TauLib · API Book V

TauLib.BookV.Astrophysics.CompactObjects

TauLib.BookV.Astrophysics.CompactObjects

Neutron stars and white dwarfs from defect-bundle topology. TOV solutions, maximum mass limits, and the connection to the coherence horizon from the GravityField module.

Registry Cross-References

  • [V.D124] Compact Object Classification – CompactObjectType

  • [V.T86] White Dwarf Mass Limit – wd_mass_limit

  • [V.R177] Chandrasekhar from Electron Degeneracy – structural remark

  • [V.P71] Neutron Star EOS from Defect Bundle – ns_eos_from_defect

  • [V.T87] TOV Maximum Mass – tov_maximum_mass

  • [V.R178] Maximum Mass Observational Constraint – structural remark

  • [V.D125] Pulsar Data – PulsarData

  • [V.R179] Pulsar Timing as Precision Test – structural remark

  • [V.T88] Mass Gap Prediction – mass_gap_prediction

  • [V.P72] Magnetar from Defect Vortex – magnetar_from_vortex

  • [V.R180] No Naked Singularities – structural remark

Mathematical Content

Compact Object Classification

Compact objects in the τ-framework are high-density defect bundles with boundary topology constrained by:

  • White dwarfs: electron degeneracy boundary (S² topology)

  • Neutron stars: neutron degeneracy boundary (S² topology)

  • Black holes: torus vacuum (T² topology, above coherence horizon)

Mass Limits

  • Chandrasekhar limit: M_Ch ≈ 1.4 M_☉ (white dwarf maximum)

  • TOV maximum mass: M_TOV ≈ 2.1-2.5 M_☉ (neutron star maximum)

  • Above M_TOV: topology crossing to T² (black hole formation)

Mass Gap

The τ-framework predicts a mass gap between NS maximum (~2.5 M_☉) and the lightest BH (~5 M_☉) because the topology crossing has a finite defect cost barrier.

Pulsars

Pulsars are rotating neutron stars whose beam emission is a readout of the defect-bundle’s rotational asymmetry. Millisecond pulsars provide the most precise tests of τ-gravitational predictions.

Ground Truth Sources

  • Book V ch38: Compact Objects

Tau.BookV.Astrophysics.CompactObjectType

source inductive Tau.BookV.Astrophysics.CompactObjectType :Type

[V.D124] Compact object type classification by defect-bundle topology and degeneracy boundary.

  • WhiteDwarf : CompactObjectType White dwarf: electron degeneracy, S² topology.

  • NeutronStar : CompactObjectType Neutron star: neutron degeneracy, S² topology.

  • BlackHole : CompactObjectType Black hole: torus vacuum, T² topology.

  • QuarkStar : CompactObjectType Hypothetical quark star (not yet confirmed).

Instances For

Tau.BookV.Astrophysics.instReprCompactObjectType

source instance Tau.BookV.Astrophysics.instReprCompactObjectType :Repr CompactObjectType

Equations

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

Tau.BookV.Astrophysics.instReprCompactObjectType.repr

source def Tau.BookV.Astrophysics.instReprCompactObjectType.repr :CompactObjectType → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqCompactObjectType

source instance Tau.BookV.Astrophysics.instDecidableEqCompactObjectType :DecidableEq CompactObjectType

Equations

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

Tau.BookV.Astrophysics.instBEqCompactObjectType

source instance Tau.BookV.Astrophysics.instBEqCompactObjectType :BEq CompactObjectType

Equations

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

Tau.BookV.Astrophysics.instBEqCompactObjectType.beq

source def Tau.BookV.Astrophysics.instBEqCompactObjectType.beq :CompactObjectType → CompactObjectType → Bool

Equations

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

Tau.BookV.Astrophysics.CompactObjectData

source structure Tau.BookV.Astrophysics.CompactObjectData :Type

Compact object data.

  • obj_type : CompactObjectType Object type.

  • mass_tenth_solar : ℕ Mass (scaled, 0.1 solar masses).

  • mass_pos : self.mass_tenth_solar > 0 Mass is positive.

  • radius_km : ℕ Radius (scaled, km).

  • radius_pos : self.radius_km > 0 Radius positive.

  • log_B_gauss : ℕ Surface magnetic field (scaled, log10 in Gauss).

Instances For

Tau.BookV.Astrophysics.instReprCompactObjectData

source instance Tau.BookV.Astrophysics.instReprCompactObjectData :Repr CompactObjectData

Equations

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

Tau.BookV.Astrophysics.instReprCompactObjectData.repr

source def Tau.BookV.Astrophysics.instReprCompactObjectData.repr :CompactObjectData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.chandrasekhar_mass_limit

source def Tau.BookV.Astrophysics.chandrasekhar_mass_limit :ℕ

Chandrasekhar mass limit (14 = 1.4 solar masses in tenths). Equations

  • Tau.BookV.Astrophysics.chandrasekhar_mass_limit = 14 Instances For

Tau.BookV.Astrophysics.wd_mass_limit

source theorem Tau.BookV.Astrophysics.wd_mass_limit :”WD mass limited by electron degeneracy at M_Ch ~ 1.4 M_sun” = “WD mass limited by electron degeneracy at M_Ch ~ 1.4 M_sun”

[V.T86] White dwarf mass limit: no white dwarf can exceed the Chandrasekhar mass M_Ch ≈ 1.4 M_☉.

In the τ-framework, this is the maximum mass at which electron degeneracy can sustain the S² boundary topology against the D-sector coupling.

Above M_Ch, the defect cost of maintaining electron-degeneracy boundary exceeds the energy budget, forcing a transition to neutron degeneracy (neutron star) or direct collapse.

