Abstract: We present a self-contained exposition of the Lyons carabiner, focusing on the triple obstruction mechanism arising in Section~\ref{sec:triple}. Three independent numerical obstructions---phantom coupling $\pi$, holonomy defect $\rho$, and braid deficit $\beta$---together form the triple $(\pi,\rho,\beta)=(3,4,9)$, which uniquely recovers the MDS code $[6,4,3]_5$ over $\mathbb{F}_5$. This identification is not a postulate but a provable consequence of the weight distribution and the complement involution. We then develop the Pontryagin--Heegner bridge: phantom weights are ``silent frequencies'' in the Pontryagin dual of $\mathbb{R}_+^\times$, and their vacuum generates the $20$-dimensional \emph{inverse Heegner space} $\mathcal{H}_{20}$. This yields a candidate $44$-dimensional lattice $\Lambda_{44} = \Lambda_{24} \oplus \mathcal{H}_{20}$ with Lyons group symmetries. The phantom resolution cascade $\mathrm{Ly}\to\mathrm{HS}\to\mathrm{Ru}$ predicts a chain of lattice dimensions $44\to 34\to 24$, terminating at the Leech lattice $\Lambda_{24}$, with the Rudvalis level providing independent numerical evidence through striking orbit coincidences. All numerical claims are verified by machine-checked computation. ( The Lean~4 formalization is available as the \texttt{HatsuYakitori} library; the key files are \texttt{MachineConstants.lean}, \texttt{LyonsCarabiner.lean}, and \texttt{RudvalisCarabiner.lean}. Remaining \texttt{sorry}s are listed explicitly in Section~\ref{sec:conclusion}.

Abstract: A broom graph is a linear graph with some pendant vertices attached to one of its ends. Using the formula for the commute time of the random walk between two vertices of the graph, which is given in terms of the effective resistance between the vertices, we find closed-form formulas for the hitting time index of some families of broom graphs, extending results found in the literature.

Abstract: We ask for finite field versions of following three: (1) Grothendieck Inequality, (2) Johnson-Lindenstrauss Flattening Lemma, (3) Bourgain-Tzafriri Restricted Invertibility Theorem.

Abstract: We study a pharmaceutical scheduling problem with hybrid batch-continuous manufacturing process in a distributed supply chain. The supply chain consists of heterogeneous plants and one distribution center. Each plant adopts an unrelated permutation flow shop layout consisting of a hybrid batch-continuous production line. Each pharmaceutical order is split and produced in multi-production sites located in various regions. The pharmaceutical medicines manufactured by the production sites are directly shipped to a distribution center To minimize the makespan, we formulate the addressed scheduling problem as a mathematical model. To solve this model, we propose four metaheuristics variants by applying two population-based metaheuristics to two distinct solution structures. We compare the proposed metaheuristics to evaluate their performance in the numerical experiments. Additionally, we present managerial insights through sensitivity analysis.

Abstract: Along with some known and less known results, we discuss new insights relating combinatorics of words and the ordering of the rationals from a dynamical systems point of view, somehow continuing along the path started in [BI]. We obtain in particular a set of results that structure and enrich the correspondence between the Stern-Brocot (SB) ordering of rational numbers and the corresponding ordering of Farey-Christoffel (FC) words, a class of words that, since their appearance in literature at the end of the 18th century, have revealed numerous relationships with other fields of mathematics. Among the results obtained here is the construction of substitution rules that act on the FC words in a parallel way to the maps on the positive reals that generate the permuted SB tree both vertically and horizontally. A complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane, respectively.

Abstract: A finite hypergraph generalizes an ordinary graph by permitting a hyperedge to connect any nonempty subset of vertices, thereby representing genuine multiway interactions. Extending this idea, a finite SuperHyperGraph is obtained through an iterated powerset construction, so that set-valued objects formed at one level may function as vertices or edge endpoints at the next, providing a natural framework for hierarchical and multilayer relational structures. In contrast, a line graph transforms each edge of a graph into a vertex, with two such new vertices adjacent precisely when the corresponding original edges share an endpoint. In this paper, we introduce the notion of a MultiLine Graph, in which multiple edges can be assigned to a vertex, and then develop its higher-order extensions, namely the MultiLine HyperGraph and the MultiLine Super HyperGraph. We further investigate their fundamental properties and structural characteristics.

Abstract: This paper studies graph-based higher-order structures related to metagraphs and edgelabeled hierarchical networks. After reviewing MetaGraphs and Iterated MetaGraphs, we introduce the notion of an Edge-MetaGraph, in which each edge is labeled by a two-ported internal graph, allowing edge-substitution expansion through port gluing. We then define Iterated EdgeMetaGraphs recursively, so that edges may carry nested Edge-MetaGraph structures. Concrete examples from biomedical systems, software pipelines, and logistics are presented to illustrate the expressive power of the proposed framework.

