Flat residual from asymptotic summation

Statement

Let $F$ be a fixed differential polynomial of finite order and degree whose coefficient jets grow at most as fixed powers of $q^{-1}$ . Suppose smooth finite states $U^{[J]}$ and a smooth realization $U$ on one domain satisfy

|U-U^{[J]}|_m\le C_J q^{h_J-L_m},\qquad h_J\to\infty,

near $q=0$ , and

|U^{[J]}|_m\le C_{J,m}q^{-K_m}(1+|\log q|)^{P_{J,m}},

|F(U^{[J]})|_m\le C_{J,m}q^{\rho_J-K_m^F}(1+|\log q|)^{Q_{J,m}}+E_{J,m},\qquad \rho_J\to\infty.

Assume $L_m,K_m,K_m^F$ are independent of $J$ , and each fixed $E_{J,m}$ is $O(q^N)$ for every $N$ . Then $F(U)$ has infinite-order decay with all derivatives.

Compare with one fixed stage

Fix an output derivative order $m$ and target decay order $N$ . If $F$ has order $s$ and degree $d$ , choose $J$ so large that the tail exponent $h_J-L_{m+s}$ exceeds $N$ plus the fixed coefficient loss and $(d-1)(K_{m+s}+1)$ . Also require $\rho_J-K_m^F>N$ , with a positive margin to absorb logarithms. Keep this $J$ fixed.

In a sufficiently small $q$ -neighborhood, the difference jet is at most one and the fixed-stage logarithms fit within the allocated power margin. The differential-polynomial difference estimate gives $|F(U)-F(U^{[J]})|_m=O(q^N)$ . The fixed-stage residual and its fixed flat remainder have the same bound. The triangle inequality proves the claim.

The order of choices is $(m,N)$ , then a finite $J$ , then a neighborhood and constants. This argument never sums the flat remainders $E_{J,m}$ over stages; arbitrary stage-dependent flat errors need not form a convergent series.