Fixed-Rank Scalar Gate-Sequence Data Admit Irreducible Quantum Realizations in All Nontrivial Finite Dimensions: Noncommuting Spin Lifts and a Level-2 Matrix-Order Separator
Consider exact scalar gate-sequence data consisting of one preparation, one binary effect, and the probability of that effect after every word over a fixed two-gate alphabet. This preprint constructs a single word law of exact noncommutative Hankel rank four that admits a control-irreducible unitary quantum realization on C^d for every finite d >= 2. In each realization the state is full rank, the binary effect is strict, and the two controls generate the full matrix algebra M_d(C), excluding direct-sum padding and invariant spectator sectors. The associated four-dimensional angular-momentum operator systems have the same first-level Lorentz cone in every dimension, while a level-2 matrix pencil is positive exactly when t >= (d+1)/(d-1). Thus matrix-level-two order distinguishes all members of the constructed family. The result is a scoped non-identifiability theorem for the Hilbert-space dimension of an arbitrary compatible irreducible realization. It does not claim that the minimum compatible dimension is unidentifiable, does not contradict informationally complete gate set tomography within a declared model dimension, and does not reconstruct a tensor-product structure. The record includes the preprint PDF, LaTeX source, bibliography, numerical verification script and output, citation metadata, and a targeted priority audit.
Consider exact scalar gate-sequence data consisting of one preparation, one binary effect, and the probability of that effect after every word over a fixed two-gate alphabet. This preprint constructs a single word law of exact noncommutative Hankel rank four that admits a control-irreducible unitary quantum realization on C^d for every finite d >= 2. In each realization the state is full rank, the binary effect is strict, and the two controls generate the full matrix algebra M_d(C), excluding direct-sum padding and invariant spectator sectors. The associated four-dimensional angular-momentum operator systems have the same first-level Lorentz cone in every dimension, while a level-2 matrix pencil is positive exactly when t >= (d+1)/(d-1). Thus matrix-level-two order distinguishes all members of the constructed family. The result is a scoped non-identifiability theorem for the Hilbert-space dimension of an arbitrary compatible irreducible realization. It does not claim that the minimum compatible dimension is unidentifiable, does not contradict informationally complete gate set tomography within a declared model dimension, and does not reconstruct a tensor-product structure. The record includes the preprint PDF, LaTeX source, bibliography, numerical verification script and output, citation metadata, and a targeted priority audit.
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.
Two finite-order quantum controls may each be compatible with a qubit while their joint algebra is not. We study a cyclic control Q and an independent involution R with target algebra C_n x C_2. For even n >= 4, no faithful projective qubit realization exists: in PU(2) ≃ SO(3), a nontrivial involution commuting with an order-n rotation collapses to Q^(n/2). An explicit qutrit construction realizes the full algebra, so the minimum projective dimension is three; n = 4 is the first obstructed even member and has an eight-ray regular orbit. The representation-theory obstruction itself lies in a mature mathematical area. The operational contribution developed here is a calibrated qubit-consistency test: for exact quarter-turn and half-turn targets, the commutator angle gives a tight state-independent fidelity bound and a tight optimized binary-contrast cap. We derive a sufficient angular-error version and extend the test to general CPTP gate deviations through diamond distance. The explicit qutrit construction reaches the algebraic maximum binary contrast of one using two control sequences and a common binary measurement. The result is a calibrated falsification test for a qubit control model, not a device-independent dimension witness, and it remains entirely within standard finite-dimensional quantum mechanics.
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.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
This deposit contains two companion technical notes developing a finite-dimensional semantic state algebra for unresolved readings, classical decisions, and order-sensitive semantic operations. The first note separates semantic states from their response formats. A response format is defined as an alphabet together with a prescribed resolution map. For a two-reading construction, four selector-polarity states are distinguished explicitly. The polarity-only yes/no map is proven non-injective, whereas the full selector-polarity map is bijective. A separate rank-nullity theorem establishes the corresponding global obstruction for linear maps from higher-dimensional semantic spaces into lower-dimensional decision spaces. A real two-dimensional space is sufficient for storage; under the additional requirements of associative and commutative multiplication, a unit, and division by every nonzero element, the unique two-dimensional real extension is the complex field. The classical real sector is embedded conservatively. The second note extends this framework to semantic operators. Questions and contextual updates generate a unital complex operator algebra, while order dependence is represented exactly by commutators and the associated derived Lie algebra. State-dependent and global order-sensitivity measures are introduced. Complex dimension two is shown to be minimal for noncommuting complex-linear dynamics, while terminal binary decisions remain separate real-valued readouts. Operator complexification preserves sums, composition, identities, and commutators. The construction is structural rather than empirical. It assumes neither quantum ontology nor a Born rule and does not claim a complete formal semantics of natural language.