Definition: A supplied first-order quantum residual interface

Assume a later construction supplies, on the same relative domain and through the same formal order:

1. all quantum products and insertion maps needed to define the first-order residual;

2. an admitted odd ghost-number-one functional ${\mathfrak{A}}_1$, including every normalization and transport contribution;

3. a convention in which adding an even ghost-number-zero counterterm $C_1$ changes that residual by ${\mathcal{Q}}^{[2]}{C}_1$;

4. the consistency identity ${\mathcal{Q}}^{[2]}{\mathfrak{A}}_1=0$ modulo ${\kappa }^3$.

These are hypotheses, not conclusions of the local fish extension.