Papers

We consider the classical additive measurement-error model $X=Y+Z$, where the latent random variable $Y$ has unknown distribution $F_Y$ and the error $Z$ has a known distribution. We develop direct estimators for three functionals of $F_Y$: (i) $F_Y(x)$ at continuity points; (ii) interval probabilities $F_Y(y)-F_Y(x)$ when $x<y$ are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require $F_Y$ to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.

We prove that the q-model category of $\mathcal P$-multipointed d-spaces for $\mathcal P\in\{\mathcal G,\mathcal M\}$ is not left proper by constructing a weak equivalence whose pushout along a q-cofibration obtained by attaching a single 1-dimensional globular cell is not a weak equivalence. The counterexample is constructed in the category of $\Delta$-Hausdorff $\Delta$-generated spaces. For $\mathcal P=\mathcal G$, all objects are automatically saturated, and for $\mathcal P=\mathcal M$, the original weak equivalence is between saturated objects. The failure is caused by a family of nonconstant execution paths that converges in the ambient mapping space to a constant map which is not an execution path; after the globular cell is attached, this degeneration causes two previously distinct path components to merge.

In this paper, we establish two improved Liouville-type theorems for the three-dimensional stationary tropical climate model without imposing any temperature assumptions. Compared with previous results, we achieve significant improvements on the growth assumptions imposed on $u$, $v$ and $\nabla v$. Owing to the special structure inherent to the equations, we fully exploit the $L^2$-estimate for the oscillation of the temperature, refined interpolation techniques and an iteration argument to establish a Liouville-type theorem under flexible and relaxed power-law growth conditions. As an interesting consequence, we obtain the triviality of solutions under the integrability conditions imposed only on $(u,\nabla v)$ or $(\nabla u,\nabla v)$. Furthermore, by developing a systematic framework to handle the energy function associated with the non-trivial solutions and with the aid of delicate ODE analysis and a contradiction argument, we refine our Liouville-type theorem by introducing logarithmic corrections to the growth rates.

Let $f$ be a non-CM newform of weight $k\geq4$, level $N$ and trivial Nebentypus. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and denote by $\boldsymbol f^{(p)}$ the $p$-adic Hida family passing through $f$. Assuming very specific instances of two general conjectures in arithmetic algebraic geometry (injectivity of $p$-adic Abel-Jacobi maps, positive definiteness of archimedean height pairings à la Gillet-Soulé), we prove that, for all but finitely many $p$ as above, if the analytic rank of $f$ is $1$, then all but finitely many specializations of $\boldsymbol f^{(p)}$ of even weight and trivial Nebentypus have analytic rank $1$. This result provides evidence for Greenberg's "minimality conjecture" on analytic ranks in families of modular forms.

We prove a Hilbert space-valued analogue of Pego's compactness theorem on compact groups. For square-integrable functions taking values in a Hilbert space rather than in the complex numbers, we show that a bounded family is precompact exactly when it is simultaneously well behaved in two complementary senses: its members do not change much under small translations of the group, and their Fourier coefficients decay uniformly across the family. This equivalence holds without restriction when the Hilbert space is finite-dimensional, and it specializes to the known scalar-valued theorem when the Hilbert space is just the complex numbers. We then construct an explicit example showing that the equivalence genuinely breaks down once the Hilbert space is allowed to be infinite-dimensional. To repair this, we introduce a uniform tightness condition and we show that under this extra hypothesis the equivalence is restored regardless of the dimension of the Hilbert space. Along the way we establish the Plancherel isometry, the Hausdorff-Young inequality and its inverse for this vector-valued Fourier transform.

We prove that the sharp $\frac{1}{2}$ lower bound for the Haraux function holds for every maximally monotone operator of type~(NI) on an arbitrary real Banach space. This extends the result established in reflexive Banach spaces to arbitrary real Banach spaces. We also establish an exact decomposition at each graph point, showing that the local contribution to the Haraux function and a nonnegative residual sum to one half of the weighted squared displacement. Due to the equivalence between type~(NI) and quasidensity, this decomposition also yields the sharp bound without requiring a graph point at which the residual vanishes. Moreover, for every operator with a nonempty graph, this decomposition yields a lower bound involving the residual infimum, and for maximally monotone operators it further yields a new characterization of type~(NI) in terms of the Haraux function. Finally, on $c_0$, we give a maximally monotone operator of type~(NI) for which the residual infimum is zero at some target but is not attained. This shows that the existence of a graph point at which the residual vanishes is strictly stronger than the vanishing of the residual infimum required in our proof.

