Four Variable Binary Affinity Geometry and Forward Reverse laws: Exact 16 sector Walsh Coordinates, Conditional Transfer Laws, Event Updates, & Prediction Boundaries

Abstract

Overview:This preprint presents a rigorous four-variable binary affinity framework, mapping 16 joint sign states exactly onto 15 nonconstant Walsh/log-linear coordinates. The work establishes an exact dual representation, derives conditional pair-affinity forward and inverse laws, and provides exact one-event update formulas without requiring full table refitting. Key Contributions:• Exact Representation: 16 joint states are exactly dual to 15 Walsh coordinates (4 fields, 6 pairs, 4 triples, 1 four-way).• Forward-Reverse Laws: Exact mathematical bridge between conditional pair affinities and higher-order interactions.• Geometry & Updates: Rigorous separation of Euclidean and Fisher-Rao geometries, with exact candidate updates. Applied Case Study:The framework is tested on a 495-session dataset (NIFTY, Reliance, HDFC Bank, Infosys). The exact reconstruction reproduces the endpoint distribution to machine precision, revealing that pair interactions dominate the squared coordinate norm. Scientific Boundaries & Open Problems:Strict walk-forward tests reveal that natural geometric and angle-based selectors collapse algebraically into existing frequency, rare-state, or repeat rules. Consequently, while the mathematical representation, conditional laws, and reverse reconstruction are substantially closed, the future-state selector remains an open problem. The paper also clearly defines the boundary of the general exact scalable inverse-Ising inference, which remains unsolved. Reproducibility:The upload includes the full raw dataset (CSV) and the complete calculation workbook (Excel) to ensure 100% reproducibility of the results.