Papers

Let $G$ be a graph. Under a strong edge coloring of $G$, every color class is an induced matching. The strong chromatic index of $G$, denoted by $\chi'_s(G)$, is the smallest integer $k$ such that $G$ admits a strong edge coloring with $k$ colors. Denote by $\Delta(G)$ the maximum degree of $G$. In this paper, we prove that every graph $G$ with $\Delta(G)\le 6$ satisfies $\chi'_s(G)\le 57$, improving the best known upper bound $60$.

In the Kourovka Notebook, V.\,D.~Mazurov asked whether an automorphism $\alpha$ of prime order $m$ of the infinite free Burnside group $G=B(m,q)$ of prime period $q$ that cyclically permutes the free generators of $G$ must have a nontrivial fixed point. In this paper, we show that for $m\ge3$ the answer is negative for all sufficiently large prime periods $q$. Precisely, we prove that for any odd $m\ge3$, there exists $q_m>0$ such that for every prime period $q>q_m$, the automorphism of $B(m,q)$ of order $m$ that cyclically permutes its free generators is fixed-point-free.

Every group $G$, of arbitrary cardinality, admits a right-invariant metric with at most five values whose isometry group is exactly the group $\Xi(G)$ of permutations preserving every inverse-pair difference. Five is sharp. Writing $N(G)$ for the least number of values, including zero, we obtain $N(G)=1$ for the trivial group, $N(G)=2$ for $C_2$ and $C_3$, $N(G)=3$ for every other group outside an explicitly listed set $\mathcal{E}$ of 24 finite groups, $N(G)=4$ on $\mathcal{E}\setminus\{C_2^3, C_3^2\}$, and $N(C_2^3)=N(C_3^2)=5$. The exact result combines Cayley-graph realization theorems with verified finite certificates and hand proofs of the lower bounds. Separately, a uniform constructive 32-value bound uses subgroup-entry ranks, sign-variation conflict labels, and short-word rigidity, but no Cayley-index input or computation. In Knill's right-translation sense, a right-invariant metric on $G$ is naturalizing when every group law on its underlying set with isometric right translations is isomorphic to $G$; a group is natural when it admits such a metric. We prove that an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. Every natural group admits a naturalizing metric with at most five values, and this bound is sharp.

A dominating set $D$ of a nontrivial connected graph $G$ is called a semitotal dominating set of $G$ if every vertex in $D$ is at distance at most two from another vertex in $D$. If, in addition, $D$ is an independent set, then $D$ is called an independent semitotal dominating set of $G$. The (independent) semitotal domination number of $G$ is the minimum cardinality among all (independent) semitotal dominating sets of $G$. In this paper, we obtain closed formulas for these parameters in the following three well-known graph operators defined from a connected graph: the subdivision, middle, and central graphs.

The one-dimensional swarmalator model is usually understood through three macroscopic states: synchrony, the phase wave, and asynchrony. We show that the identical model also contains a much larger set of hidden stable equilibria - families of q-twisted and m-clustered states - which random initial conditions almost never reach, because many are embedded in a degenerate neutral manifold of vanishing global basin. They are nonetheless locally robust: under Gaussian perturbations of per-coordinate amplitude {\epsilon} their survival radii shrink only algebraically, as 1/q and 1/m, in sharp contrast to the Gaussian basins and winding ceiling of twisted states on a nearest-neighbour Kuramoto ring, both removed by mean-field coupling. A block-circulant Jacobian fixes the stability in closed form, and far-kick experiments indicate the basins are compact cores rather than tentacled sets. Several patterns reported in extended swarmalator models thus already exist, stable or neutrally stable, in the minimal one.

