τ-Einstein Equation

The τ-Einstein equation is the boundary-character identity R^H(x) = κ_τ · T^mat(x) in H_∂[ω], where R^H is the curvature character, κ_τ = 1 − ι_τ is the gravitational coupling, and T^mat is the matter character. It is an **algebraic** identity at the categorical kernel — not a partial differential equation. The classical Einstein field equations are its chart-shadow projection.

Categorical invariant. R^H(x) = κ_τ · T^mat(x) on H_∂[ω]; κ_τ = 1 − ι_τ ≈ 0.658696.

Primary registry anchor: V.D51

Supporting items: V.D46, V.D52, V.T28

1. I.K0 — Universe Postulate
2. V.D46 — Gravitational coupling κ_τ = 1 − ι_τ — the unique D-sector depth-1 coupling
3. V.D51 — τ-Einstein equation R^H = κ_τ · T^mat — boundary-character identity
4. V.D52 — Linearized form for weak-field analysis
5. V.T28 — Newtonian limit recovery — chart shadow yields ∇²Φ = 4πGρ

Lean modules referenced: TauLib.BookV.GravityField.TauEinsteinEq

Calibration anchor: PG-P01-neutron

Calibration chain:

1. κ_τ = 1 − ι_τ from D-sector
2. G = (c³/ℏ) · ι_τ² (Newton's constant cascade)
3. SI bridge via m_n anchor for mass / curvature units

Manuscript reference: manuscript-sources/book-05/part02/ch13-tau-einstein-equation.tex

Status: Formalized

Module: TauLib.BookV.GravityField.TauEinsteinEq

Lean kind: definition

Lean symbol: Tau.BookV.GravityField.TaueinsteinEquationVd06

Related glossary entries

• PG-L01-tau-schrodinger τ-Schrödinger Equation
• PG-Q01-mass Mass

Referenced by

• PG-L05-tau-newton-gravity τ-Newton's Law of Gravity
• PG-L12-tau-gravitational-wave τ-Gravitational Wave
• PG-L13-tau-mercury-precession τ-Mercury Perihelion Precession