We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.

For the 2-D incompressible Euler equation, we prove the existence of $m$-fold symmetric uniformly rotating vortex patches with 90-degree corners for all $m\geq 12$. Properties of these patches and the induced stationary flows in the co-rotating frame are characterized. Our construction is based on a novel fixed-point method.

Neural ODE reduced-order models often achieve comparable local prediction accuracy, yet their long-horizon extrapolation behavior can differ substantially. To analyze this discrepancy, we develop a path-integral identity that separates local discrepancy injection from amplification in the learned latent dynamics. The associated multi-step Jacobian norms quantify transport sensitivity and distinguish different propagation regimes. Experiments on the Burgers and Gray–Scott systems exhibit two distinct patterns of error evolution. In Burgers systems, prediction errors remain bounded and are primarily associated with persistent local discrepancies. In contrast, Gray–Scott systems exhibit pronounced amplification during extrapolation, where Jacobian norms serve as sensitivity diagnostics rather than direct indicators of physical prediction accuracy.

We propose a robust barycenter for distribution-valued data by incorporating the Huber loss directly into the optimal transport cost. In contrast to metric-space Huber means, which apply the Huber loss to the Wasserstein distance after optimization, our construction acts on individual transport displacements, preserving quadratic behavior locally while limiting the influence of large displacements. The resulting Huber-Wasserstein barycenters form a natural interpolation between Wasserstein means and $L^1$-type Wasserstein medians. We establish the analytical and statistical foundations of this construction. For optimal transport with Huber loss, we prove regularity and uniqueness properties of dual potentials, existence of optimal transport maps, and stability as the Huber parameter varies. For the associated barycenter problem, we prove existence and characterization results, consistency of empirical plug-in estimators, and a finite-sample breakdown point essentially equal to $1/2$. In dimension one, we further derive the pointwise influence function and asymptotic distribution, quantify the associated robustness-efficiency trade-off, and show that displacement-wise Huberization can retain first-order information that is lost by distance-based Huberization under localized shape contamination. Numerical experiments on contaminated distribution-valued data demonstrate the robustness of the proposed barycenters and illustrate their interpolation between mean- and median-like behavior.

We consider the problem of shifting two tokens placed on nonadjacent vertices $u,v$ of a graph $G$ on $n$ vertices to two nonadjacent vertices $u',v'$ of $G$ using a sequence of token movements. In each step, a token is moved from the vertex it is on to a neighbour of that vertex, ensuring that the tokens remain on nonadjacent vertices after this move. We answer a question of Briański, Felsner, Hodor, and Micek [“Reconfiguring Independent Sets on Interval Graphs”, MFCS 2021] by showing that if the two tokens can be moved from their initial position to their final position, then it can be done using at most $4n$ moves.

We show the existence of ultralimits of sequences of Hajlasz-Sobolev mappings $f_i:X_i\to Y_i$ when $X_i$ converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the mappings pmG-converge to the ultralimit in a suitable sense, and obtain Arzela-Ascoli theorems in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for Hajlasz-Sobolev maps $f:(X,\mu)\to V$ into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.

In this paper, we solve risk functional minimization problems with general losses in Banach spaces within the framework of stochastic approximation. We introduce an adaptive regularization strategy for stochastic mirror descent (SMD). Specifically, using a Schauder basis, we construct nested finite-dimensional subspaces whose dimensions adapt to the sample size $n$ to balance approximation and optimization errors. We solve each SMD subproblem over the corresponding subspace and project the stochastic subgradient onto its dual space. This yields closed-form solutions and coordinate-wise coefficient updates with low computational and storage complexity. For risk minimization in $\mathcal{L}^p$ spaces with $1<p<\infty$, we construct Bregman distances adapted to the uniform convexity of the underlying Banach space using $\max\{2,p\}$-convex functionals. At the $\mathcal{L}^1$ endpoint, we instead construct a locally strongly convex functional based on the entropy function. By developing a new analytical framework, we show that the risk functional values converge to the minimum at a rate of $\mathcal{O}\left(n^{-\min\{\frac12,\frac1p\}}\right)$, up to logarithmic factors. In the misspecified setting, we prove convergence under weaker regularity assumptions and derive explicit polynomial rates under suitable regularity conditions. Finally, we apply the method to statistical inverse problems and illustrate its empirical performance through numerical experiments in both settings.

