E2: replacement contractions

Only after consistency of the forced data is established should an unordered extension be proposed. For every contraction, the construction must specify:

1. the exact inherited free distribution and slot convention;

2. the proposed time-ordered distribution;

3. its differential identity and source-contact normalization;

4. its restrictions to every ordered region;

5. the singular set requiring extension;

6. the residual actual functional after contraction.

The flat contractions mentioned in the prerequisite sheet include $K$ and

Their names do not specify a time-ordered replacement. In particular, replacing a differentiated contraction requires deciding whether differentiation is applied to a fundamental solution or whether a distinct differentiated time-ordering convention is used.

The sharp metric slot must be derived from the actual functional derivative through $L={\mathsf{E}}_{0}C$, retaining (eq:sharp). Time ordering by observation times alone is not justified by (eq:support-bounds). Nor may differentiation, sharp Cauchy inversion, or the sharp adjoint be silently commuted with time ordering.

The standard perturbative algebraic framework distinguishes time-ordering on regular functionals from the renormalized extension needed for local functionals FR. Its Feynman-kernel formulas must be used with their distributional hypotheses FR. Those results do not, without a separate verification of hypotheses, construct this nonlocal sharp sector.

A wavefront analysis must determine the actual unresolved set. Observation diagonals, the Cauchy surface, and the reflected branch of $K$ are candidates until excluded. The binary target does not require importing the degree-four triangle solely because that triangle exists in the inherited word.