Papers

Classical $(r,\delta)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,\delta)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,\delta)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N , d , k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

We study an initial-boundary value problem for a Zakharov type system in three space dimensions in a general domain. Under a smallness assumption on the initial data, we construct a unique global strong solution. The solution is obtained by a direct approach based on higher-order energy estimates and the construction of a Cauchy sequence in suitable Banach spaces, without employing compactness methods. Furthermore, we obtain estimates on the growth of higher-order Sobolev norms of solutions.

For a graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with $\mathrm{ex}(n,F)+q$ edges, where $\mathrm{ex}(n,F)$ is the maximum number of edges in an $n$-vertex $F$-free graph. Let $c(n,F)$ be the minimum number of copies obtained by adding one edge to an extremal $F$-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that $h_F(n,q)\ge q\,c(n,F)$. Ma and Yuan recently constructed stable graph counterexamples for every fixed $q\ge4$; they asked whether the one-edge equality $h_F(n,1)=c(n,F)$ might still hold for every graph $F$ containing a cycle. We give a negative answer to their question. For each integer $t\ge6$, let $H_t$ be obtained from the $t$-vertex path by replacing each edge with a $3t$-page book, using disjoint page vertices for different path edges. Then $h_{H_t}(n,1)<c(n,H_t)$ for infinitely many values of $n$. Moreover, by taking $t$ large, the ratio $h_{H_t}(n,1)/c(n,H_t)$ can be made arbitrarily small along infinitely many values of $n$.

A unitarily invariant complete Nevanlinna–Pick (CNP) kernel $k$ on the Euclidean ball gives rise to a natural class of operator tuples on Hilbert spaces, known as $1/k$-contractions. We establish a lower estimate for the Taylor joint spectrum of $1/k$-contractions with finite defect in terms of the associated characteristic function. Under the additional assumption that $k$ satisfies the Corona property, this lower estimate coincides with an upper estimate due to Clouâtre–Timko [Adv. Math., 2023], yielding an exact characterization of the Taylor joint spectrum in terms of the characteristic function. As an application, we determine the Taylor joint spectrum of quotient modules of CNP spaces. A key ingredient in the proof is a Beurling–Lax–Halmos theorem for these spaces, established in terms of characteristic functions.

We describe the construction of a database of roughly half a million abelian surfaces over Q of small conductor arising as Pryms associated to a genus 3 double cover of a genus 1 curve. Our construction uses a method that provides a degree of control over the primes of bad reduction.

The split covariance intersection filter (Split CIF) is a useful tool for general data fusion and has the potential to be applied in a variety of engineering tasks. The w-optimization problem involved in the Split CIF concerns the performance and implementation efficiency of the Split CIF. It is known that the w-optimization problem enjoys the desirable property of convexity (or more clearly, the second-order convexity in this paper's context). This paper proves that the w-optimization problem further enjoys a more desirable property namely the fourth-order convexity, thanks to which a guaranteed fast implementation of the Split CIF can be realized. The new implementation is coined as the nested Newton method, which is also presented in this paper.

This paper is devoted to the numerical analysis of a second-order bulk–surface splitting scheme for the semi-linear wave equation with kinetic boundary conditions. The construction is based on the interpretation of the equations as coupled system and the implementation of different difference formulae for the discrete states depending on their exact position in the system equations. This results in a 4-step scheme which decouples bulk and surface dynamics. We prove energy stability and second-order convergence under a weak CFL condition and validate these results also numerically.

