The Lorentzian Kernel as an Emergent Epistemic Envelope: Averaging, Resolution, and the Geometry of Distinguishability

The Lorentzian kernel appears ubiquitously across physics—from quantum resonance and statistical decay to signal filtering and neural propagation. Yet its role is usually heuristic: a convenient fitting function or minimally justified filter. In this paper, we propose a deeper origin. Using the TEQ (Total Entropic Quantity) framework, we show that the Lorentzian kernel emerges generically from entropy-weighted path suppression when resolution is finite and distinguishability becomes granular. Specifically, we derive the Lorentzian as the result of averaging Gaussian contributions over fluctuating entropy-curvature scales. This yields not a dynamical residue, but a structurally emergent boundary form. We interpret the Lorentzian kernel as the epistemological exponent of irreducible uncertainty—the universal shape taken when entropy geometry no longer supports finer resolution. Crucially, this structural boundary is not merely a limit of knowledge, but a limit of individuation: at the Lorentzian threshold, the distinction between what is and what can be known dissolves—structure itself ceases to be resolved, and thus ceases to exist within the theory. This reconceptualization suggests testable transitions between Gaussian and Lorentzian regimes and unifies a wide range of phenomena under the geometry of distinguishability.