Papers

In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in Bandi-Fregoli-Kleinbock, that is, the set of pairs of real numbers $(x,y)$ satisfying $$ \limsup_{Q\to +\infty}\ Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs $(x,y)$ satisfying $$ \liminf_{m\to +\infty}\ Q_m\cdot \min_{1\leq q\leq Q_m}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning if $Q_{m+1} \gg Q_m^{\tau}$ for some $\tau > 1$. In particular, the above sets have full Hausdorff dimension in $\mathbb{R}^2$. In addition, we prove that these sets are absolute winning on every regular $C^2$ planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.

Profitability metrics, such as the single-product gross profit margin, constitute the standard toolbox to prioritize products in sales and operations planning (S&OP). These metrics leverage the premise that production lines can be committed independently, which endows them with a simple per-ton ranking that a planner can compute and use easily in practice. However, coupled plants, especially polymer plants whose parallel lines draw simultaneously on a single bulk feed, can fundamentally undermine this premise and mislead to value-destroying production plans. We propose a framework built on what we call the Augmented Gross Profit for Product Clusters (AGPPC), which converts the per-ton ranking metric into a per-coupled-hour ranking that carries a per-instance optimality certificate. We present the theory of AGPPC, which combines the co-production column concept with a fluid relaxation of the planning problem, alongside an integrated bilinear mixed integer program of the plant and a McCormick-linearized baseline that provides certified bounds for it, and we demonstrate its effectiveness on real operational data from an Indonesian polymer producer, where the coupling-aware metric alone raises operating profit by 7.6% and the full optimization by 28% over current practice.

Integrated sensing and communication (ISAC) is a rising technology in the next wireless communication networks, enabling the simultaneous execution of communication and sensing tasks by fully utilizing limited spectrum resources. In this work, we investigate the secrecy performance of a dual-uncrewed aerial vehicle (UAV)-assisted secure ISAC system. Specifically, a base station UAV communicates with users and transmits radar signals to locate potential eavesdroppers, while simultaneously providing information to a jammer UAV to perform jamming tasks. Considering constraints such as maximum UAV velocity, transmit power, propulsion energy, and sensing thresholds, we maximize the average secrecy rate by optimizing user scheduling strategies, time allocation, transmit power, and UAV trajectories. The presence of a non-convex problem, originating from tightly coupled variables, is tackled by an efficient iterative algorithm. In particular, the original optimization problem is decomposed into six subproblems, and non-convex subproblems are transformed into approximately convex forms via successive convex approximation. Then, block coordinate descent techniques are employed to solve all subproblems sequentially. Numerical results demonstrate the convergence and effectiveness of the proposed algorithm.

Wei and Wylie asked whether a complete weighted manifold with nonnegative Bakry–Émery Ricci curvature and bounded potential must admit a Riemannian metric with nonnegative Ricci curvature. We answer this question negatively in every dimension \(8k+1\) and \(8k+2\), where \(k\geq1\). Our main geometric result is that every smooth homotopy sphere of dimension at least seven admits a weighted core metric \((g,e^{-f})\) with \(\Ric_g+\Hess_g f>0\). On the other hand, in dimensions \(8k+1\) and \(8k+2\), where \(k\geq1\), we prove that a homotopy sphere with nonzero \(\alpha\)-invariant admits no Riemannian metric with \(\Ric\geq0\). Since homotopy spheres with nonzero \(\alpha\)-invariant exist in every dimension \(8k+1\) and \(8k+2\), and since compactness makes the potential bounded, these manifolds provide the required counterexamples.

Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $\Omega \in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

We give the first known examples of Oka Calabi–Yau manifolds in any dimension: every smooth hypersurface of multidegree $(2,\ldots,2)$ in a product of projective lines is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_\delta$ subset of the base, and the sublocus of Oka fibers with vanishing Néron–Severi group is likewise a dense $G_\delta$ subset. We also obtain projective and nonprojective Oka Kummer surfaces and prove an analogous density statement in the Kummer period domain. We also prove the existence of Oka Enriques surfaces and Oka irreducible hyperkähler fourfolds.

We give a roadmap for the study of critical fractional Hardy inequalities for Dirichlet and Sobolev–Bregman forms. We focus on homogeneous forms on the real line, in particular the form of the fractional Laplacian and the Servadei–Valdinoci form. We prove criticality results in both the Dirichlet and Sobolev–Bregman settings.

