Proposition: Uniqueness of complete coefficients

Suppose two expansions of the same family have finite real negative-power logarithmic polynomials, positive logarithmic powers at exponent zero, finite distributional terms, and remainders tending strongly to zero. Their collected singular coefficients and finite terms agree.

Proof. Subtract the expansions and combine their finite exponent inventories. A nonzero singular difference would be unbounded on a fixed detecting test, whereas the remaining difference has a finite limit. All singular differences therefore vanish; taking the limit identifies the finite terms.