We develop an inhomogeneous form of Edwards' Jensen-measure duality for Perron envelopes constrained by Monge–Ampère lower bounds. The admissible subsolution families are convex but not cones; nevertheless, the dual measures remain the homogeneous Jensen measures, and the right-hand side enters through a scalar Jensen deficit \[ B_{\mathcal{A}}(x,\mu) = \inf_{u\in\mathcal{A}} \left(\int_{\partial\Omega}u\,d\mu-u(x)\right). \] Under natural structural hypotheses we prove a boundary dual formula \[ \sup\{u(x):u\in\mathcal{A},\ u\leq\varphi\text{ on }E\} = \inf_{\mu\in J_x^\partial} \left( \int_{\partial\Omega}\varphi\,d\mu - B_{\mathcal{A}}(x,\mu) \right). \] We apply the theorem to real Alexandrov subsolutions and to complex Bedford–Taylor plurisubharmonic subsolutions with continuous density. In one real dimension the deficit is the Green-potential correction; in higher dimensions it has intrinsic stress and current interpretations. On B-regular domains, a bounded Bedford–Taylor approximation theorem identifies bounded and continuous competitors and yields a duality proof of continuity for the corresponding Dirichlet solution. Finally, for smooth strictly elliptic solutions, optimal Jensen measures are the harmonic measures of the linearized Monge–Ampère operators, equivalently the boundary derivatives of the nonlinear solution map.

Motivated by the need to understand infection spreading in inhomogeneous populations, we consider a temporal version of the stochastic block model, where each edge is equipped with a random label, interpreted as a timestamp. We study the size and structure of reachable sets via increasing paths (that is, paths whose edge timestamps are strictly increasing) where the connections per node are of the order of $\log n$. We prove tight results for the first order asymptotics of the size of the reachable set from a typical vertex, and the proportions of reached vertices in each community. We also determine the typical length of the shortest and longest increasing paths with one or two fixed endpoints. In the regime where the reachable set $S$ of a typical vertex is of sublinear size, we describe the structure of the subgraph induced by $S$ by identifying an almost spanning tree (i.e., spanning all but $o(|S|)$ vertices), that is distributed as a weighted random recursive tree, whose height, profile and degree sequence are known. In particular, when $|S|\ll \sqrt{n}$, the subgraph induced by $S$ is itself distributed as this well-understood weighted random recursive tree.

The centre comonad model provided the first concrete cohesive linear $\infty$-topos, settling an open problem of Schreiber. However, the model is degenerate: the quantum modality annihilates all non-commutative algebras, and the associated linear logic collapses to classical cartesian logic. In this paper we give a complete mathematical diagnosis of this degeneracy. We prove that the centre comonad sends the representable sheaf of a simple non-commutative algebra to the empty presheaf, and that the state space of any such algebra is empty. We then prove that the Day convolution on the classical core is cartesian, forcing the Seely isomorphism to hold trivially and collapsing the linear logic. We isolate the structural reason behind this collapse: whenever the opposite of the classical core is monoidally equivalent to a cartesian monoidal category, any coreflective precomposition comonad will exhibit the same degeneracy. We conclude that a non-degenerate quantum modality must be constructed without precomposition, and we briefly discuss possible directions.

Riemannian optimization provides a powerful framework for constrained optimization by incorporating problem-specific structure directly into the geometry of the search space. In many applications, however, the explicit evaluation or application of the Riemannian metric can be computationally expensive or numerically unstable, limiting the practical efficiency of otherwise well-founded algorithms. Motivated by such settings, this work investigates to what extent classical Riemannian optimization algorithms can be reformulated without explicitly applying the metric. We show that many first-order components of Riemannian optimization only rely on the differential of the objective function and access to the Riemannian gradient, but not on explicit metric application. Based on this observation, we develop metric-free formulations and generalize optimization approaches to Finsler and Banach manifolds. Numerical experiments demonstrate that the proposed metric-free strategies retain the effectiveness of their metric-dependent counterparts while significantly reducing computational overhead. These results highlight that a substantial portion of Riemannian optimization can be carried out independently of explicit metric application, broadening its applicability to problems with expensive or implicitly defined metrics.

Algebraic Geometry (AG) codes (i.e. linear codes from algebraic function fields) in the Hamming metric were proposed by Goppa in 1980 and have been intensively studied ever since. Linearized Algebraic Geometry codes, the analogue of AG codes in the sum-rank metric, were instead introduced more recently [9], using quotients of the ring of Ore polynomials with coefficients in an algebraic function field. In this paper, we further investigate the results in [9], providing explicit, optimal and asymptotic constructions.

