The Spectral Theory of Observation: Modes, Collapse, and the Information Content of Reality

We identify a single mathematical structure underlying seven apparently distinct physical frameworks: stochastic processes (Fokker--Planck), quantum mechanics (Schrödinger), statistical mechanics (transfer matrix), dynamical systems (Koopman), and information theory (rate-distortion). In each case, the system's state is described by a density expanded in the eigenmodes of a linear generator: \(p = \sum c_k e^{\lambda_k t} \varphi_k\). Evolution is mode-independent decay or oscillation. Observation is projection: \(c_k \to c_k \cdot \varphi_k(x_0)\). "Wave--particle duality" becomes "many modes active (spread) vs few modes active (localized)." "Collapse" becomes "projection below the observer's resolution threshold." The Universal Spectral Representation Theorem (USRT) provides the quantitative backbone: \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) modes suffice for \(\varepsilon\)-accuracy, independent of dimension, for any system with smooth dynamics. We demonstrate this unity across eight realizations: (1) portfolio risk measurement, (2) the three-body problem in celestial mechanics, (3) space debris collision probability, (4) machine learning model compression, (5) DeFi liquidation risk, (6) quantum decoherence prediction, (7) quantum measurement, and (8) mathematical knowledge growth — where the Fiedler vector of a theorem corpus's Laplacian determines which theorem to prove next, making the framework self-referential: the spectral method observes its own corpus and chooses its next proof. The paper suggests that the "measurement problem" — the deepest open problem in physics — is not a problem of quantum mechanics specifically, but a universal property of spectral representations: observation is always projection, and the "collapse" is always the transition from \(N\) active modes to \(\sim 1\).