This paper develops exact target-specific identifiability criteria for linear diagnostic systems subject to sparse arbitrary corruption. The unknown parameter may remain only partially observable: the objective is not full-state reconstruction, but recovery of a prescribed protected target. Pairwise sparse-fault confusability is therefore treated as a quotient problem, yielding an exact kernel and rank criterion for whether the target remains well defined under every admitted fault support. For a finite hostile support family, the paper proves a sharp hardware law: the minimum number of unrestricted clean scalar augmentations equals the largest target-changing residual rank exposed by any one admissible support. If the new clean bank must also survive f complete erasures, the minimum increases by exactly f. A single coded augmentation can therefore protect an entire finite support library without reconstructing the full mechanism. Structural identifiability and finite-noise robustness nevertheless separate sharply. For K equally spaced hostile residual lines in ℝ², one scalar measurement remains structurally sufficient for every finite K, while the optimal worst-support gain is sin(π/(2K)) and the corresponding Gaussian repetition burden grows as Θ(K²). For dense hostile families, every sub-full-dimensional readout has zero uniform singular margin; for the complete Grassmannian, full target-dimensional readout becomes structurally necessary. A quotient-first generalized singular margin then profiles nuisance, calibration and fault directions before target normalization. The results are finite-dimensional linear geometry. Quantum Hamiltonian identification is one application, but no automatic physical realizability of the mathematically optimal measurements is assumed.
This paper develops exact target-specific identifiability criteria for linear diagnostic systems subject to sparse arbitrary corruption. The unknown parameter may remain only partially observable: the objective is not full-state reconstruction, but recovery of a prescribed protected target. Pairwise sparse-fault confusability is therefore treated as a quotient problem, yielding an exact kernel and rank criterion for whether the target remains well defined under every admitted fault support. For a finite hostile support family, the paper proves a sharp hardware law: the minimum number of unrestricted clean scalar augmentations equals the largest target-changing residual rank exposed by any one admissible support. If the new clean bank must also survive f complete erasures, the minimum increases by exactly f. A single coded augmentation can therefore protect an entire finite support library without reconstructing the full mechanism. Structural identifiability and finite-noise robustness nevertheless separate sharply. For K equally spaced hostile residual lines in ℝ², one scalar measurement remains structurally sufficient for every finite K, while the optimal worst-support gain is sin(π/(2K)) and the corresponding Gaussian repetition burden grows as Θ(K²). For dense hostile families, every sub-full-dimensional readout has zero uniform singular margin; for the complete Grassmannian, full target-dimensional readout becomes structurally necessary. A quotient-first generalized singular margin then profiles nuisance, calibration and fault directions before target normalization. The results are finite-dimensional linear geometry. Quantum Hamiltonian identification is one application, but no automatic physical realizability of the mathematically optimal measurements is assumed.
Exact identification of a target instrument need not imply a linear stability bound for nearby observations. We study fixed finite-dimensional complete instruments with rank-one target Choi blocks, all outcomes retained, a fixed diagnostic contamination path with strictly positive branch noise, and an all-device bound on Choi mass outside the target rays. Under these hypotheses, linear prescribed-target certification is equivalent to injectivity of the joint observation and completeness map on the physical target-tangent space. An invisible nonzero tangent yields an exactly feasible positive instrument pair with square-root separation; a separate target-ray condition distinguishes sharp square-root behavior from constant-order ambiguity. For two prescribed three-basis readouts on a four-dimensional incidence-record benchmark, we obtain linear and square-root compatible-instrument radii, respectively. We then distinguish arbitrary probability residuals from simultaneous physical preparation and readout errors. Positivity and normalization give a support-amplitude estimate under which both protocols have linear calibration-only sensitivity, although their responses to contamination and unrestricted probability error differ. A fixed-sample application uses additional exactly trusted references to acquire operator-error bounds and obtain a simultaneous full-instrument confidence statement. A separate named-source power calculation reaches the prescribed error threshold with a conservative sufficient allocation of approximately 6.12 × 10^21 trials. The results concern a fixed laboratory-frame target and supplied common-device and sampling assumptions; they establish neither practical efficiency nor device-independent calibration. Written proofs are supported by exact finite certificates and a returned kernel-checked Lean subset; the full argument is not formalized end to end.
