Spectral Discrepancy in k-NN Line Graphs as a Grassmannian Invariant: The Critical Principal Angle φ = π/4 and the Silver Ratio √2 − 1

We establish that spectral nodal discrepancy in cosine-weighted Line Graphs of k-nearest-neighbor graphs is a geometric invariant of the Grassmannian manifold.Each edge of the k-NN graph spans a 2-dimensional subspace — a point in the Grassmannian of 2-planes. When two edges share a vertex, the critical principal angle between the corresponding planes emerges from a fixed-point equation derived from local flow conditions on an infinite tree. This angle determines the minimum achievable spectral discrepancy for any class subgraph. The minimum discrepancy equals the reciprocal of the silver ratio, a classical constant of octagonal geometry. The maximum discrepancy equals the Shannon entropy of a fair binary channel. The width of the universal interval between these two bounds is conjectured to be a spectral freedom constant of the embedding. Empirical validation on histopathology embeddings across 32 independent spectral slices confirms the theoretical interval and demonstrates that the minimum is reached at a critical number of neighbors that depends on the embedding dimension.A convergence experiment confirms the theoretical prediction: as the number of neighbors grows, the mean cosine weight between non-shared edge endpoints decreases monotonically toward zero. At a critical embedding dimension of approximately 62,500, this weight becomes negligible by mathematical necessity — the Line Graph becomes an infinite regular tree exactly, and the minimum discrepancy is attained for any number of neighbors without algorithmic tuning.