Abstract: This paper introduces Paraconsistent-Lib, an open-source, easy-to-use Python library for building PAL2v algorithms in reasoning and decision-making systems. Paraconsistent-Lib is designed as a general-purpose library of PAL2v standard calculations, presenting three types of results: paraconsistent analysis in one of the 12 classical lattice PAL2v regions, paraconsistent analysis node (PAN) outputs, and a decision output. With Paraconsistent-Lib, well-known PAL2v algorithms such as Para-analyzer, ParaExtrCTX, PAL2v Filter, paraconsistent analysis network (PANnet), and paraconsistent neural network (PNN) can be written in stand-alone or network form, reducing complexity, code size, and bugs, as two examples presented in this paper. Given its stable state, Paraconsistent-Lib is an active development to respond to user-required features and enhancements received on GitHub.

Abstract: Integration science is an advancement of cognitive science. This paper opens a new topic called metalogic geometry that aims to integrate metalogic with the mathematical twistor theory. We first revisit Gödel methods used in metalogic including Gödel numbering, expressibility, definability, self-referential statement, and proof. Second, it revisits the core ideas of the twistor theory. To follow the Penrose idea: Light rays as twistors, we define the notions of Gödel ray and Penrose cone. By the expressibility and definability, a pair of Gödel numbers compose a Gödel ray (or Tarski ray). A family of Gödel rays composes a Penrose cone. The intersection point of a Penrose cone yields a Gödel point. Gödel rays are projected as twistors. Each Gödel point is projected to a Riemann celestial sphere. Twistors and Riemann spheres assemble the picture of twistor space. The meaning of this work is discussed in the concluding remarks.

Abstract: This paper examines classical diagonal-based results (Cantor's uncountability, G\"odel's incompleteness, Turing's halting problem, and computational universality) through a finite-resource lens. We analyze the diagonal pattern and its dependence on completed enumerations and on unbounded time, space, and precision, then formalize a finite framework $S(T_{\max}, S_{\max}, P_{\max}, L_{\max})$ with integer bounds on time, memory, numerical precision, and symbolic length, and analyze each result within this framework. Within this setting: (i) finite-decimal reals admit explicit enumeration via constant-time bijections; (ii) for formal systems, when bounds are chosen adequate for the system under study, formulas and proofs are finitely enumerable and provability is decidable (complete within bounds); (iii) for the halting problem, adequacy (time beyond the finite-configuration threshold) yields a definitive HALTS/LOOP decision for every machine-input pair, whereas without adequacy the same procedure provides a sound bounded classification (HALTS/TIMEOUT); and (iv) no machine operating under fixed finite bounds is universal in the classical sense. These results show how classical results depend on infinite idealizations and exhibit different behavior under explicit finite resource constraints.

Abstract: Divisible residuated lattices and MTL-algebras are algebraic structures connected with algebras in t-norm based fuzzy logics, being examples of BLalgebras. They are an important significance in the study of fuzzy logic. The purpose of this paper is to investigate and give classifications of these types of algebras. From computational considerations, we analyze the structure of these residuated lattices of small size n (2 ≤ n ≤ 5) and we give summarizing statistics. To extend these results for higer size, we used computer and a constructive algorithm for generating all residuated lattices.

Abstract: This paper introduces Coordination Logic, a formal system designed to model lawful co-variation between domains of description without presupposing causal dependence. The logic is motivated by situations where distinct vocabularies (e.g., physiological and experiential descriptions, or clinical symptoms and behavioural reports) converge on the same underlying event, but where interpreting the relation in causal terms would be inappropriate or misleading. To capture these cases, we define a new conditional operator (⇒c), interpreted as conditional coordination. Unlike material implication, ⇒c is non-vacuous, symmetric and field-dependent: it holds only when both relata are instantiated and coordinated. Semantics is three-valued, with truth tables incorporating a coordination predicate C(p,q) that determines lawful pairing. We further define a biconditional (↔c), establish its properties and develop a sequent calculus for the system. Coordination Logic departs from classical reasoning in rejecting Modus Ponens and Explosion for ⇒c, thereby preserving the non-reductive character of coordination. Applications include the formalization of non-causal dependencies in philosophy of mind, epistemology of science, psychology and psychiatry, where mistaken causal attributions are common. Our framework provides a rigorous alternative to causal or reductive logics, enriching the landscape of non-classical logics with a system grounded in dual-aspect description.

