Proposition: Whole-domain normalization standard

The separate pairings in (eq:individual-normalizations), imposed for every $c\in \mathcal{D}$ and every smooth spatially compact free solution, determine the two degree-one output classes uniquely.

If formal $\hslash$-dependent alternatives are allowed, this statement requires equality of the full normalization series, coefficientwise.

Normalization of the sum alone does not uniquely specify a decomposition into two summands.

Proof. For each output, subtract two candidates with the same individual pairing. Their compact test difference annihilates every spatially compact free solution. Equation (eq:separation) makes its class zero. The argument applies to each $\hslash$-coefficient.

If only the sum is fixed, one may formally add $\mathbf{\varphi }(n_c)$ to one candidate and subtract the same class from the other. Individual normalization or the coefficient recursion is needed to exclude such redistribution.