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.
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.
A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must address not only local rigidity near a structured frame but also the global possibility that a distant decomposition class could make a Hamiltonian appear equally local. This paper proves a global result in that direction for the static locality selector associated with graph-local Hamiltonians. The central exact input is exact zero-set identifiability: for an exactly graph-local Hamiltonian, the selector vanishes only on the preferred physical quotient class. Under this hypothesis, compactness of the quotient and continuity of the selector imply a strictly positive selector minimum on every closed exterior set at positive quotient distance. A Hamiltonian-uniform perturbation bound is then derived directly from the projector representation of the selector and combined with the known displacement estimate for Hamiltonians carrying small forbidden tails. The resulting perturbed exterior gap shows that every decomposition class lying a fixed positive distance from the shifted local minimizer incurs a uniform positive selector cost relative to that minimizer. Together with the local quadratic basin available in the small-tail regime, this yields conditional global uniqueness of the perturbed preferred class. The argument therefore isolates a global transfer mechanism: on any class where exact zero-set identifiability holds, no new global geometry of decomposition space is needed to obtain an exterior gap and conditional global uniqueness in the small-tail regime.
Earlier work on preferred decomposition for graph-local Hamiltonians established a layered picture rather than a single theorem: empirical structure selection under locality-sensitive selectors, identification of the correct quotient notion of uniqueness, local rigidity and local quantitative identifiability near exact realizations, stability under sufficiently small forbidden tails, and conditional global uniqueness once the exact zero set is known to be unique. The present paper shows that the unrestricted global step cannot be completed from graph locality alone. It gives an explicit connected counterexample family on star graphs: a commuting qubit Hamiltonian family preserved by an entangling star-CNOT circuit that exchanges leaf fields with edge Ising couplings while producing a genuinely inequivalent tensor-product decomposition. The same abstract Hamiltonian therefore has at least two exact admissible-support realizations, so exact zero-set identifiability fails on the physical quotient. The paper also shows why this does not contradict the earlier positive results. The counterexample lies entirely on a structurally degenerate locus excluded by the nondegeneracy hypotheses used in the local rigidity and exact reconstruction results: visible local directions are at most one-dimensional and the relevant complement algebras are abelian rather than full matrix algebras. The main foundational conclusion is sharp. Graph locality by itself does not determine a unique preferred decomposition even in the exact, static, connected setting. What remains open is a narrower problem: whether exact uniqueness holds on a suitably formulated nondegenerate locus that excludes commuting disentanglable obstruction classes of the type exhibited here.
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.
In Hilbert-space-first approaches to emergent locality or spacetime, a fundamental difficulty is that a finite-dimensional Hilbert space admits many inequivalent tensor-product decompositions, while notions such as locality, operator support, entanglement, and subsystem persistence are all defined relative to such a choice. Previous finite numerical and exact-search studies established that decomposition space is not empirically flat under sparse-support selectors, and that structured Hamiltonians can exhibit unique quotient minima within explicit finite Clifford families. What remained open was whether this selection persists as a local analytic property in the full continuous manifold of decompositions. I resolve this question for exactly graph-local Hamiltonians. For the static selector that penalizes higher-weight operator support and off-graph two-body terms, I derive the infinitesimal expansion around a structured decomposition and identify the quadratic form that governs the leading-order growth of forbidden support. Under explicit site-visibility, adjacent-edge visibility, and first-order nondegeneracy assumptions, I prove that the kernel of this quadratic form coincides exactly with the local gauge Lie algebra generated by on-site unitary rotations, while subsystem permutations remain discrete gauge directions. As a consequence, the graph-local decomposition class is a strict isolated local minimizer of the selector on decomposition space modulo local unitaries and subsystem permutations. This result shows that preferred decomposition near an exactly graph-local frame is not merely a finite-search artifact: every infinitesimal nongauge refactorization activates forbidden support at leading order. While this does not by itself reconstruct geometry or establish global uniqueness, it impacts the Hilbert-space-first emergence program by identifying preferred decomposition as an upstream structural constraint that must be derived, justified, or explicitly assumed.
A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must explain more than why a single candidate performs well in a discrete search; it must also determine whether that preference governs nearby continuous refactorizations. Earlier work in this area established three key ingredients: that decomposition space is empirically non-flat under simple support-based selectors in explicit few-qubit studies; that the physically meaningful object in exact finite searches is a quotient class rather than a uniquely labeled factorization, defined modulo local gauge transformations and subsystem permutations; and that, for exactly graph-local Hamiltonians, the associated quadratic form has a kernel precisely given by the local gauge Lie algebra under explicit visibility and genericity conditions. Building on these results, the present paper proves a local theorem showing that, in a neighborhood of the graph-local class, the static selector is quantitatively equivalent to the square of the quotient distance. As a consequence, the preferred decomposition class is a strictly isolated local minimum: small selector values enforce closeness in the quotient sense, and no sufficiently small nongauge refactorization can replicate the preferred structure. The result is intentionally local in scope. It does not assert global uniqueness, a global spectral gap, or a full reconstruction of geometry, but instead isolates the strongest quantitative statement currently supported by the available analytic framework.
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.