Papers

We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.

Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith–Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span$\{1,q(D),q(K)\}$, where $D$ and $K$ are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator $D+iK$ gives a counterexample to the Smith–Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.

We prove a canonical limit theorem for the macroscopic Feynman cycles of finite-volume ideal Bose gases. Cycles carry marks in a general Polish space $\mathsf{M}$, encoding spatial, geometric, spectral, or internal data. After removing a deterministic background density $\rho_{\mathrm{bg}}$, the marked macroscopic cycle process converges in the canonical ensemble to a marked Poisson–Kingman bridge of total mass $\rho - \rho_{\mathrm{bg}}$. The bridge is constructed from a marked Poisson point process with intensity $x^{-1}\eta_x(dm)\,dx$, conditioned on total mass~$\rho - \rho_{\mathrm{bg}}$, where the kernel $x \mapsto \eta_x$ and its total-mass profile $\phi(x) = \eta_x(\mathsf{M})$ are determined by the low-energy spectral data visible on the scale $j \sim V_L$. When $\phi$ is constant, the bridge reduces to a Gamma bridge and the ranked cycle lengths follow the Poisson–Dirichlet law. We verify this for the ideal Bose gas in dimension $d > 2$ under periodic, Dirichlet, and Neumann boundary conditions: in all three cases $\phi \equiv 1$ and the ranked lengths converge to $\mathrm{PD}(0,1)$, while the mark kernels distinguish the three models through their winding, killed-bridge, and reflected-bridge geometry. When $\phi$ is not constant, the bridge is no longer Gamma and the ranked lengths are not Poisson–Dirichlet. As a concrete example, a critical double-well potential whose tunnelling splitting satisfies $V_L \Delta_L \to \gamma$ gives $\phi_\gamma(x) = 1 + e^{-\beta\gamma x}$; more generally, a finite-type visible spectrum with $Q$ components yields $\phi(x) = \sum_{r=1}^{Q} \theta_r e^{-\beta\lambda_r x}$. These results identify Poisson–Kingman bridges as the canonical universality class for marked macroscopic Bose cycles, with the visible low-energy spectrum selecting the particular bridge.

In this paper, we follow Pan-Wu's idea [PW26] to prove local existence of the capillary wave kinetic equation. To be more specific, we will linearize the nonlinear operator and decompose it into dissipative part and bounded part. The main difficulty comes from the fact that the collision kernel has less symmetry compared with the one in gravity water wave. To overcome it, we are required to choose and pair each term delicately.

We mathematically model the dynamics of the number of migratory fish observed at a fixed location along a river in a random environment. Particularly, as a new approach, we construct a stochastic differential equation that incorporates the influence of environmental factors on the fluctuations in the start and end of migration. The model is a diffusion bridge with a non-Lipschitz diffusion coefficient, called the Cox-Ingersoll-Ross bridge, and has random initial and terminal times arising from time-change, so that the influences of environmental factors can be efficiently incorporated. The well-posedness of the model is first established, which is considered novel and significant in applied mathematics. Second, we estimate the parameters of the model based on the latest multiyear daily data set for the upstream migration of Plecoglossus altivelis altivelis (Ayu) by relying on the hypothesis that water temperature affects the migration of the fish, which has been suggested in existing studies. We also explore the application of the proposed model to the challenging task of analyzing environmental DNA data. This study advances the development of a theory of fish migration that is simple yet can take environmental factors into account.

Random exploration is usually quantified by how fast new space is found, from the range of a single walker to the territory collectively covered by many walkers. In competitive exploration, first arrival secures an exclusive resource, as when foragers compete for food items or agents capture distributed targets. It is then no longer enough to know which sites have been discovered: one must determine, for each discovered site, which searcher reached it first. We introduce the discovery share $X_n$, the fraction of the first $n$ collective discoveries secured by a tagged searcher. For two identical competitors, exchange symmetry fixes $\langle X_n\rangle=1/2$, but the central question is whether this equal split emerges in each long exploration history or only on average, i.e. whether early competitive advantages are erased or persist. Here we show that the answer is controlled by the spectral dimension $d_s$, defined by the large-time decay of the probability that a single searcher is at its starting point after $t$ steps, $p_0(t)\sim t^{-d_s/2}$. Across ordinary diffusion, long-range superdiffusion and subdiffusion induced by crowding or memory, $d_s$ separates persistent randomness in recurrent exploration $(d_s<2)$, anomalously slow non-Gaussian concentration for $2\le d_s<3$, and Gaussian concentration, logarithmically corrected at $d_s=3$, for $d_s\ge3$. For $d_s\ge2$, we derive exact asymptotic variances, including prefactors, and the discovery scale on which competitive imbalances are erased. Two-point correlations of first-discovery labels identify the memory mechanism behind these regimes. The same phase structure persists under changes in geometry, competitor heterogeneity, number of competitors and memory, revealing a general fluctuation theory of first-arrival inequalities.

Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavailable: the classical Kato and Weinstein–Temple enclosures do apply, but require a~priori information on a neighbouring eigenvalue. This paper resolves the issue by extending the author's projection-based framework for guaranteed lower eigenvalue bounds – so far realised only through finite element methods – to conforming spectral Galerkin methods. For trial spaces of exact eigenfunctions the required projection constant is the closed-form optimal value $C_N=\lambda_{M+1}^{-1/2}$, the inverse square root of the first omitted eigenvalue. For $-\Delta+V$ with $0\le V\in L^\infty$, a projection-gap estimate yields an explicit constant for the standard Galerkin matrix (exact at $V=0$), and a composite discretisation removes the $||V||_{L^\infty}$-dependence for large potentials. With Neumann domain truncation these give certified two-sided bounds on $R^d$; for two benchmark potentials on $R^2$ the spectral enclosures match or surpass certified finite element ones at two orders of magnitude fewer degrees of freedom. The same auxiliary-projector mechanism extends to singular potentials with an unbounded $L^\infty$ norm – in particular to attractive Coulomb singularities in three dimensions, via a localised Hardy inequality – which we develop in a companion paper.

We prove a new asymptotic lower bound for the number of perfect positive definite quadratic forms in n variables which is close to the optimal one.

Gradient based optimization methods are nowadays the methods of choice for training deep neural networks (DNNs) in artificial intelligence (AI) systems. In practically relevant DNN training problems, one does usually not apply the standard gradient descent (GD) optimization method but instead one employs suitable sophisticated GD optimization methods, which incorporate adaptivity and/or acceleration techniques, such as the famous Adam optimizer. It is a key contribution of this work to provide a general unified convergence analysis for GD optimization methods in the training of DNNs with analytic activations such as the softplus and the popular Gaussian error linear unit (GeLU) activation. Our general unified convergence result applies to a large class of gradient based optimization methods such as the standard GD, the momentum, the Nesterov accelerated gradient (NAG), the RMSprop, the Adam, the Adamax, the Nadam, the Nadamax, the Adan, the AdaBelief, the AMSGrad, and the Yogi optimizers. Our analysis employs the theory of Kurdyka-Łojasiewicz (KL) inequalities to establish convergence to critical points in the training of DNNs. To the best of our knowledge, the generality of our convergence analysis is also just in the special situation of the Adam optimizer a new contribution to the literature on the analysis of AI optimization algorithms.

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell(M)$ the length of the longest $M$-alternating path in $G$. We define $\alpha_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell(M) \le 2t-1$. We then prove that $$\operatorname{depth}(S/J(G)^{(t)}) \le |V(G)| - 1 - \alpha_t(G)$$ for all $t \ge 1$, where $J(G)^{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.

Future sixth generation (6G) communications are expected to support robotic control tasks in applications such as industrial automation and emergency response, where sensors, computing units, and robots are interconnected via nervous system-like networks to form sensing-communication-computing-control (SC3) closed loops. However, the limited battery capacities of devices within these SC3 loops constrain operational duration and degrade control efficiency, particularly in remote or post-disaster scenarios. To address this challenge, wireless power transfer (WPT) can be leveraged to provide continuous energy supply for SC3 closed loops. In this paper, we investigate a wireless-powered SC3 system, where a satellite transfers energy via radio frequency (RF) signals to support the communication and computing processes of multiple SC3 closed loops. By accounting for the intricate coupling among computing, communication, and energy transfer, we propose a holistic design framework to enhance overall control performance. Specifically, we adopt the linear quadratic regulator (LQR) cost as the performance metric and formulate a sum LQR cost minimization problem. The uplink/downlink transmit power, bandwidth allocation, computing capability, communication/computing time allocation, and WPT power allocation are jointly optimized. We recast the problem into a more tractable form and develop an iterative algorithm to solve it. For the special case of a single loop, we further analyze the properties of optimal solutions in energy-limited scenarios to provide insights for practical parameter configuration. Simulation results demonstrate the performance gains of the proposed scheme.

