Papers

This paper studies the finite-sample bias of plug-in estimators for joint entropy, conditional entropy, and mutual information for finitely supported discrete random variables. Using the univariate reduction framework developed in our companion works, we derive explicit first-order bias formulas. For a pair (X,Y) with support sizes r and s, the biases are respectively -(rs-1)/(2n), -r(s-1)/(2n), and (r-1)(s-1)/(2n). We then propose bias-corrected estimators for the three measures and prove that the uncorrected and corrected versions are asymptotically equivalent, differing by a deterministic term of order O(1/n); consequently, they share the same asymptotic distribution. A simulation study validates the theoretical results, including a dedicated independence study and the relationship with Wilks' theorem. An application to a motor insurance portfolio, where X is the driver's age class and Y the claim severity class, illustrates the improvement brought by the bias correction for risk classification.

A regular complex bundle on an orbispace has fibres that are positive multiples of the stabilizers' regular representations. For stabilizers of order at most two, the sharp uniform rank bound in dimension five is $16$ for paracompact orbispaces and $4$ for closed oriented smooth orbifolds. Rank $16$ occurs with dense free locus and minimum regular rank $4$ on the nonfree locus. We also prove the sharp bound $16$ for closed oriented smooth six-orbifolds. We compute the stable regular index of cyclic gerbes over finite connected CW complexes as the least common multiple of the indices of their character twists. For paracompact orbispaces, optimal integral Adams multipliers give explicit rank bounds in terms of dimension and stabilizer orders. The least positive rank of an integral exterior-power expression with regular fibre and zero reduced class for every regular rank-four bundle on $S^4\times B\mu_2$ is $32$.

We consider the one-dimensional Vlasov–Poisson system with a repulsive harmonic potential. For repulsive self-interactions, we establish modified scattering for initial data of arbitrary size in some weighted Sobolev spaces without symmetry, compact support, or spatial moment assumptions. The proof uses a Hamilton-Jacobi momentum correction and the sign of the repulsive interaction to obtain large-data bounds. We also construct modified wave operators. These results complement the higher-dimensional theory in \cite{HK26,VR24,BVR25}. We expect our methods to be useful in the study of a broader class of kinetic equations with very long-range interactions.

We present two number-theoretic compression techniques. The first is local, and compresses individual numbers. The second is global, and compresses the entire set of numbers using a prime wavelet tree, which is a data structure of independent interest. Our techniques apply to any set of numbers satisfying a Korselt-like criterion. We demonstrate our compression techniques using the recently completed tabulation of all $308{,}279{,}939$ Carmichael numbers less than $10^{24}$ that occupies $18.4$ gigabytes as text. When combined with standard compression techniques and a heuristic choice of divisors, the resulting file is $588.7$ megabytes ($31.2$ times smaller than the text file). The initial $80$-bit numbers now use about $15.3$ bits.

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. For each Dynkin diagram, this system is periodic, as a special case of Zamolodchikov periodicity. In the associated tropical dynamics, every mutation along the belt can be naturally colored red or blue, according to which term attains the maximum in the tropical exchange relation. This gives a red-blue statistic on tropical orbits. We prove that this statistic is homomesic, i.e. that the average numbers of red and blue mutations are independent of the orbit. Moreover, the number of red mutations is given by the number of roots in the associated root system. This resolves a conjecture of the first author. In the process we introduce the Zamolodchikov fan, which is closely related to the positive tropical Grassmannian, and may be of independent interest.

We study the universal isotypic block-diagonalization problem in the Type-2 theory of effectivity (TTE). We obtain a uniform computability result for the universal block diagonalizer for commutants of given finite group representations. As an application, when the given representation is orthogonal, we compute ordinary $QR$ factorizations blockwise in the isotypic basis. After transforming back to the original coordinates this yields a factorization $A=QX$, where $Q$ is orthogonal and $X$ is generally not upper triangular. Moreover, given a Wedderburn-adapted orthogonal basis, the transformed-back factors can be computed within the commutant.

It is shown that a symplectic isotopy with non-degenerate end-point on a closed symplectically Calabi–Yau manifold has simple periodic points of arbitrarily large period under a simple non-vanishing condition on the Morse–Novikov homology associated with its flux. The proof combines a sharp estimate for the Conley–Zehnder index under the assumption that the mean index is an even integer with a classical asymptotic iteration argument in the non-Hamiltonian Floer-theoretic setting. We also apply this estimate to the symplectic dynamics of the standard symplectic $4$-torus.