Most existing studies on distributionally robust optimization (DRO) with a decision-dependent ambiguity set focus on the computational and theoretical challenges posed by this class of problems. In this paper, we adopt a combined modeling and computational perspective to understand the trade-offs between modeling fidelity, solution quality, and computational effort, particularly in comparison with DRO models using decision-independent ambiguity sets. Motivated by representative applications in joint pricing-stocking newsvendor problems with price-dependent demand and facility location problems with location-dependent demand, we consider a two-stage stochastic mixed-integer program with (non)convex continuous recourse. Assuming a finite sample space, we model the decision-dependent distributional ambiguity with a polyhedral ambiguity set and reformulate the problem as a nonconvex mixed-integer nonlinear program. To efficiently solve the reformulations, we propose decomposition-based cutting-plane algorithms. Our experiments on benchmark instances indicate that a decision-dependent ambiguity set substantially reduces the postdecision disappointment for the newsvendor and out-of-sample cost for the facility location problems, up to 65% and 7% on average, respectively; thereby mitigating the optimizer's curse relative to a decision-independent DRO. However, this improvement is achieved at the expense of increased computational effort, with the absolute runtime ratio falling between 2 and 25 for medium- to large-sized newsvendor instances and between 2 and 12 for facility location instances.

We study the asymptotic and TQFT-type properties of the Kashaev–Luo–Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a possibly incomplete hyperbolic structure, we express the exponential decay rate and the 1-loop term of the partition function in terms of, respectively, the volume and the adjoint twisted Reidemeister torsion of the corresponding incomplete hyperbolic structure. Consequently, these asymptotic formulas hold for any angle structure having the same peripheral angular holonomies as the geometric one. We further introduce a Reshetikhin–Turaev-type function associated with certain admissible Neumann–Zagier data, extending the Jones function in Teichmüller TQFT previously studied by Ben Aribi and the author. We show that, after a combinatorial normalization, for any triangulation, the KLV partition function can be expressed as a weighted integral of the squared norm of the Reshetikhin–Turaev-type function, where the weight is determined by the angular holonomies of peripheral curves. In particular, for FAMED triangulations, the Reshetikhin–Turaev-type function agrees with the Jones function, so that the normalized KLV partition function is obtained by integrating the squared norm of the integrand defining the Teichmüller TQFT partition function. This provides a noncompact analogue of the relationship between Turaev–Viro and Reshetikhin–Turaev invariants.

We characterize uniform factor-balancedness in Arnoux-Rauzy words, generalizing the work of Espinoza, Popoli, and Stipulanti on ternary alphabets to finite alphabets of any size. In particular, we prove that an Arnoux-Rauzy word $\mathbf{x}$ is uniformly factor-balanced if and only if $\mathbf{x}$ has bounded weak partial quotients.

Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).

By homogenizing high-contrast periodic metric graphs, we create continuous media that admit bound states produced by local defects at energies embedded in the continuous spectrum. This is achieved by fitting two copies of a lattice graph simultaneously into space and coupling them by connecting edges. The homogenization of an ODE on a graph to a PDE in space is achieved by representing the associated operators in a common functional space through interpolation by Fourier analysis. The two coupled graphs result in an effective medium with internal degrees of freedom that allow a decomposition into two orthogonal spaces of hybrid states. Critical high contrast retains microscopic resonances in the effective medium and allows a spectral band of one hybrid state space to overlap a spectral gap of the other. This serves as the mechanism for creating bound states at frequencies embedded in the continuous spectrum of the effective medium. These bound states in the continuum persist for $\varepsilon$-scale metric graphs due to $O(\varepsilon)$ operator-norm estimates for reduced resolvents on the defect region.

A graph $G$ is det-extremal if $|\operatorname{det} A|=\operatorname{per} A$ for its adjacency matrix $A$. Det-extremal cubic bipartite graphs arise in the study of Pólya's permanent problem, and McCuaig characterized the $3$-connected ones as vertex-sums of copies of the Heawood graph. The total domatic number of a graph is the largest number of pairwise disjoint total dominating sets. Characterization of the cubic graphs with total domatic number $1$ has been a long-standing open problem. In this paper, we prove that a connected cubic graph is det-extremal if and only if its total domatic number is $1$. We further show that McCuaig's characterization extends to all $3$-connected cubic graphs, and that every connected det-extremal cubic graph has girth $3$, $5$ or $6$. We also prove that a connected det-extremal cubic non-bipartite graph has at least $28$ vertices, and that this bound is best possible. Through this correspondence, these results carry over to cubic graphs with total domatic number $1$. In addition, in the language of configurations, our results imply that every triangle-free $3$-configuration has a blocking set.