We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $\chi(F)=r+1\ge 3$, there exists a constant $\lambda_0=\lambda_0(F)$ such that if $G$ is an $F$-free graph with $\lambda (G)\ge \lambda_0$, then for every $\ell\ge 1$ with $(r,\ell )\neq (2,2)$, \[ \lambda^\ell (G) \le \Big(1-\frac1r\Big)w_\ell(G), \] with equality if and only if $G$ is a regular complete $r$-partite graph; in the case $r=2$ with $\ell$ even, equality holds for every complete bipartite graph. The pair $(r,\ell )=(2,2)$ must be excluded, since the bound fails for several forbidden graphs, e.g., $C_{2t+1}$ with $t\ge 2$. Moreover, walks counts may also be replaced by the homomorphism counts of unbalanced trees. As further applications, we extend a spectral supersaturation result of Bollobás and Nikiforov [J. Combin. Theory Ser. B (2007)], and we also extend the entropic Turán theorem of Chao and Yu [J. London Math. Soc. (2026)] from $K_{r+1}$-free graphs to $F$-free graphs with $F$ color-critical. We provide a framework by passing through weighted Turá theorems of independent interest. If $G$ is $F$-free and $\mathbf{p}$ is a probability vector on $V(G)$ with $\lVert \mathbf{p}\rVert_\infty$ sufficiently small, then \[ 2\sum_{uv\in E(G)}p_up_v \le 1-\frac1r + o(1), \] and the error term $o(1)$ can be removed if and only if $F$ is color-critical. This is a Motzkin-Straus-type inequality in which the clique number of $G$ is replaced by $\chi (F)-1$. The proof of this weighted result combines a blow-up argument, the graph removal lemma, the Erdős-Simonovits stability theorem, a probabilistic sampling argument, and an exact estimate near a complete $r$-partite graph. The bridge linking spectral inequalities to weighted inequalities is based on the entropy method for the Markov chain attached to the Perron vector.

In this manuscript, we are concerned with the hierarchical control of a non-degenerate quasi-linear parabolic system, posed on a bounded interval. We use the Stackelberg-Nash strategy with a single control called leader and two controls called followers. First, we study the controllability for the linear system, where we will demonstrate a Carleman inequality associated with the adjoint problem. Then, using the results obtained for the linear case and applying Liusternik's Inverse Function Theorem, we obtain local null controllability for the quasi-linear problem.

At an $O(2)$-equivariant Hopf bifurcation a spatially periodic system selects between traveling waves and standing waves, and which branch appears and whether it is stable is decided by two cubic normal form coefficients. Obtaining these coefficients for a given PDE has required a derivation carried out afresh for each model. We remove that step for two-component systems with conservative polynomial differential nonlinearities on a one-dimensional periodic domain, deriving closed form formulas for both coefficients. They are expressed directly in terms of the array of nonlinear PDE coefficients and the spectral data of the linearization, its critical eigenvectors, adjoint eigenvectors, and non-critical resolvents. We give verifiable conditions under which the underlying center manifold reduction is valid. A structural condition on the principal part of the linearization, together with sign conditions on the trace and determinant of its Fourier symbol, yields the resolvent estimate and the spectral gap that the reduction requires. A companion implementation evaluates the two normal form coefficients from the coefficient array and verifies the assumptions for a given system. We specialize the formulas to two applications: a strain-gradient regularization of the nonlinear $p$-system, whose quadratic case contains a previously studied model as a special case, and a first order conservative bilinear family, not previously analyzed, in which every bifurcation scenario allowed by the general classification is realized.

We give a short proof of global regularity for the critical dissipative SQG equation in two dimensions. The argument is based on a pointwise representation of the Leibniz commutator for the half-Laplacian, recently obtained by the author with Luca Gennaioli, which simplifies and unifies the nonlinear estimates used in previous maximum principle proofs.

A decade ago, locally harmonic Maaß forms were first established by Bringmann, Kane, and Kohnen in negative weights and independently by Hövel in weight $0$. Since then, they have seen important applications related to twisted central $L$-values of newforms. However, there are just a few examples of these forms known so far, and in particular there is no unifying theory behind these examples. Towards this direction, we show that the representations of Zagier's $f_{k,D}$ and Mono's $g_{k+1,D}$ functions as traces of cycle integrals are not unique.