Motivated by scenarios in control systems where decision making is informed by only partial information about the states, we introduce a novel formulation of the Schrödinger bridge problem for inertial interacting-agent systems. Within our setting, the agent ensemble is steered to match partial specifications of the collective configuration. Our analysis pertains to a mean-field framework in which the collective dynamics are described by a nonlinear Vlasov–Fokker–Planck integro-PDE with nonlocal interactions. We assume that only distributions over positions are specified at the endpoints of a finite time horizon, representing aggregate population snapshots, while the corresponding velocity degrees of freedom remain unspecified. We derive the associated optimality system, identifying the optimal controlled evolution as jointly informed by the initial and terminal specifications through forward and backward propagation in time. We also show that the partial endpoint specifications bring about distinct conditions on the optimal control and the consistent evolution. Subsequently, to compute solutions to the optimality system, we develop a nested fixed-point algorithm inspired by the classical Fortet–Sinkhorn iteration, but extended to handle the nonlinearities arising from nonlocal interactions and temporal coupling.

We compare five natural classes of birational automorphisms of an algebraic variety: projectively regularizable, regularizable, projectively pseudo-regularizable, pseudo-regularizable, and those admitting an algebraically stable model. These classes provide different geometric frameworks in which the dynamics of a birational automorphism, and in particular its first dynamical degree, can be studied. We determine the general inclusion relations among them and show that, for $\mathbb{P}^k$ with $k\geqslant 3$, all the resulting non-trivial inclusions are strict; we prove this by giving explicit examples.

We show that $\text{gr}(\text{Av}(1324))\leq 13.167248$. This is achieved by combining the approach of Bevan, Brignall, Elvey Price and Pantone using the `domino', with ideas from earlier upper bounds by Bóna using pairs of decorated words with additional restrictions. More specifically, we replace the arbitrary interleavings of Bevan et. al. with interleavings more like those of Bóna. Our method uses shuffle automata to handle the interleavings, and this method has the potential to provide better upper bounds than the one established here. To estimate how many shuffles are possible, we use two key statistics on dominoes: the number of internal points (that is, points that are neither left-to-right minima not right-to-left maxima), and the number of runs of internal points.

We study rumor-spreading models on the one-dimensional integer lattice, with particular emphasis on the effect of forgetting mechanisms on the long-term behavior of the rumor. We consider the spatial version of the classical Maki–Thompson model together with two variants that incorporate forgetting. Each model is formulated as an interacting particle system and describes the propagation of information through local interactions between neighboring individuals. Our results show that the introduction of forgetting fundamentally changes the behavior of the process. In the classical Maki–Thompson model, the rumor becomes extinct, whereas the two models with forgetting exhibit nontrivial phase transitions between extinction and survival. More precisely, we establish the existence of critical transmission rates that separate regimes in which the rumor dies out almost surely from regimes in which it survives with positive probability. We also investigate the dependence of these critical thresholds on the stifling rate. Combining rigorous arguments with numerical simulations, we prove the existence of the phase transitions, obtain estimates for the critical values, and explore the corresponding phase diagrams. These findings show that forgetting, despite acting as a mechanism that removes spreaders, can qualitatively alter the dynamics of rumor propagation by creating regimes of sustained information spread. Our work provides new insight into the critical behavior of spatial rumor models and raises several questions for future research.

We prove a large deviation principle for the area of a two-pointed weight-$W$ quantum disk conditioned to have left boundary length one, where $0<W<2$ is fixed and $\gamma\downarrow0$ without a subsequence restriction. Conformal welding and the one-block mating-of-trees law give independent Brownian blocks and a recursive area decomposition. Growing moments and negative Laplace transforms lead to two Hamilton–Jacobi equations, whose characteristic solutions determine the limiting transforms. Direct comparison proves moment convergence; compact-interval comparison and a change of measure prove Laplace convergence. The resulting good rate function is specified by scalar endpoint equations and is elementary when $W=1$.

We study finite-horizon stochastic linear-quadratic (LQ) control for finite-activity jump diffusions with bounded random coefficients, indefinite weights and a linear recursive cost. The cost is defined by a backward stochastic differential equation (BSDE) whose generator has bounded deterministic coefficients multiplying the Brownian and jump integrands. A Brownian-Poisson change of measure and an integrating factor reduce the criterion to a nonrecursive LQ functional. On the intersection of the physical and transformed square-integrable control spaces, we establish recursive evaluation for L^1 quadratic data under the transformed measure and equality with the full transformed value. Under transformed uniform convexity and invertibility of the uncontrolled jump map, we identify the stochastic Riccati equation from the nonrecursive value kernel. Conditional energy estimates justify transferring its martingale coefficients to the physical measure. Completion of squares gives feedback representation, comparison and uniqueness in the feedback-admissible class. For each initial pair, the recursive value is attained exactly when the transformed optimizer has finite physical control energy. Analytical examples illustrate nonzero feedback and uniform convexity with a negative control weight.