Many economic applications establish stability of Markov dynamics using the monotone mixing condition (MMC) of Hopenhayn and Prescott (1992). Working in the same setting as that paper, we introduce a weak monotone mixing condition (WMMC), phrased in terms of the reversal of rank between populations, and show that it is strictly weaker than the MMC and both necessary and sufficient for global stability under the Kolmogorov metric. We apply these results to a small open economy version of the Aiyagari model, showing how the WMMC can be used to establish global stability in a setting where the MMC fails. In addition, we show that, when the state space is one-dimensional, the MMC and WMMC coincide, implying that the original MMC is necessary as well as sufficient in this setting.

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis–Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka–Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson–Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

We show that the percolation components on a square grid can be $5$-colored as a factor of iid so that neighboring components get different colors. Above the critical probability, we observe that these regions can be 4-colored. Along the way, we extend the classical correspondence between fiid process and measurable labellings of the Bernoulli shift to this quotient setting, prove a general coloring result about hyperfinite pmp planar graphs, and clarify some foundational issues about how to define planarity for Borel graphs.

Let $X$ be a smooth projective complex surface with numerically trivial canonical class. With the help of GPT-5.6 Sol, we prove Qin's conjecture that the reduced generating series of intersection numbers of Chern characters of tautological bundles against the total Chern class of $X^{[n]}$ is a quasimodular form with the predicted mixed-weight bound. The main ingredient is a Wick's theorem-type formula for computing traces of normally ordered products of Nakajima operators on $\bigoplus_n H^*(X^{[n]})$. Following the argument of Li-Qin-Wang, the computation of the reduced generating series is reduced to computing the constant term of the supertrace of a product of certain operator-valued currents and the Carlsson-Okounkov operator. Our trace formula shows that this supertrace can be expressed in terms of two quasi-elliptic functions $\widehat Z$, $P$ and a quasimodular function $T$, whose constant terms are quasimodular forms by the theorem of Goujard-Moller. Finally, we give an explicit algorithm for computing the general reduced generating series.

Neural surrogates can accelerate large-scale optimization by replacing expensive or intractable model components with efficient learned approximations, but solving the resulting embedded problems can remain prohibitively costly. For instance, standard exact encodings of neural networks with ReLU activations allow the problem to be solved by mixed-integer solvers, but add large numbers of binary variables to accommodate the nonlinearity of the activations, which can render the problem computationally prohibitive. To address this, we propose DCEmbed, a heuristic for optimization problems with embedded neural surrogates that leverages the difference-of-convex (DC) representation of the network and avoids adding activation binaries. Exploiting shared structure within the DC representation of a ReLU network, we derive a reduced-size, exact formulation for its convex components that can be embedded in optimization problems using just two linear inequalities and one continuous auxiliary variable per hidden neuron. Using this, the problem is solved via an iterative penalty convex-concave procedure, where only the concave portions of the neural terms are approximated at each stage. The original objective, constraints, and any discrete decisions are retained, allowing standard convex or mixed-integer optimization solvers to optimize the host and surrogate jointly at each iteration. In experiments on quadratic programs, mixed-integer resource allocation, and neural two-stage stochastic programming, our method demonstrates much faster progress toward high-quality feasible solutions than approaches using exact mixed-integer embeddings. In particular, DCEmbed achieves $4\times$ lower normalized primal integral than the best exact baseline on the resource allocation problem, while in two-stage stochastic programming it reaches the global surrogate optimum $\sim 5\times$ faster than Gurobi ML.

We establish Bourgain–Brezis selection maps for Hodge systems in new ranges of Besov and Triebel–Lizorkin spaces. Our approach combines the trace-free Beurling–Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz'ya's $\Phi$-inequalities. These methods yield selection maps in Besov spaces $\dot B^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1\leq q\leq p\leq 2$, and in Triebel–Lizorkin spaces $\dot F^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1<p\leq q\leq 2$. The theorems hold for every dimension $n\geq 2$ and every form degree $1\leq l\leq n-1$. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain–Brezis problem and underscore the importance of the trace-free Beurling–Ahlfors transform in its resolution.

