Abstract

We replace an unrestricted deterministic state-functional coupling by a finite classical–quantum transition instrument. Classical labels contain the geometric interface and every controller variable used to select future operations. Matter occupies a fixed finite-dimensional Hilbert space, and the unconditional evolution is linear, completely positive, and trace preserving.

We give an exact observation criterion in the direct-sum state algebra and strengthen it to agreement of stochastic history laws and history-conditioned accessible residuals. A finite backward construction identifies the smallest observable linear spaces needed for a declared continuation. Arbitrary entangled spectator extensions preserve every unnormalized geometric-history block after restriction, provided the complete controller and its certificates are spectator blind. Implementation equivalence is characterized by selective adjoint maps and is stable under compatible continuations.

Two limitations are proved independently: a nonconstant deterministic packet selection cannot be the classical output of an affine channel on a convex state domain, and an informative classical record is incompatible with exact recovery of every input state. Resource and energy balance are stated as operator identities, with resource accounting kept distinct from physical energy. Choi bounds yield checkable assisted stability estimates for both terminal states and complete histories.

A resource-budgeted unsharp qubit instrument selects strictly ordered horizontal slices of a fixed spinless cone. Every branch has an explicit certificate; its classical sector description and a quantitative dephasing approximation are calculated exactly. This example is matter-controlled slicing, not matter-driven gravitational dynamics. No quantum-field source, changing deficit, moving particle, or gravitational field equation is constructed.