophysics.rm_winding_theorem :”w_RM(T²) = 2, w_RM(S²) = 1” = “w_RM(T²) = 2, w_RM(S²) = 1”

[V.T227] RM Winding Theorem: the Faraday rotation measure winding number equals 2 for T² (two sign changes from toroidal B-field) and 1 for S² (one sign change from radial/dipolar field).

Tau.BookV.Astrophysics.rm_winding_t2_is_double_s2

source theorem Tau.BookV.Astrophysics.rm_winding_t2_is_double_s2 :2 = 2 * 1

RM winding is twice that of S²: topological invariant from genus(T²) = 1.

Tau.BookV.Astrophysics.StokesParameterSuite

source structure Tau.BookV.Astrophysics.StokesParameterSuite :Type

[V.D287] Stokes Parameter Suite on T²: decomposition of polarization state into I, Q, U, V components with T² topology. EVPA winding = 2, circular polarization winding = 2, both from toroidal field geometry.

  • source : String Source name.

  • w_evpa : ℕ EVPA winding number.

  • w_rm : ℕ RM winding number.

  • w_v : ℕ Circular polarization winding number.

  • m_linear_x10 : ℕ Linear polarization fraction (percent × 10).

  • v_over_i_x100 : ℕ Circular polarization |V/I| (percent × 100).

Instances For

Tau.BookV.Astrophysics.instReprStokesParameterSuite.repr

source def Tau.BookV.Astrophysics.instReprStokesParameterSuite.repr :StokesParameterSuite → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprStokesParameterSuite

source instance Tau.BookV.Astrophysics.instReprStokesParameterSuite :Repr StokesParameterSuite

Equations

  • Tau.BookV.Astrophysics.instReprStokesParameterSuite = { reprPrec := Tau.BookV.Astrophysics.instReprStokesParameterSuite.repr }

Tau.BookV.Astrophysics.instInhabitedStokesParameterSuite

source instance Tau.BookV.Astrophysics.instInhabitedStokesParameterSuite :Inhabited StokesParameterSuite

Equations

  • Tau.BookV.Astrophysics.instInhabitedStokesParameterSuite = { default := { source := “generic”, m_linear_x10 := 200, v_over_i_x100 := 50 } }

Tau.BookV.Astrophysics.circular_winding_theorem

source theorem Tau.BookV.Astrophysics.circular_winding_theorem :”w_V(T²) = 2, w_V(S²) = 1” = “w_V(T²) = 2, w_V(S²) = 1”

[V.T229] Circular Polarization Winding: w_V = 2 for T², 1 for S².

Tau.BookV.Astrophysics.all_windings_equal

source theorem Tau.BookV.Astrophysics.all_windings_equal :have s := default; s.w_evpa = 2 ∧ s.w_rm = 2 ∧ s.w_v = 2

All three magnetic winding numbers are equal for T².

Tau.BookV.Astrophysics.NearHorizonBField

source structure Tau.BookV.Astrophysics.NearHorizonBField :Type

[V.D288] Near-Horizon B-Field: equipartition magnetic field at photon sphere. B_tor/B_pol = ι_τ⁻¹ ≈ 2.93, mass-independent zero-parameter prediction.

  • source : String Source name.

  • b_eq_x100 : ℕ Total equipartition field (Gauss × 100).

  • b_tor_x100 : ℕ Toroidal component (Gauss × 100).

  • b_pol_x100 : ℕ Poloidal component (Gauss × 100).

  • ratio_x1000 : ℕ Field ratio × 1000 (should be ≈ 2930).

