Paper 17- Four Variable Binary Affinity Geometry and Forward Reverse laws: Exact 16 sector Walsh Coordinates, Conditional Transfer Laws, Event Updates, & Prediction Boundaries
Year: 2026
Type: preprint
Abstract: Overview This preprint develops a rigorous and fully reproducible mathematical framework for analyzing four binary variables through an exact Walsh/log-linear representation. The 16 possible joint sign states are mapped exactly onto 15 nonconstant interaction coordinates, providing a compact and interpretable description of fields, pair interactions, triple interactions, and the four-way interaction. The framework also derives exact forward and reverse laws, conditional pair-affinity relations, and one-event update formulas that can be evaluated without repeatedly refitting the complete joint table. Key Contributions Exact 16-State Representation: The full four-variable distribution is represented exactly by 15 Walsh/log-linear coordinates: 4 fields, 6 pair terms, 4 triple terms, and 1 four-way term. Forward-Reverse Laws: Exact mathematical relations connect conditional pair affinities with higher-order interaction structure. Exact Reconstruction: The original joint distribution can be reconstructed from the coordinate representation. Geometry: Euclidean and Fisher-Rao viewpoints are clearly separated, allowing different aspects of interaction structure to be studied without mixing their interpretations. Efficient Updates: Exact formulas describe how the coordinates change after adding a single new observation. Applied Case Study The framework is applied to a 495-session dataset containing NIFTY, Reliance, HDFC Bank, and Infosys. The exact reconstruction reproduces the endpoint distribution to machine precision. The analysis shows that the observed coordinate energy is strongly concentrated in the pair-interaction sector, while fields, triple interactions, and the four-way interaction provide additional structured information. This example demonstrates how the framework can turn a complex four-variable binary dataset into a transparent and interpretable interaction geometry. Scientific Boundary and Open Direction Walk-forward testing also provides an important result: several natural geometric and angle-based prediction rules reduce algebraically to already existing frequency, rare-state, or repeat-state rules. This establishes a useful boundary between what the current mathematics can determine exactly and what remains genuinely unresolved. The representation, conditional laws, interaction geometry, and reverse reconstruction are substantially developed, while a genuinely new future-state selector remains an open research problem. General scalable exact inverse-Ising inference is also kept clearly outside the claims of the present work. Reproducibility The Zenodo record includes the full raw CSV dataset and the complete calculation workbook, allowing the reported calculations and endpoint reconstruction to be independently checked and reproduced. Overall, the work provides an exact, transparent, and reusable framework for studying four-variable binary interactions while clearly separating established mathematical results from the remaining open prediction problem.
Author: chandaliya, Roshankumar
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7212356530
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