Computer Science and Mathematics

Abstract: Recommender systems increasingly incorporate graph embeddings and graph neural networks to capture high-order relationships between users and items. However, the additional complexity of these approaches does not necessarily guarantee better recommendation quality than strong classical and latent-factor baselines. This study presents a reproducible comparison of six recommendation models representing four methodological families: Logistic Regression, Random Forest, Matrix Factorization with Bayesian Personalized Ranking, DeepWalk, node2vec, and LightGCN. The experiments were conducted on the MovieLens 1M dataset using a per-user temporal split. For each user, the most recent positive interaction was assigned to testing, the preceding interaction to validation, and all earlier positive interactions to training. All models were evaluated using identical candidate sets containing one held-out positive movie and 99 sampled unobserved movies. Performance was measured using Recall, Precision, Hit Rate, and NDCG at multiple cutoffs, complemented by bootstrap confidence intervals, paired statistical tests, computational-efficiency measurements, and analyses by user activity and movie popularity. Matrix Factorization achieved the best overall performance, reaching a Recall@10 of 0.7458 and an NDCG@10 of 0.4558, representing an approximately 56% improvement in NDCG@10 over Random Forest, the strongest classical baseline. LightGCN did not significantly outperform Logistic Regression and remained below Random Forest, despite its higher computational cost. DeepWalk and node2vec obtained similar and substantially lower aggregate results. Popularity-based analysis further revealed that classical models and LightGCN achieved substantially higher ranking effectiveness for popular movies, whereas Matrix Factorization maintained comparatively stronger performance for less-popular items. These findings demonstrate that model complexity alone is not a reliable indicator of recommendation effectiveness and highlight the importance of strong baselines, standardized evaluation, and reproducible experimental protocols.

Abstract: We prove that every finite two-player game \(G\) with entangled value \(\omega^*(G)=1-\epsilon\) satisfies \[ \omega^*(G^{\otimes n}) \le\exp(-\Omega(\frac{\epsilon^3}{\epsilon+ \ell }n)) \] for every $n\ge1$, where \(\ell:=\log(|A| |B|)\), and \(A\) and \(B\) are the answer alphabets. Compared with Chapter~6 of the OpenAI report [Ope26], this improves the gap exponent from thirteen to three and matches the cubic gap dependence in Holenstein's general classical bound [Hol09]: \[ \omega(G^{\otimes n})\le\exp(-\Omega( \frac{ (1-\omega(G))^3}{1+\ell} n)). \] The proof replaces the randomly shifted logarithmic grid used in quantum correlated sampling by smooth soft labels. This makes the relevant label infidelity quadratic in the distance between state descriptions and avoids a Jensen loss when averaging over questions. Together with the postselection argument, these improvements yield the cubic gap dependence stated above.

Abstract: We prove two deterministic inapproximability results. First, for every fixed \(\epsilon>0\), Euclidean \(\mathrm{GapCVP}^{(2)}\) is NP-hard with gap factor \(n^{1/8-\epsilon}\) under deterministic polynomial-time many-one reductions, where \(n\) denotes the lattice rank. Consequently, the Euclidean closest vector problem is NP-hard to approximate within the same factor. This improves the previous \(n^{1/400}\) hardness factor in Chapter 7 of the OpenAI report [1]. Second, for every fixed \(\epsilon>0\), binary nearest codeword and binary syndrome decoding are NP-hard to approximate within \(n^{1/4-\epsilon}\) under deterministic polynomial-time many-one reductions, where \(n\) denotes the binary block length. This improves the previous \(n^{1/200}\) hardness factor in Chapter 7 of the OpenAI report [1].

