Lemma: Exact crossed-order-two reduction

At sufficiently small scales, the complete $5+$ seed numerator can be written as a sum of crossed-order-two terms

and a finite sum of crossed-order-zero and crossed-order-one terms. The relative arguments of ${d_{\delta }}^{\rho }$ are $(y-c,y-c-s)$.

Proof. On every required compact relative neighborhood, eventually

Indeed, $P_{m,2}(W_0\otimes {\Delta }_m)=0$. The cutoff commutator is supported away from the relative image, so its convolution vanishes there at small scales.

The kinetic part after the $t$-derivatives is

Because ${\partial }_s=-{\partial }_2$ on the crossed factor, its six Leibniz terms consist of:

1. the apparent crossed-order-three term $f{\partial }_{2,\alpha }{\partial }_{2,\mu }{\partial }_{2,\kappa }{d_{\delta }}^{\rho }T$;

2. two terms combining to $-{\partial }_{2,\mu }{\partial }_{2,\kappa }{d_{\delta }}^{\rho }{\partial }_{s^{\alpha }}(fT)$;

3. a term $-f{\partial }_{2,\alpha }{\partial }_{2,\kappa }{d_{\delta }}^{\rho }{\partial }_{s^{\mu }}T$;

4. two terms of crossed order one.

After contraction by ${\mathsf{B}}_{\mathrm{kin}}$, item (i) is crossed order one by (eq:Bkin-box) and the local Klein–Gordon equation. Item (iii) is crossed order zero. Item (ii) gives the first line of (eq:positive-q2).

For the mass vertex,

Its crossed-order-two term is the second line of (eq:positive-q2); all its remaining terms have crossed order at most one. This proves an exact reduction, not merely a leading-symbol cancellation.