Papers

We present a constructive atlas of finite simple groups with proofs of their orders, simplicity, and structural properties. The eight completed families are cyclic groups of prime order, alternating groups, the classical series $A_r(q)$, $B_r(q)$, $C_r(q)$, $D_r(q)$, and the exceptional series $G_2(q)$ and the small Ree groups $ {}^2G_2(3^{2m+1})$, $m\geq1$. The fifteen sporadic entries are $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $\mathrm{Co}_1$, $\mathrm{Co}_2$, $\mathrm{Co}_3$, $\mathrm{McL}$, $\mathrm{HS}$, $\mathrm{Suz}$, $J_2$, and $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$, $\mathrm{Fi}_{24}'$. The simple parameter ranges and exceptional cases are stated explicitly. The models arise from codes, lattices, forms, algebras, and finite geometries, including the split octonions and the Conway–Parker algebra. They retain natural actions, stabilizers, central quotients, and comparison maps for subsequent group theory. Several classical comparison isomorphisms relate the models, while involution-class counts distinguish the equal-order orthogonal and symplectic families in odd characteristic and rank at least three. Drawing on classical sources and companion mathematical work, the project develops the construction side of finite simple group theory, not the exhaustiveness proof of the classification. The completed models and stated proofs are formalized in Lean for verification and reuse.

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

The 10th Working Formal Methods Symposium (FROM 2026) was held in Timişoara, Romania on September 15-17, 2026. It was organized jointly by the Faculty of Informatics of West University of Timişoara, the e-Austria institute, and the Institute for Logic and Data Science (ILDS). This event was co-located with the 28th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2026). This volume contains the texts of the 14 papers accepted by the Program Committee for presentation.

Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.

We study how determinant recurrences of banded Toeplitz matrices behave when their Laurent symbols are multiplied. Two classical structures underlie the problem. Clean banded Toeplitz determinants have a root-product description of their recurrences, while a product of two finite Toeplitz sections differs from the finite section of the product by finite-rank corner corrections. We connect these structures by identifying multiplication-induced boundary defects with exterior-degree sectors of the determinant recurrence. To each polynomial core we associate recurrence polynomials for all exterior degrees. Multiplication of cores becomes a graded convolution via composed products, while complementary degrees are related by a scaled reciprocal duality. On the matrix side, one-sided factors produce a defect filtration whose successive pieces are exactly these sectors. For general two-sided factors, independently weighting the two corner corrections gives an interval filtration: increasing either defect order adds one adjacent sector, and the cumulative recurrences are generically minimal. The multiplication-induced same-corner minors used in these filtrations lie in the standard row-column state module, whereas cross-corner imbalance corresponds instead to Laurent recentering. Opposite Laurent recenterings realize the individual sectors directly, and for several factors the admissible recenterings form an explicit lattice polytope. The pentadiagonal case gives the basic $1+4+1$ decomposition. The resulting framework gives a factor-level description of how finite-section boundary effects generate the recurrence spectrum of structured banded Toeplitz products.

Organizations increasingly use LLMs for policy, compliance, risk, and operational decision support, requiring evidence validation, review routing, version control, and auditability. We introduce Policy-as-Skill (PaS), a modular runtime that packages these functions as executable, versioned policy capabilities. Thirteen methods are evaluated with a fixed Gemma4 backend on 600 development tasks. PaS+Audit achieves 53.8% exact accuracy, macro-F1 0.346, review F1 0.854, citation precision 1.000, policy-reference recall 0.984, and audit completeness 1.000, outperforming LLM+RAG on most governance and review metrics. Deterministic control raises aggregate accuracy to 61.2% but is strongly task dependent, supporting selective rather than universal rule-based intervention.

We evaluate a SymPy implementation of the parallel Risch-Norman method using Charlwood's 2008 suite of 50 challenging indefinite integrals. The system automatically builds integrand towers, verifies answers by differentiation, and returns correct, verified integrals for 49 problems with zero errors. The single failure, $\int\arcsin(x\sqrt{1-x^2})\,dx$, is proven non-elementary using a holomorphic-remainder certificate, though limited by an unverified completeness hypothesis on a genus-three curve. Part II's degree bounds successfully reduce classical ansatz sizes by two-thirds without impacting running time. The paper details algorithmic mechanisms like $S'$-units, Pell units, and residue computing in tower coordinates. Compared to mature implementations, this untuned SymPy prototype is slower than FriCAS (by a factor of six on the median integral) but more accurate than AXIOM (which returned three wrong answers). Profiling pinpoints performance bottlenecks in nonlinear norm searches and nested number fields, outlining clear targets for optimisation.

