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Paper 18- Universal Binary Affinity Framework: Binary Balance Law, Universal Mathematical Law, Reverse Universal Affinity Law, and Cross-System Validation

Year: 2026
Type: preprint
Abstract: Overview This work presents a simple and unified mathematical framework for binary affinity. For a binary variable with states (+1) and (-1), the observable response (X) and the affinity coordinate (\theta) are linked exactly by [X=\tanh(\theta),\qquad\theta=\operatorname{atanh}(X).] Within this defined binary exponential-affinity family, these forward and reverse relations are called the Universal Mathematical Law and the Reverse Universal Affinity Law. Main Results The paper also derives an exact balanced-count probability law for binary trials and a related normalized quantity whose large-sample limit approaches (\pi). The (\pi) limit comes from standard central-binomial asymptotics and is not claimed as a new physical derivation of (\pi). Physical Checks The same tanh structure is compared with: thermal spin polarization, the one-dimensional Ising model, a probabilistic Ising-machine system, and a 2026 photonic quantum-switch dataset. The quantum-switch reanalysis reconstructs a Bell-like causal witness of 1.808954375, while six spacelike-marginal affinity diagnostics remain statistically consistent with zero. This shows an important point: a strong correlation or causal witness should not automatically be interpreted as a generic affinity or new causal effect. In Simple Terms Paper 18 provides a clear forward-and-reverse mathematical language for binary relationships. It shows how to convert between an underlying affinity (\theta) and an observed normalized response (X), compares the same structure across several physical systems, and clearly separates exact mathematics from broader physical claims.
DOI: 10.5281/zenodo.22308531
Language: English
Open Access status: green
OpenAlex ID: W7208785678
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