Variance Saturation as a Diagnostic of Phase Transitions: Variance Saturation, Derivative Cascades, and Encoding-Invariant Diagnostics Near Phase Transitions

This work extracts and formalizes the minimal diagnostic core of the Relational Thermodynamic Layered Index (RTLI) framework. Stripped of domain-specific constants and interpretive overlays, the framework is shown to rest on a small set of general assumptions: boundedness of a variance-like response, monotonic approach to saturation, and smoothness near transition regions. From these assumptions alone follow several robust, coordinate-independent features: the existence of a sensitivity corridor near saturation, a characteristic derivative cascade in which higher-order derivatives peak before lower-order ones, and a natural separation between intrinsic (structural) and extrinsic (algorithm-dependent) variance effects. We demonstrate that these features are not tied to particular numerical constants, problem encodings, or physical interpretations. Instead, they represent generic properties of systems undergoing smooth transitions with bounded response. The framework is positioned as a diagnostic and falsification tool rather than a generative theory: it provides criteria for identifying genuine transition structure, calibrating resolution, testing encoding invariance, and distinguishing geometric variability from algorithmic masking. The resulting “RTLI minimal core” is best understood as a synthesis of finite-size scaling intuition, sensitivity analysis, and information-geometric perspectives, applicable across computational complexity, statistical physics, and related domains. No claims are made regarding complexity-class separation or derivation of universal constants. The contribution lies in methodology: clarifying what can be robustly inferred from variance behavior near transitions, and how such inferences can be systematically tested and potentially falsified.