One Operator, Many Worlds: Elastic Net Reconstruction of Hidden Structure Across Scales From Quasicrystals to Financial Markets and Legislative Voting

We argue that the phase problem—the loss of information when only magnitudes of a transform are measured—is not a peculiarity of crystallography but a universal structural feature of inverse problems across scales. We demonstrate that a single reconstruction operator, based on Elastic Net with non-negativity constraints, augmented by a phase-retrieval loop (FISTA + Charge Flipping in 1D, HIO in 2D, and temporal ordering in macro-systems), recovers hidden structure in four domains that share no physical substrate: (i) the 1D Fibonacci quasicrystal, (ii) the 2D Penrose tiling, (iii) the directed influence network of S&P 500 equities, and (iv) the factional structure of the 118th US House of Representatives. In each case, the raw observable is symmetric and directionless (diffraction intensities, correlation matrices, roll-call agreement), and in each case the operator recovers an asymmetric, sparse, non-negative structure. We also map the limits of the operator: a phase-transition analysis shows precisely where noise and data scarcity destroy reconstructability, and we locate the "Benard cell" of optimal regularization that rescues the hardest cases. The unifying claim is that what looks like domain-specific knowledge is often the same inverse problem in disguise. We conclude by proposing a prospective blind test using the 2026 US midterm elections to validate the operator's predictive power beyond retrospective calibration.