Option A: the encoding-independent negative closure

No Born rule, dimensional bound, continuity, linearity, or amplitude encoding enters this proof. It applies equally to an encoding by density operators. A stochastic encoding, after averaging its input-dependent preparation distribution, still gives a state-valued function of $a$; its output law is still a function of $a$. Retaining a preparation label does not alter the argument if that label is itself supplied only as a function of $a$.

Consequently, exact readout closure for the convolution coefficient $a$ alone is closed in the negative. Any informative invariant input must be supplied upstream by changing the input object, not by choosing a different readout of the same scalar. This conclusion does not authorize a coordinate counterterm.

The lemma does not obstruct functions of several coefficients whose gauge orbit occupies only part of their joint domain. Option B changes the input in precisely that way. Nor does the lemma force a function of an already invariant scalar to be constant across different invariant values: such a scalar does not satisfy the lemma's surjective gauge-orbit hypothesis.