Proposition: Support and reference agreement

Suppose Assumption ass:expansion holds and the complete raw family converges distributionally to $t^{\circ }$ on $V_{\mathrm{ref}}$. Then every nonsummable coefficient vanishes there,

including the degree-zero logarithmic coefficients, and

Proof. A nonzero restricted nonsummable coefficient would produce an unbounded fixed pairing supported in $V_{\mathrm{ref}}$, contradicting the proved reference convergence. After these coefficients vanish, the remainder bound identifies the restricted limit.