Same Orbit Implies Same Response: Strict Boundaries of Response Equivalence in Quantum Graph Heisenberg Models

Abstract

In quantum many-body systems, the response of a defect is often determined by its position: two edges give the same response when they belong to the same edge automorphism orbit. We ask whether different orbits can also give the same response. We study the isotropic Heisenberg model on all connected graphs with N=6 nodes (112 graphs) and N=7 nodes (853 graphs), using the standard deviation of all edge correlation functions as a function of the coupling of a single defect edge. Same orbit implies same response with zero counterexamples, for both N=6 and N=7, and for all four interaction rules. Different orbits can also give the same response: 47 pairs for N=6 and 382 pairs for N=7, not generated by involutive automorphisms. The mechanism is layered. The total spin S of the two defect-edge states must be equal (zero false negatives). The absolute value of the Heisenberg coupling expectation at s=1/2 must be equal. The sign rule is set by S. Full p_S(s)-curve equality is sufficient and selects 29 pairs (100% precision, 61.7% recall). The singlet probability is p_S(s)=(1-dE_0/ds)/4 by the Hellmann-Feynman theorem, and equal-value windows end exactly at s=0 and s=1. Data and verification scripts are public.