In this paper we clarify the relation between Geometric Thermodynamics and Information Geometry based on the Fisher matrix. On the macroscopic odd-dimensional contact manifold of thermodynamic variables, we introduce for the first time a metric, whose pull-back on the isoentropic symplectic submanifolds transverse to the Reeb field is Kählerian. The pull-back of such metric on equilibrium states, that are lagrangian submanifolds, is the Fisher Hessian. Then we consider the Souriau-like Thermodynamics that uses Calabi-Vesentini (CV) manifolds as Kaehlerian microscopic event manifolds and the Killing moment maps as observable functions. A systematic use of the theory of compact abelian structures and the setup of Special Kähler Geometry in which CV manifolds are encoded allows us to perform the explicit integration defining the partition function for any entry in the CV Tits Satake universality class. The additional actions completing the abelian structure are non linear Casimir functions of the Killing moment-maps and suggest a generalization of Souriau thermodynamics that partially breaks the isometry group symmetry by means of the non vanishing mean values of the Casimir functions in a manner similar to the spontaneous magnetization in ferromagnetism. Our new exact Gibbs distributions provide the analogue for Cartan Neural Networks of the Gaussian probability distributions in flat space used in conventional Machine Learning.

Let $f(n,H)$ be the maximum number of copies of a graph $H$ in a planar graph of order $n$. When $H$ is a connected graph on four vertices, $f(n,H)$ has been completely determined except for two cases: $K_{1,3}^+$ (the claw graph $K_{1,3}$ with one additional edge) and $K_4^-$ (the complete graph $K_4$ with one edge removed). Here, we address these two cases and establish that for all $n\ge4$, $$f(n,K_{1,3}^+) = 4n^2-12n-4 \textrm{ and } f(n,K_4^-) =\frac{1}{2}(n^2+9n-40).$$

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t \psi = D\psi - \mathcal N(\psi), \qquad \psi(0)=\psi_0, \] where $\psi=(\psi_1,\psi_2)^{\mathsf T}:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, \[ \mathcal N(\psi)=\mathcal B\bigl(\Pi_+\psi,\Pi_-\psi\bigr), \] where $\Pi_\pm$ are the spectral projections of $D$ and $\mathcal B$ is a fixed bilinear map on $\mathbb{C}^2$ applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of $D$ and a mixed-sign bilinear estimate on $N$-star graphs, we prove local well-posedness in the operator Sobolev space $H_D^s(G)$ for \(s>-\frac18\). We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.

Benchmarking forecasting architectures for daily equity portfolios is not just a prediction exercise. It also asks which model remains usable after preferences, costs, and portfolio constraints are imposed. We build a CRSP daily-stock benchmark for 15 deep and statistical time-series architectures over 2018–2024. The protocol combines common-window decile portfolios, stochastic multi-criteria acceptability analysis, a deployment-adjusted acceptability index, and a constrained quadratic portfolio layer with capacity, beta, industry, risk, leverage, and turnover controls. The index starts from the SMAA rank-acceptability distribution and downweights models whose criteria-level wins produce high portfolio regret; its Gibbs form is characterized as an entropic update from the SMAA prior. Empirically, no architecture dominates the raw benchmark: TransEnc-8 has the largest rank-1 acceptability, 0.352, and no model exceeds about 0.36. Rankings vary with preferences, market state, feature universe, and transaction costs. In the promoted five-model constrained-portfolio comparison, TransEnc-8 is selected throughout, while return-oriented raw rankings can favor TS-RIDGE. Broad-universe decile signals can survive costs, but the baseline constrained-QP net Sharpe at 20 bps is negative for every promoted model. The benchmark supports model selection and diagnosis rather than a standalone trading-strategy claim.

Given a principally polarized abelian variety, we give a sufficient condition for the van Geemen-van der Geer locus $\Gamma_{00}$ to be positive dimensional. Along the way, we prove a sharp Castelnuovo-type inequality for hyperelliptic curves on abelian varieties, and characterize the cases in which we have equality. These results are consequences of a general relation between Seshadri constants and the surjectivity of high order Gauß-Wahl maps on curves inside abelian varieties, which we also prove. Applying such relation to Abel-Jacobi curves we obtain new results about the surjectivity of higher Gauß-Wahl maps on smooth curves, sharpening a previous result of Bertram, Ein and Lazarsfeld. In the process, we discuss further questions and conjectures.

