The Bogoliubov bridge has separate hypotheses

A proposed free-field bridge must provide:

1. the spacetime and state preparation, field algebra, mode normalizations, and incoming and outgoing access;

2. a canonical transformation with its complete partner and purifier accounting;

3. an implementability statement in the chosen representation;

4. a fixed-code complete-channel comparison to its finite proxy;

5. moment or flux control for the unbounded observables used in energy predictions.

Unitary implementability of bosonic canonical transformations is a nontrivial issue in infinite dimensions BruneauDerezinski. It is not supplied by choosing a lattice.

Nor can finite matrices satisfy the exact oscillator relation $[a,a^{†}]=I$, because the trace of a finite commutator is zero. Finite truncation therefore requires a stated approximation and cannot be described as an exact finite representation of the full CCR.

Even trace-distance convergence does not alone control an unbounded number observable. For example,

converges in trace distance to vacuum, but $\mathrm{Tr}(N{\rho }_n)=n$. Equation (eq:thermal-moment) supplies moment control for its particular geometric preparation; a collapse model must supply its own.