Hilbert-Space-First Local Rigidity of Preferred Decomposition for Graph-Local Hamiltonians

Patrick Panella

Published Apr 15, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.19591931

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