Despite the success of neural models in natural language processing, their black-box nature limits interpretability and conceals the linguistic phenomena underlying their predictions. We present SLITE, an explainable hybrid model for Recognizing Textual Entailment that integrates two complementary layers of semantic analysis: a structural-relational layer, based on semantic compatibility and incompatibility between compositional entities, and a distributional-informational layer, based on structured patterns of information change between embedding-based representations of the premise and the hypothesis. We propose 17 features that combine entity-level semantic relations, polarity-sensitive lexical matching, and alignment measures over semantic sub-representations of the similarity matrix, including measures based on entropy and transfer entropy. A logistic regression trained on these features achieves an accuracy of 83% on three-class SICK and 96% on SICK-CE, outperforming IsoLex by 4 percentage points and falling within 2 percentage points of RoBERTa with a fraction of its computational complexity. Ablation studies and SHAP analysis confirm that structural-relational features are the primary drivers of classification, while distributional-informational features provide essential complementary contributions, particularly for detecting neutrality and contradiction. Our results demonstrate that further exploration of hybrid approaches is a viable and scientifically productive alternative to massive neural architectures, and we hope they will strengthen the dialogue between linguistic theory and computational modeling of inference

$R\Pi\Sigma^*$-extensions form a rich class of difference rings that provide a unified algebraic framework for modeling indefinite nested sums, transcendental products, and nested products over roots of unity structures that frequently appear in combinatorics, number theory, and particle physics. For a large subclass of these extensions whose ring of constants is a field, we introduce a complete reduction approach to resolve the telescoping problem without solving any difference equations. More precisely, we explicitly construct a complement to the subspace of differences over the constant field and develop an algorithm that decomposes any element of the extension into the sum of a difference and a component lying in this complement. Consequently, summability holds if and only if this complementary component is zero. This structural approach yields significant speed-ups for parameterized telescoping and, notably, creative telescoping for deriving linear recurrences of definite sums. Finally, we compute an explicit idempotent representation that extends existing telescoping algorithms and our complete reduction framework to the general class of $R\Pi\Sigma^*$-extensions, opening up previously untreatable classes of sums and products.

Massive three-loop form factors for vector, axial-vector, scalar, and pseudoscalar currents are vital for precision collider phenomenology. Extending the quarkonic framework to the more complex gluonic case, in Ref. (arXiv:2609.22034 [hep-ph]) we computed these contributions widely using advanced computer algebra, automated guessing, and large-scale PSLQ searches. Here, we outline the core strategy of the large-moment method and illustrate its main challenges, including solving record-sized recurrences and differential equations. Our high-precision evaluation provides an exact analytic representation around $s=0$ in terms of MZVs. We achieved a complete analytic series expansion around $s=\pm\infty$ for the first time, which depends on MZVs and three constants out of higher number spaces, along with reliable analytic continuations across $s \in (-\infty, \infty)$, serving as a landmark benchmark for modern symbolic computation.

Chain-of-Thought (CoT) reasoning has been shown to improve the performance of large language models (LLMs), yet existing optimization methods largely rely on outcome-based feedback, leaving the logical validity of intermediate reasoning steps largely unverified. To address the gap whereby LLMs arrive at correct final answers through logically flawed intermediate reasoning chains, we propose LogicTrack, a neuro-symbolic framework that audits reasoning trajectories by auto-formalizing each reasoning step into symbolic representations and verifying it with automated theorem provers. LogicTrack introduces Solver-Based Backtracking Reward (SBR), a step-wise scoring mechanism that quantifies logical soundness and guides backtracking tree search at inference time. We further extend LogicTrack to construct supervised fine-tuning (SFT) data with backtracking traces from its trajectories, enabling fine-tuned models to internalize step-wise auditing as an intrinsic capability. Extensive experiments across 8 reasoning benchmarks and 7 LLMs demonstrate that LogicTrack effectively improves both the verifiability of reasoning chains and final answer pass rate, thereby enhancing overall CoT quality and trustworthiness in high-stakes domains.

