Spectral Nodal Discrepancy in Weighted Line Graphs: A Universal Interval [√2 − 1, ln 2] with Log-Uniform Distribution

We identify a universal interval [sqrt(2)-1, ln 2] for the spectral nodal discrepancy of the dominant eigenvector in cosine-weighted Line Graphs constructed from k-nearest-neighbor graphs over high-dimensional embeddings. The lower bound sqrt(2)-1 ≈ 0.4142 arises from the relative excess path length (triangle inequality slack) in the isosceles right triangles that dominate local k-NN topology when the embedding dimension D → ∞. The upper bound ln 2 ≈ 0.6931 is the Shannon entropy of a fair binary channel, bounding the maximum discrepancy achievable under the spectral energy constraint imposed by orthogonality to the constant function. Empirical validation on 999 histopathology patches (NCT-CRC-HE-100K, Phikon embeddings, D=768, k=8) across 32 independent spectral slices yields bounds and a mean of 0.5365, consistent with the theoretical geometric mean sqrt((sqrt(2)-1)*ln 2) ≈ 0.53585 to within 0.12%. We demonstrate that the log-transformed discrepancy follows a standard uniform distribution U(0,1), representing the maximum-entropy state on the logarithmic scale. From these bounds, we derive closed-form Hamming-distance guarantees for binary spectral hashing (W1H-SPQ), establishing that no class subgraph can be spectrally invisible to the hash code.