Why Squared Norms Dominate: Sharpe Ratios, Signal-to-Noise, and Structural Stability in Linear Regimes (IM-K004, v0.3)

This document is part of the Informational Mechanics (IM) framework and is published in the IM-K (Essays & Position Papers) series within the IM v0.3 corpus. The paper examines a structural analogy between quadratic persistence in Hilbert-space formulations of physical theory and stability measures used in financial risk-adjusted performance metrics, such as the Sharpe ratio. The analysis is interpretive and comparative rather than foundational or predictive. Building on the mathematical admissibility results of IM-B002 and IM-B003, the discussion explores how squared-norm structures arise in both domains as stability-preserving constructs under invariance and disturbance conditions. The claim is structural and limited: no derivation of financial theory from physics is asserted, and no ontological identity between domains is proposed. The document introduces no new axioms, physical models, financial models, or empirical claims. Its purpose is to clarify structural similarities and to illustrate how admissibility-based reasoning can reveal cross-domain stability patterns without collapsing disciplinary boundaries.