We can associate a compactness principle $\mathcal K_{\mathcal{D}}$ to the Constraint Satisfaction Problem (CSP) with constraint library $\mathcal{D}$. These principles were studied by Katáy, Tóth, and Vidnyánszky and by Rorabaugh, Tardif, and Wehlau. We expand their work comparing the strength of $K_{\mathcal{D}}$ for varying structures $\mathcal{D}$. We characterize the structures whose compactness principles are provable from ZF; these turn out to be the width-1 structures. We compare some important compactness principles, namely those of $2\text{SAT}$, $3\text{LIN}2$, and $K_2$, settling a question of Katáy, Tóth, and Vidnyánszky. And, we find an infinite chain and an infinite anti-chain of compactness principles related to directed cycles and finite choice.

Conformal prediction provides marginal coverage guarantees, yet practitioners may wonder if the observed coverage is truly abnormal or consistent with sampling variation. Inference for realized coverage has received comparatively little attention, especially for time series. We study split conformal prediction with adjacent calibration and test sets of temporally dependent data. Using the functional dependence measure, we derive non-asymptotic bounds on marginal coverage error without mixing assumptions, which can be difficult to verify and may fail even for simple short-memory models. We establish a Bahadur representation to derive, to our knowledge, the first central limit theorem for realized coverage of split conformal prediction under temporal dependence. A consistent block-based estimator of the standard error yields an asymptotically justified test. We further study long-memory time series, which remain understudied in conformal prediction. For Gaussian linear processes, we show how very strong temporal dependence can lead to a non-Gaussian limiting law of realized coverage and establish block-sampling inference with an estimated normalization. The resulting theory explains how temporal dependence changes coverage uncertainty.

We study conformalized quantile regression and conformalized median regression that fit a model on one block of a time series and calibrate the conformal interval on the adjacent block. The existing theory of conformal prediction for time series rests largely on mixing conditions, which are hard to verify from a time-series model and fail for many standard processes, including simple ones with short memory. We replace this theoretical toolbox with the functional dependence measure, which in principle accommodates long-memory observations. The accuracy of the conformal interval length for time series has been understudied. To the best of our knowledge, this paper is the first work that establishes non-asymptotic coverage guarantees and accuracy of interval length simultaneously for split conformal regression on time series. Furthermore, for Gaussian linear processes with long memory, where both the estimation of the center and its calibration converge slowly, we establish a sharper rate for the length error. We show that the calibrated length converges faster than the estimated center itself, and provide a matching lower bound for the usual centers when the calibration block is sufficiently large relative to the training block. To our knowledge, this is the first theoretical analysis of conformal interval length dedicated to long memory.

We present an interactive computational framework for exploring the torus parametrization of the one-dimensional Fibonacci quasicrystal and its local hull. While cut-and-project schemes and Fourier analysis of local functions have been actively explored, a purely algebraic treatment still requires deeper geometric intuition regarding the underlying dynamical systems to advance these methods further. We develop an open-source interactive Python/Jupyter tool that links coordinates in the fundamental domain of the embedding lattice directly to the physical tiling and word generation, offering a transparent visual analysis of translation orbits and singular boundary cases. The tool is the basis for our further computational work, and it can also be used in teaching and to illustrate the geometry of the local hull.

We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $\Delta$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\partial$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $\nu$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1/2$ but $\gamma_{p}=\frac{d_f+p-1}{pd_w}$, where $d_f=\log_{3}5$ and $d_w=d_f+1$. For $p\in[1,2]$, we prove the weak-type reverse Riesz inequality $\|(-\Delta)^{\gamma_p}f\|_{L^{p,\infty}(m)}\lesssim\|\partial f\|_{L^p(\nu)}$. The corresponding strong estimate fails for $1\le p<2$, whereas the weak estimate fails for $2<p<\infty$, even after heat regularization. For the Riesz transform $\mathcal{R}_p=\partial(-\Delta)^{-\gamma_p}$, we show that $\mathcal{R}_1$ is not of weak type $(1,1)$ but that $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^{p,\infty}(\nu)$ for $p\in(1,2)$, and from $L^{p,1}(m)$ to $L^p(\nu)$ for $p\in(2,\infty)$. However, if $p \ge 1$, $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^p(\nu)$ only for $p=2$.