When computing with univariate polynomials, two fundamental and related problems are the XGCD and rational reconstruction, classically solved in quasi-linear complexity using the half-gcd algorithm. These problems have various applications in algebraic computations and bear strong connections to linearly recurrent sequences, structured matrices, and continued fractions. This article first gives a collection of algorithmic reductions, showing that rational reconstruction and XGCD can be solved via the computation of bases of relations modulo a freely-chosen polynomial $M(x)$. In particular, one recovers the folklore idea that bases of Padé approximants (i.e., $M(x) = x^d$) can be used to perform quasi-linear rational reconstruction or XGCD, extending to fast algorithms the well-known link between the Berlekamp-Massey algorithm and the extended Euclidean algorithm. One highlight of these reductions is that, instead of approximants, one may rely on Cauchy interpolants (i.e., $M(x)$ vanishes at chosen points). In a second part, this article describes divide-and-conquer algorithms for approximants and interpolants along with complexity analyses showing an explicit leading constant in front of the dominant term. For interpolants, the best leading constant is obtained through a variant that stores polynomials represented by evaluations, and exploits fast extrapolation in order to avoid repeated conversions to the monomial basis; this requires special points, in geometric or arithmetic progression, or FFT points when the base field allows them. Combining the analyses with the reductions leads to the best complexity bounds we are aware of for rational reconstruction and XGCD. Perhaps surprisingly, even Padé approximants or Berlekamp-Massey-like computations, which intrinsically involve $M(x) = x^d$, are accelerated by reducing them to Cauchy interpolation at well-chosen points.

We compute the convex envelope of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm is composed of the following steps: (1) compute the convex envelope of each quadratic piece obtaining piecewise rational functions (quadratic divided by linear function) defined over a polyhedral subdivision; (2) compute the (Legendre-Fenchel) conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision; (3) compute the maximum of all those functions to obtain the conjugate of the original PLQ function as a piecewise quadratic function defined on a parabolic subdivision; (4) compute the conjugate of each resulting piece; and finally (5) compute the maximum over all those functions to obtain the convex envelope (biconjugate) as rational functions (quadratic divided by linear function) defined over a polyhedral subdivision. Our contribution includes a practical algorithm running in linear time, and proving that the convex envelope is a piecewise rational function.

Region-of-attraction (ROA) certificates for polynomial systems become expensive as state dimension and degree grow: direct sum-of-squares (SOS) formulations require combinatorially growing monomial bases. Quadratization represents a polynomial vector field exactly on an invariant manifold of a quadratic system, allowing a quadratic Lyapunov function to certify the ROA. For a fixed lift, stabilizer gains shape the off-manifold extension and transverse dynamics, while representation gauges change the matrix representation but not the vector field. Both affect the spectral-norm certificate, yet prior work fixes the gain by a feasibility heuristic before optimizing the gauge. We formulate optimal dissipative quadratization (ODQ), jointly designing gains and gauges for a fixed monomial lift, reference extension, stabilizer factorization, and Lyapunov weight $Q=I$. Gains lie in a prescribed compact Hurwitz box. At each gain, an exact semidefinite program globally minimizes the spectral-norm bound over the gauge. Residual-aware bounds yield a certified closed Lyapunov sublevel set, accounting for the floating-point Lyapunov residual. Under our stated assumptions, every accumulation point of the idealized outer search is box-Clarke stationary. A finite run returns the best independently verified candidate; global optimality of the gain search is not claimed. On a planar quintic, optimizing the gain increases the certified area by a factor of $2.238$ over a matched zero-gain gauge. Across 16 heterogeneous polynomial systems with stabilizer freedom, ODQ improves on both fixed-gain lifted baselines. All 36 ODQ runs on the relay benchmark complete, and all 27 repeat-level comparisons across nine fully paired cases favor ODQ over an SOS baseline with a fixed quadratic Lyapunov function in both the fixed-direction proxy and construction time. Broader comparisons with direct SOS methods remain mixed.