Abstract: Assembly theory measures a string by the shortest history that builds it. We sharpen this picture in two directions. First, we attach a discrete Dirichlet energy to an assembly space and show that it splits into the size of the space and a secondary energy charging each step vertex the square of its secondary (pathway depth) increment. The assembly index does not imply minimum energy, and the assembly depth and the energy are independent complexity measures. Second, we ask how an ensemble T of strings co-assembles within a single space. Individuating step vertices by their formation histories rather than by the strings they carry yields three joint assembly spaces: the collectively (\( \omega_T \)), multiplicity (\( \mu_T \)), and singly (\( \pi_T \)) optimal ones, where the latter need not exist for certain ensembles and \( |\omega_T| \le |\mu_T| \le |\pi_T| \). All three sizes can differ, and multiplicity can lower the singly optimal size only for ensembles with \( |\pi_T| - |\omega_T| \ge 2 \). Collective optimization is NP-complete, while singly and multiplicity optimizations are NP-hard and lie a priori in the second level of the polynomial hierarchy.

Abstract: Global network alignment maps nodes of one network onto nodes of another, preserving as much topology as possible. When two or more alignments are produced for the same pair of networks---by different algorithms or different random seeds of the same algorithm---a natural question arise: show much do these alignments agree? The standard answer, the Jaccard index over the sets of aligned node pairs, captures only set overlap and is blind to the local geometric structure that each alignment induces. We introduce a family of curvature-based agreement measures that compare alignments through the discrete graph curvatures they induce on the two networks. For each of three discrete Ricci-type curvatures—Forman–Ricci (k_FR), Ollivier–Ricci (k_OR), and Cayley–Menger (k_CM)—we define two complementary agreement scores: a distribution-based \( \chi^2 \) distance and a correspondence-aware Spearman correlation. We evaluate these measures on 22 pairwise comparisons across 10 organism pairs (6 bacterial, 4 eukaryotic), using alignments produced by SANA and MAGNA++. We find that (i) no single curvature dominates across all comparisons (k_CM is highest in 11/22, k_FR in 8/22, k_OR in 3/22); (ii) the Jaccard index replicates the finding of Guerra et al. (2020) that real alignments match the random null; (iii) the curvature-based Spearman measures detect agreement variation that Jaccard cannot, particularly in eukaryote pairs where the range widens to \( \rho \in [0.26, 0.93] \); and (iv) an approximate k_CM based on truncated eigendecomposition scales to 13,276-node networks in \( \sim \)3.4 min while retaining Spearman \( \rho = 0.92 \) against the exact computation on a benchmark graph. These results establish discrete curvature as a viable and informative lens for alignment agreement, complementary to the established set-overlap paradigm.

Abstract: (1) Background: Characterizing causal structures in multivariate time series is a central challenge across neuroscience, economics, and environmental sciences; classical estimators identify which causal relations exist but lack compact descriptors capable of detecting how these relations organize themselves topologically over time. (2) Methods: We propose a multimodal framework that integrates sliding-window Granger causality estimation, the Vector Visibility Graph (VVG) representation, and three complementary topological descriptors—betweenness centrality, the Network Laplacian Spectral Descriptor (NetLSD), and the Deterministic Tourist Walk (DTW). The framework is evaluated on synthetic coupled autoregressive processes, coupled chaotic logistic maps, and a real-world case study with meteorological and rice price data from Uruguaiana, Brazil. (3) Results: Despite their mathematical independence, the three descriptors consistently identify a common critical coupling threshold λ∗ ≈ 0.9–1.0, providing internal cross-validation of the methodology. Experiments reveal pronounced topological reorganization near the synchronization regime, including attractor collapse, abrupt regime transitions detectable via bimodal degree distributions, and changes in graph connectivity patterns. The real-world case study demonstrates perfect non-parametric separation between distinct meteorological combinations. (4) Conclusions: The proposed multimodal framework provides interpretable, compact, and statistically robust topological descriptors of dynamic causal networks, complementing existing causal discovery methods.

Abstract: The P versus NP problem is a cornerstone of theoretical computer science: it asks whether every problem whose solution can be checked quickly can also be solved quickly. Here, “quickly” means in polynomial time, where the running time grows as a fixed power of the input size. The question traces back to John Nash’s 1955 letter to the National Security Agency, and was later formalized by Stephen Cook and Leonid Levin. Despite decades of research, it remains open. At the heart of the problem lies the notion of NP-completeness: if any single NP-complete problem, such as SAT, admitted a polynomial-time algorithm, then every problem in NP would, proving P = NP. This paper presents such an algorithm for SAT, thereby establishing that P equals NP.

