Prism: Structural Symmetry Scanning via Duality-Constrained Laplacian Projection

We introduce Prism, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian L and a duality operator P (a symmetric involution), Prism computes the duality defect δ(L,P) = |LP - PL|F / |L|F -- a scalar measuring how far the network deviates from structural self-consistency. When P encodes the network's true symmetry, δ starts near zero and rises monotonically as structure degrades; an arbitrary P gives noise. We prove that the optimal L' satisfying [L', P] = 0 is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns P from the graph's own Fiedler vector. Experiments on synthetic networks show the true-P defect is 3.38× more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves 94.5% community detection accuracy at 5% noise versus 76.6% for the raw Laplacian baseline. Applied to live S&P~500 data (2026-05-17), Prism detects rising structural stress (defect 0.43 → 0.73 over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels before the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.