RTLI: Testing Exponent-Ladder Constants in Holographic Systems: A Pre-Registered Hypothesis Framework

This technical note introduces a pre-registered diagnostic framework for testing the appearance of specific numerical thresholds—particularly ratios involving e, \sqrt{e}, and related partitions—within holographic and AdS/CFT-motivated observables. The work does not propose a new dynamical model, nor does it modify General Relativity or quantum field theory. Instead, it provides a geometry-based, variance-oriented tool designed to assess whether apparent constant clustering reflects genuine structural organization or arises from statistical or methodological artifacts. The framework formalizes a variance-geometry diagnostic, centered on eigenvalue-based measures and scale-separation metrics, that can be applied uniformly across different observables and datasets. Explicit null models, tolerance definitions, and failure modes are specified in advance, allowing both positive and negative outcomes to be scientifically meaningful. The approach emphasizes observable stratification—the idea that the same physical system may yield distinct effective ratios depending on whether it is probed through scalar, projected, or tensorial measurements. By framing constant-like behavior as a structural diagnostic rather than a fundamental law, this note establishes a reusable methodology for auditing claims of universality in holographic and cosmological contexts. The framework is intended as an open analytical tool that can be adopted, stress-tested, or falsified by independent researchers using existing data.