Mathematics

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

A team at Eindhoven University of Technology organized LLM-assisted Dafny verification into a ten-step workflow in which Claude Code acts as a verification construction agent, producing specifications, implementations, invariants and proofs under human supervision. The workflow yielded machine-checked Dafny implementations for 37 algorithms from graph theory, labelled transition systems and computational geometry, including the Hopcroft–Tarjan triconnectivity test; the authors report many verifications within hours to a few days, but ran no controlled comparison against human-led verification.

A team at Eindhoven University of Technology organized LLM-assisted Dafny verification into a ten-step workflow in which Claude Code acts as a verification construction agent, producing specifications, implementations, invariants and proofs under human supervision. The workflow yielded machine-checked Dafny implementations for 37 algorithms from graph theory, labelled transition systems and computational geometry, including the Hopcroft–Tarjan triconnectivity test; the authors report many verifications within hours to a few days, but ran no controlled comparison against human-led verification.

Planning and control of differentially flat systems rests on a flat output, yet no general procedure writes one down for arbitrary nonlinear dynamics. FlatSeek reverses the reconstruction relation in the definition of flatness: fix an output and its reconstruction maps, close the dynamics differentially, and obtain system–output pairs with known certificates at scale. A Qwen3.5-2B model fine-tuned with LoRA proposes candidates that a symbolic verifier accepts or rejects. FlatSeek solves 74 literature and 171 synthetic systems, against 58 and 127 for the strongest traditional symbolic method.

Planning and control of differentially flat systems rests on a flat output, yet no general procedure writes one down for arbitrary nonlinear dynamics. FlatSeek reverses the reconstruction relation in the definition of flatness: fix an output and its reconstruction maps, close the dynamics differentially, and obtain system–output pairs with known certificates at scale. A Qwen3.5-2B model fine-tuned with LoRA proposes candidates that a symbolic verifier accepts or rejects. FlatSeek solves 74 literature and 171 synthetic systems, against 58 and 127 for the strongest traditional symbolic method.

A non-convex objective is written into the potential term of a photonic waveguide array, where light evolves by quantum tunneling and the output intensity distribution marks where the minima sit. Experiments ran on one- and two-dimensional waveguide arrays, preparing a low-energy input state with incomplete local annealing and then adding a varying Hamiltonian to speed the search and resolve the global minimum. The authors report quantum speedup over classical stochastic gradient descent, though the main text frames that comparison as an illustrative exploration-behavior baseline.

A non-convex objective is written into the potential term of a photonic waveguide array, where light evolves by quantum tunneling and the output intensity distribution marks where the minima sit. Experiments ran on one- and two-dimensional waveguide arrays, preparing a low-energy input state with incomplete local annealing and then adding a varying Hamiltonian to speed the search and resolve the global minimum. The authors report quantum speedup over classical stochastic gradient descent, though the main text frames that comparison as an illustrative exploration-behavior baseline.

When a pretrained flow-matching model must satisfy an equality constraint supplied only at inference time, common practice repeatedly corrects predicted endpoints, and those discrete corrections can disturb the learned physical structure. MAFM instead evolves a persistent endpoint anchor continuously under an ordinary differential equation, separating model-guided tangential motion from normal attraction toward the constraint. On five PDE benchmarks—Poisson, Burgers, reaction–diffusion, Euler and Navier–Stokes—MAFM meets the prescribed constraint tolerance and achieves lower sample-level relative error than baselines that meet the same tolerance.

When a pretrained flow-matching model must satisfy an equality constraint supplied only at inference time, common practice repeatedly corrects predicted endpoints, and those discrete corrections can disturb the learned physical structure. MAFM instead evolves a persistent endpoint anchor continuously under an ordinary differential equation, separating model-guided tangential motion from normal attraction toward the constraint. On five PDE benchmarks—Poisson, Burgers, reaction–diffusion, Euler and Navier–Stokes—MAFM meets the prescribed constraint tolerance and achieves lower sample-level relative error than baselines that meet the same tolerance.