We introduce an autoregressive differentiable method to solve 0-1 integer programs. We fix an arbitrary order of the binary variables and we train a transformer to predict the next bit while remaining in the feasible set. Our method is first trained on feasible incumbents provided by any solver, thus allowing us to initialize the transformer in the feasible set. Our procedure then implements a Lagrangian penalty to penalize infeasible solutions, and the transformer is further trained to explore the feasible set using Gumbel-softmax activations on the relaxed objective. We have tested our method on non-convex instances of quadratic knapsack problem and demonstrated consistent improvement upon state-of-the-art open-source solvers for dense problems up to 10,000 binary variables. In particular, we empirically demonstrate a phenomenon akin to a tunneling effect where the effective change of variables from binary variable to the continuous weights of the transformer that the method implements enables crossing barriers in the relaxed objective landscape.

We study permutations $\pi \in S_n$ that avoid $123$ while satisfying the adjacency constraint $\lvert \pi_{i+1}-\pi_i\rvert \le m$ for some $m\in\mathbb{Z}^{+}$. We first show that every admissible permutation is localized within vertical distance $m$ of the reverse diagonal, revealing a strong global restriction imposed by the interaction between pattern avoidance and bounded adjacency. For every fixed $m$, we then construct an exact finite-state description of the family, with precisely $2^{m+1}-m-1$ realizable states. This yields an explicit transfer-matrix enumeration and, in particular, a rational generating function for every fixed $m$; consequently, the enumeration sequence satisfies an eventual constant-coefficient linear recurrence. We further identify the exponential growth rate with the spectral radius $\rho_m$ of the corresponding transfer matrix and show that $\rho_m<4$, is nondecreasing in $m$, and converges to the Catalan growth constant $4$ as $m\to\infty$. Finally, we show that the bounded-adjacency constraint breaks the ordinary length-three Wilf equivalence, and we develop algorithmic and graph-theoretic interpretations of the family, including a correspondence with constrained Hamiltonian paths in powers of the path graph.

We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut–Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut–Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut–Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including $S^1\times S^2\times S^3$ and $S^3\times S^3$. In these families, the metric is fixed and the compatible almost complex structures form an $S^2$-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

These notes present the construction in \cite{euler} of finite-time blowup for the three-dimensional incompressible, unforced Euler equations on $\mathbb R^3$: \begin{equation} \partial_tu+(u\cdot\nabla)u+\nabla p=0,\qquad \nabla\cdot u=0. \end{equation} The construction successively adds oscillatory layers to smooth parent solutions. Each layer is constructed by a finite expansion and an exact correction, so that the resulting child solves Euler.

We introduce and analyze a new model of graph burning, in which two competing fires (coloured yellow and green) ignite vertices of a given signed graph and propagate along its edges. The boolean sign of an edge determines whether a fire spreading along the edge changes colour or not. In each step, a player ignites a new vertex in a colour of their choice, while previously activated fires continue to spread. Given a signed graph $\Gamma$, the objective is to burn the maximum possible number of vertices in a single colour; the optimal achievable value is called the plurality number of $\Gamma$. We express the plurality number through Hamming distances to a binary code associated with $\Gamma$. In particular, the minimum plurality number over the switching class of $\Gamma$ equals the number of vertices minus the covering radius of this code. Under certain conditions on the signature, we determine exact values of the plurality number of signed paths. By contrast, we prove hardness results for multiple variants of the problem of determining or approximating the plurality number.

We introduce SemiBase, a project that computes and formally certifies finite identity bases for small semigroups. Deciding finite basability is undecidable for finite algebras and remains open for finite semigroups. The task requires a proof that a candidate basis is complete, or a proof that none exists, rather than a single first-order validity query. LLM-guided agents search for these proofs; a referee agent rebuilds them from source, and the Lean kernel checks the resulting corpus in a final audit. Humans choose targets and approve final outcomes. We certify every semigroup of order at most 6: all 1309 semigroups of order at most 5 and all 15973 of order 6, including proofs that the four known nonfinitely based semigroups have no finite basis. The bases for order 6 define 505 distinct varieties, whose inclusion order Vampire determines except for four pairs. The resulting catalogue is a machine-checked account of results scattered across the literature and a tested foundation for order 7.