We study the triangular array $T(n,k;\mu)$ defined by the Graham–Knuth–Patashnik recurrences $$ T(n,k) \;=\; (\alpha n + \beta k + \gamma) \, T(n-1,k) + (\alpha' n + \beta' k + \gamma') \, T(n-1,k-1) $$ with initial condition $T(0,k)=\delta_{k,0}$ and parameters $\mu=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$, which are considered to be indeterminates. We first prove that, for any fixed $n\ge 0$, the sequence $(T(n,k;\mu))_{k\ge 0}$ is strongly log-concave with the coefficientwise partial order in the variables $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. Moreover, we show that the sequence of the corresponding row-generating polynomials $(P_n(x;\mu))_{n\ge 0}$ is strongly log-convex with the coefficientwise partial order in the variables $x$ and $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. Finally, we show that this sequence is coefficientwise Hankel-totally positive of order 2 with the same partial order.

A central challenge in many mobile multi-robot applications is that communication topologies are inherently time-varying. Agents may enter or exit the network and such changes cannot generally be restricted a priori. This work introduces a distributed multi-agent control algorithm based on local communication that supports anytime agent joining and leaving the communication network without centralized coordination. The method scales efficiently with the number of agents by relying on a distance-based neighbor definition and on contracts derived from predicted trajectories. The resulting contract constraints guarantee collision avoidance and constraint satisfaction. We validate the proposed method in an autonomous multi-agent driving scenario, demonstrating effective collision avoidance in high-speed, dynamic environments with agents moving in opposite directions, in both simulated and real-world experiments.

The below-threshold scattering problem is considered for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. A main non-radial obstruction is the possible drift of the concentration center of a soliton-like critical element, which prevents a fixed-center localized virial estimate from closing directly. It is shown that such a compact critical solution must vanish if it has an $L_t^\infty L_x^q$ bound and its center drifts sufficiently slowly. The result covers the pure-energy, endpoint $L^4$, and finite-mass regimes and gives a corresponding conditional scattering criterion.

We study a receding horizon game in which multiple agents drive linear systems subject to additive disturbances, private state and input constraints, and shared coupling constraints. We propose a robust game-theoretic control framework that combines tube-based constraint tightening with a finite-horizon generalized Nash equilibrium problem (GNEP), equipped with a discrete algebraic Riccati equation (DARE)-based terminal cost and a decoupled positively invariant terminal set. The framework guarantees recursive feasibility for every bounded disturbance realization. Exploiting the potential-game structure induced by tracking costs, we further establish asymptotic convergence of each agent's nominal state to a steady-state variational generalized Nash equilibrium (vGNE), and show that each agent's actual state converges to a neighborhood of the vGNE determined by the minimal robust positively invariant set.

A Killing tensor field on a Riemannian manifold $(M,g)$ is a covariant symmetric tensor field whose contraction with the velocity vector along a geodesic produces a homogeneous polynomial first integral of the geodesic flow. Such a tensor is called decomposable if it lies in the subalgebra generated by Killing vector fields; equivalently, the corresponding polynomial integral is then a polynomial in the linear integrals coming from infinitesimal isometries. On spaces of constant sectional curvature and on the complex projective space, every Killing tensor field is decomposable. By contrast, the quaternionic projective spaces and the Cayley projective plane admit indecomposable quadratic Killing tensor fields. We prove that every quadratic Killing tensor field on the compact classical Lie groups $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, equipped with a bi-invariant Riemannian metric, is decomposable; equivalently, every quadratic first integral of the geodesic flow on these groups is a quadratic polynomial in the linear first integrals.

We study continuous-time quantum walks on cycles. We prove two complementary results. Firstly, the cycle $C_9$ does not admit uniform mixing at any time. Using the similar idea and Dickson polynomials, we prove that $C_{15}$ does not admit uniform mixing at any time neither. Secondly, for every prime $p$, we show that the cycle $C_{p^2}$ admits $\epsilon$-uniform mixing.

We introduce abstract measure grand Lebesgue spaces endowed with ball basis structures and investigate their fundamental properties, together with the behavior of $\mathfrak{BO}$ operators. By exploiting sparse domination for $\mathfrak{BO}$ operators and norm inequalities for sparse operators, we derive norm estimates for this class of operators. As applications, we establish that $\mathfrak{BO}$ operators encompass the maximal operators, Calderón-Zygmund operators on homogeneous spaces and Carleson operators.

