Spectral Separation and the Non-Symmetric Strong Spectral Property for Looped Double Paths

We classify exactly when a looped double-path zero--nonzero pattern allows the non-symmetric strong spectral property (nSSP). The structural core proves that a real symmetric irreducible tridiagonal matrix with one nonzero diagonal entry has the nSSP exactly when the two Jacobi arms obtained by deleting that vertex have disjoint spectra. Together with scaling, the Superpattern Lemma, and the known negative cases, this resolves the allowance question for every loop set. The result classifies allowance, not the finer require-versus-allow distinction. Status: Public Beta v0.2; internally verified candidate proof; external mathematical review pending.