We investigate an algebraic approach to the Syndrome Decoding Problem, based on a reformulation of the Hamming weight constraint and its integration with the Information Set Decoding paradigm. We begin with a systematic analysis of the Hamming variety, deriving its defining equations in terms of elementary symmetric functions. Since these equations may have high degree, we exploit convolution identities for elementary symmetric functions, together with factorizations based on Lucas' identity, to derive an equivalent formulation with auxiliary variables and equations of bounded degree. Building on this modeling, we generalize the ISD paradigm through an ISD-like decoding strategy, implemented by the GBDecode algorithm, in which only a subset of an information set is fixed. This approach reduces the size of the combinatorial search space at the cost of solving the associated multivariate nonlinear systems. To handle this algebraic component, we employ the MultiSolve algorithm, which replaces a single Grobner basis computation with a collection of computations on simpler systems, obtained by exhaustively assigning a varying number of indeterminates over the finite field. This provides a tunable balance between combinatorial search and algebraic solving. We evaluate the resulting approach experimentally on instances of the Syndrome Decoding Problem for random binary linear codes, using parameters corresponding to the NIST Security Category 1 parameter set of the Classic McEliece cryptosystem. The experiments assess the feasibility of this combinatorial-algebraic approach and provide insights into the practical behavior of Grobner basis techniques within an ISD-like decoding framework.

We prove that the tensor rank of $3\times3$ matrix multiplication over $\mathbb F_2$ is at least $21$. The structural proof, independently developed by Qiushi Engine, converts occupation constraints on a single tensor factor into algebraic relations coupling all three factors. Certified quotient-rank bounds and finite geometry force any hypothetical $20$-term decomposition to have first-factor matrix-rank profile $(16,1,3)$. The ranks of the corresponding split-flattened summands therefore sum to $27$, exactly the rank of the full split flattening. Equality in rank subadditivity forces their images to form a direct sum; normalization by the inverse flattening then makes the summands pairwise annihilating idempotents. An explicit product identity for matrix multiplication implies that at most one first factor can be invertible, contradicting the three forced by the profile. The same obstruction constrains $22$-term decompositions attaining the split-rank bound. The complete proof, including the finite quotient bounds, is formalized in Lean. The accompanying research trajectory records Qiushi Engine's long-horizon autonomous research, from numerical experiments and quotient constructions to the structural proof.

The computation of row echelon form is one of the main bottlenecks in the F4 algorithm. Several state of the art implementations use a probabilistic algorithm attributed to Monagan, Pearce, and Steel to accelerate this computation. Despite this, no bound on the probability that the algorithm returns an incorrect result appears to be available. In this paper, we provide such a bound. Furthermore, building on this result, we propose a Las-Vegas variant of the F4 algorithm and show experimentally that it can outperform deterministic F4 on some classical examples.

In Part I we extended the structure theorems underlying the Risch–Norman (parallel Risch) method to a simple radical extension $L=K(y)$, $y^m=q$, of a differential field $K=F(t_1,\dots,t_n)$ closed under the derivation. Here we remove the closure hypothesis: the radical may occupy any position in the tower, so that the derivatives of the generators above it involve $y$ — the setting of Bronstein's algorithm for mixed elementary functions. The working ring is $\cA=\cO[t_{j+1},\dots,t_n]$, the integral closure of $F[t_1,\dots,t_n]$ in $L$: a Krull domain, free over the polynomial ring on Trager's basis, so that all factorisation remains in a unique factorisation domain. The denominator of the derivation is no longer an element but a divisor $\fd_D$ on $\cA$, and the valuation lemma takes the unified form $v_P(Dg)=v_P(g)-(1+v_P(\fd_D))$ at normal height-one primes, subsuming the shifts $\{1,e_P\}$ of Part I; the proof localises and requires no cancellation analysis. Stability of the class group and the unit group, $\Cl(\cA)\cong\Cl(\cO)$ and $\cA^*=\cO^*$, splits the admissible logands into $S$-units of $\cO$ — computed by the machinery of Part I — and irreducible polynomials moving in the upper variables, whose residues must be constants. We prove degree bounds in the top variable and describe the resulting algorithm, which — unlike the classical parallel method, whose failure proves nothing — returns certificates of non-elementarity in two situations: a residue outside the constant field, and, when every bound in force is proved, a residue-free remainder that the linear system shows to be non-exact.