Abstract: The circular economy is a major manufacturing sector of the economy and the ongoing proliferation of advance digital technologies and digital transformation is revolutionizing manufacturing industries and enabling more sustainable, data-driven, digitally transparent and efficient processes. Smart manufacturing technologies, such as the Internet of Things (IoT), artificial intelligence (AI), and digital twins, offer unprecedented opportunities to optimize resource usage and reduce environmental impact. Concurrently, the circular economy model promotes resource optimization, material repurpose, and sustainability by emphasizing reuse, recycling, and regeneration of materials. This study explores the various opportunities inherent in the current digital technologies and how it certainly impacts on the circular economy, it also explores the intersection of digitalization and circular economy principles, focusing on how smart manufacturing facilitates resource optimization. Through a comprehensive analysis of various digital technological advancements and industrial case studies, this paper highlights how these various digital technologies can enhance lifecycle management, improve waste reduction, and drive circular innovation towards a more sustainable society. The study adopted systemic literature review SLR in relation to the Dynamic Capability Theory DCT and Resource-Based View RBV theory to analyze existing scholarly studies on digitalization in the Circular Economy CE. This research investigate how digitalization in manufacturing can optimize resource use, reduce environmental impact, and promote circular economic practices. The study analyzed current technologies, implementation challenges, and the potential for future development in this field. The paper explored how the circular economy can benefit from digital transformation that supports sustainable practices in manufacturing, focusing on how smart technologies contribute to resource optimization within the framework of the Circular Economy. The study concludes with recommendations for stakeholders seeking to adopt sustainable practices within the Circular Economy CE in the digital age.

Abstract: The advent of quantum computing threatens the long-term security of classical public-key cryptosystems: RSA and standard elliptic curve cryptography are vulnerable to Shor's algorithm, making quantum-resistant alternatives a priority for modern cryptography. Code-based constructions are among the strongest candidates in post-quantum cryptography. The McEliece and Niederreiter cryptosystems remain competitive after decades of study but suffer from large public keys and reliance on high-order Galois fields, imposing heavy overhead on resource-constrained platforms such as mobile and embedded devices. The symmetric Rao-Nam (RN) construction offers a more compact alternative by operating over smaller fields. This article develops a family of post-quantum cryptographic code constructions (CCC) combining asymmetric McEliece and Niederreiter cryptography with the symmetric Rao-Nam scheme, based on algebraic-geometric codes derived from elliptic curves. It examines systems synthesizing crypto-code constructions with flawed codes to reduce energy costs to GF(2²) while preserving cryptographic strength, alongside an alternative approach using hyperelliptic curves, and CCC built for mobile protocols using LDPC codes. These code families differ structurally: shortened codes tighten parameter control and reduce key material, extended codes raise the minimum distance and robustness, and higher-genus hyperelliptic curves provide richer algebraic structure and stronger security for comparable parameter sizes.

Abstract: Hyperspectral remote sensing instruments record hundreds of narrow, contiguous spectral bands, and the resulting data volumes make compression essential for storage and transmission. Lossless compression preserves this data exactly, which matters for archival and reanalysis, but existing methods largely target redundancy in the original image, leaving the redundancy in residual data after an initial compression pass largely unexploited. The proposed Multi-level Clustering Lossless Compression Algorithm (MLCLCA) is built on an Improved K-means Algorithm (IKA) that exploits the spatial geometry of remotely sensed pixels to prune distance comparisons among distant or already-stable classes, accelerating convergence relative to standard K-means. MLCLCA re-clusters the residue produced at each level and searches, at every level, for the class count that maximizes the lossless compression ratio (LCR). On 220-band AVIRIS hyperspectral data, rounding class centers to the nearest integer outperforms nearest-pixel selection, and differencing class centers across bands and classes reduces inter-band redundancy by roughly two orders of magnitude. Extending clustering from one level to nineteen raises the LCR from 2.51 to 2.84, a 13.1% relative gain, though about 89% of it is already realized within the first eight levels, showing that residue redundancy, while substantial, is finite. These results indicate MLCLCA is best suited to archival and reprocessing scenarios that prioritize compression ratio over real-time throughput.