We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the $\kappa$-Hurewicz Conjecture, and related questions on $\kappa$-Menger and $K_\kappa$-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact $\kappa$, we show that there is a model of ZFC in which the $\kappa$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $\kappa$. Moreover, any model in which $\kappa$ is weakly compact and the $\kappa$-Hurewicz Conjecture holds must satisfy $2^{\mathfrak{b}_\kappa} = 2^\kappa$. We also study the existence of $\mathfrak{b}_\kappa$-scale sets which are $K_\kappa$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no $\mathfrak{b}_\kappa$-scale set is a $K_\kappa$-space. Consequently, under this assumption, every $\mathfrak{b}_\kappa$-scale set yields an alternative example of a $\kappa$-Menger space which is not a $K_\kappa$-space.

Mean-field control barrier functions (MF-CBFs) enforce swarm safety through constraints on the agents' distribution. Previous formulations, including applications to stochastic coverage and shepherding, use first-order differential inequalities. These cannot directly enforce constraints whose barrier functions yield control-free first derivatives. Thus, we develop the theory of higher-order MF-CBFs, which ensures the positivity of the barrier functionals through higher-order differential inequalities, as in their finite-dimensional analogs. More specifically, we introduce the notion of relative degree for mean-field safety functionals and develop an analytic framework for computing their higher-order derivatives. Moreover, for practical examples of cross-correlation and self-correlation functionals, we show that repeated differentiation along mean-field dynamics is structure-preserving and reduces to kernel recursions. As an illustration, we apply higher-order MF-CBFs to double-integrator swarm tracking and avoidance problems, where position-only constraints lead to MF-CBFs of relative degree two.

We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.

A submetric space is a topological space equipped with a continuous metric that generates a metric topology weaker than the original one (e.g., a separable Hilbert space with the weak topology). In this paper, we systematically investigate the compactness-related properties of submetric spaces, which make this class of topological spaces a particularly suitable tool for probabilistic applications. As an illustration, we discuss in detail the compactness of the closure of a càdlàg trajectory taking values in a submetric space.

Guessing Random Additive Noise Decoding-Aided Macrosymbol (GRAND-AM) provides a coding-based solution for non-orthogonal multiple access (NOMA) systems, enabling joint multiuser detection and error correction. GRAND-AM outperforms prior methods in symbol error rates while providing hard-detection output. We introduce soft-output GRAND-AM (SOGRAND-AM), a macrosymbol-level joint multiuser detection and decoding framework which generates calibrated bitwise a posteriori error probabilities. SOGRAND-AM extends GRAND-AM by enabling soft-decision outer forward error correction decoders, transparently integrating with existing decoders. SOGRAND-AM achieves approximately 2 dB improvement in bit error rate at the output of the outer soft-input decoder compared to hard input with GRAND-AM.

For cocompact Fuchsian groups $\Gamma \leq \hbox{PSL}(2, \mathbb{R})$ we improve Selberg's bound for the error term of the counting function in the hyperbolic lattice counting problem, achieving an essentially optimal upper bound for the error term \begin{equation*} E(X;z, w) = O(X^{1/2+\varepsilon}) \end{equation*} for almost every pair of points $z,w$. We extend this pointwise result for cocompact lattices acting on the $n$-dimensional real hyperbolic space $\mathbb{H}^n$. Moreover, in the $2$-dimensional case we study the second moment of the error term of the counting function. We improve the upper bounds of Chamizo and Cherubini for almost all pairs $z,w$, but we disprove a conjecture of Phillips and Rudnick for the case of cocompact Fuchsian groups.

Optimization layers enable the incorporation of structured constraints and decision problems into learning systems. Training such systems requires differentiating through the embedded optimization problem, which can be challenging for general conic programs. We introduce dOPT, a solver-agnostic framework that, rather than differentiating the full conic formulation, reduces it at a computed primal-dual solution to an equality-constrained quadratic program that preserves the reference solution and its first-order sensitivity. The reduction captures the local first- and second-order conic geometry relevant to differentiation and remains well defined at singular configurations. Computing solution derivatives then requires a single symmetric linear solve, independently of the forward solver. We derive explicit reductions for convex NLPs, QPs, SOCPs, and SDPs. Numerical experiments validate the computed gradients and show favorable backward-pass scalability, with substantial speedups over existing differentiable conic optimization methods as problem size increases.