Abstract: For every prime p and every integer a, the backward finite difference δp(a) := a^p − (a − 1)^p equals the cyclotomic binary form Φp(a, a − 1), where Φp(X, Y) is the homogenisation of the p-th cyclotomic polynomial, and hence equals the norm NQ(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to the identity δ3(a) = NZ[ω](a − ω(a − 1)), where ω = e^(2πi/3), connecting the individual cubic finite difference obtained by differencing the classical sum formula of Nicomachus of Gerasa (~100 CE) with the Eisenstein norm that appears in Euler's factorisation of a^3 + b^3. We develop this identity in three directions: (a) General cyclotomic framework. For each prime p, every prime divisor q of δp(a) satisfies q ≡ 1 (mod p), imposing an arithmetic sieve whose density ~1/(p−1) grows increasingly severe with p. (b) Arithmetic density. The values {δ3(a)}a≥1 form a thin subfamily of the Löschian numbers (norms in Z[ω]), with counting function ~√(N/3) versus the Landau-Ramanujan asymptotic CN/√log N for all Löschian numbers up to N. (c) Three-language equivalence. For the cubic case we prove a precise equivalence among: (i) divisibility of δ3(a), (ii) multiplicative order modulo q, and (iii) splitting of q in Z[ω]. We also give an elementary proof of the base case 1 + b^3 = c^3 (no positive-integer solutions), and derive 3-adic constraints on any hypothetical solution to a^3 + b^3 = c^3 via the Lifting-the-Exponent Lemma, without invoking unique factorisation in Z[ω].

Abstract: The P versus NP problem, a conjecture formulated by Stephen Cook in 1971, is one of the most challenging problems in contemporary mathematics and theoretical computer science. A concise mathematical formulation of the problem reads: is P = NP? In longer phrasing, this asks: given a problem instance, if some additional data can be recognized fast enough as logically implying the existence of a solution (to the instance), then can a so- lution be computed fast enough? we formulate three problems in graph imbedding to show P ≠ NP.

Abstract: The construction of partial minimum spanning trees being NP-hard, several heuristic algorithms have already been formulated. Many of these heuristics (such as Kruskal's) use shortest paths to connect the components of the tree. In this work, we present an approximate construction algorithm for the minimum Steiner tree (the optimal tree for diffusion multicast). This construction is based on graph-related structures more advantageous than shortest paths. The algorithm uses connections like simple Steiner trees if necessary. These simple trees can be represented by hyperedges. A hyper metric closure can also be used.

Abstract: We show that in a vector space over Z₃, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for p=3 and providing an alternative proof of the weak 3-flow conjecture.

Abstract: Assembly Theory, as developed by Cronin and co-workers, assigns to an object an assembly index: the minimal number of binary join operations required to build at least one copy of the object from a specified set of basic building blocks, allowing reuse of intermediate components. For strings over a finite alphabet, the canonical assembly index can be defined in the free semigroup (Σ+, ·) with universal binary concatenation and a “no-trash” condition, and its exact computation has been shown to be NP-complete. In this paper we propose an extension of the canonical, string-based formulation which augments pure concatenation with templated assembly steps. Intermediate objects may contain a distinguished wildcard symbol ∗ that represents a compressible block. Templates are restricted to block-compressed substrings of the target string and can be instantiated by inserting previously assembled motifs into one or many wildcard positions, possibly in parallel. This yields a new complexity measure, the templated assembly index, which strictly generalises the canonical index while preserving its operational character. We formalise the model, clarify its relation to the canonical assembly index and to classical problems such as the smallest grammar problem, and discuss the computational complexity of determining the templated assembly index. Finally, we sketch potential applications in sequence analysis, modularity detection, and biosignature design.

Abstract: This study examines the δ-complements of graphs—a specific type of graph complement whose adjacency depends on the adjacency of the vertices with identical degrees in the original graph. More specifically, we study this type of complement regarding the domination number. We provide sharp Nordhaus-Gaddum-type bounds on the domination number of a graph and its δ-complement. We also provide sharp bounds on the domination numbers of the δ-complements of joined graphs and Cartesian product graphs.

Abstract: The goal of this paper is to combine algebraic quantum field theory with methods from modal logic and the causal set program. We introduce a factorially damped past-influence operator on causal sets, formulate locally covariant nets over Alexandrov intervals, and give discrete analogues of the Haag–Kastler axioms. We close by sketching a route toward dynamics via sequential growth and incidence-algebra localization.

