Inner Heisenberg dynamics is insensitive to central generator shifts, yet finite control data can expose and progressively refine the responses of the common invariant sector. We develop a center-specific finite-data inverse protocol in which reversible processing suppresses noninvariant transients while repertoire enrichment changes which invariant response relations are explicitly distinguished. The main theorem turns finite-depth noisy data into certified complement intervals for candidate responses: a positive lower endpoint guarantees a genuine rank increase, a quantitative post-extension singular gap, and an explicit processing-depth requirement. The recursion certifies its current gap from the measured table, admits seedless initialization, supports adaptive confidence spending, and separates the value of deeper processing from improved measurement precision. Two-sided extensions show that response novelty can belong to an input–readout relation rather than to either candidate alone. A block residual theorem identifies all joint rank novelty inside the fixed linear response model, while independent rank-budget and common-target tests distinguish incomplete articulation from structural incompatibility. For finite-dimensional complex C*-algebras, the invariant target specializes to the center under center-separating inner controls. We retain the exact central factorization C_Z = P Q, its alignment defect and completion counts, finite-time aliasing boundaries, and one scalable U(1) charge family because they determine interpretation, resources, or falsifiability. In that family, normalized local controls exhibit an O(n^-4) visibility obstruction, uniformly mixed product preparations have an exponentially weak charge direction per normalized preparation budget, whereas a rank-minimal classically correlated separable family retains a dimension-independent raw singular gap. Standard commutant, observability, Schur-complement, perturbation, concentration, model-invalidation, and experimental-design tools are treated as prior art. The contribution is their coherent center-response composition into explicit continuation, relational-enrichment, diagnosis, and stopping logic.
Inner Heisenberg dynamics is insensitive to central generator shifts, yet finite control data can expose and progressively refine the responses of the common invariant sector. We develop a center-specific finite-data inverse protocol in which reversible processing suppresses noninvariant transients while repertoire enrichment changes which invariant response relations are explicitly distinguished. The main theorem turns finite-depth noisy data into certified complement intervals for candidate responses: a positive lower endpoint guarantees a genuine rank increase, a quantitative post-extension singular gap, and an explicit processing-depth requirement. The recursion certifies its current gap from the measured table, admits seedless initialization, supports adaptive confidence spending, and separates the value of deeper processing from improved measurement precision. Two-sided extensions show that response novelty can belong to an input–readout relation rather than to either candidate alone. A block residual theorem identifies all joint rank novelty inside the fixed linear response model, while independent rank-budget and common-target tests distinguish incomplete articulation from structural incompatibility. For finite-dimensional complex C*-algebras, the invariant target specializes to the center under center-separating inner controls. We retain the exact central factorization C_Z = P Q, its alignment defect and completion counts, finite-time aliasing boundaries, and one scalable U(1) charge family because they determine interpretation, resources, or falsifiability. In that family, normalized local controls exhibit an O(n^-4) visibility obstruction, uniformly mixed product preparations have an exponentially weak charge direction per normalized preparation budget, whereas a rank-minimal classically correlated separable family retains a dimension-independent raw singular gap. Standard commutant, observability, Schur-complement, perturbation, concentration, model-invalidation, and experimental-design tools are treated as prior art. The contribution is their coherent center-response composition into explicit continuation, relational-enrichment, diagnosis, and stopping logic.
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.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.
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.
Relational quantum causal processes formulate finite operational contexts as normal positive functionals on local completely positive maps. Response differences generate an influence algebra, and its central projections define jointly readable Boolean events. We develop this starting point through a sequence of exact and controlled models. Fresh-environment unitary collision circuits produce dephasing-exchange kinetics with an exact charge-center fixed algebra, a uniform finite-step limit at fixed response order, and graph-controlled metastable Markov dynamics. An absorbing-state model exhibits a sharp transition between non-Abelian quantum memory and Boolean records. A reversal-covariant defect dynamics generates a locally finite partial order on a restricted graph family without assuming a Lyapunov time. Conditional on a certified order, a positive additive record measure, compactness, and identifiability, we prove subsequential convergence to a Lorentzian metric-measure space, finite reconstruction bounds, and uniqueness of admissible smooth limits. Complementary finite regulators provide controlled tests of modular-to-boost response, null tomography, same-update variational identities, induced quadratic gravity, and compatible common-refinement limits. These results are exact or controlled within their stated models, but they do not yet constitute a single background-independent microscopic law that jointly generates adjacency, time, volume normalization, dimension, signature, nonlinear Einstein constraints, and quantum matter. We therefore present RQCP-QG as a theorem-indexed framework that separates established mechanisms, conditional compositions, and open assumptions.
