Finite-Shot Certification of Hidden Quantum Factor Structure from Calibration-Free Sequence Statistics

Abstract

This preprint studies finite-shot reconstruction of hidden quantum factor structure from global calibration-free sequence statistics without assuming an internal tensor-product cut in advance. A probability-only complex-projective 3-design self-test is used as an imported anchor for the active Hilbert/Jordan geometry. From this anchor, global control maps are reconstructed with explicit confidence bounds, including a weighted estimator for unequal shot counts and missing reference settings whenever the surviving frame remains full rank. The main certification result is a whole-commutant-energy lower bound. For reconstructed controls, the sum of the lowest commutator singular energies yields a basis-free lower bound on the aggregate projective Hilbert–Schmidt distance to every common hidden matrix factor of a specified multiplicity. Combined with exact Choi-to-projective-unitary conversion and finite-shot confidence balls, this produces a one-sided lower bound on distance to the complete common-factor model class. An explicit exact factor witness gives the complementary PROXIMAL-RECOVER certificate; otherwise the procedure returns ABSTAIN. The paper also proves that exact generated-algebra / tensor-product-structure type is not uniformly identifiable from finite samples without a structural separation margin. For the fixed two-qubit 60-state stabilizer benchmark, the paper provides an anchor-inclusive raw-count validation and an exact-rational / outward-rounded one-sided trust path from integer counts through anchor and control confidence radii to factor RECOVER/REJECT. The scope is deliberately operational: the result certifies active factor structure supported by the augmented data and does not claim identification of arbitrary inactive spectator degrees of freedom or a uniquely true microscopic decomposition.