A finite Hilbert space admits many inequivalent ways of being partitioned into subsystems, so any Hilbert-space–first account of locality must do more than identify a preferred structure in the perfectly local case. It must also address how that structure behaves when the dynamics are only approximately local. This paper establishes such a stability result at the perturbative level. Starting from a reference decomposition in which the dynamics are exactly local, we consider small deviations that introduce weak nonlocal interactions. By constructing an explicit locality diagnostic and analyzing its behavior in a neighborhood of the reference structure, we derive quantitative bounds on its variation and curvature within a suitably defined space of decompositions, modulo local redundancies. These bounds demonstrate that the strong local convexity present in the exactly local case persists under sufficiently small perturbations. As a result, there exists a unique nearby decomposition that remains optimally local, and its deviation from the reference structure is controlled by the size of the nonlocal contribution. Moreover, the locality measure continues to behave comparably to a squared distance in the reduced space, ensuring robust identifiability. The result is formulated at the level of physically relevant equivalence classes, accounting for local basis changes and subsystem relabelings. The theorem is explicitly local in scope: it does not address global uniqueness, spectral gaps, or emergent geometry. Its central conclusion is that local, quantitative notions of subsystem structure remain well-defined and stable under small violations of exact locality.
A finite Hilbert space admits many inequivalent ways of being partitioned into subsystems, so any Hilbert-space–first account of locality must do more than identify a preferred structure in the perfectly local case. It must also address how that structure behaves when the dynamics are only approximately local. This paper establishes such a stability result at the perturbative level. Starting from a reference decomposition in which the dynamics are exactly local, we consider small deviations that introduce weak nonlocal interactions. By constructing an explicit locality diagnostic and analyzing its behavior in a neighborhood of the reference structure, we derive quantitative bounds on its variation and curvature within a suitably defined space of decompositions, modulo local redundancies. These bounds demonstrate that the strong local convexity present in the exactly local case persists under sufficiently small perturbations. As a result, there exists a unique nearby decomposition that remains optimally local, and its deviation from the reference structure is controlled by the size of the nonlocal contribution. Moreover, the locality measure continues to behave comparably to a squared distance in the reduced space, ensuring robust identifiability. The result is formulated at the level of physically relevant equivalence classes, accounting for local basis changes and subsystem relabelings. The theorem is explicitly local in scope: it does not address global uniqueness, spectral gaps, or emergent geometry. Its central conclusion is that local, quantitative notions of subsystem structure remain well-defined and stable under small violations of exact locality.
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.
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 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 study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form H=∑X⊆ΛΦ(X) acting on a tensor product Hilbert space, we associate to each subset S⊆Λ a subsystem Hamiltonian HS and its spectrum S(S)=σ(HS). This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: HS can be approximated by operators supported on finite neighborhoods with an error bounded by ∥HS−HS,r∥≤∣S∣e−μr∥Φ∥μ. As a consequence, subsystem spectra are stable under truncation in the sense that dH(S(S),σ(HS,r))≤∣S∣e−μr∥Φ∥μ. We then prove that for disjoint subsets S1,S2⊆Λ, the subsystem spectrum is approximately additive: dH(S(S1∪S2),S(S1)+S(S2))≤(∣S1∣+∣S2∣)e−μD∥Φ∥μ, where D=d(S1,S2). In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.
This paper studies local generic exact zero-set identifiability for preferred Hilbert-space decompositions associated with graph-local Hamiltonians on the reference exact coefficient slice of a deletion-robust connected graph. Starting from a reference-regular base Hamiltonian with finite exact zero set, and assuming the relevant branch-codimension condition at each competing exact zero through the base point, the paper proves that in a sufficiently small neighborhood each nearby competitor survives only on a semialgebraic subset of positive codimension in coefficient space. The argument formulates competitor survival as a finite real algebraic incidence problem in reference coefficients, competitor coefficients, and a unitary matrix representing the competing Hilbert-space decomposition. Projection then produces semialgebraic survival sets, and the branch-codimension hypothesis ensures that their local closures remain lower dimensional. Consequently, exact zero-set identifiability is characterized by exclusion from finitely many semialgebraic survival sets and is certified on an open dense semialgebraic region obtained by removing their closures. On this certified region, the Hamiltonian has unique admissible-support realization modulo local basis changes and subsystem permutation.