Abstract: The word agent now names systems as different as a tool-using language model and a chemotactic cell, with no shared definition across the fields that use it. We argue that these uses converge on one substrate-independent structure—the algorithmic agent: model-mediated regulation built from an implicit or explicit Modeling Engine, a scalar Objective Function, and a Planning Engine. Its root is algorithmic persistence: a pattern whose compressed identity survives the filter of time. Holding a pattern bounded under perturbation takes a load-bearing regulator somewhere in the pattern–world system—a structure that absorbs, cancels, or exports what interaction would otherwise let accumulate. By the Algorithmic Regulator Theorem, this regulator shares mutual algorithmic information with the world: it carries a model of what it regulates and can be read as-if acting through an objective and a planner. The pattern is an agent only when this regulator is localized within it (self-regulation), and telehomeostatic only when the regulator’s objective is the pattern’s own persistence. Regulation in macroscopic systems acts on coarse-grained, many-to-one variables. It is therefore irreversible andexacts a Landauer cost — a price that scales with the coarse-graining and vanishes for reversible, equilibrium persistence (e.g., an isolated atom). Hence the thesis: a macroscopic agent is a persistent pattern that conserves its own bounded code through a thin, thermodynamically costly boundary, in a world that, when closed and reversible, conserves algorithmic information up to the fixed description of its law and time index. Its conserved quantity is therefore algorithmic—program information, not probability mass—so Shannon entropy, free energy, and heat enter as its coarse-grained faces. We use it to reframe the free-energy principle and evolution: in a collective the parts need not share an objective, so alignment is an objective-distribution problem—the design of local objectives and of the constraints that bound them so that the whole persists, not the search for one correct reward. Whether a persistent collective is itself one agent or many is then read from a landscape of its regulation—distributed agency, not mere integration.

Abstract: Enterprise management systems have historically served as an externalized Master (in McGilchrist's sense): they hold the memory of business acts and their relational logic, but through a procedural paradigm dating to Pacioli (1494) marked by redundancy, rigidity, vulnerability, and reliance on external verification. Until now, humans have supplied the missing Emissary function, correcting inconsistencies through judgment the system itself lacks. Generative AI represents the first externalization of the Emissary, but lacking access to a Master, it invents data to produce coherent output. Rather than equipping AI agents to reconstruct an internal World Model at each query — a computationally expensive approach — this work proposes building an explicit, external Master accessible to generative AI. We formalize this as Modelo G, an algebraic framework separating a declarative Concrete System, which encodes business entities, documents, data, and rules as a relational structure, from an Abstract System, the generative AI, which acts as a grammatical translator interpreting natural language and documents against that structure without needing domain-specific training. Because logic resides in the Concrete System rather than in the model or the prompt, governance, operation, and audit collapse into a single continuous cycle, reversing the usual growth of the gap between formal rules and actual practice. Developed and validated in the domain of enterprise management systems, the model generalizes to other domains of structured information, and achieves through external declarative structure what architectures such as DeepSeek's Engram pursue internally: separating static knowledge from reasoning to avoid its constant reconstruction.

