Quantum Metric Correspondence: Physical Substrate for Homeostatic Basin Geometry

ABSTRACTRecent experimental confirmation of the quantum metric tensor in common metallic materials (Sala etal., 2026) provides an opportunity to examine correspondence with theoretical predictions generatedindependently by the Bonding Relationships framework. Paper 4 of this series (deposited Zenodo 11March 2026, prior to Sala et al.) proposed that homeostatic basin geometry must have a physicalsubstrate at quantum scale — that the H coupling parameter of the Homeostatic Quality Coefficient(HQC) framework describes coupling to geometric curvature in state space, and that a recoherenceterm, if derivable, would operate through this geometric structure. This correspondence note mapsthe confirmed quantum metric against the framework's prior geometric claims. The mapping isinterpretive rather than demonstrated: the quantum metric is proposed as a candidate physicalsubstrate for basin geometry, not as a confirmed identification. Three points of structuralcorrespondence are identified, two explicit limitations are stated — including the unresolved scalebridging problem spanning fifteen or more orders of magnitude — and a mathematically well-definedcandidate form for the recoherence term is proposed using the Bures distance and a memory kernel.Three falsification conditions are specified with operational measurement procedures. The purpose ofthis note is to motivate a research programme, not to claim a result.Keywords: quantum metric tensor, homeostatic basin geometry, recoherence term, Bures distance, bond signmanagement, HQC framework, non-Markovian dynamics, quantum geometry, history manifold