We prove uniqueness in the Lorentzian Calderón problem in a semiglobal setting, comparing a potentially large perturbation of the Minkowski metric against a small one. Our proof uses a reduced amount of data, solving a formally determined version of the problem. In contrast to traditional approaches, it employs neither control-theoretic arguments nor a reduction to a geometric inverse problem via microlocal methods or high-frequency solutions. Instead, it relies on distorted plane waves and weighted $L^2$-estimates.

We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de~Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de~Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect \(n|h|^2-H^2\) with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.

The main goal of tomography is the reconstruction of density function out of its line integrals (integral measurements of this density along x-ray lines). Such construction is possible and known as inverse x-ray/Radon transformations. We are interested in generalization of this problem to the case of particles. For this we need to limit the speed of these particles, otherwise the problem is reduced to the previous one. The question we discuss in this paper is how low can this speed be depending on the parameters of the studied potential. We study this problem using simple theoretical examples and Machine Learning pipeline to restore the potential.

Physical neural networks perform learning through the intrinsic nonlinear dynamics of matter. Optimising their design presents a considerable challenge: complex many-body physics can provide powerful computation, but are expensive to simulate and large experimental optimisation runs are impractical to fabricate. Hence, the high-dimensional space of possible network topologies cannot be effectively directly searched. Here, we show that this search can be efficiently performed in an abstract graph space that is vastly cheaper to explore. Using random lasing networks – composed of interconnected nanoscale waveguides and hosting strongly coupled lasing modes – as an exemplar physical vision system, we establish a quantitative three-layer link: simple graph-theoretic metrics predict the nonlinear lasing physics, which in turn predicts vision performance. After validating this relationship using physical simulations, we exploit it to drive an evolutionary algorithm using graph metrics, producing network topologies that outperform random designs at a fraction of the computational cost (3000$\times$ speed-up compared to physical simulation). On simulated image-classification tasks, graph-optimised networks substantially improve classification accuracy. As our framework operates on network topology rather than substrate-specific physics, we anticipate it can transfer to other network-based physical learning systems, providing an efficient route for the directed design and optimisation of complex, strongly-interacting physical neural networks.

A Deza graph is a regular graph in which the number of common neighbours of two distinct vertices takes at most two values, regardless of adjacency. Its children are the graphs on the same vertex set in which adjacency is determined by these two common-neighbour counts. A Deza graph is called strongly Deza if both children are strongly regular. We construct an infinite family of edge-regular strongly Deza graphs using non-degenerate quadratic forms over the field with five elements. For every odd dimension greater than three and each of the two determinant square classes, we obtain a graph on the projective points represented by vectors of norm one. Its children are complementary strongly regular graphs with the parameters of the corresponding orthogonality graph on non-isotropic points and its complement. We determine the parameters by counting solutions to systems involving the associated bilinear form. These counts also yield symmetric association schemes over finite fields of odd characteristic. Further constructions include Deza graphs in dimension four, orthogonality graphs in odd dimensions, unions of relations in even dimensions over the field with nine elements, and an odd-dimensional family over the field with thirteen elements. We also give low-dimensional examples, including one whose children are a triangular graph and its complement.

