Finite Certification of Quantum Subsystem Cuts from Unpartitioned Control Data

Version 1.2 presents a finite sound certification procedure for operationally exposed quantum subsystem cuts without supplying subsystem labels to the factor-search stage. An exact finite control presentation with empirically confirmed interchange squares is reduced to unlabelled control-direction classes. The algorithm searches used direction subsets, forms noncommutation components, and tests the generated effect algebras for commuting matrix-factor witnesses. It distinguishes a unique finest full cut, partial factor structure with explicit residual multiplicity, and the absence of a uniquely certified cut, including ambiguity between inequivalent embedded full frames. Unused controls are localized only after certification by algebraic support. Version 1.2 closes the uniqueness proof for the global finest-frame claim, gives an exact finite tomographic characterization of terminal equality in the interchange-square condition, explicitly distinguishes the framework from Achenbach et al.'s multimeter-map factorization, fixes and reports the floating-point reference tolerance at 1e-8, and adds an independent exact 120-trial audit comparing the component reduction with exhaustive set-partition search. Five Pauli stress tests remain independently verified by tolerance-free binary-symplectic arithmetic. The operator-algebraic factorization itself is treated as standard; the priority claim remains restricted to the reverse unlabelled witness-search and certificate architecture.