We develop a finite operational response theory for controlled quantum systems in which observer closure, hidden microscopic structure, and higher-order response are treated within a single algebraic framework. The central distinction is between generic higher-order observability and Closure–Residue (CR)-constrained generative response. Standard finite-dimensional quantum theory does not, by itself, impose the specific CR accessibility condition introduced here; physically realizable controls are ordinarily determined by the microscopic Hamiltonian, hardware connectivity, conservation laws, and environmental couplings of the system under study. We therefore formulate First-Residue Accessibility as an additional finite selection principle for the coherent-control sector studied here: the infinitesimal deformation of the observer closure algebra generated by any admissible tangent in the declared target control manifold must lie in the first-residue derivation space inherited from an underlying microscopic CR sector. The algebraic consequences are rigid. Within this coherent-control sector, the maximal abelianness of the population algebra implies that the infinitesimal action of a control on the closure algebra determines its anti-Hermitian generator uniquely modulo a diagonal population gauge. First-Residue Accessibility therefore forces the declared coherent control class, up to this gauge, to lie in a canonical graph-internal subtype extracted from the microscopic CR sector. For diagonal prepared states, the visible first-order population response is silent, while the second variation factorizes through a graph-divergence map of a microscopic bilinear edge flow. A three-site single-excitation benchmark realizes the factorization exactly and yields an even response law of the form Sigma(epsilon X) = epsilon^2 Q_CR(X,X) + O(epsilon^4). We also separate exact discrete support transport from continuous-time quasilocality. A hard one-step support cone has a singular continuous-time limit whenever the limiting generator genuinely spreads local observables; finite propagation in the continuous limit must therefore be formulated by soft tails rather than exact support. We introduce a normalized-trace visible-leakage profile and derive a soft causal readout bound. In the coherent single-excitation sector this profile is uniformly equivalent, up to fixed observer-volume constants, to intrinsic operator-norm quasilocality. For finite-range exchange with simultaneous site dephasing we prove an N-uniform normalized-trace quasilocality estimate without using rank preservation. This genuinely nonunitary extension establishes a common finite-time exponential locality class with standard open-system quasilocality, while the stronger two-sided intrinsic equivalence is claimed only for the coherent unitary subclass. We further show that divergence-image membership is non-discriminating for connected full-population readout, construct an explicit standard CPTP control family that is first-order silent but violates the CR component constraint on a disconnected graph, and isolate experimentally measurable null tests. We distinguish a declared target-control manifold from the larger independently calibrated manifold of physically available coherent controls, thereby separating a conditional declared-manifold accessibility hypothesis from the stronger Physical First-Residue Accessibility statement. Under star-compatible oriented provenance we also derive an observable cross-prediction bound that links independently fixed static CR data to a measurable quadratic response coefficient. A framework-discriminating test requires both that the CR structure be fixed without using the target Hessian data and that the resulting constraint be stronger than what follows from the same calibrated microscopic information in a standard quantum model. We further exhibit a connected three-site collective-mode candidate in which CR Accessibility predicts an exact left–right quadratic null, while a same-support standard comparator constrained only by independently fixed aggregate tangent norms permits a nonzero response envelope. This yields a strict prediction-set separation under an explicit Dataset-A information condition, while also making clear that the separation disappears if standard microscopic calibration independently fixes the same collective tangent relation. We map this strict comparator onto a single-electron linear triple quantum dot, construct a four-point mixed-Hessian estimator, and propagate swap-odd instrumental asymmetry, finite-difference bias, and shot noise into a preregistered experimental envelope. The deeper derivation of the accessibility principle from the full Generative Map is left to the subsequent axiomatic stage of the program.
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.