Finite experiments on countable substrates

We work with finite time windows and finitely many active registers. A countable site set may have an unlimited reserve of names, but every experiment declares the finite registers it actually uses. Multiple declared registers may occupy one site.

This is an explicit capacity convention. It is not the injective one-qubit-per-site convention of one supplied draft. Arbitrarily many names cannot be gathered into a uniformly bounded-size block under that injective convention.

At a site and tick, a finite hybrid observable algebra has the form

where $L_{x,s}$ contains classical record, controller, occupation, and placement labels. Different labels may have different quantum dimensions. An inactive reserved slot is not an accessible surviving measured system.

For finite $I\subseteq \Lambda$, set ${\mathcal{A}}_s(I)={⨂}_{x\in I}{\mathcal{A}}_{x,s}$. States are positive arrays

Compatible finite-region states suffice on a countable substrate; an unspecified infinite tensor-product density operator is not used.