In this paper, we derive the Onsager–Machlup functional for a class of degenerate stochastic differential equations on $\mathbb{R}^{m+n}$ driven by $n$-dimensional fractional Brownian motion with time-dependent diffusion coefficients, where the Hurst parameter satisfies $H\in(1/4,1)$. The main difficulty arises from the interaction between the degenerate structure and the multidimensional fractional noise, which produces nontrivial coupling terms under small-ball conditioning. By combining the Gaussian correlation inequality with an approximation argument for infinite-dimensional convex sets, we establish a decoupling mechanism that allows these coupling effects to be controlled by unconditional Gaussian expectations. Furthermore, through regularity estimates for the degenerate components and sharp Hölder norm estimates, we eliminate the remaining coupling contributions and obtain an explicit expression for the Onsager–Machlup functional. As applications of the derived functional, we obtain the corresponding constrained Euler–Lagrange equations characterizing the most probable transition paths of the system, and establish a sufficient condition under which the stochastic differential equation preserves the most probable path.

Krajči, Liu, Mikeš, and Moser proved in 2015 that the redundancy of binary Shannon-Fano coding is always below one bit. We sharpen this to a bound depending on the largest source probability $p_1$: an explicit seven-piece envelope $R<f(p_1)$. The envelope equals the exact supremum of $R$ given $p_1$ for every $p_1\ge\tfrac12$ and on a subinterval below $\tfrac13$, and gives the cap $R<\tfrac52-\tfrac56\log_2 5=0.5651$ for $p_1<\tfrac12$. It is the first $p_1$-dependent redundancy bound for Fano codes. The method is more sophisticated than the approach typical for Huffman codes: Fano trees are built top-down by contiguous balanced splits and lack the sibling property. From the $R<1$ theorem the rest follows from the Fano recursion, through a min-corrected affine potential and a no-burial lemma. Every scalar inequality in the proof reduces to a comparison of integer powers.

In this paper, we show that for any $S$-unimodal map $T$ on $[0,1]$ with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant $\lambda_T$. Moreover, $\lambda_T=0$ if and only if $T$ admits neither an absolutely continuous $T$-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an $S$-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every $x\in [0,1]$ the Lyapunov exponent along the orbit of $x$ exists and is equal to $0$. A key ingredient is the following result of independent interest. If an $S$-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every $x\in [0,1]$ the lower Lyapunov exponent along the orbit of $x$ is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.

For graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is weakly $H$-saturated in $F$ if the edges in $E(F)\setminus E(G)$ can be added one at a time, each addition creating a new copy of $H$. Recently, Tancer and Tyomkyn proved that, given an $n$-vertex graph $F$, deciding whether $\mathrm{wsat}(F,K_3)=n-1$ is NP-hard. In this paper, we study the decision version of the weak saturation problem and show that, for every fixed integer $r\ge 3$, given a graph $F$ and an integer $k$, deciding whether $\mathrm{wsat}(F,H)\le k$ is NP-complete when $H\in\{K_r,K_{r,r}\}$. Our approach uses novel graph-theoretic and topological ideas and techniques, yielding new constructions that build on the construction of Tancer and Tyomkyn. In particular, our proofs bring the flag-no-square property, a fundamental property in topology that is of independent interest, into the study of weak saturation problem.

We prove a priori exponential growth and decay estimates for locally square integrable eigenfunctions with complex eigenvalues of the non-self-adjoint Schrödinger operator. There appear two natural critical exponents, given by the imaginary parts of square roots of twice an eigenvalue. Any associated eigenfunction has the same exponential growth or decay rate as one of these, or greater than the upper one. The results may be seen as extensions of the Phragmén–Lindelöf theorem, the Agmon estimates and Rellich's theorem to complex eigenvalues. For the proofs we adapt a commutator scheme of Ito–Skibsted~(2020) to the non-self-adjoint framework, presenting a unified approach to all of them. In order to control contributions from the imaginary parts of a potential and an eigenvalue, an additional commutator estimate is employed.

In this paper, we study the algebraic degree and distinctness of a kind of generalized $n$-dimensional Kloosterman sums over finite field $F_q$ by Galois theory and the Stickelberger theorem on $p$-adic Gauss sums.