Three large language models independently reproved all 465 propositions of Euclid's Elements, and their citation choices differ systematically from Heath's human-proof baseline. Mean pairwise Jaccard similarity among three Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs, so the difference is stable and reproducible. Its concrete signature is that about 40% of AI citations point to foundational elements such as definitions, common notions and postulates, against only 13% in human proofs.

Three large language models independently reproved all 465 propositions of Euclid's Elements, and their citation choices differ systematically from Heath's human-proof baseline. Mean pairwise Jaccard similarity among three Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs, so the difference is stable and reproducible. Its concrete signature is that about 40% of AI citations point to foundational elements such as definitions, common notions and postulates, against only 13% in human proofs.

Financial scenario pipelines often run diffusion samplers at very few function evaluations, where time-grid discretization error can quietly distort terminal reports such as covariances, tails and drawdown proxies. TAPES treats grid construction as a goal-oriented numerical problem: Richardson comparisons estimate each step's signed local defect, a terminal-report adjoint transports those defects through the remaining discrete flow, and the squared norm of their sum is minimized before a capped direct refinement. Across six budgets on Regime-SV and Mag7, TAPES gives the lowest design- and audit-report errors in all 24 dataset–budget–report rows within the reported five-pipeline panel, with cross-budget mean relative audit-error reductions of about 10–11% and about 1%.

Financial scenario pipelines often run diffusion samplers at very few function evaluations, where time-grid discretization error can quietly distort terminal reports such as covariances, tails and drawdown proxies. TAPES treats grid construction as a goal-oriented numerical problem: Richardson comparisons estimate each step's signed local defect, a terminal-report adjoint transports those defects through the remaining discrete flow, and the squared norm of their sum is minimized before a capped direct refinement. Across six budgets on Regime-SV and Mag7, TAPES gives the lowest design- and audit-report errors in all 24 dataset–budget–report rows within the reported five-pipeline panel, with cross-budget mean relative audit-error reductions of about 10–11% and about 1%.

Lower bounds on mutually orthogonal Latin squares (MOLS) have long lagged far behind upper bounds at orders that are not prime powers. Using established techniques including difference matrices, quasi-difference matrices, orthogonal-array lifting and projective-plane point selection, OpenAI's GPT-6 Astra constructed larger MOLS sets at 33 orders between 22 and 494, raising their lower bounds; at order 22 it produced 4 MOLS for the first time, so combined with existing results every order from n=22 upward is known to admit at least 4.

Lower bounds on mutually orthogonal Latin squares (MOLS) have long lagged far behind upper bounds at orders that are not prime powers. Using established techniques including difference matrices, quasi-difference matrices, orthogonal-array lifting and projective-plane point selection, OpenAI's GPT-6 Astra constructed larger MOLS sets at 33 orders between 22 and 494, raising their lower bounds; at order 22 it produced 4 MOLS for the first time, so combined with existing results every order from n=22 upward is known to admit at least 4.

High-accuracy gravitational-wave source modeling has long avoided machine learning because neural networks carry a noise floor. Researchers trained a graph neural network as a preconditioner for an iterative solver, leaving the traditional GMRES to enforce final accuracy while the network only accelerates convergence. On the high-frequency elliptic equations for asymmetric-mass black hole binaries, the hardest parameter regions need 2 to 15 fewer GMRES iterations than basic ILU and 2 to 4 times less wall time than the next best method.

High-accuracy gravitational-wave source modeling has long avoided machine learning because neural networks carry a noise floor. Researchers trained a graph neural network as a preconditioner for an iterative solver, leaving the traditional GMRES to enforce final accuracy while the network only accelerates convergence. On the high-frequency elliptic equations for asymmetric-mass black hole binaries, the hardest parameter regions need 2 to 15 fewer GMRES iterations than basic ILU and 2 to 4 times less wall time than the next best method.