Abstract: The maximum independent set problem asks for the largest set of pairwise non-adjacent vertices in an undirected graph and is NP-hard in general. This paper describes Esperanza, an approximation algorithm that runs in \( O(n + m) \) time on a graph with \( n \) vertices and \( m \) edges. The starting point is a small observation about the bipartite double cover of the input graph: its maximum cut is trivially exact, and the freedom left in choosing that cut, namely flipping whole components, can be spent so that one side of the assignment becomes as small as possible. Projecting the opposite, large side back onto the original vertex set produces a generous candidate solution. The candidate is then repaired: on every edge whose two endpoints both survive in the candidate, the endpoint carrying more conflicts is deleted, with conflict counters updated on the fly. A final pass grows the repaired set into a maximal independent set, taking vertices in increasing order of degree. We prove that the output is always a maximal independent set and that on bipartite graphs its size is at least half the independence number, which gives a ratio of \( 2 \) there. Beyond the bipartite case we prove nothing about the ratio, and we say so plainly. What we offer instead is evidence. In experiments where every optimum was certified by integer programming, a reproducible run over \( 30{,}000 \) instances never produced a ratio above \( 2 \) (the mean was \( 1.049 \)), and the worst ratio we have ever seen is \( 2.5 \), on a small hand-crafted instance whose ratio survives, without growing, two natural amplification constructions. We conjecture that the approximation ratio is bounded by a universal constant, keep \( 5/2 \) only as the current record, examine how each family of graphs behaves relative to the conjectured bound, and sketch linear-time modifications that would eliminate every worst case we know of.

Abstract: The Minimum Vertex Cover problem is NP-hard. Its classical polynomial-time approximation ratio is 2, and Khot and Regev showed that, assuming the Unique Games Conjecture (UGC), no polynomial-time algorithm can achieve a factor 2 − ε for any fixed ε > 0. We present FINDVERTEXCOVER, an algorithm that reduces any graph to a linear-size planar forest core via a weighted Minimum Independent Dominating Set (MIDS) gadget and solves that gadget with an accuracy-controlled Baker-style PTAS whose layering width is k = ⌈1/ε⌉; non-core edges are then covered by a greedy repair step and a final redundancy-pruning p ass. For any fixed ε the algorithm runs in near-linear time and always returns a valid vertex cover. We include a reproducible experiment, stored in the car/ folder, that computes exact optima without MILP through a branch-and-bound maximum-independent-set solver, with K˝onig/maximum-matching certificates on bipartite i nstances. Under the default call ε = 0.1, across 1,718 feasible graphs—the graph atlas through seven vertices, random and structured bipartite families, grids, and random general graphs—every returned cover is valid and the maximum ratio is 7/4 = 2 − 1/4, attained by an explicit eleven-vertex bipartite graph; no instance exceeds 7/4. We therefore conjecture a universal 7/4 approximation ratio, a bound that the experiment shows is tight. If this 7/4 bound were proved, then, under the standard assumption P ≠ NP, the Khot–Regev UGC-based hardness theorem for Vertex Cover would force the Unique Games Conjecture to be false. An open-source implementation is released as the Salvador package (v0.0.6).

Abstract: We present Furones, a linear-time candidate-comparison algorithm for the Minimum Dominating Set (MDS) problem on undirected graphs. The algorithm (version v0.3.8) applies a TSCC-style pendant cascade, solves the reduced instance by a Baker-style routine used only as a validated candidate generator, and compares the lifted candidate against several original-graph candidates: closed-degree coverage, witness sweeps, order ownership, seed completion, a Salvador-style auxiliary, a max-cut double-cover auxiliary, and reverse-delete scans. We prove two unconditional guarantees. First, every normal return is a valid dominating set. Second, because the portfolio contains the dynamic greedy maximum-coverage dominator, every returned set D satisfies |D| ≤ H(∆ + 1) γ(G) ≤ (1 + ln(∆ + 1)) γ(G), where ∆ is the maximum degree and H(k) is the k-th harmonic number; this is a constant factor on bounded-degree graphs and sub-logarithmic in n for sub-polynomial degree. We also record the near-threshold ratio hypothesis that Furones meets ratio max{4, ln n} on all graphs; proving it would imply P = NP. The hypothesis is already proved for all graphs with ∆ ≤ √n/e. Exact benchmarks on 11,000 small instances with exhaustive optimum certificates record zero violations of the conjectured bound.