Resolving in a strong sense a problem of Gyárfás on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs. We also show that the chromatic number can be replaced by the fractional chromatic number or $\frac{|V(G)|}{\alpha(G)}$.

Large language models can be fine-tuned to encode prompt-borne secrets into fluent, seemingly benign outputs. This creates a steganographic exfiltration risk that is difficult to detect with output-level steganalysis. Recent work proposes mechanistic detection using linear probes that recover the secret from internal activations. We show that this defense can be systematically evaded, but that detectability can be recovered through a targeted data-level intervention. First, we extend the detection setup to include a non-linear MLP probe. We then adversarially fine-tune steganographic trojans across five base models: Qwen3-8B, Llama-3.1-8B, Ministral-8B, Qwen3-14B, and Phi-4-14B. The resulting models retain $58$–$79\%$ exact-match secret recovery while evading both ridge and held-out MLP probes, with $1$–$8\%$ average capability degradation across six benchmarks. We then give an information-theoretic characterization of this evasion. Successful evasion preserves recoverability while reducing low-order extractability of the secret from the content-aligned representation, forcing the payload into synergistic interaction with residual degrees of freedom. This motivates a recontextualization dataset that restricts these residual degrees of freedom. On this distribution, both ridge and MLP detectability are restored across all five evasive trojans. Overall, our findings show that activation-based steganography detection is vulnerable to adaptive evasion, but also that theory-guided evaluation distributions can expose otherwise hidden payloads.

In the paper, we study the problem of recovering the Sturm-Liouville operator on a star-graph from the Weyl vector. It generalizes the problem of recovering the classical Sturm-Liouville operator on an interval from the Weyl function, and the problems of recovering from other spectral data can be reduced to this problem. The uniqueness and the constructive method for solving the problem under study were previously obtained by V.A. Yurko in the case of a tree (Inverse Problems, 2005). Here, we prove its uniform stability, which includes Lipschitz estimates with a constant depending only on the number bounding the norms of the potentials. Stability results are necessary for justifying the well-posedness of the problem statement, and they are important for developing numerical methods. As auxiliary results, we obtain the uniform stability of the direct problem, as well as the uniform stability of the partial derivatives of the transmutation operator kernel related to the classical Sturm-Liouville operator.

Let $A$ be a bounded non positive commutative differential graded algebra $A$. Let $\mathbf{D}(A)$ its derived category of DG-modules. If $\mathbf{D}(A)$ is generated by the DG-modules corresponding to the residue fields of the ordinary ring $H^0(A)$ then its localizing subcategories and its colocalizing subcategories are in bijection with the subsets of $\textrm{Spec}(H^0(A))$. These results generalize well-known theorems by A. Neeman (from 1992 and 2011, respectively), because any Noetherian ring satisfies this condition.

Kling-Gupta efficiency ($\mathrm{KGE}$) is a model performance evaluation metric widely used in hydrology, but its properties as a statistical estimator have remained unexplored. We formalize the Kling-Gupta loss $L_\mathrm{KG} = (1 - \mathrm{KGE})^2$ in an extremum estimation framework (maximizing $\mathrm{KGE}$) for multiple linear regression. We give explicit formulas showing that Kling-Gupta regression scales the ordinary least squares (OLS) coefficient vector by a variance-inflation factor depending on sample variances and covariances. Its predictions reproduce the training set response variance, unlike OLS's variance reduction, while both maintain the response mean and achieve the same sample correlation. We prove that no estimator simultaneously maximizes Nash-Sutcliffe efficiency ($\mathrm{NSE}$) and $\mathrm{KGE}$: OLS maximizes $\mathrm{NSE}$ but not $\mathrm{KGE}$, whereas Kling-Gupta regression maximizes $\mathrm{KGE}$ at the expense of $\mathrm{NSE}$. We establish almost-sure convergence of the Kling-Gupta estimator to well-defined population limits. The training and test set performance metrics for both estimators converge asymptotically to identical limits (different for OLS vs. Kling-Gupta). In a single-predictor model with fixed intercept, we identify conditions where a global minimum of $L_\mathrm{KG}$ does not exist because of discontinuity at zero slope. This work establishes a mathematical foundation for $\mathrm{KGE}$-based estimation and clarifies its effects on predictive performance in hydrologic modeling.

