Theorem: Conditional quantum admission

Suppose the two-line graph distributions of the selected pair admit a joint extension with all the following properties:

1. agreement with the defined off-collision contractions and with the corrected ordered-support values;

2. the frozen ambiguity class C_{tot};

3. support preservation and compatibility with the inherited root branch;

4. a balanced wavefront bound on the complete graph variables, or an independently proved microcausal bound after pushforward.

Then the resulting ${R_1}^{(1)}$ is a compact even, ghost-number-zero polynomial of degree two, and

Proof. Each double contraction leaves two field slots. Properness follows from Proposition prop:properness. A balanced complete graph bound, after setting internal pushforward covectors to zero, leaves a sum-zero bound on the two external covectors. That bound is microcausal. Alternatively, the separately proved output bound in (iv) gives the same admission directly.

Allowed counterterms are already admitted by Definition def:counterterms. The inherited quotient map then gives the stated class. All entries and lines are even in the relevant sector, and no ghost slots are generated.