Theorem: Exact result of the selected attempt

For the fixed selected pair and the copied class ${\mathfrak{C}}_{\mathrm{part}}$:

1. The common-frequency regularization defines compact admitted quadratic graph functionals at every positive regulator.

2. Its numerical graph products converge to the prescribed products on the microlocally admissible open sets. Auxiliary compact-localization changes have a remainder smaller than every regulator power.

3. Contacts in ${\mathfrak{C}}_{\mathrm{part}}$ cannot alter a prescription on the collision complement. Agreement only on the admissible subset does not prove that two prescriptions differ by permitted contacts.

4. The selected characteristic-set limit, the uniform transverse estimates, the contact overlaps, and the complete output theorem have not all been established.

The requested coefficient therefore retains the ending Unresolved, with E1 the earliest unfinished gate. No assertion excludes every extension in ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. Proposition prop:finite, Lemma lem:smoothing, Propositions prop:localization and prop:good-limit, Lemma lem:rigidity, and Proposition prop:agreement prove the affirmative statements.

Definition def:E1-missing identifies the missing characteristic-set limit and its prescription independence. Sections sec:E2–sec:E4 state the subsequent obligations without asserting that they have been discharged. Under the declared release criteria, the absence of those proofs permits neither Constructed nor Proved obstructed for the full coefficient.