We study replacements in fixed bilinear tensor decompositions, counting changes to complete rank-one summands, including output factors. The shortening frontier records the maximum rank defect of a fixed-size subset and determines the minimum length attainable within a change budget. For the rational 48-term Li–Wang–Hu decomposition \(D(2)\) of \(4\times4\) matrix multiplication over \(\mathbb{C}\), we prove rank radius at least 12, strong radius exactly 11, and border radius at least 8. Every shorter complex decomposition therefore changes at least thirteen original summands. An exact rational twelve-term replacement attains the equal-length barrier. The proofs combine exhaustive support reductions with saturated projected kernels and zero-corner completion arguments controlling arbitrary minimal competitors. A reduced-incidence argument transfers kernel certificates to tensor-space neighborhoods. A Laurent normal form gives strong radius exactly 11 for the sixteen-term core at every nonzero complex parameter. On a nonempty Zariski-open subset of the actual parameter curve, the rank radius is at least 12, the strong radius exactly 11, and the border radius at least 8. We also prove incomparability of the full Kothari–Moitra–Wein sufficient criterion and the Sylvester-equipped kernel criterion. These results describe local decomposition structure rather than a new rank bound for full matrix multiplication.

As the use of Large Language Models moves from chatbots into agentic systems, where outputs become actions with irreversible consequences on reality, the existing paradigm on AI Interpretability research, post-hoc interpretability, is structurally inadequate for safe and trustworthy model deployment: it explains behavior after the fact but cannot audit or intervene in an inference computation before it commits to an output. We therefore argue for a shift toward generative interpretability, an architectural property under which a model's inference pass natively exposes semantically meaningful checkpoints that are human-understandable and amenable to causal intervention. We show the merits of generative interpretability as comparison to other interpretability research paradigms, and propose Neuro-Symbolic Models as a concrete instantiation.

We study the rational quadratic differential equation \[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay^2+by+cx}{y^2}, \qquad a,b,c\in\C, \] under the non-degeneracy assumptions \[ c\neq 0,\qquad 2ay+b\not\equiv 0. \] Equivalently, after clearing the denominator, we consider the polynomial vector field \[ \dot{x}=y^2,\qquad \dot{y}=ay^2+by+cx. \] We give a complete, directly checkable classification of its Darboux curves. If $a=0$, no non-constant Darboux polynomial exists. If $a\neq 0$, a non-constant Darboux polynomial exists if and only if \[ c=-ab\qquad\text{or}\qquad c=-2ab. \] In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively \[ y-ax,\qquad y^2-2bx. \] Consequently every non-constant Darboux polynomial is a non-zero constant multiple of a positive integral power of the corresponding irreducible factor. We then place this classification in the framework of Riccati (R-)integrability and rational potentials. Excluding the exceptional branch $c=-2ab$, the analytic integrability theorem identifies $c=-ab$ as the branch admitting a global Riccati-type analytic first integral. For $c=-2ab$, we compute the first four transverse variational groups along the transformed Darboux divisor over the rational function field. Their dimensions are $1,2,3,4$; the fourth is the full group of invertible fourth-order transverse jets. We prove that this exceptional branch admits no Riccati first integral and hence is not R-integrable. Finally, we relate the branch $c=-ab$ to the Quartic Inverse Riccati (QIR) class and discuss an invariant-based classification problem for quartic Abel equations.

We present a fast algorithm for multiplying two 4x4 matrices using 48 multiplications and 204 other operations (addition, subtraction or scaling by a constant) over any ring containing an inverse of 2. Applied recursively, this algorithm reaches a cost bound with leading term (7+3/8)n^{\log_4(48)}. Alternative basis decompositions of this algorithm further reduce this to (6+13/32)n^{\log_4(48)}, with square change of bases, and then to (5+2/3)n^{\log_4(48)}, using rectangular encodings with inner dimensions 24 and 32. The conversion costs of these alternative basis variants are included as lower-order terms.

