Proposition: Uniqueness of the extracted asymptotics

For finite real negative degrees, nonnegative integer logarithmic orders, a degree-zero logarithmic part, and a remainder tending to zero on every test, the nonsummable coefficients and the finite distribution are unique.

Consequently, the coefficients in Theorem thm:main are intrinsic to the fixed regularized family, even if the origin-germ decomposition itself is not unique.

Proof. Subtract two expansions and combine equal exponents. If a nonsummable coefficient in the difference is nonzero, the dominance argument above gives one unbounded fixed pairing, contradicting the vanishing remainder. Thus all nonsummable differences are zero. Taking the limit then makes the finite distributions equal.