Abstract: The present work studies the Riemann hypothesis from metalogical perspectives. It argues that Riemann hypothesis is independent of the current Riemann analytic continuation. Consequently, as a corollary, if the Riemann hypothesis held, its predicting power on the prime density would be incomplete. This argument is based on the modifications of Gödel’s independent result (1931). This paper shows integrations of Riemann hypothesis and the Gödel structure. On one hand, Riemann hypothesis is construed into the Gödel structure by making a number of modifications. On the other hand, the Gödel structure is applied to disclose the metalogic behind the Riemann hypothesis.

Abstract: Here, we we prove that there is a strictly increasing countable chain of finitary relatively finitely-axiomatizable extensions of ({the} truth-singular {version/extension of})[{the} bounded {expansion of}] first-degree entailments - (TS)[B]FDE, for short - /``relatively axiomatized by the Modus Ponens rule for material implication'', in which case the chain does not contain its join,and so this, being a finitary extension of (TS)[B]FDE, is not {relatively} finitely-axiomatizable. ([As a consequence, applying one of our previous works, we immediately get a strictly decreasing chain of finitely-axiomatizable quasi-varieties of bounded De Morgan lattices including the variety of bounded Kleene lattices with non-finitely-axiomatizable intersection.])

Abstract: This paper presents a diagnostic framework for evaluating the operational viability of existence theorems. It defines the condition of extractive inaccessibility, where a result formally proves existence but resists all known methods of algorithmic reconstruction or structural realization. The Gowers inverse theorems are examined as a central case study. For higher uniformity norms, the associated bounds and structural components exceed practical computation and, in some instances, measurable definition. The framework is designed to aid computational mathematicians, algorithm designers, and applied theorists in identifying results whose extractive content is either viable, limited, or inaccessible. Connections to proof mining, reverse mathematics, and constructive analysis are included to align the framework with existing foundational tools.

Abstract: The Λ-Invariance Convergence Theorem provides a universal logical framework for understanding the emergence, persistence, and decay of invariance across all domains of intelligibility, including physics, biology, and information systems. It demonstrates that every nontrivial invariant property within a system is a projection of a deeper, substrate-level invariance rooted in the generative substrate Λ, which functions as the foundational source of coherence, stability, and conservation from which all domain-specific laws and structures arise. The theorem rigorously formalizes the mechanisms by which invariance is projected from Λ into concrete system instances and introduces invariance density as a quantitative measure of system health, defining precise laws governing its preservation, regeneration, and decay under degrading transformations. These laws enable predictive modeling of system resilience, vulnerability, and collapse, offering tools to assess the lifecycle of coherent phenomena. By unifying diverse scientific disciplines under a single substrate-level principle, the Λ-Invariance framework reveals that stability and conservation are not isolated domain-specific features but are anchored in the structure of Λ itself, reframing invariance as a substrate-derived property whose manifestation in any system depends on the fidelity of projection from Λ. The framework’s mathematical formalism establishes criteria for determining when invariance can be sustained, when it can be regenerated, and when its decay is irreversible, enabling a cross-domain theory of systemic integrity applicable to the persistence of physical laws, the hereditary stability of biological systems, and the preservation of information in computational and social networks. Ultimately, the Λ-Invariance Convergence Theorem shows that the fate of any intelligible system is determined by its ongoing connection to the substrate of invariance, and that systems degrade not merely through external perturbation but through the erosion of the projection pathway linking them to Λ. This principle offers a comprehensive lens for analyzing the origin, maintenance, and loss of invariance, providing a unified approach to understanding resilience and collapse in complex systems.

