Conditional entropy in algebraic statistical physics

İbrahim Şentürk, Metin Bilge, Tahsin Öner

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.

Lev B. Levitin

Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk

Rostislav Stepanik

Description QSTH M.1B is the operational continuation of QSTH M.1A, Genealogy of Alpha-I-Dim and Equations E1–E10. M.1A established the algebraic genealogy of the reference Alpha-I-Dim construction and identified effective entropy S_eff and the structural parameter κ(Γ) as imported quantities whose physical meaning could not be supplied by algebra alone. M.1B addresses that open requirement by constructing a first explicit finite operational framework for structural configurations, equivalence, representation, statistical measure, dynamics, entropy and independent classification. The study defines typed relational configurations with a decidable admissibility predicate and equivalence under relabelling; constructs a faithful incidence representation; distinguishes representational equivalence, physical equivalence and indistinguishability under a limited readout; and states the conditions under which deterministic and probabilistic dynamics descend consistently to configuration classes. A finite operational statistical entropy is then constructed from a prespecified state space, measure and readout. Independently, a dynamical classifier based on closed recurrent sectors is introduced, together with an audit of exact charge protection versus finite escape time. The central integration result is deliberately restrictive. The independently defined sector count is not a universal entropic coefficient for Equation E10. The same entropy may coexist with different sector structures, while genuine internal degrees of freedom may increase entropy without creating an additional recurrent sector. The independent test is therefore allowed to reject the proposed identification rather than being normalized to reproduce it. Accordingly, the formal mathematical and statistical stage B1-B7 is complete within its declared domain, while physical horizon identification, M-independence and the continuation to E11+ remain open. This publication does not claim a new universal constant, a microscopic theory of horizon entropy, or empirical confirmation of QSTH. Its contribution is an auditable operational framework, explicit counterexamples, reproducible finite tests, and a precise boundary between mathematical construction and physical identification. A separate companion study, QSTH M.1B-SR1, examines whether a richer spectral response can provide a conditional bridge between an independently defined observable and the thermal entropy of specified interface carriers.

Maxim Raginsky

We derive, for a bistochastic strictly contractive quantum channel on a matrix algebra, a relation between the contraction rate and the rate of entropy production. We also sketch some applications of our result to the statistical physics of irreversible processes and to quantum information processing.

В. М. Максимов