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g=af^p+b$ with $a,b\in\mathbb{C}$, $p\in\mathbb{N}$, and $a\neq 0,1$. We examine how singular values and postsingular sets behave under composition. Within this framework, we show that if one of the functions is postsingularly finite (respectively, postsingularly bounded, hyperbolic), then the other function also has this property, and so do their compositions. As an application, we derive several results concerning transcendental semigroups, including situations in which Eremenko's conjecture is satisfied.

This paper investigates the causal rate-distortion function for networked control systems with multiple encoders and a single decoder, a longstanding open problem in information and control theory. While previous work has explored the causal rate-distortion function for single-encoder and feedback-enabled networked settings, the case of networks without feedback remains unaddressed. We establish a novel directed information lower bound, the first derived for the networked control setting. We further demonstrate the optimality of linear, independent encoders and linear decoders for optimizing this lower bound for Linear Quadratic Gaussian (LQG) plant and quadratic cost, with the condition that the full plant state is observed when sensors are sitting together. By reducing the original infinite-dimensional optimization problem to a finite-dimensional one, our approach simplifies the analysis. Additionally, our directed information lower bound provides an alternate proof for the sufficiency of linear encoders in the single encoder and single decoder setting with side information, extending prior results in the literature. We present Semidefinite Programming formulations for the causal rate distortion function of Gaussian-Markov sources with linear side information and the singular noise matrix.

If a finite collection (with repetitions allowed) of unit fractions has sum larger than $K$, how close can a sum of some of them come to $1$ from below? Improving a bound of Erdős and Graham, we prove that the gap is at most $\exp(-c\sqrt{K\log K})$ for an absolute constant $c>0$ and all sufficiently large $K$. The proof combines equal fractions and applies a Fourier estimate to groups of denominators with comparable contributions. Elementary divisor counts then give the bound.

We introduce split-free cable expressions and their sequential restriction. Live cables are vertex blocks that future operations cannot split. The main result identifies sequential split-free cable width, up to an additive two, with active neighborhood-width, the minimum number of distinct nonzero future-neighborhood classes across a linear layout. The lower bound extracts such a layout from every cable play and includes a parity argument for prefixes inside a birth group. The upper bound compiles any active-neighborhood layout into a singleton-birth play. This gives an exponential upper bound and a linear lower bound in terms of linear rank-width. We prove that the complement of the seven-vertex cycle has linear rank-width two and sequential cable width exactly eight. This disproves the proposed affine upper estimate already at rank two. We also prove an unbounded separation between branching expression width and sequential width on trees.

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \[ X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'), \] where \(D,D'\) are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition \[ \frac{u}{w^2}=\frac{u'}{(w')^2}, \] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \[ a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2, \] then the produced hive density $q_n$ satisfies \[ D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n). \] The third scale $c_{**}$ is determined by a limiting tetrahedral optimization problem; equivalently, writing \(\delta=u+u'\), \[ \delta^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }. \] Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The introduction records two explicit surface-tension approximations and numerical comparisons which motivated the construction.

This second part of the series develops the statistical and applied layer of the theory built in Part I [1]. Unlike current Hilbert diffusion models [2-5], we focus not only on well-posedness of the infinite-dimensional generative dynamics, but on the noncommutative $A$-geometry, explicit entropy constants, and checkable lower bounds. The analytic estimate of Part I reduces stability of the backward evolution to control of a validation error $\mathcal{E}_{val}$. We prove three results. First, we construct a cylindrically weak denoising label for the infinite-dimensional score field, consistent with the exact logarithmic gradient. Second, the local parametric closure is replaced by a global nonparametric score closure, its complexity controlled via the Dudley entropy integral and uniform empirical bounds in $L^2(\nu_*;A)$. Third, for the trace-smoothed nonlinear class we construct minimax bounds by the Le Cam-Assouad method, showing the root statistical rate cannot be improved without additional quadratic structure. The final section applies the theory to anisotropic diffusion in a nanosystem model and checks the constants $c_A,C_A,C_{LSI}^A$ independently of any smallness condition, establishing explicit accuracy orders: the parametric trace-smoothed minimax score risk is $p/M$, while the uniform error of the risk functional is $\sqrt{p/M}$, showing the statistical plateau of order $p/M$ is not a proof artifact. The applied layer closes with an independent analytic benchmark for the isotropic case, a comparison with classical cell homogenization, and an approximation theorem for smooth logarithmic gradients by $A$-adapted spectral networks.