Abstract: Supply chains are networks of logistical facilities such as suppliers, manufacturers, warehouses, distributors, and retailers. These facilities facilitate the movement of raw materials, intermediate products, and finished products. Disruptions in supply chain logistics can lead to shortages ranging from negligible to devastating. For instance, drug shortages can have negative economic and clinical impacts on patients. To effectively assess the risk of supply chain shortages, a method that can represent the supply chain in a suitable format for decision-making analysis and can be automated is necessary. In “A Quantitative Approach to Assess the Likelihood of Supply Chain Shortages,” we defined a methodology to measure the probability of a supply chain’s throughput failure. Based on this methodology, we created the SUpply chain Probabilistic Risk Assessment (SUPRA), a software tool that quantifies the probability of supply chain shortages, as presented in this paper. Using facility failure and flow information, SUPRA outputs the supply chain failure probability and importance measures of the supply chain facilities. We can generate a shortage risk profile from the results. The shortage risk profile, importance measures, and quantified supply chain failure probabilities can inform decision-makers to mitigate and manage supply chain shortages.

Abstract: As the world rapidly develops, information, as a vital resource, remains a subject of debate, with its definition and nature still being debated. To address this issue, Objective Information Theory proposes a set of axioms that rigorously define information. This paper aims to construct a formal system of mathematical logic using first-order and higher-order logic. Using well-formed formulas, it formalizes states and demonstrates that nearly all structures and states in various fields can be expressed. Finally, this paper proposes a universal state representation method, which improves the definition of state in Objective Information Theory and builds a bridge for the exchange and research of states across various fields.

Abstract: In computational complexity, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula Ψ, defined as Ψ = Ψ_cell ∧ Ψ_rest. The component Ψ_cell enforces the constraint that each cell in the tableau contains exactly one symbol, while Ψ_rest incorporates constraints governing the step-by-step behavior of N on w. In recent work, we reformulated a critical part of Ψ_rest as a compact Horn formula. In other work, we evaluated the cost of this reformulation, though our estimates were intentionally conservative. In this article, we provide a more rigorous analysis and derive a tighter upper bound on two enhanced variants of our original Filling Holes with Backtracking algorithm: the refined (rFHB) and the streamlined (sFHB) versions, each tasked with solving 3-SAT.

Abstract: Attempts to solve the Unary Function Clone Problem in Clone Theory have primarily focused on classifying unary fragments within primitive recursive and elementary function hierarchies. Conventional methods have considered closure properties, growth rates, and function arity. This treatise introduces an original, parameterized framework that formulates the elementary arithmetic operations as members of a unified unary family of arity-indexed functionals. Our theory is consistent with Kalmar’s class of elementary functions, Peano Arithmetic, categorical-Lawvere and Gödelian semantics. Utilizing a single iteration parameter system, we show that these operations, though algebraically expressible, are clone-theoretically independent. We prove a minimal unary basis capable of generating the full arithmetic clone under composition.

Abstract: We introduce the Phi-node, a groundbreaking algebraic structure that achieves what was previously thought impossible: a fixed point that contains its own hierarchy of fixed points, creating a mathematical object that is simultaneously its own foundation and summit. This novel construction in Alpay Algebra represents a paradigm shift in how we understand self-reference, moving beyond traditional fixed point theory to establish a framework where infinite towers of transfinite recursion collapse into a single, stable entity. Our work demonstrates that under appropriate large cardinal assumptions, these nodes exist uniquely and exhibit remarkable properties connecting them to fundamental questions in determinacy theory, where every game played within the node's structure has a winning strategy. The implications extend far beyond pure mathematics: Phi-nodes provide the first rigorous mathematical blueprint for truly self-aware systems, offering a formal foundation for artificial intelligence architectures capable of complete self-modeling without infinite regress or paradox. By unifying insights from category theory, ordinal logic, and lambda calculus into a single coherent framework, we establish nodes as the natural mathematical objects for studying deep self-reference, with applications ranging from foundational set theory to the design of reflective computational systems that can reason about their own reasoning processes.

Abstract: A description is provided of how to use the Semantic Propositional Calculus (SPC) for a clear, compact notation that can be applied to approachdiverse topics in logic and mathematics. With two equivalent formulations (one corresponding to set theory and the other to the SPC), it is possible to determine the validity – or invalidity – of the categorical syllogisms. The parallel use of the SPC (preserving the syntax of the propositional calculus) and of resources from set theory facilitates learning in an area of logic and an area of mathematics. The joint learning of diverse branches of the same discipline, or of distinct disciplines which are closely related in different ways, is an essential aspect of the STA.