We consider an optimal control problem governed by a stationary linear transport equation in two or three spatial dimensions, motivated by photon transport in radiotherapy treatment planning. The control is an inflow boundary source, and the observation operator may track either the angular flux or its angular integral, the latter serving as a simplified dose surrogate. The associated optimality system consists of a transport–adjoint pair with angular dependence, coupled through a boundary optimality condition. We formulate a residual least-squares functional whose minimisers coincide with solutions of the Karush–Kuhn–Tucker system and show that the functional controls the $L^2$ errors in the state and adjoint and the flux-weighted $L^2$ error in the control. This yields a continuum-level a posteriori estimate without recourse to discretisation. An additional unweighted optimality residual certifies the control in the norm of the objective. The estimate holds for every admissible approximation and accounts for the different boundary measures in the control penalty and transport traces, without a positive lower bound on $|\omega\cdot n|$. We then parameterise the state, adjoint and control using neural networks and minimise over a neural network space using appropriate quadrature. The Monte Carlo formulation replaces each residual integral by a weighted sample mean, whose quadrature error enters the a posteriori estimate. We prove convergence under explicit assumptions on approximation, quadrature and optimisation and derive bounds on the total error in terms of approximation, quadrature and optimisation error. Numerical examples in two and three dimensions illustrate the residual–error relation and the flexibility of the residual framework for field- and dose-tracking control problems, including regular manufactured solutions.

Let \[ S_h=\langle a,ah+d,ah+2d,\ldots,ah+kd\rangle, \qquad \gcd(a,d)=1,\quad 1<k<a,\quad h> 1, \] be a numerical monoid generated by a generalized arithmetic sequence. In this paper, we determine the delta set \[\Delta(S_h),\] resolving an open problem posed by Christopher O'Neill and Roberto Pelayo in 2014 \cite{ONeill}. Writing $e=h-1$ and letting $E(e,d)$ denote the set of values arising from the subtractive Euclidean algorithm applied to $e$ and $d$, we prove that \[ \Delta(S_h) = E(h-1,d)\cup \{M-r(h-1):M-r(h-1)>0,\ r\geq0\}, \] where \[ M=\left\lceil\frac{a}{k}\right\rceil(h-1)+d \] is the maximal element of $\Delta(S_h)$. In particular, $d$ and $h-1$ always belong to $\Delta(S_h)$. We further establish a reduction theorem showing that $\Delta(S_h)$ is a multiple of the delta set of a primitive monoid satisfying $\gcd(d,h-1)=1$, and deduce that the minimum delta is $\gcd(d,h-1)$. We investigate the relationship between delta sets and catenary degrees for these monoids, obtaining an explicit connection between their maximal values and formulating several conjectures concerning the catenary degree set. We also find an interesting way to generate metallic fibonacci sequences through Delta Sets. These results provide a structural description of factorization invariants for generalized arithmetic numerical monoids and suggest several directions for further study.

A parking function is said to have $\ell$-limited preferences if there are at most $\ell$ copies of each element in the parking function. In this paper, we define a notion of pattern avoidance for parking functions and enumerate those $2$-limited parking functions that avoid certain patterns.

This paper investigates the connection between mean-field systems and forward-backward stochastic systems arising from the mean-variance portfolio selection problem. By introducing the conditional expectation process of terminal wealth, the mean-field term is represented by a backward state, leading to a class of control problems for forward-backward stochastic systems with coupled initial-terminal values. The coupling between the terminal value of the forward state and the initial value of the backward state in the cost functional prevents classical methods for forward-backward stochastic optimal control from being directly applicable. To overcome this difficulty, we establish a stochastic maximum principle for control problems with running costs and coupled initial-terminal values, and provide sufficient conditions for optimality. Furthermore, we investigate a class of generalized stochastic linear-quadratic control problems with initial-terminal coupling, allowing the control weight in the running cost to be indefinite, and derive the optimal feedback representation via a Riccati equation. The classical mean-variance portfolio selection model is shown to be a special case, for which the corresponding efficient investment strategy and efficient frontier are obtained explicitly, and numerical simulations are presented to illustrate the results and examine parameter sensitivity and its financial implications.

Let a normalized continuous Parseval frame describe the phase space of a reproducing kernel Hilbert space. The frame coefficients of an orthonormal family have total mass equal to the number of states and pointwise mass at most one. This elementary exclusion principle gives explicit lower bounds for the total cost of a family in terms of the volume of low-cost regions. For radial costs, it yields collective mean-dispersion, umbrella, and transport inequalities. The same argument applies to normalized Bessel families, with the Bessel bound measuring the permitted overlap. We also consider semiclassical families for which the frame measures vary. Weak convergence of the normalized measures, together with averaged localization of the normalized kernels, yields continuous-symbol Szegő limits and sharp asymptotics for the lowest Toeplitz energies. The results apply to Bargmann–Fock and weighted Bergman spaces, Gabor and Paley–Wiener spaces, and orthogonal-polynomial and determinantal ensembles, including the GUE.

