Abstract

We construct the coupled gauge-fixed free input for the original-rod lapse variant explicitly frozen in XX-A. The background, scalar calibration, positive lapse branch, slice-harmonic rods, decaying spatial boundary condition, and factorial perturbation convention are retained and reproduced.

The Einstein–scalar Euler linearization is converted to its correctly normalized action Hessian. A generalized de Donder gauge, with a specified auxiliary-field normalization, gives a reduced operator whose principal part is an invertible fiber matrix times the background wave operator. After fiber normalization it is normally hyperbolic. Retarded and advanced inverses are constructed on the open finite-time slab, and the complete auxiliary–ghost inverse is obtained by an exact block calculation. Causal intertwining identities and wavefront bounds are proved.

A nonempty class of normalized Hadamard contraction matrices is constructed using exact Feynman inverses for the resulting bundle operators. The construction is algebraic: positivity on the physical quotient, spatial-tail estimates for the lapse, and an exact BRST-invariant symmetric covariance are not claimed. The complete second-order classical field equations and the embedding, induced-metric, clock, and rod contributions to the lapse are displayed. The second spatial rod is constructed and its absence from the quadratic lapse is retained.

Q1 is closed for the compact-test, gauge-fixed free-input target defined in this paper. This does not close Q2–Q5 for the nonlocal relational lapse, identify the variant with every native Em02-E convention, or admit an inherited characteristic quantum coefficient. No quantum evolution without external time, regulator removal, bulk symmetry restoration, or compilation is asserted.