We study a population model with non-overlapping generations, where individuals have a type and retain the memory of the types of all their ancestors. We assume that for each individual, the reproduction law depends on the empirical distribution of ancestral types, unlike for a classical multi-type branching process where it hinges on the parental type only. We further assume the mean reproduction matrix has rank one. Building on techniques from previous works with Bastien Mallein, which themselves draw on foundational contributions by Philippe Flajolet and collaborators in the context of analytic urns, we analyze the population at large generations. In particular, we determine the Malthusian parameter and estimate the expected size of the sub-population whose empirical distribution of ancestral types lies within a given set.

This work extends an earlier proposed universal random-coding ensemble for sample-wise lossy compression of individual sequences to the two-stage (successive-refinement) setting — an extension that is not quite straightforward, as explained in the sequel. The construction has three layers. First, a complete, self-contained direct/converse match in the language of type classes and typical sequences, a two-stage analogue of Rimoldi's classical characterization (applied in the superalphabet of blocks). Second, a realization of the same rates via a random-coding scheme built from the Lempel–Ziv (LZ78) complexity rather than the type-class uniform measure the first layer relies on, since an LZ-based ensemble is universal across source statistics and distortion measures, carries no block-length parameter in the codebook itself, and is directly implementable. Third, we give a fully type-free achievability construction (no joint types of $\ell$-vectors appear anywhere in its search criterion) and show it matches the same converse exactly, reaching every point of the achievable region: the same converse from the first layer already binds any code, type-free schemes included, so no separate argument is needed to certify the match. We also describe (without proof) how to extend the type-free construction to any fixed number $r>2$ of stages. Finally, we show this paper's achievable region is a superset of the one obtained by a companion paper, which is based on a finite-state modeling approach to a similar two-stage problem, extending a known single-stage domination result to two stages.

In this short note, we exhibit a Legendrian simple knot that is not detected by Khovanov homology. This disproves a conjecture of Chernov and Maguire.

We compute the second mixed moment of elementary symmetric polynomials in the eigenvalues of the random matrix model $AUBU^*$, where $A$ and $B$ are positive, and $U$ is a Haar-distributed random matrix, associated with finite free multiplicative convolution.

We consider $k$-graphs, $\mathcal{F}\subset \binom{[n]}{k}$, $k\geq 3$. A $k$-graph is called intersecting if any two of its edges have non-empty intersection. It is called a star if all its edges share a common vertex. The $k$-graph $\mathcal{F}$ is called Helly if all its intersecting subfamilies are stars. If the same is required only for subfamilies consisting of three edges, it is called triangle-free. It is well known that for $n\geq 3k/2$, the full star is the unique largest triangle-free family whence the largest Helly family as well. In 1984 Tuza proved the best possible bound $|\mathcal{F}|\leq \binom{n-k-1}{k-1}+\binom{n-2}{k-2}+1$ for Helly families that are not stars, albeit only for some unspecified $n>n_0(k)$. The aim of this paper is twofold. First we establish the same bound for $n>2k$. Second we show that for $n>12k^2$ the same upper bound holds for triangle-free families. It is shown as well that it is not true for $2k<n\leq 3k-4$.

We consider the absolute value equation (AVE) $Ax-|x|=b$ under residual monotonicity. We characterize this property through the symmetric part of the coefficient matrix, thereby allowing nonsymmetry, and propose a coordinate proximal predictor-corrector (CPPC) method based on an AVE-specific forward-proximal decomposition. An exact scalar proximal update on the largest proximal-residual coordinate generates a predictor point, while a positive-alignment identity certifies the ensuing full-residual separating-hyperplane correction without backtracking. With the current matrix product cached, each iteration requires one new full matrix-vector product. For a nonempty solution set, we prove Fejér monotonicity, whole-sequence convergence, and an $O(K^{-1/2})$ best-iterate residual bound. A positive monotonicity margin further ensures unique solvability for every right-hand side and global linear convergence. Numerical results identify regimes in which the reduced per-iteration work yields shorter solution times.

We present a conjectural construction of Arthur packets within Fargues-Scholze's framework for the categorical local Langlands correspondence (CLLC). This construction consists of three parts. We first provide an overview the main statement of the CLLC, the underlying moduli stacks – $\mathrm{Par}_G$ of parameters and $\mathrm{Bun}_G$ of $G$-bundles – on the two sides of the correspondence, and its relation to representations of reductive p-adic groups. We then review the geometric Satake correspondence in order to define Hecke operators and the spectral action of sheaves on $\mathrm{Par}_G$ on sheaves on $\mathrm{Bun}_G$, and to construct semisimple parameters using excursion data. Finally, we generalize the geometric construction of Arthur packets from pushing-forward skyscraper sheaves on the regular conormal bundle of $\mathrm{Par}_G$ over $\mathbb C$ to a conjectural analogous operation on the stack of singularities on $\mathrm{Par}_G$ over $\overline{\mathbb Q}_\ell$.