Neural combinatorial optimization (NCO) solvers report the best of many sampled solutions per instance, and the sample count is, by convention, identical for every instance. Whether a non-uniform allocation of a fixed total budget would buy anything has not been measured. We measure it, and we audit the measurement itself. First, on in-distribution workloads the allocation headroom is not detectable. Across three pretrained solvers (POMO, AM, SymNCO) on uniform TSP-100, an oracle allocation computed and evaluated on the same stored samples reports a 2.2-2.6% gain with intervals excluding zero; measured out of sample the same gain is indistinguishable from zero (0.457, 0.015, -0.512 percent). Following the customary in-sample procedure, all three solvers would have supported a published 2%-level gain that does not exist. We calibrate this bias against an instance-wise null in which the true gain is zero by construction; over the ranges we test it does not shrink with more samples or more instances. Second, the same correction that removes the phantom gains preserves a real one. Under distribution shift (a workload mixing uniform and clustered instances), a pre-registered confirmatory experiment finds that allocation guided by held-out sample statistics improves best-of-k by 11.5% (AM, primary endpoint; 95% CI [7.4, 19.7]) and 12.0% (SymNCO, replication) at equal evaluation budget, with the signal-acquisition cost not charged; a pre-registered negative control (POMO, an order of magnitude more robust to shift) shows -0.3% [-0.7, 0.24]. The gain exceeds a frozen distribution-label baseline by 4.2 points [1.9, 7.7]. An exploratory policy charging a 20-sample probe against the same budget retains 3.4% (AM) and 4.6% (SymNCO). We give a correction procedure and a reporting checklist, and release all data, code, and the pre-registration record.

For a $T_1$-space $X$, let $Cld(X)$ denote all its nonempty closed subsets and $T_4(X)=\{(F_1,F_2)\in Cld(X)^2:F_1\cap F_2=\emptyset\}$. In terms of closed sets, $X$ is $T_4$ iff for each $(F_1,F_2)\in T_4(X)$, there is a pair of closed sets $\phi_1(F_1,F_2)$ and $\phi_2(F_1,F_2)$ such that their union is $X$ and $F_j\cap \phi_j(F_1,F_2)=\emptyset$ for $j=1,2$. Thus, in this paper, for a topology $\tau$ on $Cld(X)$, we introduce the definition: A $T_1$-space $X$ is called $CT_4$ with respect to $\tau$ if the above maps $\phi_j:T_4(X)\to Cld(X)$ are continuous on $\tau$. Similarly, for $i=1,2,3$, we can define a $T_1$-space to be $CT_i$ for $\tau$. We only consider the Vietoris topology on $Cld(X)$ and show that every $CT_4$-space is countably compact, and every $CT_3$-space is a Fréchet-Urysohn space, every separable subspace of a $CT_3$-space is metrizable. We give relevant examples. Any finite-dimensional cubes, the infinite-dimensional cube, any finite-dimensional spheres, and all 0-dimensional compact metrizable spaces are $CT_4$. All infinite discrete spaces, all finite-dimensional Euclidean spaces and all countable limit ordinal spaces are $CT_3$ but not $CT_4$. Also, each metrizable space with a unique non-isolated point is $CT_3$, and is $CT_4$ if it is compact. Moreover, the infinite sum of $CT_3$ spaces is $CT_3$ but not $CT_4$. Every countable space with a unique non-isolated point is $CT_2$, and it is $CT_3$ if and only if it is metrizable. All subfields of real numbers and their complement spaces are $CT_2$ but not $CT_4$; and their $CT_3$ status remains unclear. All uncountable ordinal spaces are not $CT_2$. The one-point compactification of any uncountable discrete space is not $CT_2$. $CT_1$ and $T_1$ are equivalent, hence all spaces above are $CT_1$. Open Problems: is there a non-metrizable $CT_3$ or $CT_4$ space? Is any compact $CT_2$ space $CT_3$ or $CT_4$?

We consider a branching random walk on \(\Z^d\) in a random environment given by Bernoulli site percolation with parameter \(p\in (0,1)\). In this model, each particle located at an open site reproduces according to a law \(\mu_\circ\), whereas a particle at a closed site reproduces according to another law \(\mu_\bullet\). Each newly born child performs an independent simple random walk jump from the position of its parent. We study the quenched survival probability under various assumptions on \((\mu_\circ, \mu_\bullet)\) and establish a Yaglom theorem when both offspring distributions are critical.

We tropically enumerate planar well-spaced elliptic curves with a fixed $j$-invariant. Combined with the recent genus 1 correspondence theorem of Cela–Koyama, this yields an alternative proof of Pandharipande's algebraic enumeration.