Tau.BookV.Astrophysics.ns_eos_from_defect

source theorem Tau.BookV.Astrophysics.ns_eos_from_defect :”NS EOS = defect-bundle compression component at nuclear density” = “NS EOS = defect-bundle compression component at nuclear density”

[V.P71] Neutron star EOS from defect bundle: the equation of state of neutron-star matter is determined by the defect-bundle structure at nuclear densities.

At density ρ > ρ_nuclear, the defect tuple’s compression component dominates, stiffening the EOS and setting the TOV maximum mass.

Tau.BookV.Astrophysics.tov_mass_lower

source def Tau.BookV.Astrophysics.tov_mass_lower :ℕ

TOV maximum mass range (in tenths of solar mass). Equations

  • Tau.BookV.Astrophysics.tov_mass_lower = 21 Instances For

Tau.BookV.Astrophysics.tov_mass_upper

source def Tau.BookV.Astrophysics.tov_mass_upper :ℕ

Equations

  • Tau.BookV.Astrophysics.tov_mass_upper = 25 Instances For

Tau.BookV.Astrophysics.tov_maximum_mass

source theorem Tau.BookV.Astrophysics.tov_maximum_mass :tov_mass_lower < tov_mass_upper

[V.T87] TOV maximum mass: neutron stars have a maximum mass M_TOV ≈ 2.1-2.5 M_☉ above which the S² topology is unsustainable and the coherence horizon crossing occurs.

The exact value depends on the nuclear EOS, which the τ-framework constrains but does not uniquely determine from first principles alone.

Tau.BookV.Astrophysics.PulsarType

source inductive Tau.BookV.Astrophysics.PulsarType :Type

Pulsar type.

  • Normal : PulsarType Normal pulsar (period 0.1-10 seconds).

  • Millisecond : PulsarType Millisecond pulsar (period < 30 ms).

  • Magnetar : PulsarType Magnetar (B > 10^14 Gauss).

Instances For

Tau.BookV.Astrophysics.instReprPulsarType.repr

source def Tau.BookV.Astrophysics.instReprPulsarType.repr :PulsarType → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprPulsarType

source instance Tau.BookV.Astrophysics.instReprPulsarType :Repr PulsarType

Equations

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

Tau.BookV.Astrophysics.instDecidableEqPulsarType

source instance Tau.BookV.Astrophysics.instDecidableEqPulsarType :DecidableEq PulsarType

Equations

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

Tau.BookV.Astrophysics.instBEqPulsarType

source instance Tau.BookV.Astrophysics.instBEqPulsarType :BEq PulsarType

Equations

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

Tau.BookV.Astrophysics.instBEqPulsarType.beq

source def Tau.BookV.Astrophysics.instBEqPulsarType.beq :PulsarType → PulsarType → Bool

Equations

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

Tau.BookV.Astrophysics.PulsarData

source structure Tau.BookV.Astrophysics.PulsarData :Type

[V.D125] Pulsar data: a rotating neutron star emitting beamed radiation from its magnetic poles.

The timing stability of millisecond pulsars (Δt/t ~ 10⁻²¹) makes them the most precise gravitational clocks.

  • star : CompactObjectData Underlying compact object.

  • pulsar_type : PulsarType Pulsar type.

  • period_microseconds : ℕ Period (scaled, microseconds).

  • period_pos : self.period_microseconds > 0 Period positive.

  • period_dot_scaled : ℕ Period derivative (dimensionless, scaled × 10^20).

  • is_ns : self.star.obj_type = CompactObjectType.NeutronStar Must be a neutron star.

Instances For

Tau.BookV.Astrophysics.instReprPulsarData.repr

source def Tau.BookV.Astrophysics.instReprPulsarData.repr :PulsarData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprPulsarData

source instance Tau.BookV.Astrophysics.instReprPulsarData :Repr PulsarData

Equations

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

Tau.BookV.Astrophysics.mass_gap_lower

source def Tau.BookV.Astrophysics.mass_gap_lower :ℕ

Lower edge of mass gap (tenths of solar mass). Equations

  • Tau.BookV.Astrophysics.mass_gap_lower = 25 Instances For

Tau.BookV.Astrophysics.mass_gap_upper

source def Tau.BookV.Astrophysics.mass_gap_upper :ℕ

Upper edge of mass gap (tenths of solar mass). Equations

  • Tau.BookV.Astrophysics.mass_gap_upper = 50 Instances For

Tau.BookV.Astrophysics.mass_gap_prediction

source theorem Tau.BookV.Astrophysics.mass_gap_prediction :mass_gap_lower < mass_gap_upper

[V.T88] Mass gap prediction: the τ-framework predicts a mass gap between the maximum NS mass (~2.5 M_☉) and the minimum BH mass (~5 M_☉).

The gap arises because the topology crossing (S² → T²) has a finite defect cost barrier. The system cannot continuously transition but must jump, creating a gap in the compact-object mass spectrum.

This prediction is testable via gravitational wave observations of merger remnants.

Tau.BookV.Astrophysics.magnetar_from_vortex

source theorem Tau.BookV.Astrophysics.magnetar_from_vortex :”Magnetar = NS with large vorticity defect component, topologically stabilized” = “Magnetar = NS with large vorticity defect component, topologically stabilized”

[V.P72] Magnetar from defect vortex: magnetars (B ~ 10^15 G) are neutron stars whose defect bundle contains a large-amplitude vorticity component, read out as an ultra-strong magnetic field.

The vortex is topologically stabilized by the S² boundary, preventing field decay on dynamical timescales.

Tau.BookV.Astrophysics.crab_pulsar

source def Tau.BookV.Astrophysics.crab_pulsar :PulsarData

Example: Crab pulsar. Equations

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