Direct evaluation of the defining product in the real Cayley–Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow–Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow–Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

The discovery of the EML operator, sufficient to evaluate the standard explicit purely transcendental elementary functions, has led to considerable interest and discussion across multiple scientific disciplines. However, most authors have focused on the binary EML itself, while numerous similar variants with slightly different properties are now known. This article attempts to close this gap by enumerating and classifying them. We also take this opportunity to clarify common misconceptions related to the EML operator. The principal goal, symbolic regression within an architecture as close as possible to proven neural networks which combine matrix multiplication with a single univariate non-linear activation function, remains beyond reach. Instead, we propose a Möbius layer, with rational functions replacing matrix operations, and showcase the recently discovered activation function eml(x,1/x), which allows exp(x) and ln(x) to be recovered separately, and hence all elementary functions to be evaluated within a rational generalization of the neural network.

Polynomial modular reduction is central to binary finite-field arithmetic and repeated Frobenius powering. Top-down shift/XOR folding can use few operations when the nonleading support is sparse, but a tap near the leading term creates a long feedback chain. We formulate this recurrence as inversion of a nilpotent shift operator and factor its inverse by characteristic-two Frobenius powers. The resulting Frobenius-factorized reduction (FFR) applies to every monic binary modulus without materializing a reciprocal or dense reduction matrix, and its shifts can be generated online without a persistent modulus-specific schedule. For degree $m$ and nonempty nonleading support of size $s$, with nearest-tap distance $\Delta_{\min}$, FFR uses exactly $\lceil\log_2(m/\Delta_{\min})\rceil$ sequential feedback stages and has scheduled work $O(ms(1+\log(m/s)))$. A portable-C evaluation on 1,096 supports through degree $131072$ identifies distinct FFR, López–Dahab, and gf2x-backed Barrett regions. On four certified irreducible moduli, FFR makes complete Rabin irreducibility testing $1.35$–$8.04$ times faster than NTL and $1.60$–$6.81$ times faster than the matched Barrett implementation.

Every data-driven equation discovery method assumes (implicitly and without verification) that the target system obeys a single governing law ($K{=}1$). Here we show that this assumption is the primary bottleneck limiting scientific discovery in multi-mechanism systems, and introduce Physical Law Ecology, a framework that makes $K^*$ (the number of coexisting independent mechanisms) itself the first quantity to be determined from data. The framework automatically mines a pool of topologically distinct candidate equations, constructs a continuous dominance weight field across parameter space, and discovers analytic evolution laws governing mechanism succession—with optional monotonicity constraints encoding irreversible physics. Across four unrelated systems (elastomer mechanics, pool boiling, galactic dynamics, and droplet evaporation), BIC consistently identifies $K^*{=}3$ independent governing topologies. Applied to 163 SPARC galaxies (3,269 spatially resolved measurements), the framework autonomously recovers three gravitational laws whose coexistence provides evidence against the single-universal-acceleration hypothesis of MOND ($p<10^{-34}$). In engineering applications, multi-law weighted prediction reduces error by 67-72% over single-equation baselines while retaining full interpretability. By establishing the determination of $K^*$ as the zeroth step of scientific discovery-prior to and independent of equation search—this work opens a direction orthogonal to existing symbolic regression: not finding better equations, but mapping the ecology of mechanisms that govern complex systems.

Consider a parametric ODE control model. A function of the states and parameters is called observable if its value can in principle be reconstructed from input-output data. The observable functions form a field, called the observation field, represented naturally by a set of generators. Even when the model is not fully observable, this field captures the information still accessible from input-output data. We present an algorithm for computing a concise generating set for the observation field of a model with rational dynamics. The algorithm relies on two new results: one allows observable functions to be extracted from the coefficients of repeated Lie derivatives of the outputs, while the other reduces the required orders of differentiation by exploiting identifiable parameter combinations. We implement the resulting algorithm in StructuralIdentifiability$.$jl (https://github.com/SciML/StructuralIdentifiability.jl). For computational efficiency, we employ recent techniques for differential elimination and rational function field simplification. Using models from epidemiology, chemical kinetics, and cancer modeling, we show that the algorithm produces generators with domain-specific interpretations that can inform model analysis and development.

We determine the maximum rank of $4\times 4\times 4$ and $k\times 4\times 3$ multidimensional arrays over the finite field $\mathbb{F}_2$. These results are made possible by a new method of enumerating non-isomorphic tensors.

Humans can learn and generalize novel concepts from sparse data because they express knowledge in rich structural formats. In this paper, we propose that programs are a strong candidate for universal representation of concepts. We review computational models of concept learning that use programs as their concept representation and evaluate their contribution toward a universal representational language.

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.