Using patch-matrix-product methods, we study two reversible three-state interaction-round-a-face cellular automata introduced by Klobas and Prosen [J. Phys. A 55, 094003 (2022)] - the species-preserving and species-flipping rules - and a coherent quantum deformation interpolating between them. Within an explicit translationally invariant ansatz, the two classical rules share a one-parameter family of quasi-local conservation laws with an auxiliary-dimension-three realization. Every regular member is also the first centered logarithmic derivative of an analytic auxiliary-dimension-two family passing through the identity observable. On the closed span of this family and the three elementary local charges, the only nonzero Euler velocities are $\pm\sqrt{3/23}$. Consequently, $\sqrt{3/23}$ is a rigorous lower bound on the maximal Euler speed in any larger conserved sector. For the species-flipping rule, this value agrees with the reported extrapolated value $0.361$ of Klobas and Prosen, strongly supporting completeness of the known sound-active charges. The species-preserving rule admits, in addition, a staggered generating family whose logarithmic derivatives form an infinite tower of strictly local charges. In the quantum deformation, the same algebraic structures yield exact low-bond-dimension invariant states (scar candidates), exponentially long-lived quasi-local quasimodes, and diagonal quasi-local charges that produce a nonzero Mazur bound after projection away from the three elementary local charges.

Zero-knowledge (ZK) proofs certify that a message belongs to an allowed semantic class without revealing the message, but the certificate compares a high-dimensional embedding against class centroids, so its cost grows with the embedding dimension $d$. A Johnson–Lindenstrauss (JL) projection lowers $d$ to $m\ll d$ while preserving pairwise distances, yet a random JL matrix must be committed and its sampling proved inside the circuit, which is costly and a leakage risk. We construct a public deterministic projection from the standardized orbit of a Pisot $\beta$-transformation, analyzed through the spectral gap of the $\beta$-map, the geometric decay of its correlations, rather than equidistribution. We prove that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is $V_0/m$ with a constant $V_0$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory. A single public seed preserving all pairwise centroid distances therefore exists and is found by search. Against six standard projections, including the chaotic-sequence matrix of Yu et al., the construction matches statistical quality to within measurement noise, and it is the only one simultaneously free of in-circuit randomness and exactly reproducible in a fixed finite field at a per-step cost $\log_2\beta$ rather than $2^{k}$.

This paper derives the second-order positive Burgers' equation from the standard Burgers' hierarchy to explore its complete integrability and exact analytical solutions. We construct this higher-order nonlinear evolution equation by systematically applying the recursion operator to the classical Burgers' equation. Expanding on this structural framework, we then derive the explicit third-order equation and use Complete Bell Polynomials to generalize the $n$-th order hierarchy. Through the Cole-Hopf transformation, the second-order nonlinear equation rigorously maps to the linear third-order dispersion equation. We establish complete integrability by explicitly formulating the scalar Lax pair (zero-curvature representation), which allows us to directly derive the associated differential and algebraic Darboux transformations. To systematically classify explicit stationary, time-dependent traveling wave, and self-similar profiles, we apply several techniques to the linear domain: separation of variables, the Hirota perturbation method, traveling wave reduction via the Complete Discrimination System for Polynomial Method (CDSPM), and Lie similarity reduction. Crucially, our analysis demonstrates the exact finite truncation of the Hirota perturbation series. We conclude by outlining how this equation impacts the theoretical understanding of fluid dynamics, nonlinear transport phenomena, and higher-order wave propagation.

We study a cooperative search game with $N$ lockers, $N-1$ labelled objects, one empty locker, and $N-1$ players. Player $i$ seeks object $i$ and may inspect at most two lockers; the second inspection may depend on the content of the first. The players may coordinate beforehand but receive no information about the searches of other players after play begins. We prove that the maximum probability that every player finds the assigned object is $I_N/N!$, where $I_N$ is the number of involutions of $N$ elements. An optimal strategy represents the blank by the common fictitious symbol $N$ and follows pointers: player $i$ first opens locker $i$, then opens the locker whose number was observed. The upper bound holds for every deterministic or randomized adaptive strategy and follows from a deletion lemma and two recurrences. We give explicit examples, identify the three-door case with an isomorphic "Return of Monty Hall" game, and report a reproducible mixed-integer verification for $N=2,3,4,5$. The computation is independent of, and not needed for, the proof.

