A finite observation schedule cannot bound coupling error in an unbounded linear oscillator model

We ask if time-lagged activity resolves coupling that zero-lag functional connectivity cannot identify. For a fully observed stable two-state linear oscillator with known white drive, we derive the exact relative entropy between complete Gaussian paths at any fixed finite observation schedule. Simultaneous rational approximation gives arbitrarily large coupling differences whose path laws are arbitrarily close. Thus worst-case mean squared coupling error is infinite on this unbounded class. Two gaps with irrational ratio give exact uniqueness within the model but do not give uniform statistical stability. Local Fisher information cannot detect the remote near-aliases. With 1,000 gaps alternating between 1 and sqrt(2) leak-time units, a coupling difference of 2563.54 gives path divergence 0.000931 and a proved testing-error lower bound of 48.92 percent. Optimal decisions on 10,000 simulated paths gave 49.80 percent error. A fixed change of coordinates extends the result to nonzero stationary FC; a four-state example also preserves diagonal decay. System aliasing is established work. Our contribution is the finite-path error bound and its scope test for neural inference. No animal data were fitted. The result needs an unbounded signed coupling class and fixed instantaneous observations. Biological frequency bounds or other measurements can remove this obstruction.