Relational Anchors for Quantum Subsystem Identification: Minimal Separation and Robust Rigidity Subtitle: Algebra rounding, synchronized separation, and finite-shot two-qubit certification

Abstract

We develop a finite-dimensional reconstruction framework for identifying tensor-product structure (TPS) from noisy relational localization data without assuming exact local algebras. A unital completely positive localizer is rounded to a nearby exact algebra using explicit idempotent spectral rounding plus finite-dimensional C*-algebra stability; a central-derivation-gap criterion certifies matrix-factor structure, and near-commuting complementary factors are rectified to an exact TPS. Relational uniqueness is formulated with a single synchronized residual gauge across all anchor coordinates, turning minimal robust experiment selection into a test-cover problem on candidate-alignment triples. For D=4 we give an explicit Pauli-specialized branch and finite-shot confidence theorems. Independent multi-context pilot data recover a rank-four Choi support and an exact candidate factor, while a three-setting validation test treats the discarded-subsystem state as a nuisance parameter and certifies proximity to the full family of UCP retractions onto that factor. The resulting split-sample theorem gives radius-valued RECOVER/ABSTAIN/REJECT guarantees for a declared localizer/anchor interface. The result does not infer a TPS from a Hilbert space, Hamiltonian and state alone, nor does it construct localizer channels from arbitrary structure-free laboratory records.