Abstract: In the Kolmogorov Theory (KT) of consciousness, an algorithmic agent is an information-processing system that compresses sensory data into simpler models to plan actions that optimize an objective function, while operating under limited data access, finite computational resources, and the fundamental limits of algorithmic information theory (AIT). We show how these limitations naturally give rise to probability, Bayesian inference, precision, and emergence. Using a toy example of an agent compressing pages fromalarge library, we recover a weighted multi-model strategy in which probabilistic reasoning and Occam’s razor appear as the agent navigates between models. We then introduce precision—the confidence the agent assigns to its model relative to noisy data—as the second-order quantity that arbitrates the trade-off between trusting the prediction and trusting the observation. We formalize precision as inverse-variance weighting of prediction errors at the Comparator and show what it gives the agent: a principled model-updating process carried out by the Updater (a submodule of the Modeling Engine), in which a confidence-dependent gain determines how much each prediction error revises the model — so that reliable, persistent errors reshape the model while structureless errors are retained as residual noise, and structural learning saturates once the compressible regularity has been captured. We then connect the picture to Karl Friston’s Free Energy Principle and Active Inference, which appear as the variational-Bayesian special case of the bounded-agent story, and flag the main differences rather than collapsing the two. Finally, we propose a formal, agent-centric definition of emergence in terms of coarse-graining and Kolmogorov complexity, and connect it to cellular automata, the renormalization group, and partial models. The result is a unified account in which probability, precision, and emergence are all consequences of an agent’s drive to compress and model a noisy world under bounded resources.

Abstract: Schrödinger (What Is Life?, 1944) and Anderson (More Is Different, 1972) argued thathigher levels of organization obey novel laws not straightforwardly derivable from microscopicones. We make this precise in Kolmogorov Theory (KT), where agents model the world bycompressing coarse-grained data. Emergence here is agent-relative: algorithmic emergenceoccurs when an agent empirically finds a concise, predictive macro-model that it could not havealgorithmically derived from the micro-rules alone. The emergent entity is that macro-model.Beyond a trivial resource barrier (o)—simulation is possible but infeasible—three barriersseparate micro-knowledge from macro-models. (i) A weak barrier: for bounded finite-statesystems an agent can simulate step by step but cannot in general shortcut the simulation.(ii) A strong barrier: with unbounded size, coarse-grained questions encode the haltingproblem and become undecidable. (iii) Our main result, the algorithmic barrier, in twoparts: for generic data no concise macro-model exists (most trajectories are Kolmogorov-random), and even when one exists no algorithm can find it (the structure function isuncomputable). Concise macro-laws are guaranteed neither to exist nor, where they exist, tobe derivable—even for bounded systems. Anderson’s “reduction ̸= construction” is thus acorollary of uncomputability: knowing the micro-laws rarely yields the compressed macro-laws.Effectivemacro-modelingstaysempirical. Favorablesymmetries—renormalization-groupflows,hydrodynamics, some elementary cellular automata—sometimes permit concise descriptions,but as exceptions, not algorithmic guarantees.

Abstract: The Minimum Independent Dominating Set problem (MIDS), equivalently the problem of finding a smallest maximal independent set, is strongly inapproximable: unless P = NP, no polynomial-time algorithm approximates it within any fixed constant factor. This rules out a single universal constant, but it does not rule out constant factors that are specific to structured graph classes. This manuscript studies Siriaisa, the unweighted MIDS algorithm distributed as the siriaisa package, and proves such family-restricted guarantees. Siriaisa removes isolated vertices, processes each component independently, solves an LP relaxation whose values serve only as priority scores for a maximal-independent-set sweep; for every component it assembles a deterministic pool of candidates (two lifted degree-four seeds, the direct LP sweep, six fixed orderings, and single- and paired-seed sweeps) and shrinks each feasible candidate by a bounded local-exchange compression built from reverse-delete and one-add/two-add exchanges, returning the smallest verified independent dominating set. We give two unconditional guarantees. First, every returned set is independent and dominating. Second, because every returned component set is a maximal independent set, it is a ∆-approximation of the component optimum; hence Siriaisa is a constant-factor approximation on every bounded-degree family (factor 2 for paths and cycles, 3 for ladders, 4 for grids, r for r-regular graphs), and it returns the exact optimum, ratio 1, on structurally rigid families of unbounded degree: cliques, stars, complete bipartite graphs, and double stars for all parameters, and crown graphs within the seed-pool range. A separate consequence of the compression is that every returned set is reverse-delete minimal and locally optimal under the implemented one-add and two-add exchanges. We make explicit what is not claimed: the LP ordering carries no universal approximation guarantee, and a universal constant factor would contradict Irving’s theorem unless P = NP, so we claim no graph-independent constant. We validate the theory against exact SciPy MILP optima on a small adversarial DIMACS suite and on a large-scale reproducible harness, the car suite, that runs ten thousand instances across the structured families and random graphs: every instance stays within its family constant, 99.85% are solved exactly, and the largest ratio across the whole suite is 1.20.

