Proposition: Residual identifiability ledger

The following statements hold for the frozen acquisition:

1. Any parameter absent from the source states, generators, selector law, detector, and attempted-run model has an independent record law; every Fisher-information entry involving it is zero wherever derivatives exist.

2. The included direction and kick nuisance affect the likelihood through $u,v,\gamma$. They are not generally zero-information parameters.

3. At fixed angles and direction, distinct finite kick-weight vectors with the same $u,v$ have identical records. For $m\ge 4$ distinct transition values, such distinct positive weight vectors exist locally around every strictly positive weight vector.

4. The occupation records do not certify inversion of the local kick. Replacing ${V_k}^{†}$ on $A$ by $V_k$ leaves the record law on the promised source family unchanged.

Proof. For (i), the complete likelihood contains no such parameter. Any post-processing of equal record laws again has equal laws.

Statement (ii) is the explicit dependence in (eq:moments)–(eq:likelihood). For (iii), the weight constraints depend only on the three linear functionals

The matrix with rows $1,t_k,{t_k}^2$ has rank three when at least three transition values are distinct. With $m\ge 4$ it has a nonzero kernel vector $z$. For a strictly positive $w$, both $w+ϵz$ and $w-ϵz$ remain positive for sufficiently small $ϵ>0$, preserve normalization and both moments, and are distinct.

For (iv), $V_k$ and ${V_k}^{†}$ have the same computational transition probabilities. The source is diagonal, CZ changes no populations, and the relaxation population rule is unchanged. The proof of Lemma lem:law therefore gives the same likelihood.