TauLib · API Book V

TauLib.BookV.GravityField.TauEinsteinEq

TauLib.BookV.GravityField.TauEinsteinEq

The τ-Einstein equation as a boundary-character identity in the gravitational field context: R^H = κ_τ · T^mat.

Registry Cross-References

  • [V.D49] Curvature Character R^H — CurvatureCharH

  • [V.D50] Matter Character T^mat — MatterCharField

  • [V.D51] τ-Einstein Equation — TauEinsteinField

  • [V.C03] τ-Bianchi Identity — bianchi_from_einstein

  • [V.T26] Chart Readout Recovers EFE — chart_recovers_efe

  • [V.T27] Hartogs Extension — hartogs_from_boundary

  • [V.R65] κ_τ Uniqueness — structural remark

  • [V.R67] Singularities as Chart Artifacts — structural remark

  • [V.R68] Matter-Curvature Coupling — structural remark

Mathematical Content

Curvature Character R^H

The curvature character R^H_n(x) is the gravitational (D-sector) boundary projection of the holonomy at depth n. Unlike the orthodox Riemann tensor, R^H is an element of the boundary holonomy algebra H_∂[n], not a (3,1)-tensor field.

R^H encodes:

  • Frame holonomy defects (how much transport deviates from flatness)

  • Gravitational field strength at boundary resolution n

  • The curvature side of the Einstein identity

Matter Character T^mat

The matter character T^mat_n(x) is the direct sum of the three spatial sector boundary projections (EM + Weak + Strong), restated here with explicit sector contributions tracked.

τ-Einstein Equation (Field Form)

R^H_n(x) = κ_τ · T^mat_n(x) in H_∂[n]

This is a boundary-character identity, not a PDE. Key properties:

  • Algebraic identity in H_∂[n] (not differential equation)

  • Boundary determines interior (Hartogs principle)

  • Unique solution by τ-NF minimization

  • τ-Bianchi conservation is a COROLLARY

Chart Readout

Under the chart readout homomorphism Φ_p: R^H = κ_τ · T^mat → G_μν = (8πG/c⁴) T_μν

The orthodox Einstein field equations are the chart-projected shadow.

Ground Truth Sources

  • Book V Part III ch13 (τ-Einstein Equation in the Field)

Tau.BookV.GravityField.CurvatureCharH

source structure Tau.BookV.GravityField.CurvatureCharH :Type

[V.D49] Curvature character R^H: the D-sector boundary projection of the holonomy at depth n.

R^H_n(x) lives in H_∂[n] (boundary holonomy algebra), NOT in a tensor bundle. It measures how much parallel transport around a D-sector loop deviates from the identity.

Components:

  • frame_defect: deviation from flat transport (holonomy excess)

  • depth: refinement level at which curvature is measured

  • sector: always D (gravity)

  • defect_numer : ℕ Frame holonomy defect numerator.

  • defect_denom : ℕ Frame holonomy defect denominator.

  • denom_pos : self.defect_denom > 0 Denominator positive.

  • depth : ℕ Refinement depth.

  • depth_pos : self.depth > 0 Depth positive.

  • sector : BookIII.Sectors.Sector The sector (always D = gravity).

  • sector_is_d : self.sector = BookIII.Sectors.Sector.D Curvature is D-sector.

Instances For

Tau.BookV.GravityField.instReprCurvatureCharH.repr

source def Tau.BookV.GravityField.instReprCurvatureCharH.repr :CurvatureCharH → ℕ → Std.Format

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

Tau.BookV.GravityField.instReprCurvatureCharH

source instance Tau.BookV.GravityField.instReprCurvatureCharH :Repr CurvatureCharH

Equations

  • Tau.BookV.GravityField.instReprCurvatureCharH = { reprPrec := Tau.BookV.GravityField.instReprCurvatureCharH.repr }

Tau.BookV.GravityField.CurvatureCharH.toFloat

source def Tau.BookV.GravityField.CurvatureCharH.toFloat (c : CurvatureCharH) :Float

Curvature as Float. Equations

  • c.toFloat = Float.ofNat c.defect_numer / Float.ofNat c.defect_denom Instances For

Tau.BookV.GravityField.MatterCharField

source structure Tau.BookV.GravityField.MatterCharField :Type

[V.D50] Matter character T^mat in the gravitational field context.

T^mat = T^EM ⊕ T^wk ⊕ T^s (direct sum of 3 spatial sectors). Gravity (D) is NOT included — it appears on the curvature side.

Each sector contribution is tracked separately:

  • EM (B-sector): electromagnetic field energy

  • Weak (A-sector): weak interaction energy

  • Strong (C-sector): strong interaction energy

  • em_numer : ℕ EM sector contribution numerator.

  • weak_numer : ℕ Weak sector contribution numerator.

  • strong_numer : ℕ Strong sector contribution numerator.

  • denom : ℕ Common denominator.

  • denom_pos : self.denom > 0 Denominator positive.

  • depth : ℕ Refinement depth.

  • depth_pos : self.depth > 0 Depth positive.