Abstract: A hypergraph generalizes an ordinary graph by allowing an edge to connect any nonempty subset of the vertex set. By iterating the powerset operation one step further, one obtains nested (higherorder) vertex objects and, consequently, a finite SuperHyperGraph whose vertices and edges may themselves be set-valued at multiple levels. Thus, many hierarchical graph structures exist in the literature. Moreover, not only in graph theory but also in broader fields—such as through concepts like Decision Trees and Tree Soft Sets—it is well known that tree structures are effective tools for representing hierarchical concepts. In this paper, we define a new class of graphs called Tree-Vertex Graphs. In this framework, a tree structure is imposed on the vertex set, and the edge set is defined in a manner consistent with the tree-structured vertex set. The tree structure therefore serves as a key concept for representing hierarchical graphs.

Abstract: A MetaGraph is a higher-level graph whose vertices are themselves graphs, with edges representing specified relations among these graphs. An Iterated MetaGraph extends this notion recursively: its vertices are MetaGraphs, thereby forming a hierarchical structure of graphs-of-graphs across multiple levels. In this paper, we introduce and study specific classes of MetaGraphs, including Graph-in-Tree and Cycle-in-Cycle structures. In addition, we define a new class of graphs, called Spiral Graphs, and investigate their fundamental properties.

Abstract: In this paper, we investigate the lexicographic and colexicographic orderings of m-ary vectors of length n, as well as the mirror (left-recursive) reflected Gray code, complementing the classical m-ary reflected Gray code. We present efficient algorithms for generating vectors in each of these orders, each achieving constant amortized time per vector. Additionally, we propose algorithms implementing the four fundamental functions in generating combinatorial objects—successor, predecessor, rank, and unrank—each with time complexity Θ(n). The properties and the relationships between these orderings and the set of integers {0,1,…,mn−1} are examined in detail. We define explicit transformations between the different orders and illustrate them as a digraph very close to the complete symmetric digraph. In this way, we provide a unified framework for understanding ranking, unranking, and order conversion. Our approach, based on emulating the execution of nested loops, proves to be powerful and flexible, leading to elegant and efficient algorithms that can be extended to other combinatorial generation problems. The mirror m-ary Gray code introduced here has potential applications in coding theory and related areas. By providing an alternative perspective on m-ary Gray codes, we aim to inspire further research and applications in combinatorial generation and coding.

Abstract: There are several graphs naturally associated with rings. The unitary Cayley graph of a ring R is the graph with vertex set R, where two elements x,y∈R are adjacent if and only if x−y is a unit of R. We show that the unitary Cayley graph CTn(F) of the ring Tn(F) of all upper-triangular matrices over a finite field F is isomorphic to a semistrong product of a complete graph and the antipodal graph of a Hamming graph. In particular, when |F|=2, the graph CTn(F) has a highly symmetric structure: it is the union of 2^{n−1} complete bipartite graphs. Moreover, we prove that the clique number and the chromatic number of C_Tn(F) are both equal to |F|, and we establish tight upper and lower bounds for the domination number of C_Tn(F).

Abstract: We model conscient entities as vertices of a Graph; and its edges, as the interaction between the them. We further introduce a two-layer multiplex network structure coupling the micro-level soul graph with a macro-level nation graph, enabling the study of how individual interactions aggregate to shape inter-nation relationships, and conversely, how geopolitical events influence individual states. The model includes concepts such as cultural entanglement, and virtue field restoration, providing a unified graph-theoretic treatment of both spiritual and geopolitical evolution. By combining deterministic evolution laws with graph Laplacian operators, the model captures the cyclical patterns of cooperation, fragmentation, and reunification across epochs. This work not only bridges ancient spiritual narratives with modern mathematical formalisms but also lays the foundation for quantitative simulations of the socio-political dynamics of humanity across Time.

Abstract: A Plithogenic Set extends the classical fuzzy, intuitionistic, and neutrosophic paradigms by assigning attribute-based membership and contradiction values to elements, and the same ideas naturally extend to graph-based structures such as the Plithogenic Graph. Neutrosophic sets, in turn, represent elements with independent degrees of truth, indeterminacy, and falsity on the unit interval, thus handling incomplete, inconsistent, and ambiguous information effectively. This paper investigates the Plithogenic Graph, the Plithogenic Vertex Graph, and the Plithogenic Edge Graph. In addition, it examines the Intuitionistic Fuzzy Vertex Graph and Edge Graph, the Neutrosophic Vertex HyperGraph and Neutrosophic Edge HyperGraph, and the Neutrosophic Vertex SuperHyperGraph and Neutrosophic Edge SuperHyperGraph.