Abstract: We prove that the set \( T=\Bigl\{n\in\mathbb{N}: \exists p,q\in\mathbb{N}\;\Bigl((2n=(p+q)(p+q+1)+2q)\;\wedge\ \) \( \forall (x_0,\ldots,x_p)\in\mathbb{N}^{p+1}\;\exists (y_0,\ldots,y_p)\in\{0,\ldots,q\}^{p+1}\ \) \( \bigl((\forall k\in\{0,\ldots,p\}\;(1=x_k \Rightarrow 1=y_k))\;\wedge\ \) \( (\forall i,j,k\in\{0,\ldots,p\}\;(x_i+x_j=x_k \Rightarrow y_i+y_j=y_k))\;\wedge\ \) \( (\forall i,j,k\in\{0,\ldots,p\}\;(x_i\cdot x_j=x_k \Rightarrow y_i\cdot y_j=y_k))\bigr)\Bigr)\Bigr\}\ \) is not recursively enumerable. By using Gödel's \( \beta \) function, we prove that the formula that defines the set T can be easily translated into a first-order formula which uses only + and \( \cdot \). The same properties has the set \( \Bigl\{n\in\mathbb{N} : \exists p,q\in\mathbb{N}\;\Bigl((2n=(p+q)(p+q+1)+2q)\;\wedge\ \) \( \forall (x_0,\ldots,x_p)\in\mathbb{N}^{p+1}\;\exists (y_0,\ldots,y_p)\in\{0,\ldots,q\}^{p+1}\ \) \( \bigl((\forall j,k\in\{0,\ldots,p\}\;(x_j+1=x_k \Rightarrow y_j+1=y_k))\;\wedge\ \) \( (\forall i,j,k\in\{0,\ldots,p\}\;(x_i\cdot x_j=x_k \Rightarrow y_i\cdot y_j=y_k))\bigr)\Bigr)\Bigr\}\ \).

Abstract: This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency N(μ)∈[0,1], defined for any non-constant fuzzy Lie subalgebra μ of a Lie algebra g. This invariant equals 1 precisely when μ is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies N(μ)=1, but that the converse fails in general, as witnessed by an explicit counterexample.

Abstract: This work traces a philosophical and mathematical thread from ancient Greek mathematics to modern foundational logic. The Greeks maintained a sharp distinction between ἀριθμητική (arithmetic as the theoretical science of numbers) and λογιστική (calculation as a practical art), while also separating arithmetic from formal logic. This separation, grounded in ontological and epistemological considerations, allowed Greek mathematics to avoid the foundational crises that would emerge two millennia later. The development of formal logic in the late nineteenth and early twentieth centuries—particularly through the work of Frege, Russell, and Hilbert—sought to unify arithmetic and logic within a single syntactic framework. Gödel's incompleteness theorems (1931) demonstrated the impossibility of this project, showing that any consistent, recursively axiomatizable theory strong enough to encode arithmetic must be incomplete and cannot prove its own consistency. Furthermore, phenomena such as Tarski's undefinability of truth and the existence of non-standard models demonstrate that pure syntax faces a total epistemological collapse. This work argues that these metamathematical limits can be synthesized into a "Semantic Necessity Theorem": a complete, consistent, arithmetically strong theory cannot be purely syntactic. The Greek separation of arithmetic from formal logic thus appears not merely as a historical curiosity, but as a mathematically prescient framework that anticipates the structural necessity of ontology in modern mathematics.

Abstract: Mathematics, as actually practiced, operates as a federated system: practitioners work within autonomous domain-specific axiomatizations (geometry, algebra, analysis) and construct explicit bridges only when cross-domain reasoning is required. This organization is not accidental; it is a structural adaptation that safeguards local decidability and algorithmic efficiency.Yet the dominant foundational narrative still operates on the Compiler Myth—the belief that all mathematics must theoretically compile down into ZFC set theory to achieve rigor. We argue that this monolithic reductionism confuses representational universality with logical priority. Embedding a decidable (tame) domain into an undecidable (wild) one does not clarify foundations; it imposes a crippling epistemic overhead. It buries efficient, domain-specific decision procedures under general proof search and destroys the native structural immunities of the object.We introduce the Decidability Threshold — a litmus test based on Negation, Representability, and Discrete Unboundedness — to explain why mathematicians instinctively isolate tame domains from wild ones. Finally, we distinguish the Mathematician (builder of formal systems) from the Scientist (consumer modeling reality). We argue that federalism, through explicit bridges and domain autonomy, is not a failure of unification, but the primary safeguard preventing the scientist from inadvertently importing wild, undecidable paradoxes into physical theories.

Abstract: In this exercise, we will discuss quantum entanglement in an intuitionistic context and its evolution. This requires a definition of the tensor product, as well as the introduction of a Hamiltonian.

Abstract: Being either true or false, 1 or 0, the standard logic and the Boolean algebra traditionality never rotate. Thus, it can only account for polarization states but not superposition states. This paper proves the Boolean rotation theorem through complexifications. This result allows us to formulate polarization spinors as well as superpositions spinors. It provides a new understanding of the Riemann sphere of two-state systems. It also provides an alternative solution to the measurement of wavefunctions, which accounts for both the U-process and the R-process. The work reported in this paper formulates the Penrose twistor geometry of the polarization spinors and the superposition spinors.

Abstract: This paper investigates cooperative advertising decisions in production–retailing chan-nels for seasonal products under demand seasonality. We develop analytical game-theoretic models to examine how advertising cooperation influences channel coor-dination and profit distribution between manufacturers and retailers. Two channel struc-tures are considered: a single-manufacturer–single-retailer channel and a sin-gle-manufacturer channel with two competing retailers. For each structure, Stackelberg and Nash equilibrium settings are analyzed and compared. Our results show that coop-erative advertising can serve as an effective coordination mechanism by increasing adver-tising intensity and improving channel efficiency. Retailers always benefit from manu-facturer-supported advertising through cost sharing and higher profitability, whereas the manufacturer’s incentive to participate depends on whether demand expansion out-weighs shared advertising costs. Importantly, we demonstrate that channel leadership plays a critical role: the Stackelberg equilibrium consistently dominates the Nash equilib-rium in terms of total channel profit. This study contributes to the cooperative advertising literature by explicitly incorporating demand seasonality and competing retailers, and by clarifying when cooperative advertising leads to Pareto improvements in seasonal supply chains.

Abstract: In the framework of (fuzzy) Multi-Criteria Decision-Making, we propose a method that1 allows the decision maker to subjectively approach the problem by suitably modifying the decision matrix. We consider a decision problem related to a random quantity X with set of values {x₁, x₂, . . . , x_n}, and a set of properties {C₁, C₂, . . . , Cm}of X. In this setting, the properties C_j are the criteria of the decision problem, the alternatives represent the events A_i = (X= x_i), for i= 1, . . . , n, and the criteria’s weights w_j, for j= 1, 2, . . . , m, are seen as the probabilities of the events “C_j is relevant with respect to the decision problem”. For each i= 1, . . . , n and j= 1, 2, . . . , m, we interpret the scores aij as membership functions representing “how much alternative A_i satisfies criterion C_j”. By adopting the interpretation of membership functions as suitable conditional probabilities, together with the theory of logical operations among conditional events, we allow logical operations among criteria and consistently apply this interpretation to the corresponding scores. In particular, when considering the complement, conjunction, and disjunction of criteria, the resulting scores are the coherent) previsions of the respective compound conditionals within the framework of conditional random quantities.

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.