Observer-Dependence Index and Coherence Corridors in Holographic Sound Speed: A Methodological Diagnostic

This technical note introduces the Observer-Dependence Index (ODI) as a diagnostic framework for quantifying how empirical conclusions depend on methodological choices, particularly the choice of null model. Rather than treating observer dependence as a weakness to be eliminated, ODI treats it as measurable information about the scale and type of structure present in data. The framework compares classifications of a statistic (e.g., structural vs. baseline) across a hierarchy of null models, each preserving different forms of structure (such as marginal distributions, short-range correlations, or block-level coherence). The ODI quantifies the degree of disagreement among these classifications, while the pattern of classification flips identifies which structural scales drive the observed signal. Applied across multiple domains—including time-series dynamics, number-theoretic data (Goldbach partition counts), and holographic sound speed in gauge–gravity models—the ODI reveals a consistent pattern: structure dominated by multiplicative constraints tends to exhibit smoother behavior, stronger coherence, and lower observer dependence, while structure dominated by additive degrees of freedom tends to be looser, noisier, and more sensitive to null-model choice. In the holographic sound-speed example, ODI identifies a coherence corridor in chemical potential, beyond which the conformal value v_s^2 = 1/2 becomes observer-independent. In the Goldbach analysis, ODI shows that the magnitude of partition counts is explained by local prime gap structure, while their smoothness remains unexplained by tested nulls. Across cases, ODI converts ambiguity about “which null is correct” into a quantified scale decomposition. The ODI is presented as a diagnostic tool, not a theory of underlying mechanisms. It does not establish causality or universality but provides a structured method for reporting and interpreting methodological sensitivity in empirical inference.