Classical observable coalescence as a discriminating diagnostic for spectral degeneracies in non-Hermitian systems

Exceptional points (EPs) are conventionally identified through eigenvector coalescence, inaccessible in classical platforms lacking a Hamiltonian spectrum. We introduce classical observable coalescence as a diagnostic identifying EPs directly from physical configurations, and demonstrate it in a known non-Hermitian Ising chain with coupling $J$ and staggered imaginary field $\gamma$, chosen for verifiability against established quantum results. A mean-field reduction of the Heisenberg dynamics yields equilibrium branches organising into two symmetry-related families whose fusion at the boundary of the EP supersurface $\mathrm{\mid }\gamma \mathrm{\mid }=\mathrm{\mid }J\mathrm{\mid }$ exhibits the characteristic $\sqrt{J^2-{\gamma }^2}$ branch-point structure and two-sheeted Riemann topology. Crucially, the degeneracy at $\gamma =0,\eta =0$ is shown to be a diabolic point, producing no observable coalescence, establishing the diagnostic as discriminating rather than merely suggestive. An exact Jordan-Wigner and Bogoliubov-de Gennes treatment confirms eigenvector coalescence at the same threshold, and a coupled gain-loss circuit realisation is proposed, establishing classical observables as experimentally accessible signatures of non-Hermitian criticality.