This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.
This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.
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.
This preprint develops a finite-dimensional, proof-audited route from noisy relational localization certificates to exact tensor-product-structure (TPS) envelopes and robust subsystem identification. It separates the pre-Jordan gauge-calibration problem from the post-calibration algebra-rounding problem, proves a quadratic non-identifiability obstruction, and gives a finite cubic relation criterion for Wigner/Jordan gauge collapse. The cubic action is quantitatively analyzed through an exact transverse singular-value identity, nonlinear geodesic bounds, spectral recovery of the Wigner derivation algebra, a two-qubit Pfaffian model, and structured far-sector certificates. For two qubits, the derivation-normalizer problem is reduced to a quartic Grassmann objective and then closed analytically by an explicit Chevalley–Eilenberg/Hodge coercivity estimate on the 90-dimensional normal SU(4) module. This yields a rigorous global entry bound beta_4(pi/(2 sqrt(2))) > sqrt(2/37) and an unconditional cubic-to-Wigner threshold Delta_3^0 < 4/(3 sqrt(185)), eliminating the previous global far-region certification gate in this dimension. The post-calibration branch proves exact Wedderburn articulation for bistochastic idempotent anchors, Choi-rank and Choi-spectrum diagnostics, a central-derivation factor certificate, complementary-factor rectification, complete rounding-set ambiguity control, and stabilizer-synchronized relational-anchor separation. The resulting theorem is a conditional, radius-valued robustness statement inside a finite-dimensional full matrix algebra; it does not claim raw-frequency sample complexity, a dimension-free algebra-rounding constant, or preferred subsystems from a Hamiltonian and state alone.
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.
Operational data can be complete inside a chosen subsystem frame while leaving the frame itself non-unique. A previous finite certification construction exhibits this phenomenon explicitly: an unpartitioned three-qubit Pauli presentation supports three inequivalent finest full tensor frames even though its generated observable algebra is all of M_8(\mathbb C). We ask what additional information can select one frame without treating that information as merely another unlabelled control direction. We introduce relational anchors: gauge-equivariant constraints on a finite refinement poset of direction-generated frames. A general finite-poset theorem characterizes exact anchor collapse and turns target certification into a hitting-set problem, while global frame identification becomes a test-cover problem. In the three-qubit presentation we enumerate the complete nontrivial full witness poset, prove that one two-direction locality anchor separates all three finest frames, and determine exactly twelve single anchors of this type that pass the full-poset test. For robustness, exact support is replaced by conditional-expectation signatures, yielding a sharp \gamma/2 separation criterion. We prove a zero-radius result for the exact witness poset under unrestricted raw perturbations and derive a sharp projective Pauli decoding radius. Six Hamiltonian-derivation squares reconstruct the projective Pauli geometry. A residual synchronization analysis separates ordinary graph cycles from gauge-invariant parity constraints, and four atomic Jordan–Lie anchors are necessary and sufficient, within the declared anchor class, to reduce the remaining scale and sign freedom to Clifford gauge. An explicit relative-defect rigidity modulus and a protocol-conditioned end-to-end error certificate are obtained. The result does not claim that probabilities alone determine a physical tensor-product structure. Rather, unpartitioned process data together with a finite relational anchor package robustly identify a subsystem frame up to the declared gauge.
We study the reconstruction of a qubit factor from global operator relations without presupposing a tensor-product structure. Given two Hermitian operators (A,B) on an unpartitioned finite-dimensional Hilbert space, we quantify how approximate involution and anticommutation relations,[|A^2-I|,\qquad |B^2-I|,\qquad |{A,B}|,]control the existence of a nearby exact same-space qubit representation. The construction is explicit and yields quantitative upper and lower bounds relating observable relation defects to distance from an exact marked qubit factor. A central obstruction is then exhibited: arbitrarily small perturbations of an exact qubit-factor pair can generate the full ambient matrix algebra, causing the exact commutant of the raw operators to collapse discontinuously from the complementary matrix factor to the scalars. Thus exact-algebra or exact-commutant reconstruction from noisy operators is intrinsically unstable. Despite this discontinuity, the complementary factor remains quantitatively recoverable from a soft commutant. For the commutator map associated with an exact rounded pair, we determine the complete singular-value structure and use its fixed spectral separation to control the low-singular-value subspace of the original unrounded operators. If[\rho=2\sqrt{p^2+q^2}<1,]where (p) and (q) bound the same-space rounding errors, the reconstructed complementary subspace satisfies[\left|P_{\widehat{\mathcal C}}-P_{\mathcal A’}\right|_{2\to2}\le\frac{\rho}{2-\rho},]while the observed spectral gap is at least (2-2\rho). Composing the rounding bounds with the relation defects gives an explicit relation-defect-to-complement certificate. The results separate three notions that coincide only in the noiseless idealization: exact generated algebra, exact commutant, and robustly recoverable complementary factor structure. The paper also clarifies the distinction between factor existence and tensor-product-structure uniqueness and positions the result relative to stability of approximate qubit relations, almost-commuting operator algebras, finite-dimensional (C^*)-algebra stability, and numerical commutant reconstruction.
Version 1.1 of a finite-dimensional mathematical preprint on robust quantum subsystem identification from relational operator estimates in a common physical matrix-algebra gauge. The paper gives spectral obstruction and localization certificates from a normalized commutator Laplacian, simple matrix-factor discrimination, generic matrix-unit witness construction, a witness-producing REJECT/ABSTAIN/RECOVER procedure using one uncertainty set throughout, explicit finite-error ambiguity radii, outward-rounded post-gap certification, a one-sided relative-expander robustness design principle, a Weyl-versus-expander asymptotic separation, and a first-order tangency theorem explaining quadratic normal distance to the matrix-factor orbit. Standard commutant, JBD, approximate-C*-algebra, TPS-geometry, and expander ingredients are not claimed as new individually.