Required specification packet

An obligation is not mathematically testable until the relevant objects and operations are identified. For each $j\in \{4,5,6\}$, a future proof must provide the following packet.

1. Exact objects. Give the distributions or kernels to be estimated, including their dependence on $J_D$, their tensor or graded slots, symmetry, density conventions, and relevant formal orders. Names such as “the required kernel” are insufficient.

2. Spaces and maps. Give every input and output manifold, bundle, test space, pullback map, diagonal embedding, and projection used by the slot.

3. Operation recipe. Specify the ordered contractions, products, pullbacks, pairings, and integrations. State whether any kernel is initially defined only away from a diagonal.

4. Exact covector statements. State the actual wavefront inclusions to be proved. For each operation, identify the corresponding normal-set, opposite-covector, or projection test. Specify the final allowed cone rather than referring to “the usual estimate.”

5. Support and extension. Prove the support condition for each integration and the declared retarded support condition. Construct any needed extension and state its normalization. Compact tests do not automatically prove every internal projection to be proper.

6. Terms retained. Show how the full correction $D_{\sigma }$ enters. Retain the gauge companion and the Einstein-shell term whenever the relevant derivation produces them. A cancellation may be used only after it has been proved.

7. Parameters and scope. State that $\epsilon$ is fixed. Specify any other parameter dependence and any topology needed for a claimed functional or parameter-family result. Pointwise estimates are not silently promoted to uniform estimates or algebraic closure.

8. Discharge statement. State exactly which operation and output estimate the proof establishes. Distinguish that result from construction, normalization, covariance, variational compatibility, and quantum admission.

The supplied material does not contain a completed packet for any of obligations 4–6. Accordingly, no slot-specific covector estimate is fabricated here.