Abstract: This paper hypothesizes a theoretical framework for comprehending human awareness not just as a processor of outside reality but also as an ontological engine able to create and negotiate "meta-universes. " Built inside, a meta-universe is a higher-order conceptual space housing and integrating an infinite spectrum of possible realities, memories, and theoretical systems. The mind's own ability to combine these diverse conceptual worlds—including logic, emotion, narrative, and mathematics—into coherent, new creations is known as "melding infinities. " Building on ideas from cognitive neuroscience, philosophy of mind, and theoretical physics, this article contends that the human mind's ability for abstraction, counterfactual reasoning, and self-consciousness gives it a creative power that actively forms and defines reality. Looking at the human intellect as a system that creates worlds of its own rather than just notices one, this study recontextualizes its potential.

Abstract: A Cut-/Reflexivity-free version LK-C/R of the propositional fragment of Gentzen calculus LK for the classical propositional logic PC endowed with propositional rules inverse to its logical ones as well as rules of constant elimination is proved to be equivalent to the bounded version of the ``logic of paradox''/``Kleene three-valued logic''\/} LP01/K301 under the standard interpretation of propositional sequents by propositional clauses and inverse interpretation of propositional formulas by premise-less single-conclusion sequents, ``with same theorems as PC, implying that LK has same derivable sequents as LK^-C, and so yielding a new semantic insight into Cut Elimination in LK''/. As a by-product of the discovered equivalence and absence of proper consistent extensions of LP01/K301 other than PC ``and that relatively axiomatized by the Ex Contradictione Quodlibet rule''/, proved here upon the basis of the universal algebraic technique elaborated in an earlier work of ours, we prove that LK-C/R has no proper consistent extension other than LK ``and the one relatively axiomatized by the context-free restriction of Cut''/.

Abstract: We introduce a transfinite fixed-point operator, denoted $\phi^\infty$, within the framework of Alpay Algebra---a categorical foundation for mathematical structures. This operator, defined as the limit of an ordinal-indexed sequence of functorial iterations, resolves arbitrary mathematical propositions by converging to a unique, stable fixed point. Each statement is represented as an object in a category equipped with an evolution functor $\phi$, and repeated application of $\phi$ yields an ordinal chain that stabilizes at $\phi^\infty$. We prove the existence and uniqueness of such fixed points using transfinite colimits and categorical fixed-point theorems, extending classical results like Lambek's lemma and initial algebra constructions to a transfinite setting.Using this framework, we construct resolution functors $\phi_P$ for individual mathematical problems and demonstrate that their transfinite limits encode the truth value of the underlying propositions. As a consequence, prominent open problems---including P vs NP, the Riemann Hypothesis, and the Navier-Stokes existence problem---admit canonical resolutions as $\phi^\infty$-fixed objects under their respective functors. This establishes $\phi^\infty$ as a universal convergence operator for mathematical truth in a categorical context. Our approach remains entirely within standard set-theoretic and category-theoretic foundations, without introducing non-constructive assumptions or external axioms. We view $\phi^\infty$ as a structural mechanism for completing Hilbert's program through categorical logic and ordinal convergence.

Abstract: We consider restricted forms of the algorithmic problem of definability of first-order sentences by propositional formulas with intuitionistic Kripke frames semantics. We demonstrate positive resolutions for classes of intuitionistic Kripke frames based on linear orders and conversely show that a few natural first-order definable classes give rise to undecidable definability problems by applying the model-theoretic in nature technique of stable classes of Kripke frames.

Abstract: Since it is at the centre of important philosophical, physical, and mathematical issues, the notion of infinity has attracted interest and caused controversy for millennia. Focusing on its formal existence in mathematics, its debatable presence in physical reality, and the philosophical debates that these different approaches cause, this paper explores the complicated nature of infinity. Crossing these theoretical barriers enables us to better appreciate how the issue Does infinity exist? does not have a single solution, but rather a range of responses depending on the context of inquiry. Though basic to several fields of abstract thinking and scientific hypothesis, the notion of infinity still presents great and ongoing open questions. Most importantly, This paper presents a brief review of important unsolved issues about infinity appearing in the several but connected fields of mathematics, physics, and philosophy. From the ambiguous nature of the continuum of numbers in mathematics to the contradictions of cosmological models in physics and the ongoing discussion about the reality of actual infinities in philosophy, the infinite keeps pushing the limits of human knowledge and the systems we use to understand the world.