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Foundations, Volume 5, Issue 4 (December 2025) – 6 articles

Cover Story (view full-size image): Perceptual Control Theory (PCT) and the Free Energy Principle (FEP) are two foundational, principle-based frameworks originally developed to explain brain function and the behavior of living systems more broadly. In this article, we provide the first systematic full stack comparison of the philosophy, mathematics, and modeling methodology behind PCT and FEP concretely in the context of bacterial chemotaxis. With these foundations in place, we use tools from category theory to argue that PCT can be formally understood as a subset of the FEP framework; however, we note that the mathematical machinery unique to FEP is not required to successfully model bacterial chemotaxis. Finally, we conclude with a proposal for a mathematical synthesis where each framework plays an orthogonal yet complementary role. View this paper

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21 pages, 351 KB

Open AccessArticle

Existence and Uniqueness Results for (k, ψ)-Caputo Fractional Boundary Value Problems Involving Multi-Point Closed Boundary Conditions

by Furkan Erkan, Nuket Aykut Hamal, Sotiris K. Ntouyas and Bashir Ahmad

Foundations 2025, 5(4), 37; https://doi.org/10.3390/foundations5040037 - 8 Dec 2025

Viewed by 174

Abstract

In this paper, we investigate a new class of nonlinear fractional boundary value problems (BVPs) involving

(k, ψ)

-Caputo fractional derivative operators subject to multipoint closed boundary conditions. Such a formulation of boundary data generalizes classical closure constraints in terms [...] Read more.

In this paper, we investigate a new class of nonlinear fractional boundary value problems (BVPs) involving

(k, ψ)

-Caputo fractional derivative operators subject to multipoint closed boundary conditions. Such a formulation of boundary data generalizes classical closure constraints in terms of nonlocal dependence of the unknown function at several interior points, giving rise to a flexible mechanism for describing physical and engineering phenomena governed by nonlocal and memory effects. The proposed problem is first transformed into an equivalent fixed-point formulation, enabling the application of standard analytical tools. Results concerning the existence and uniqueness of solutions to the problem are obtained through the application of fixed-point principles, specifically those of Banach, Krasnosel’skiĭ, and the Leray–Schauder nonlinear alternative. The obtained results extend and generalize several known findings. Illustrative examples are presented to demonstrate the applicability of the theoretical findings. Moreover, the introduction incorporates a succinct review of boundary value problems associated with fractional differential equations and inclusions subject to closed boundary conditions. Full article

(This article belongs to the Section Mathematical Sciences)

21 pages, 479 KB

Open AccessArticle

Bias-Corrected Root Mean Square Deviation Estimators

by Alexander Robitzsch

Foundations 2025, 5(4), 36; https://doi.org/10.3390/foundations5040036 - 28 Nov 2025

Viewed by 171

Abstract

The root mean square deviation (RMSD) is a widely used item fit statistic in item response models. However, the sample RMSD is known to exhibit positive bias in small samples. To address this, seven alternative bias-corrected RMSD estimators are proposed and evaluated in a simulation study involving items with uniform differential item functioning (DIF). The results demonstrate that the proposed estimators effectively reduce the bias of the original RMSD statistic. Their performance is compared, and the most favorable estimators are highlighted for empirical research. Finally, the application of the various RMSD statistics is illustrated using PISA 2006 reading data. Full article

(This article belongs to the Section Mathematical Sciences)

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31 pages, 4933 KB

Open AccessArticle

PCT vs. FEP: A Comparison Between Reorganization Theory and Bayesian Inference

by Ty Roachford, Warren Mansell and Rodrigo Pena

Foundations 2025, 5(4), 35; https://doi.org/10.3390/foundations5040035 - 27 Oct 2025

Viewed by 1009

Abstract

Perceptual Control Theory (PCT) and the Free Energy Principle (FEP) are two foundational, principle-based frameworks originally developed to explain brain function. However, since their initial proposals, both frameworks have been generalized to account for the behavior of living systems more broadly. Despite their conceptual overlap and practical successes, a mathematical comparison of the two frameworks has yet to be undertaken. In this article, we briefly introduce and compare the philosophical foundations underlying PCT and FEP. We then introduce and compare their experimental and mathematical foundations concretely in the context of bacterial chemotaxis. With these foundations in place, we can use tools from category theory to argue that PCT can be formally understood as a subset of the FEP framework; however, it is worth noting that the mathematical machinery unique to FEP is not required to successfully model bacterial chemotaxis. Finally, we conclude with a proposal for a mathematical synthesis where each framework plays an orthogonal yet complementary role. Full article

(This article belongs to the Section Mathematical Sciences)

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14 pages, 376 KB

Open AccessArticle

Probabilistic Geometry Based on the Fuzzy Playfair Axiom

by Edward Bormashenko

Foundations 2025, 5(4), 34; https://doi.org/10.3390/foundations5040034 - 1 Oct 2025

Viewed by 915

Abstract

A probabilistic version of geometry is introduced. The fifth postulate of Euclid (Playfair’s axiom) is adopted in the following probabilistic form: consider a line and a point not on the line—there is exactly one line through the point with probability P, where [...] Read more.

0 \leq P \leq 1

. Playfair’s axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms.

P = 1

corresponds to the standard Euclidean geometry;

P = 0

corresponds to the elliptic- and hyperbolic-like geometries.

0 < P < 1

corresponds to the introduced probabilistic geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the probabilistic geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the probabilistic geometry. Probabilistic geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean geometry is addressed. Applications of the probabilistic geometry are discussed. Full article

(This article belongs to the Section Mathematical Sciences)

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7 pages, 224 KB

Open AccessArticle

On Relative Stability for Strongly Mixing Sequences

by Adam Jakubowski and Zbigniew Stanisław Szewczak

Foundations 2025, 5(4), 33; https://doi.org/10.3390/foundations5040033 - 25 Sep 2025

Viewed by 566

Abstract

We consider a class of strongly mixing sequences with infinite second moment. This class contains important GARCH processes that are applied in econometrics. We show the relative stability for such processes and construct a counterexample. We apply these results and obtain a new [...] Read more.

(This article belongs to the Section Mathematical Sciences)

14 pages, 1990 KB

Open AccessHypothesis

From Magnetic Field Seeds to Planetary and Galactic Magnetism

by Angelo De Santis, Roberto Dini and Gianfranco Cianchini

Foundations 2025, 5(4), 32; https://doi.org/10.3390/foundations5040032 - 23 Sep 2025

Viewed by 765

Abstract

This study investigates the origin and amplification of magnetic fields in planets and galaxies, emphasizing the foundational role of a seed magnetic field (SMF) in enabling dynamo processes. We propose a universal mechanism whereby an SMF arises naturally in systems where an orbiting body rotates non-synchronously with respect to its central mass. Based on this premise, we derive a general equation for the SMF applicable to both planetary and galactic scales. Incorporating parameters such as orbital distance, rotational velocity, and core radius, we then introduce a dimensionless factor to characterize the amplification of this seed field via dynamo processes. By comparing model predictions with magnetic field data from the solar system and the Milky Way, we find that the observed magnetic fields can be interpreted as the product of a universal gravitationally induced SMF and a body-specific amplification factor. Our results offer a novel perspective on the generation of magnetic fields in a wide range of astrophysical contexts and suggest new directions for theoretical investigation, including the environments surrounding black holes. Full article

(This article belongs to the Section Physical Sciences)

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