This research manuscript develops exact measurement-resource formulas for quantum erasure correction with stabilizer codes and an optimal sparse algorithm for selecting native surface-code checks. It studies known erasure locations, where the stabilizer observables may be chosen after the erased sites are identified. The central question is how many independent syndrome outcomes are necessary to distinguish Pauli errors on those sites, modulo stabilizers. For every prime-dimensional [[n,k,d]]_p stabilizer code encoding at least one logical qudit, the manuscript proves that the worst-case syndrome-resolving budget for δ erased sites is exactly 2δ whenever 0 ≤ δ < d. This includes degenerate codes. The result concerns the number of independent p-valued outcomes in the stated noiseless measurement model. The work is directly connected to the measurement-reduction setting studied by Ryutaroh Matsumoto in “Reducing measurements in quantum erasure correction by quantum local recovery,” Computational and Applied Mathematics, DOI: 10.1007/s40314-026-03881-4. It addresses the relationship between erasure-dependent measurement costs and code invariants, together with efficient selection of sufficient surface-code checks. The manuscript presents four connected results: Exact paired-column rank profile. For any matrix G over any field whose columns are partitioned into n labeled pairs, let r be its rank and ν the largest number of complete pairs whose columns are jointly independent. The maximum projected rank over all sets of δ pairs is exactly R(δ) = min{2δ, δ + ν, r}. A single ordering of the pairs realizes every maximum through its successive prefixes. The full profile therefore follows from two invariants, rather than requiring a separate optimization at every erasure size. Distance bounds and the exact worst-case syndrome budget. For an isotropic stabilizer space with k > 0, every inclusionwise maximal complete-pair independent set A satisfies d ≤ |A| + 1. Consequently, ν ≥ d − 1, yielding the exact 2δ worst-case budget below distance. The argument also gives a constructive extension property: every independent-pair set smaller than d − 1 can be enlarged by adding another pair. Particular correctable erasure patterns can require fewer outcomes because of stabilizers supported inside the erased region. Checkable CSS certificates. For Calderbank–Shor–Steane codes, complete-pair independence becomes simultaneous column independence in the X-check and Z-check matrices. Classical matroid intersection supplies an exact rank-partition certificate for ν. A matching independent set and partition provide lower and upper certificates that can be verified using finite-field elimination. Optimal native surface-code check selection. For binary check matrices with at most two nonzero entries per column, the restricted matrix is represented by a sparse graph with boundary pins. Connected components determine a minimum-cardinality native check basis in O(v_I + e_I) time. A pinned component retains every check; an unpinned component omits one check. With nonnegative additive costs, omitting a most expensive check in each unpinned component also minimizes total cost. The two CSS check systems are processed independently. The accompanying verification uses exact arithmetic over GF(2), GF(3), and GF(5). The algebra audit covers 120 general paired matrices and 12,276 subset-rank evaluations, including exact distance checks for 137 positive-logical-dimension code instances and 401 maximal-set distance checks. The graph audit covers 1,785 exhaustive sparse column multisets, 5,355 matrix/cost combinations, 500 additional seeded instances, and named boundary regressions. Both audit programs passed again during preparation of the release. These finite checks support implementation correctness; the general mathematical statements are supported by the supplied proofs. The release includes the complete manuscript, editable LaTeX source, Python native-check selector, exact audit programs, recorded results, generator data, figures, reproduction instructions, and an explicit claim-status record. The selector and audits require only Python 3.10 or later and its standard library; matplotlib is needed only to regenerate the figure. The results apply to independent syndrome information and classical check selection. They do not establish bounds for arbitrary coherent recovery, noisy fault-tolerant extraction, physical circuit depth, or one fixed measurement schedule that must serve every erasure pattern. At or above distance, the general formula remains an algebraic rank profile rather than a universal recovery guarantee. Classical matroid parity, matroid intersection, and incidence-rank methods are acknowledged foundations; historical priority for the presented combination is not certified. Ōrynthéa ∿ Qhuil is relevant to research on quantum error correction, erasure decoding, stabilizer and CSS codes, surface codes, quantum local recovery, measurement complexity, and matroid optimization. This mathematical research under the author designation Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
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.
A finite-dimensional Hilbert space admits many inequivalent tensor-product decompositions, so exact uniqueness of subsystem structure must be formulated on a specified class of Hamiltonians and graphs. In the graph-local setting, exact zero-set identifiability means that the static selector vanishes on a single physical quotient class, or equivalently that the Hamiltonian has a unique admissible-support realization modulo local basis changes and subsystem permutation. On the deletion-robust graph class, defined by the requirement that deleting any vertex leave no isolated vertex component, the analysis gives a finite-certificate description of that question. The reference complement-generation locus in exact graph-local coefficient space is nonempty exactly on this graph class, and an explicit family of one-site fields and mixed edge couplings realizes it. Two finite certificates are then introduced on the reference exact slice: a bounded-length omitted-site word-algebra rank test and a quadratic-rigidity determinant for the first forbidden-support quadratic form. On deletion-robust graphs the resulting reference-regular locus is nonempty, Zariski-open, and Euclidean dense. Assuming the exact zero set of a base Hamiltonian is finite, each competitor exact zero determines a local branch whose intersection with the reference exact slice is governed by an explicit Jacobian-rank criterion. Near a regular-transverse base point, exact zero-set identifiability is therefore controlled by finitely many explicit obstruction manifolds. The graph boundary, the reference certificates, and the structural failure modes are all explicit within this nondegenerate exact-uniqueness framework.
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.
We study the local identifiability of continuous parameters describing intermediate history interventions in a graph-local quantum system from a single final-time measurement architecture in the presence of nuisance parameters. For a fixed-axis commuting Hamiltonian, we prove that back-propagated single-site observables obey an exact one-hop support closure and derive a topology-dependent ceiling on the rank of the history Jacobian. Under A3, which provides one single-site scalar readout per site, a chain geometry confines a five-dimensional history space to at most four history-sensitive observation channels, yielding exact local non-identifiability. We then introduce weak noncommuting perturbations. Although the deficient raw history direction opens at first order in the perturbation strength ε, under the fixed A4 row partition and the baseline exact structural zeros of A5, graph-local support structure and nuisance-space placement force that first-order correction to remain inside the baseline joint history–nuisance observation space, J_h,1v₀ ∈ col 𝓙₀. Hence no new independent nuisance-quotient history direction can open at first order. Combining rigorously interval-certified nonzero second-order coefficients with real-analytic genericity, we obtain, outside a proper analytic exceptional set within a fixed right-analytic structural component, σ_raw=Θ(ε) and σ_quot=Θ(ε²). Unknown-input observability, nuisance projection, graph distance–relative degree, graph-local perturbative suppression, second-order identification, and singular-value perturbation are established prior results. To our knowledge, within the prior art we examined, the narrow contribution not directly anticipated is the sufficient-condition chain by which graph-local support placement structurally traps an already first-order-visible raw history tangent inside the baseline joint history–nuisance observation space, delaying only the first new quotient-identifiable direction to second order, together with the resulting strict raw/quotient order split.