TauLib · API Book V

TauLib.BookV.Gravity.GravitationalConstant

TauLib.BookV.Gravity.GravitationalConstant

The gravitational constant G_τ derived from torus vacuum geometry.

Registry Cross-References

  • [V.D01] Torus Vacuum — TorusVacuum

  • [V.D02] Gravitational Constant — GravConstant

  • [V.T01] Vacuum Shape Ratio — vacuum_shape_ratio_holds

  • [V.P01] G_τ well-defined — structural remark

Mathematical Content

Torus Vacuum Shape Ratio

Every mature black hole state is a stabilized torus vacuum with a fixed shape ratio:

r_n(x) / R_n(x) = ι_τ ∀ mature BH states x

where r_n(x) is the minor radius index and R_n(x) is the major radius index.

This ratio is fixed by refinement coherence beyond the maturity horizon: ι_τ is the canonical shape invariant of the mature torus vacuum. Only the scale degree of freedom (R) remains as free parameter.

Gravitational Constant G_τ

G_τ is defined from the minimal mature BH state:

G_τ := R_n(x_min) / (2 · M_n(x_min))

where x_min is the minimal mature BH state. This is well-defined by refinement coherence: both R_n and M_n are readouts of a single surviving scale parameter on the stabilized torus.

The linear coupling R = 2G_τM is the canonical invariance structure from the τ-kernel (Schwarzschild theorem, see Schwarzschild.lean).

Physical Interpretation

  • G_τ is the τ-derived gravitational constant

  • In orthodox physics: G = (c³/ℏ) · ι_τ² (from sector self-coupling)

  • The gravity sector self-coupling κ(D;1) = 1−ι_τ determines the gravitational coupling strength

Ground Truth Sources

  • gravity-einstein.json: torus-vacuum-shape-ratio, gravitational-constant

  • holonomy-sectors.json: gr-sector-coupling

Tau.BookV.Gravity.TorusVacuum

source structure Tau.BookV.Gravity.TorusVacuum :Type

[V.D01] Torus vacuum: the stabilized torus configuration of a mature black hole state.

The shape ratio r/R = ι_τ is fixed by refinement coherence:

  • r = minor radius index (fiber dimension)

  • R = major radius index (base dimension)

  • Only scale (R) remains as free parameter

The BH topology is a 2-torus T² (NOT a 3-ball). This is the unique stabilized topology from τ-NF convergence.

  • minor_numer : ℕ Minor radius index numerator (r).

  • minor_denom : ℕ Minor radius index denominator.

  • major_numer : ℕ Major radius index numerator (R).

  • major_denom : ℕ Major radius index denominator.

  • minor_denom_pos : self.minor_denom > 0 Both denominators positive.

  • major_denom_pos : self.major_denom > 0

  • major_positive : self.major_numer > 0 Major radius positive (physical).

  • shape_ratio : self.minor_numer * self.major_denom * BookIV.Sectors.iotaD = BookIV.Sectors.iota * self.minor_denom * self.major_numer Shape ratio r/R = ι_τ = iota/iotaD (cross-multiplied).

Instances For

Tau.BookV.Gravity.instReprTorusVacuum

source instance Tau.BookV.Gravity.instReprTorusVacuum :Repr TorusVacuum

Equations

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

Tau.BookV.Gravity.instReprTorusVacuum.repr

source def Tau.BookV.Gravity.instReprTorusVacuum.repr :TorusVacuum → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.TorusVacuum.minorFloat

source def Tau.BookV.Gravity.TorusVacuum.minorFloat (v : TorusVacuum) :Float

Minor radius as Float. Equations

  • v.minorFloat = Float.ofNat v.minor_numer / Float.ofNat v.minor_denom Instances For

Tau.BookV.Gravity.TorusVacuum.majorFloat

source def Tau.BookV.Gravity.TorusVacuum.majorFloat (v : TorusVacuum) :Float

Major radius as Float. Equations

  • v.majorFloat = Float.ofNat v.major_numer / Float.ofNat v.major_denom Instances For

Tau.BookV.Gravity.TorusVacuum.ratioFloat

source def Tau.BookV.Gravity.TorusVacuum.ratioFloat (v : TorusVacuum) :Float

Shape ratio r/R as Float (should be ≈ 0.341304). Equations

  • v.ratioFloat = v.minorFloat / v.majorFloat Instances For

Tau.BookV.Gravity.unit_torus_vacuum

source def Tau.BookV.Gravity.unit_torus_vacuum :TorusVacuum

Example torus vacuum with r = ι_τ, R = 1 (unit major radius). Shape ratio: ι_τ/1 = ι_τ. Equations

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

Tau.BookV.Gravity.GravConstant

source structure Tau.BookV.Gravity.GravConstant :Type

[V.D02] Gravitational constant G_τ.