We survey maximal directional averages and directional singular integrals, and their connections with Kakeya–Nikodym phenomena, differentiation theory and the Carleson operator. The exposition is organized by the structure of the set of directions: lacunary and arbitrary finite sets, vector fields as in the conjectures of Zygmund and Stein, and, in higher dimensions, algebraic sets of directions and subspaces. The unifying tool is almost orthogonality, which rests on the ordering of the circle in the plane and on polynomial partitioning in higher dimensions. We explain where it yields sharp or nearly sharp bounds and where new ideas are needed, notably for multiscale averages along algebraic curves and for vector-valued single-annulus estimates, and we collect open problems.

For a primitive symplectic variety $X$ admitting a holomorphic Lagrangian fibration $f: X \to B$, we classify the possible Mumford–Tate groups of the very general fiber. We extract as a consequence that the discriminant of $f$ has codimension one in $B$. In the special case when $X$ is a hyper-Kähler manifold which is general among those admitting a Lagrangian fibration, and $b_2(X) \geq 7$, we further show that the very general fiber of $f$ is Hodge-generic. Finally, as an application of our classification, we confirm an expectation of Li–Tosatti that when $f$ is not isotrivial, the special Kähler metric on $B$ has positive holomorphic sectional curvature on the regular values of $f$.

We first revisit the Nevanlinna condition for representing measures of Herglotz-Nevanlinna functions in two variables and give a general operator-theoretic criterion for extremality. We then derive an explicit formula for the representing measure of a real rational Herglotz-Nevanlinna function in two variables, and combine it with the extremality criterion to obtain a family of extreme elements in the full cone of Nevanlinna measures. The family includes examples with reducible denominators, and these yield counterexamples to the irreducibility assertion in a conjecture of Knese. We then characterize reducibility within this family both algebraically and geometrically.

Electroconvective (EC) instability near ion-selective surfaces plays a key role in electro-chemical transport beyond the diffusion-limited regime. While EC instability in Newtonian electrolytes has been extensively investigated, the influence of viscoelasticity on the instability mechanism remains less understood. In the present study, we investigate the viscoelastic modulation of EC instability using linear stability, eigenspectrum, and mechanical-work analyses, supported by asymptotic pathway ordering. The analysis reveals a non-monotonic dependence on the Weissenberg number Wi, with the maximum growth rate and minimum critical voltage occurring near Wi ~ O(1). The mechanical-work balance separates electric, solvent-viscous, and polymeric contributions, and further decomposes the polymeric work into normal-stress-difference and shear-stress pathways according to their constitutive origins. For Wi << 1, the polymer stress recovers Newtonian viscous dissipation at leading order. As Wi approaches unity, the phase efficiencies of the Newtonian-like polymeric pathways decrease substantially, causing the negative polymeric work to collapse and the instability to intensify. At intermediate Wi, an additional minimum in the critical voltage appears at low wavenumbers. Eigenspectrum analysis attributes this feature to the selection of a slowly propagating mode that retains substantial negative polymeric work. At large Wi, this mode no longer determines the instability threshold, and the low-wavenumber minimum disappears. The normal-stress-difference and shear-stress work then nearly cancel, accompanied by stronger ESC-localized shear deformation and predominantly solvent-viscous dissipation. Thus, the non-monotonic response is governed by stress-deformation phase relations, solvent dissipation, and eigenmode selection rather than stress magnitude or a single polymeric contribution.

This paper establishes sufficient conditions for finite-horizon solvability of fully coupled nonlinear mean-field forward–backward stochastic difference equations with dependence on unconditional first moments. The forward recursion uses conditional projections of the next backward state and its product with the innovation. Building on existing domination–monotonicity methods, we formulate deterministic matrix combinations in centered and mean coordinates and prove a continuation estimate uniform in the homotopy parameter. Global Lipschitz continuity and one active domination–coercivity direction yield a unique square-integrable adapted solution, an a priori bound, and coefficient stability. The active parameter can be normalized without imposing an additional smallness restriction on the original coefficients. A sign transformation handles the opposite monotonicity orientation. A nonlinear example with saturating state and mean interactions verifies the assumptions, including degenerate domination directions. A scalar mean-field LQ buffer-adjustment model illustrates the theorem: its Hamiltonian solution characterizes the unique open-loop optimizer. An exact finite scenario-tree calculation checks this characterization against direct quadratic optimization.