Abstract

We complete the algebraic descent and overlap tests left open in IX-Free for the ordered word $B_f,W_g,B_{f^{\prime }}$. The previously constructed graph outputs are used as inherited values.

A compact spatial-diagonal metric direction, obtained by applying the wave operator to a test tensor, proves sharp-generator separation even when the $00$-metric component is constrained to vanish. Slot isolation then proves injectivity of both degree-four presentation maps. Consequently all four prescribed degree-four coefficients descend to actual functionals.

An explicit advanced-trace witness proves that every nonzero sharp generator has nonzero intrinsic metric-derivative charge at some smooth configuration. The partial contractions have zero charge. The old microcausal algebra also has zero charge: a nonzero boundary coefficient would produce a timelike conormal singularity. These facts eliminate the live new-domain intersection and separate the sharp space from the old algebra. An algebraic anchored-domain lemma eliminates the old overlap of the entire new-domain sum, including possible multi-domain relations.

For every real common parent parameter, the prescriptions therefore glue to one truncated partial product preserving the locked first stages and the old product. The two associations of the selected word agree through order ${\hslash }^3$. Reversal, presentation independence, and fixed-support test-family continuity hold. Actual-stage continuity, associativity on unspecified triples, and the IX-D and G2 constructions are not asserted.