Defined from the minimal mature BH state: G_τ := R_min / (2 · M_min)

Both R and M are readouts of a single surviving scale parameter on the stabilized torus vacuum. The factor of 2 comes from the canonical Schwarzschild form.

Properties:

  • Well-defined by refinement coherence beyond maturity horizon

  • G_τ > 0 (positive gravitational coupling)

  • In orthodox units: G = (c³/ℏ) · ι_τ² (sector self-coupling readout)

  • g_numer : ℕ G_τ numerator.

  • g_denom : ℕ G_τ denominator.

  • denom_pos : self.g_denom > 0 Denominator positive.

  • g_positive : self.g_numer > 0 G_τ is positive (gravitational attraction).

Instances For

Tau.BookV.Gravity.instReprGravConstant.repr

source def Tau.BookV.Gravity.instReprGravConstant.repr :GravConstant → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprGravConstant

source instance Tau.BookV.Gravity.instReprGravConstant :Repr GravConstant

Equations

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

Tau.BookV.Gravity.GravConstant.toFloat

source def Tau.BookV.Gravity.GravConstant.toFloat (g : GravConstant) :Float

Float display for gravitational constant. Equations

  • g.toFloat = Float.ofNat g.g_numer / Float.ofNat g.g_denom Instances For

Tau.BookV.Gravity.g_tau_iota_factor_numer

source def Tau.BookV.Gravity.g_tau_iota_factor_numer :ℕ

G_τ is proportional to ι_τ² (from the gravity sector self-coupling). In orthodox units: G = (c³/ℏ) · ι_τ². Here we record the ι_τ² factor as the structural core. Equations

  • Tau.BookV.Gravity.g_tau_iota_factor_numer = Tau.BookIV.Sectors.iota_sq_numer Instances For

Tau.BookV.Gravity.g_tau_iota_factor_denom

source def Tau.BookV.Gravity.g_tau_iota_factor_denom :ℕ

Equations

  • Tau.BookV.Gravity.g_tau_iota_factor_denom = Tau.BookIV.Sectors.iota_sq_denom Instances For

Tau.BookV.Gravity.gravity_self_coupling_numer

source def Tau.BookV.Gravity.gravity_self_coupling_numer :ℕ

The gravity sector self-coupling κ(D;1) = 1 − ι_τ connects to G_τ. Equations

  • Tau.BookV.Gravity.gravity_self_coupling_numer = Tau.BookIV.Sectors.iotaD - Tau.BookIV.Sectors.iota Instances For

Tau.BookV.Gravity.gravity_self_coupling_denom

source def Tau.BookV.Gravity.gravity_self_coupling_denom :ℕ

Equations

  • Tau.BookV.Gravity.gravity_self_coupling_denom = Tau.BookIV.Sectors.iotaD Instances For

Tau.BookV.Gravity.vacuum_shape_ratio_holds

source theorem Tau.BookV.Gravity.vacuum_shape_ratio_holds (v : TorusVacuum) :v.minor_numer * v.major_denom * BookIV.Sectors.iotaD = BookIV.Sectors.iota * v.minor_denom * v.major_numer

[V.T01] The torus vacuum shape ratio r/R = ι_τ is encoded in the shape_ratio field of every TorusVacuum.

Tau.BookV.Gravity.unit_torus_has_iota_ratio

source theorem Tau.BookV.Gravity.unit_torus_has_iota_ratio :unit_torus_vacuum.minor_numer = BookIV.Sectors.iota ∧ unit_torus_vacuum.major_numer = BookIV.Sectors.iotaD

The unit torus vacuum has shape ratio ι_τ.

Tau.BookV.Gravity.g_tau_well_defined

source theorem Tau.BookV.Gravity.g_tau_well_defined (g : GravConstant) :g.g_numer > 0 ∧ g.g_denom > 0

[V.P01] G_τ is well-defined: the gravitational constant structure requires a positive numerator and denominator.

Tau.BookV.Gravity.gravity_coupling_positive

source theorem Tau.BookV.Gravity.gravity_coupling_positive :gravity_self_coupling_numer > 0

The gravity self-coupling is positive (1 − ι_τ > 0 since ι_τ < 1).

Tau.BookV.Gravity.g_tau_factor_positive

source theorem Tau.BookV.Gravity.g_tau_factor_positive :g_tau_iota_factor_numer > 0

The ι_τ² factor for G_τ is positive.