The static selector for graph-local Hamiltonians vanishes exactly at decomposition classes in which the Hamiltonian is a sum of one-site terms and graph-supported two-site terms. We give a sufficient operator-algebraic criterion for uniqueness of such an admissible-support realization, and hence for exact identification of the underlying Hilbert-space decomposition on the physical quotient. For an exact graph-local realization and an omitted site, we study the unital *-algebra generated by all one-site and graph-supported two-site terms that avoid that site. A seed-and-propagation theorem shows that if full one-site matrix control is available at a seed vertex in each connected component of the omitted-site complement and operator-Schmidt coefficients propagate that control along complement edges, then this complement algebra is the full matrix algebra on the omitted-site complement, so the omitted one-site algebra is recovered exactly as a commutant. A pairwise rigidity theorem then shows that two complement-generated exact realizations with the same recovered site-family differ only by local basis changes and subsystem permutation. Consequently, any exactly graph-local Hamiltonian with frame-stable site reconstruction has a unique admissible-support realization and exact zero-set identifiability. The hypotheses are sufficient and are stated entirely in terms of the exact graph-local realization.
The static selector for graph-local Hamiltonians vanishes exactly at decomposition classes in which the Hamiltonian is a sum of one-site terms and graph-supported two-site terms. We give a sufficient operator-algebraic criterion for uniqueness of such an admissible-support realization, and hence for exact identification of the underlying Hilbert-space decomposition on the physical quotient. For an exact graph-local realization and an omitted site, we study the unital *-algebra generated by all one-site and graph-supported two-site terms that avoid that site. A seed-and-propagation theorem shows that if full one-site matrix control is available at a seed vertex in each connected component of the omitted-site complement and operator-Schmidt coefficients propagate that control along complement edges, then this complement algebra is the full matrix algebra on the omitted-site complement, so the omitted one-site algebra is recovered exactly as a commutant. A pairwise rigidity theorem then shows that two complement-generated exact realizations with the same recovered site-family differ only by local basis changes and subsystem permutation. Consequently, any exactly graph-local Hamiltonian with frame-stable site reconstruction has a unique admissible-support realization and exact zero-set identifiability. The hypotheses are sufficient and are stated entirely in terms of the exact graph-local realization.
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.
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.
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.
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 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.