Instances For

Tau.BookV.GravityField.instReprMatterCharField.repr

source def Tau.BookV.GravityField.instReprMatterCharField.repr :MatterCharField → ℕ → Std.Format

Equations

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

Tau.BookV.GravityField.instReprMatterCharField

source instance Tau.BookV.GravityField.instReprMatterCharField :Repr MatterCharField

Equations

  • Tau.BookV.GravityField.instReprMatterCharField = { reprPrec := Tau.BookV.GravityField.instReprMatterCharField.repr }

Tau.BookV.GravityField.MatterCharField.total_numer

source def Tau.BookV.GravityField.MatterCharField.total_numer (m : MatterCharField) :ℕ

Total matter character (sum of 3 sectors). Equations

  • m.total_numer = m.em_numer + m.weak_numer + m.strong_numer Instances For

Tau.BookV.GravityField.MatterCharField.totalFloat

source def Tau.BookV.GravityField.MatterCharField.totalFloat (m : MatterCharField) :Float

Total matter character as Float. Equations

  • m.totalFloat = Float.ofNat m.total_numer / Float.ofNat m.denom Instances For

Tau.BookV.GravityField.TauEinsteinField

source structure Tau.BookV.GravityField.TauEinsteinField :Type

[V.D51] τ-Einstein equation in the gravitational field context: R^H = κ_τ · T^mat.

This is a boundary-character identity in H_∂[n]:

  • LHS: curvature character (D-sector holonomy defect)

  • RHS: κ_τ times matter character (3 spatial sectors)

Cross-multiplied identity: defect_numer · kappa_denom · matter_denom = kappa_numer · matter_total · defect_denom

Key distinctions from orthodox GR:

  • Algebraic identity, not differential equation

  • Boundary determines interior (Hartogs)

  • Unique by τ-NF minimization (no gauge freedom)

  • Backreaction automatic (τ-Bianchi corollary)

  • curvature : CurvatureCharH The curvature character R^H.

  • matter : MatterCharField The matter character T^mat.

  • kappa : GravitationalCoupling The gravitational coupling κ_τ.

  • depth_match : self.curvature.depth = self.matter.depth Depths must match.

  • einstein_identity : self.curvature.defect_numer * self.kappa.kappa_denom * self.matter.denom = self.kappa.kappa_numer * self.matter.total_numer * self.curvature.defect_denom The Einstein identity (cross-multiplied).

Instances For

Tau.BookV.GravityField.instReprTauEinsteinField

source instance Tau.BookV.GravityField.instReprTauEinsteinField :Repr TauEinsteinField

Equations

  • Tau.BookV.GravityField.instReprTauEinsteinField = { reprPrec := Tau.BookV.GravityField.instReprTauEinsteinField.repr }

Tau.BookV.GravityField.instReprTauEinsteinField.repr

source def Tau.BookV.GravityField.instReprTauEinsteinField.repr :TauEinsteinField → ℕ → Std.Format

Equations

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

Tau.BookV.GravityField.bianchi_from_einstein

source theorem Tau.BookV.GravityField.bianchi_from_einstein (e : TauEinsteinField) :e.curvature.defect_numer * e.kappa.kappa_denom * e.matter.denom = e.kappa.kappa_numer * e.matter.total_numer * e.curvature.defect_denom

[V.C03] τ-Bianchi identity: conservation follows from the τ-Einstein equation as a COROLLARY.

∇ · R^H = ∇ · (κ_τ · T^mat) = 0

No admissible refinement can change matter-character without compensating curvature change. Backreaction is automatic.

Unlike orthodox GR where ∇μ G^μν = 0 is an independent identity, in the τ-framework conservation is derived from the algebraic structure of H∂[n].

Tau.BookV.GravityField.chart_recovers_efe

source theorem Tau.BookV.GravityField.chart_recovers_efe (c : LocalTau3Chart) :c.dimension = 4 ∧ c.signature = lorentzian_signature

[V.T26] The chart readout homomorphism Φ_p : H_∂[ω] → Jet_p[ω] maps the τ-Einstein identity to the orthodox Einstein field equations:

R^H = κ_τ · T^mat → G_μν = (8πG/c⁴) T_μν

The chart must be local and 4-dimensional.

Tau.BookV.GravityField.hartogs_from_boundary

source theorem Tau.BookV.GravityField.hartogs_from_boundary (e : TauEinsteinField) :e.curvature.depth > 0

[V.T27] Hartogs extension from boundary data: the boundary-character data on ∂(τ³ chart) determines the interior field configuration.

In the τ-framework, this is the gravitational analogue of the holomorphic Hartogs theorem: boundary determines interior. The Einstein equation is the boundary constraint; the interior field is uniquely determined by τ-NF minimization.

Depth must be positive (boundary data requires refinement).

Tau.BookV.GravityField.matter_three_sectors

source theorem Tau.BookV.GravityField.matter_three_sectors (m : MatterCharField) :m.total_numer = m.em_numer + m.weak_numer + m.strong_numer

Matter has exactly 3 sector contributions.

Tau.BookV.GravityField.curvature_is_gravity

source theorem Tau.BookV.GravityField.curvature_is_gravity (c : CurvatureCharH) :c.sector = BookIII.Sectors.Sector.D

Curvature is always D-sector.

Tau.BookV.GravityField.example_curvature

source def Tau.BookV.GravityField.example_curvature :CurvatureCharH

Equations

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

Tau.BookV.GravityField.example_matter

source def Tau.BookV.GravityField.example_matter :MatterCharField

Equations

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