Metric bases of graphs have been widely studied since their introduction in the 1970's by Slater and, independently, by Harary and Melter. In this paper, we concentrate on the existence of vertices in a graph $G$ that belong to all metric bases of $G$. We call these basis forced vertices, and denote the number of them by $\mathrm{bf}(G)$. We show that $\mathrm{bf}(G)\le 2/3(n-k-1)$ for any connected nontrivial graph $G$ of order $n$ having $k$ vertices in each metric basis. In addition, we show that this bound can be attained. Furthermore, the previous result implies the bound $\mathrm{bf}(G)\le 2/5(n-1)$ formulated in terms of the order $n$ of the graph for any nontrivial connected graph $G$. This result answers a question posed by Bagheri et al. in 2016. Moreover, we provide a complete realization of the parameters $n$, $\dim(G)$ and $\mathrm{bf}(G) \ge 1$ within the previous bounds. We consider some extremal cases related to basis forced vertices in a graph, in particular, we give a full characterization of the graphs with $\mathrm{bf}(G) = 2$ and $\dim(G) = n-4$.

Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A}) \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A} \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's theorem which preserves the overall strategy of his approach.

In this paper, we propose a novel physical model for traffic flow incorporating nonlinear anticipation effects and an intelligent-control boundary, mathematically formulated as a damping boundary condition: \begin{align*} \begin{cases} u_t^\tau+v_x^\tau=0, & x\in(0,1),\; t>0,\\[1mm] v_t^\tau+g(u_x^\tau)u_x^\tau=\dfrac{f(u^\tau)-v^\tau}{\tau}, & x\in(0,1),\; t>0,\\[1mm] (u^\tau,v^\tau)(x,0)=(u_0^\tau(x),v_0^\tau(x)), & x\in(0,1),\\[1mm] u_x^\tau(0,t)=0,\quad u_x^\tau(1,t)=-ku_t^\tau(1,t), & k>0,\; t>0, \end{cases} \end{align*} where $f$ and $g$ are smooth functions satisfying suitable structural assumptions, and $\tau>0$ is the relaxation time. The primary objective is to rigorously investigate how the intelligent-control boundary suppresses the stop-and-go phenomenon in the large-relaxation-time regime-a mechanism that has not been mathematically addressed in previous studies. Utilizing the energy method, we establish the global well-posedness and exponential time-decay of solutions for the original system under arbitrarily large initial data with small spatial derivatives. Furthermore, we analyze the asymptotic behavior in the large-relaxation-time limit $\tau\to\infty$, by introducing a novel technique that incorporates constant shifts into the initial data of the limiting system. This approach enables us to construct a modified auxiliary system, through which we successfully obtain the global convergence of the original solutions to the asymptotic profiles for all time $t$ as relaxation time $\tau\to\infty$. Numerical simulations further demonstrate that in the large-relaxation-time regime, the stop-and-go density waves emerging at the early stage are gradually suppressed by the damping boundary. This leads the traffic stream to eventually evolve into an essentially uniform profile, which perfectly validates our theoretical results.

We develop a systematic method for computing traces of products of Möbius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in ${SL}_2(\mathbb R)$ which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we derive explicit geometric formulas for the Poisson brackets of trace functions associated with closed geodesics. More precisely, we express the Poisson bracket of two trace functions and the iterated Poisson bracket of three trace functions in terms of the hyperbolic lengths of the corresponding geodesics, their intersection angles, and the signed distances between intersection points. The results naturally lead to a unified perspective for revisiting Wolpert's cosine and sine formulas and deriving new proofs of them.

In this paper, for quadratic polynomials with Cremer points or Siegel disks whose boundaries don't contain critical points, we give an arithmetic condition for their Julia sets to have positive area. Our proof relies on the Siegel Return Machinery as well as the pseudo-Siegel disk theory developed by Dudko and Lyubich.

