Target-Relative Identification in Finite Controlled Systems: Refinement Depth, Probe-Action Cost, and Set-Valued Observation

A bounded observation interface need not identify a complete dynamical system to certify a task-specific invariant. We examine three finite consequences of this distinction. For a deterministic controlled 𝑛-state system possessing exactly one proper nontrivial common congruence 𝑇, any nonconstant initial observer partition with 𝑘blocks refines 𝑇after at most 𝑛−𝑘controlled-history rounds. This follows from classical partition refinement and is not proposed as a new general automata theorem. For all 48 = 65,536 four-state, two-input transition tables, an independent census finds 1,236 with a unique balanced 2 + 2 congruence. Relative to a catalogue of eight one-step and sixteen two-step endpoint probes, every such target requires exactly eight controlled actions to identify: 35 symmetry representatives have weighted hitting-set optimum eight, matching an eight-probe upper bound. For three-state systems and 64 conservative set-valued observation channels, an independent exhaustive census finds only one channel with guaranteed target identification across all compatible reports, while 37 channels admit at least one certifying complete report. These results separate history refinement, action accounting, and guarantee-versus- contingent certification. The target and probe languages are restricted finite constructs; the results neither establish generic physical observability nor identify conscious subjects.