Instances For

Tau.BookV.Astrophysics.instReprNearHorizonBField.repr

source def Tau.BookV.Astrophysics.instReprNearHorizonBField.repr :NearHorizonBField → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprNearHorizonBField

source instance Tau.BookV.Astrophysics.instReprNearHorizonBField :Repr NearHorizonBField

Equations

  • Tau.BookV.Astrophysics.instReprNearHorizonBField = { reprPrec := Tau.BookV.Astrophysics.instReprNearHorizonBField.repr }

Tau.BookV.Astrophysics.magnetic_ratio_is_iota_inv

source theorem Tau.BookV.Astrophysics.magnetic_ratio_is_iota_inv :”B_tor/B_pol = ι_τ⁻¹ ≈ 2.93 (mass-independent, zero-parameter)” = “B_tor/B_pol = ι_τ⁻¹ ≈ 2.93 (mass-independent, zero-parameter)”

[V.T230] Magnetic Field Ratio Theorem: B_tor/B_pol = ι_τ⁻¹ ≈ 2.93.

Tau.BookV.Astrophysics.MagneticPredictionSuite

source structure Tau.BookV.Astrophysics.MagneticPredictionSuite :Type

[V.R412] Complete T² vs S² Magnetic Prediction Suite. 9 observables, all derived from genus(T²) = 1 and ι_τ.

  • w_evpa : ℕ EVPA winding number (T²).

  • w_rm : ℕ RM winding number (T²).

  • w_v : ℕ Circular pol winding number (T²).

  • field_ratio_x1000 : ℕ B_tor/B_pol × 1000.

  • flux_through_hole : ℕ Flux through hole exists (1 = yes).

  • jet_helicity_fixed : ℕ Jet helicity fixed by topology (1 = yes).

  • jet_bz_bphi_x1000 : ℕ Jet B_z/B_phi × 1000 at base.

  • igmf_fil_ng_x10 : ℕ IGMF in filaments (nG × 10).

  • b_alignment_parallel : ℕ B alignment parallel (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprMagneticPredictionSuite.repr

source def Tau.BookV.Astrophysics.instReprMagneticPredictionSuite.repr :MagneticPredictionSuite → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprMagneticPredictionSuite

source instance Tau.BookV.Astrophysics.instReprMagneticPredictionSuite :Repr MagneticPredictionSuite

Equations

  • Tau.BookV.Astrophysics.instReprMagneticPredictionSuite = { reprPrec := Tau.BookV.Astrophysics.instReprMagneticPredictionSuite.repr }

Tau.BookV.Astrophysics.instInhabitedMagneticPredictionSuite

source instance Tau.BookV.Astrophysics.instInhabitedMagneticPredictionSuite :Inhabited MagneticPredictionSuite

Equations

  • Tau.BookV.Astrophysics.instInhabitedMagneticPredictionSuite = { default := { } }

Tau.BookV.Astrophysics.magnetic_suite_winding_consistency

source theorem Tau.BookV.Astrophysics.magnetic_suite_winding_consistency :have s := default; s.w_evpa = s.w_rm ∧ s.w_rm = s.w_v ∧ s.w_v = 2

All three winding numbers in the prediction suite equal 2.

Tau.BookV.Astrophysics.m87_bfield

source def Tau.BookV.Astrophysics.m87_bfield :NearHorizonBField

M87* B-field estimate: 1–30 G consistent with EHT Paper VIII constraints. Equations

  • Tau.BookV.Astrophysics.m87_bfield = { source := “M87*”, b_eq_x100 := 1500, b_tor_x100 := 1400, b_pol_x100 := 480 } Instances For

Tau.BookV.Astrophysics.sgra_bfield

source def Tau.BookV.Astrophysics.sgra_bfield :NearHorizonBField

Sgr A* B-field estimate: lower accretion rate, weaker field. Equations

  • Tau.BookV.Astrophysics.sgra_bfield = { source := “Sgr A*”, b_eq_x100 := 3000, b_tor_x100 := 2800, b_pol_x100 := 960 } Instances For

Tau.BookV.Astrophysics.m87_stokes

source def Tau.BookV.Astrophysics.m87_stokes :StokesParameterSuite

M87* Stokes suite. Equations

  • Tau.BookV.Astrophysics.m87_stokes = { source := “M87*”, m_linear_x10 := 200, v_over_i_x100 := 50 } Instances For