In this work, we develop an accurate numerical homogenization method for computing effective Hamiltonians of quasiperiodic Hamilton–Jacobi equations (QHJEs) with convex Hamiltonians of the form $H(x,p) = |p|^k/{k}-f(x), ~k>1$, where $f$ is quasiperiodic. Computing effective Hamiltonians in the quasiperiodic setting requires solving QHJEs posed on the whole space. Their solutions generally possess neither translational symmetry nor decay and may exhibit low regularity. These features pose substantial challenges for numerical computation. To address these difficulties, we introduce a quasiperiodic boundary condition, which allows the original whole-space problem to be treated on a bounded domain while preserving quasiperiodicity at the boundary. We then propose an SL–FPR scheme that combines a semi-Lagrangian (SL) approximation with the finite points recovery (FPR) method and establish stability and error estimates for the resulting scheme. We also extend the quasiperiodic homogenization result from the quadratic case to general $k>1$ and apply the proposed method to accurately approximate the corresponding effective Hamiltonians. Numerical experiments illustrate the convergence and applicability of the method and validate the extended homogenization results.

We study a broad family of active scalar equations with fractional dissipation on $\mathbb{R}^2$ and prove that their entropy diverges to $-\infty$ at a sharp rate of order $\log t$. The proof is based on a suitable regularization scheme, uniform entropy estimates, and a limiting argument via the Vitali convergence theorem. Our analysis introduces new fractional logarithmic Sobolev inequalities, weighted commutator estimates, and fractional moment bounds. These results provide a general framework for studying spatial decay and long-time dynamics of nonlocal nonlinear partial differential equations.

For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact $\Theta(N^{-2})$ bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic–combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.

We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous Lévy–Itô diffusions driven by a Brownian motion and an independent non-truncated pure-jump Lévy process. Using Bismut's directional Malliavin calculus on the Wiener–Poisson space together with the running-maximum criteria of Song–Xie and Nakagawa–Suzuki, we reduce the problem to constructing an admissible direction $\Theta$. The key is to ensure that the directional derivative $D_\Theta X_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_\Theta X_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $\kappa(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t \kappa(s)^2\,ds>0$ for every $t>0$ and infinite activity of the Lévy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom–density decomposition.

We study three-dimensional wave scattering by high-index dielectric resonators with Kerr-type nonlinearity under plane-wave incidence, together with the associated nonlinear dielectric scattering resonances, through a nonlinear Lippmann-Schwinger equation. Using a Lyapunov-Schmidt reduction near a simple eigenmode of the Newtonian potential, we obtain a local decomposition of the scattered wave into a resonant contribution and a controlled remainder and derive an explicit nonlinear equation for the resonant modal coefficient for sufficiently small contrast parameter and locally small resonant amplitude and incident field. A second reduction covers the regime of incident fields of order one and weak nonlinearity. Our results extend for the first time the linear modal decomposition to nonlinear wave-scattering problems. We complement them by developing a numerical framework based on Nyström discretization, real Newton iteration, and pseudo-arclength continuation. For single resonators of several geometries, our computations confirm the predicted high-contrast resonance scaling and the nonlinear modal approximation, and exhibit the multivalued incident-wave response. For a mirror-symmetric resonator dimer, we track symmetric, antisymmetric, and symmetry-broken resonance branches over a broad range of separations. Our computations show that the symmetry-breaking threshold increases as the gap between the resonators decreases, in agreement with the leading-order bifurcation theory.

We study abstract wave equations with time-dependent propagation speed under strict hyperbolicity. Our starting point is the observation that a substantial part of the classical energy theory, developed through different constructions, can be organized around a common variational mechanism. Approximate energies naturally lead to regularity–fidelity problems in which a smooth approximation of the reciprocal propagation speed is chosen by balancing derivative size against distance from the original coefficient. We combine this approximation mechanism with higher-order energy corrections and obtain a hierarchy of variational problems together with a cascade of almost conserved energies of arbitrary order. The resulting variational problems are then studied independently of the evolution equation. A qualitative distinction emerges between the first and higher orders: for arbitrary prescribed growth, all higher orders generate the same classes, while the first-order problem may behave differently. We also establish connections with fractional and generalized fractional Sobolev regularity. This framework provides a unified interpretation of several classical sufficient conditions for energy control and derivative-loss estimates. At the same time, it identifies a natural limitation of the traditional approach, namely the use of absolute integrability of energy errors. Once this limitation is recognized, a different regime becomes accessible: by retaining their oscillatory structure and exploiting cancellations, we construct strictly hyperbolic propagation speeds beyond the regularity thresholds detected by the variational theory, for which uniform energy estimates and hence no derivative loss nevertheless hold. Thus the variational viewpoint reveals an unexpected common structure behind a substantial part of the classical theory and indicates a genuinely different regime beyond it.