The Entropic Scree: An Information-Theoretic Diagnostic Framework for Intrinsic Rank and Informational Gravity in Tabular Systems

Determining the intrinsic dimensionality of complex datasets is a foundational prerequisite for multivariate statistics, manifold learning, and predictive modeling. While standard Principal Component Analysis (PCA) and its variance-based scree plot remain the universal diagnostic standards, PCA suffers from structural limitations: it deflates correlations in mixed-type data, ignores non-linear interactions, caps extractable rank at $N-1$ in sample-starved ($m > N$) regimes, and geometrically fragments non-linear dependencies into Spurious Orthogonal Dimensions, driving Dimensional Inflation. This paper introduces the Entropic Scree, an information-theoretic diagnostic framework designed to evaluate intrinsic dimensionality through the eigendecomposition of Normalized Mutual Information, independent of underlying generative assumptions. By utilizing Information-Theoretic Jaccard Similarity (Variation of Information), the methodology produces a geometrically coherent metric space that resolves the deflation of shape mismatch and unifies continuous, ordinal, and discrete variables. Furthermore, by evaluating pure probability mass rather than linear variance, monotonic rank, or distance metrics, the methodology aims to map true overlapping redundancy to compress fragmented dimensions back towards their Intrinsic Generative Rank. Crucially, it maintains a rigid structural boundary under root entanglement, where non-linear baselines like Kernel PCA systematically degrade. Evaluating the Entropic Scree's decay on a logarithmic scale separates the primary generative drivers from both their Extended Signal Tail and the continuous bulk of unstructured Idiosyncratic Informational Variance. Additionally, the framework decouples the concept of representational rank into discrete geometric counts and continuous probabilistic volumes, introducing Average Informational Gravity (AIG) and Factor-Specific Informational Gravity (FSIG). By systematically rebundling residual expansion variance, these metrics translate abstract matrix properties into actionable, variable-equivalent engineering footprints. Finally, the framework serves as an exploratory map that separates unrelated clusters of variables, allowing for the identification of decoupled sub-networks. Ultimately, this framework provides a non-parametric, model-agnostic diagnostic whose comparative advantage compounds at scale and with system complexity.