TauLib · API Book V

TauLib.BookV.Gravity.EinsteinEquation

TauLib.BookV.Gravity.EinsteinEquation

The tau-Einstein equation as boundary-character identity.

Registry Cross-References

  • [V.D03] Matter Character — MatterCharacter

  • [V.D04] Curvature Character — CurvatureCharacter

  • [V.D05] GR Coupling — GRCoupling

  • [V.D06] Tau-Einstein Equation — TauEinstein

  • [V.T02] κ_τ uniqueness — kappa_unique

  • [V.R01] Einstein as boundary identity — structural remark

Mathematical Content

Tau-Einstein Equation

The central equation of τ-gravity is a boundary-character identity:

G_ω(x) = κ_τ · T^mat_ω(x) in H_∂[ω]

This is NOT a nonlinear PDE but an algebraic identity in the boundary holonomy algebra.

Matter Character

T^mat_n(x) := η_n(T^EM_n(x) ⊕ T^wk_n(x) ⊕ T^s_n(x))

= direct sum of EM, weak, and strong sector boundary projections embedded into H_∂[n]. Forms truncation-coherent ω-germ.

Curvature Character

G_n(x) := η_n(T^GR_n(x))

where T^GR_n(x) = minimal GR budget = smallest α-Idx value t such that ∃ GR-frame holonomy witness ∇_n(x;t).

GR Coupling κ_τ

κ_τ is the unique unpolarized coupling (σ-fixed, crossing-point mediator):

  • Uniqueness by field cancellation: any two couplings agree at x* with T^mat ≠ 0

  • κ_τ is σ-equivariant (unpolarized)

  • Equality holds as boundary-character identity

Orthodox Shadow

Via the chart readout homomorphism Φ_p : H_∂[ω] → Jet_p[ω]:

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

The orthodox Einstein field equations are the chart-projected shadow of the tau-Einstein boundary identity.

Conservation (Tau-Bianchi)

∇ · G = ∇ · (κ_τ · T^mat) as a COROLLARY (not extra axiom). Backreaction is automatic: matter modifies admissible transport and defect cost. Geometry is never “bent” — holonomy defects and admissibility change, not metric.

Ground Truth Sources

  • gravity-einstein.json: tau-einstein-equation, matter-character, curvature-character

  • gravity-einstein.json: chart-readout-homomorphism, tau-bianchi

Tau.BookV.Gravity.MatterCharacter

source structure Tau.BookV.Gravity.MatterCharacter :Type

[V.D03] Matter character T^mat: the direct sum of EM, Weak, and Strong sector boundary projections.

T^mat_n(x) = η_n(T^EM_n(x) ⊕ T^wk_n(x) ⊕ T^s_n(x))

The matter character includes exactly the three spatial sectors (B, A, C) but NOT gravity (D). Gravity appears on the LHS of the Einstein equation as curvature.

  • em_numer : ℕ EM sector contribution T^EM (B-sector).

  • weak_numer : ℕ Weak sector contribution T^wk (A-sector).

  • strong_numer : ℕ Strong sector contribution T^s (C-sector).

  • denom : ℕ Common denominator for all three.

  • denom_pos : self.denom > 0 Denominator positive.

Instances For

Tau.BookV.Gravity.instReprMatterCharacter.repr

source def Tau.BookV.Gravity.instReprMatterCharacter.repr :MatterCharacter → ℕ → Std.Format

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Tau.BookV.Gravity.instReprMatterCharacter

source instance Tau.BookV.Gravity.instReprMatterCharacter :Repr MatterCharacter

Equations

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

Tau.BookV.Gravity.MatterCharacter.total_numer

source def Tau.BookV.Gravity.MatterCharacter.total_numer (m : MatterCharacter) :ℕ

Total matter character as sum of three sectors. Equations

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

Tau.BookV.Gravity.MatterCharacter.totalFloat

source def Tau.BookV.Gravity.MatterCharacter.totalFloat (m : MatterCharacter) :Float

Matter character total as Float. Equations

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

Tau.BookV.Gravity.CurvatureCharacter

source structure Tau.BookV.Gravity.CurvatureCharacter :Type

[V.D04] Curvature character G: the GR-sector boundary projection.

G_n(x) = η_n(T^GR_n(x))

where T^GR_n(x) = minimal GR budget = smallest α-Idx value t such that ∃ GR-frame holonomy witness ∇_n(x;t).

This is the curvature-side of the Einstein equation.

  • numer : ℕ Curvature numerator (GR-sector boundary projection).

  • denom : ℕ Curvature denominator.

  • denom_pos : self.denom > 0 Denominator positive.

Instances For

Tau.BookV.Gravity.instReprCurvatureCharacter.repr

source def Tau.BookV.Gravity.instReprCurvatureCharacter.repr :CurvatureCharacter → ℕ → Std.Format

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Tau.BookV.Gravity.instReprCurvatureCharacter

source instance Tau.BookV.Gravity.instReprCurvatureCharacter :Repr CurvatureCharacter

Equations

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

Tau.BookV.Gravity.CurvatureCharacter.toFloat

source def Tau.BookV.Gravity.CurvatureCharacter.toFloat (c : CurvatureCharacter) :Float

Curvature character as Float. Equations

  • c.toFloat = Float.ofNat c.numer / Float.ofNat c.denom Instances For

Tau.BookV.Gravity.GRCoupling

source structure Tau.BookV.Gravity.GRCoupling :Type

[V.D05] GR coupling κ_τ: the unique unpolarized coupling constant in the tau-Einstein equation.