For an $n$-vertex graph $G$ and a permutation $\pi$ of its vertex set, let \[ I_G(\pi)=|E(G)\cap E(G_{\pi})|,\qquad \mu(G)=\min_{\pi} I_G(\pi), \] where $G_{\pi}$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $\pi(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $\mu(G)\ge k$. Erdős recorded a construction of Mullin showing $f(n,5)\le 2n-2$ and asked whether equality holds for sufficiently large $n$. We prove that it does: \[ f(n,5)=2n-2 \] for all sufficiently large $n$. The proof strategy is a core–buffer–completion framework: it moves the few high-degree vertices into carefully chosen low-degree positions, confines the allowed overlap to this bounded part, and then relabels the sparse remainder without creating any additional common edge.

We audit the exact-arithmetic certificate in arXiv:2310.19781v2, its 2025 Communications in Mathematical Physics version, and their shared MATLAB archive. CMP corrects none of the audited items; we therefore treat both versions together and use the openly accessible arXiv source for exact provenance. We found 11 proof-affecting defects (alongside typographical slips) and give reproducible exact counterexamples to several implemented bounds and reconstruction steps. To the best of our knowledge, this is the first independent, version-pinned, source-level audit of a published computer-assisted proof (CAP) in analysis to identify multiple proof-affecting defects in its released computational certificate. The findings do not refute the intended theorem or its analytic reduction, but they show that the published certificate does not prove the claimed conclusion. Future work will give a corrected, structurally simpler proof with substantially less machine assistance.

Machine learning theory studies learning procedures through mathematical setups in which the data model, training protocol, oracle access, loss, metric, and randomness define the phenomenon that a theorem is meant to explain. Solving an open problem therefore requires the problem formulation, theorem target, and proof mechanism to be developed in concert. Researchers formulate hypotheses, test them through preliminary theoretical or empirical analysis, and refine both assumptions and proofs. We investigate whether this process can be organized as an autonomous agentic workflow for ML theory research. We develop VALG, an agentic system that combines multi-level Verification, Adaptive formulation of Learning-theory problems, and Graph-structured proof development. Within each source-relative theorem branch, VALG maintains a fixed mathematical specification, checks the theorem-level composition of a typed proof-dependency graph, and constructs and reviews local proofs in dependency order. When a proof attempt fails, VALG identifies whether the obstruction lies in a derivation, the proof structure, or the theorem formulation and routes the next attempt accordingly. Formulation-level obstructions initiate an explicitly related variant or relaxation, preserving the mathematical relation between the resulting theorem and the source problem. We evaluate VALG on nine subproblems from five COLT 2026 open problems. Two runs produce internally finalized theorem candidates that match the scope of their source briefs; the remaining seven yield restricted-method results, special cases, or conditional theorems. These case studies show how VALG keeps source-scope matches, relaxations, conditional results, and blocked attempts mathematically distinct. VALG is open source at https://github.com/DechenZhang/VALG-ML-Theory-Agent.

Scheduling follow-up Computed Tomography (CT) examinations requires balancing two competing objectives: assigning patients as close as possible to their recommended examination dates while ensuring an equitable distribution of radiologists' workload. Existing approaches optimize scanner utilization or patient waiting times, overlooking reporting activities and the need to balance fairness across multiple stakeholders. This paper proposes a predictive-prescriptive framework for fairness-aware follow-up CT scheduling. Patient-specific Machine Learning (ML) models are first developed to predict both examination and reporting durations. These predictions are then embedded into a multi-objective Mixed-Integer Linear Programming (MILP) model that simultaneously minimizes deviations from patients' preferred examination dates and balances radiologists' reporting workloads through a lexicographic min-max fairness criterion. We derive a dominance reduction property within an $\varepsilon$-constraint framework that substantially reduces the number of optimization problems required to generate the Pareto frontier. Computational experiments based on data from a real-world emergency radiology department show that allowing patients a scheduling flexibility of only one to two days is sufficient to substantially improve workload equity among radiologists while preserving timely access to follow-up examinations. The proposed dominance reduction strategy eliminates most $\varepsilon$-constraint evaluations without affecting the Pareto frontier. Finally, evaluating predictive models through downstream optimization regret demonstrates that XGBoost provides the most effective support for scheduling decisions, outperforming models that achieve lower prediction errors according to conventional predictive metrics.