Chapter 11: The τ-Einstein Equation: Boundary-Character Equality

The τ-Einstein equation is the central equation of τ-gravity. It is not a nonlinear partial differential equation on a background manifold. It is a boundary-character identity in the holonomy algebra H_∂[ω]:

stating that the curvature character R^H equals the gravitational coupling κ_τ times the matter character T^mat. Both sides are ω-germs—elements of the boundary algebra—and their equality is an algebraic identity, not a differential equation. The orthodox Einstein field equations einstein1915field,einstein1916foundation G_{μν} = (8π G/c⁴) T_{μν} are recovered as the chart shadow of this boundary identity under the local readout functors of the relevant chapter. Conservation (∇ · T = 0) is a corollary, not an extra axiom. Backreaction is automatic. Well-posedness follows from the Hartogs principle: the boundary determines the interior.