Pseudo-Additive Tsallis Entropy and Non-Factorizing Joint Statistics in Product Sheffer Stroke Basic Algebras

This paper addresses the problem of formulating generalized, non-extensive information-theoretic measures on finite non-distributive algebraic structures equipped with Riečan states, with particular emphasis on product Sheffer stroke basic algebras. Our approach formalizes finite summations, admissible partitions, refinement relations, and Sheffer stroke joint refinement candidates by using the primitive Sheffer stroke operation, with partition and marginalization properties imposed under the stated product and admissibility assumptions. By leveraging the state-theoretic properties of Riečan states, we construct baseline Shannon and logical entropies alongside algorithmic procedures for their computational evaluation. As the main result, we introduce and analytically characterize a parametric Tsallis entropy functional over these basic algebras. We prove its fundamental properties, including bounding inequalities, state concavity, monotonicity under refinement, subadditivity (for α>1), conditional chain-type identities under the relevant joint refinement marginalization assumptions, and exact analytical convergence to the classical Shannon limit as the entropic index α→1. Furthermore, under a state-dependent statistical independence condition, we show that the joint Tsallis entropy satisfies a pseudo-additive relation. By defining the Tsallis mutual information and the associated pseudo-additive residual, we isolate the deviation of a joint Sheffer stroke refinement from the factorized model determined by its marginal Riečan-state distributions. This residual is intended as a state-dependent algebraic indicator of deviations from the factorized Tsallis pseudo-additive model; it is not claimed to be an operational contextuality witness, a contextuality inequality, an entanglement measure, or a physical implementation criterion.