Abstract: We present Aegypti, a triangle-detection framework for an undirected simple graph \( G=(V,E) \) with \( n=|V| \) vertices and \( m=|E| \) edges, built on the duality between triangles and independent sets: a triangle of \( G \) is exactly a three-vertex independent set of the complement \( \overline{G} \), which is what a small vertex cover of \( \overline{G} \) leaves uncovered. Aegypti dispatches on density at \( \lceil n^{4/3}\rceil \). When \( m \le \lceil n^{4/3}\rceil \) it runs the exact Chiba--Nishizeki routine, whose \( \mathcal{O}(m^{3/2}) \) cost is \( \mathcal{O}(n^{2}) \) on inputs this sparse. When \( m > \lceil n^{4/3}\rceil \) it covers \( \overline{G} \) with the linear-time Hvala algorithm and reads three uncovered vertices, certified as a triangle in \( \mathcal{O}(1) \). We prove unconditional soundness for both variants. The fast variant is a quadratic-time, one-sided certificate procedure: in the dense regime it is complete exactly when the Hvala cover of \( \overline{G} \) leaves at least three vertices uncovered, a condition the factor-\( 2 \) bound guarantees only for graphs with\( \omega(G) \ge \lceil(n+3)/2\rceil \); its failure to return a triangle is not, by itself, evidence of triangle-freeness. The safe variant adds a Chiba--Nishizeki fallback and is therefore unconditionally complete, with worst-case running time \( \mathcal{O}(n + m^{3/2}) \). We make no claim against any fine-grained lower bound. Empirically, across a deterministic benchmark---random and structured families, adversarial dense families with small clique number, and an exhaustive sweep of all graphs on at most seven vertices---all routines agreed with an exact oracle and the fast dense branch recorded no miss. A public reference implementation is provided in the aegypti Python package, which depends on hvala.

Abstract: We study synchronization transitions in financial markets via persistent homology applied to time-varying correlation networks. Vietoris–Rips filtrations on rolling Mantegna distance matrices (49 Fama–French industry portfolios, 1976–2026) capture one-dimensional homological cycles (H₁) that reflect intransitive sectoral triples—configurations where two pairwise correlations are strong but the third is weak. Exact analytical null distributions for the persistence and count of such intransitive precursors in random metric spaces are derived. The data reveal a two-scale topological response: stress amplifies intransitivity among the most strongly correlated industries while dissolving it among weakly and moderately correlated ones. Because static topological summaries are highly collinear with average correlation, we assess the predictive ability of the momentum of topological reorganization for market stress onset. At short horizons (5–20 days) the momentum of average pairwise correlation dominates, but at 80–100 day horizons the standardized rate of change of the persistence-weighted mean cycle birth (dAvgBirth) significantly outperforms both a correlation-only control and the momentum of average correlation itself. Decomposing cycles into sectoral triples maps abstract topology onto interpretable linkages. The findings show that higher-order topological momentum captures a slow, structural component of stress build-up complementary to fast-moving correlation dynamics.