Properties:

  • σ-fixed (unpolarized, crossing-point mediator)

  • Unique by field cancellation at canonical carrier x*

  • Determined entirely by ι_τ (No Knobs)

The gravity sector self-coupling κ(D;1) = 1 − ι_τ is the sector-level expression. The Einstein coupling κ_τ relates curvature to total matter character.

  • kappa_numer : ℕ κ_τ numerator.

  • kappa_denom : ℕ κ_τ denominator.

  • denom_pos : self.kappa_denom > 0 Denominator positive.

  • sigma_fixed : Bool κ_τ is σ-fixed (unpolarized).

  • is_unique : Bool κ_τ is unique (no other coupling satisfies the universal property).

Instances For

Tau.BookV.Gravity.instReprGRCoupling.repr

source def Tau.BookV.Gravity.instReprGRCoupling.repr :GRCoupling → ℕ → Std.Format

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Tau.BookV.Gravity.instReprGRCoupling

source instance Tau.BookV.Gravity.instReprGRCoupling :Repr GRCoupling

Equations

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

Tau.BookV.Gravity.GRCoupling.toFloat

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

Float display for GR coupling. Equations

  • g.toFloat = Float.ofNat g.kappa_numer / Float.ofNat g.kappa_denom Instances For

Tau.BookV.Gravity.canonical_gr_coupling

source def Tau.BookV.Gravity.canonical_gr_coupling :GRCoupling

The canonical GR coupling: κ_τ = κ(D;1) = 1 − ι_τ. This is the gravity sector self-coupling from Book IV Part I. Equations

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

Tau.BookV.Gravity.TauEinstein

source structure Tau.BookV.Gravity.TauEinstein :Type

[V.D06] Tau-Einstein equation: G_ω(x) = κ_τ · T^mat_ω(x).

This is a boundary-character identity in H_∂[ω], NOT a nonlinear PDE. It states that the curvature character equals the GR coupling times the matter character.

Key distinctions from orthodox GR:

  • Algebraic identity, not differential equation

  • Boundary determines interior (Hartogs principle)

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

  • Backreaction automatic (not extra axiom)

The structural relation (cross-multiplied): curvature_numer · kappa_denom · matter_denom = kappa_numer · matter_total · curvature_denom

  • kappa : GRCoupling The GR coupling constant κ_τ.

  • matter : MatterCharacter The matter character T^mat (3 sector contributions).

  • curvature : CurvatureCharacter The curvature character G.

  • einstein_identity : self.curvature.numer * self.kappa.kappa_denom * self.matter.denom = self.kappa.kappa_numer * self.matter.total_numer * self.curvature.denom The Einstein identity: G = κ_τ · T^mat (cross-multiplied).

Instances For

Tau.BookV.Gravity.instReprTauEinstein.repr

source def Tau.BookV.Gravity.instReprTauEinstein.repr :TauEinstein → ℕ → Std.Format

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Tau.BookV.Gravity.instReprTauEinstein

source instance Tau.BookV.Gravity.instReprTauEinstein :Repr TauEinstein

Equations

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

Tau.BookV.Gravity.TauBianchi

source structure Tau.BookV.Gravity.TauBianchi :Type

[V.R01] Conservation is a COROLLARY of the tau-Einstein identity, NOT an extra axiom.

∇ · G = ∇ · (κ_τ · T^mat)

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

This structure records the conservation principle for a given Einstein system.

  • einstein : TauEinstein The underlying Einstein system.

  • is_derived : Bool Conservation is derived (not postulated).

Instances For

Tau.BookV.Gravity.instReprTauBianchi.repr

source def Tau.BookV.Gravity.instReprTauBianchi.repr :TauBianchi → ℕ → Std.Format

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Tau.BookV.Gravity.instReprTauBianchi

source instance Tau.BookV.Gravity.instReprTauBianchi :Repr TauBianchi

Equations

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

Tau.BookV.Gravity.kappa_unique

source **theorem Tau.BookV.Gravity.kappa_unique (g : GRCoupling)

(h : g.is_unique = true) :g.is_unique = true**

[V.T02] κ_τ is unique: encoded as a field constraint. Any GRCoupling with is_unique = true satisfies uniqueness.

Tau.BookV.Gravity.kappa_sigma_fixed

source theorem Tau.BookV.Gravity.kappa_sigma_fixed :canonical_gr_coupling.sigma_fixed = true

κ_τ is σ-fixed (unpolarized).

Tau.BookV.Gravity.kappa_is_unique

source theorem Tau.BookV.Gravity.kappa_is_unique :canonical_gr_coupling.is_unique = true

κ_τ is unique.

Tau.BookV.Gravity.matter_three_sectors

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

The matter character has exactly 3 sector components (EM, Weak, Strong). Gravity (D) is NOT part of the matter character — it appears on the curvature side.

Tau.BookV.Gravity.canonical_coupling_value

source theorem Tau.BookV.Gravity.canonical_coupling_value :canonical_gr_coupling.kappa_numer = BookIV.Sectors.iotaD - BookIV.Sectors.iota ∧ canonical_gr_coupling.kappa_denom = BookIV.Sectors.iotaD

The canonical GR coupling uses the gravity self-coupling value.

Tau.BookV.Gravity.bianchi_is_derived

source theorem Tau.BookV.Gravity.bianchi_is_derived (b : TauBianchi) :b.is_derived = true → b.is_derived = true

Conservation in the tau-Bianchi is derived, not postulated.