We study uniform recursive directed acyclic graphs with freezing. Here, a graph is built by adding vertices one-by-one and connecting a new vertex to $m\in\mathbb N$ uniformly selected vertices already present. At certain steps vertices can also be frozen, and arriving vertices are not allowed to connect to frozen vertices. This model generalises the uniform attachment tree with freezing, introduced by Bellin et. al (which corresponds to the case $m=1$) as well as the uniform recursive directed acyclic graph model (where no vertices are frozen). Under mild assumptions on when vertices are frozen, we study the empirical degree distribution, large degrees in the graph, and other properties of large-degree vertices such as their label and distance to the first vertex in the graph. Our work improves and/or extends various results from the literature on uniform attachment trees (with freezing) and uniform recursive directed acyclic graphs without freezing. In particular, our results show that statistics that are determined `locally' (e.g. the empirical degree distribution and maximum degree) are essentially unaffected by the freezing of vertices, whereas statistics that are determined `globally' (e.g. the length of paths between vertices) are highly affected by introducing freezing. The analysis relies on adapting the Kingman coalescent construction for uniform attachment trees to the non-tree setting.

Momentum-based acceleration of iterative hard thresholding (IHT) can dramatically speed up sparse signal recovery from linear measurements, but its effectiveness hinges on careful parameter tuning – a task complicated by the frequent support changes inherent to hard thresholding. We propose CosMIHT(Consecutive Support Matching Induced Momentum IHT), which resolves this difficulty through a simple adaptive rule: start with the conservative parameters and whenever two consecutive iterates share the same support, estimate the extreme eigenvalues of the support restricted Gram matrix via a lightweight power method and switch to the corresponding optimal heavy-ball parameters. This mechanism allows CosMIHT to automatically interpolate between cautious MIHT-like behavior during support discovery and near-optimal accelerated convergence after support identification. Under standard restricted isometry assumptions, we develop a two-phase convergence theory. In the wandering phase, we establish a linear contraction of the recovery error up to a noise floor and derive an explicit upper bound on the number of iterations required to identify the correct support. In the lock-in phase, we establish that, with a randomly initialized power method based eigenvalue estimates that depend on the number of power iterations, the algorithm enjoys, with high probability, a near-optimal accelerated convergence rate akin to the heavy ball method. We corroborate the theoretical findings with extensive numerical experiments on both noiseless and noisy measurements demonstrating that CosMIHT achieves faster convergence than state-of-the-art iterative sparse recovery techniques without compromising the recovery performance.

We investigate the asymptotic behavior of the complex roots of Rayleigh-Lamb dispersion relations arising in elastic waveguides. We consider Neumann, Dirichlet, and fluid boundary conditions and derive explicit asymptotic expansions of the associated wavenumbers as their modulus tends to infinity. In the Neumann case, we provide a rigorous justification of asymptotic formulas that have long been used in the literature without proof, together with higher-order terms and explicit constants. Similar results are obtained for Dirichlet and fluid boundary conditions. The analysis relies on a reformulation of the dispersion relations as zeros of holomorphic functions and on asymptotic properties of the Lambert function. We also show how these asymptotic expansions can be used to establish modal decompositions and well-posedness results for elastic waveguide problems. Numerical experiments confirm the accuracy of the proposed formulas.

We study optimal quantization of probability measures on Carnot groups equipped with a left-invariant homogeneous distance. We prove two main results. First, we establish a Zador-type asymptotic formula for the quantization error: after the natural rescaling, the error converges as the number of centers tends to infinity, and the exponent is determined by the homogeneous dimension of the group. The limiting constant is a Carnot cell constant, defined through the quantization problem on a reference cell. Second, we prove weak convergence of the empirical measures associated with optimal centers, and describe the limit in terms of the density of the absolutely continuous part of the measure. The proof combines a tiling of Carnot groups by exponential cubes with a sub-Riemannian version of Pierce's lemma, which allows us to treat measures with non-compact support.

We introduce and study the class of $t$-$g$-radical supplemented modules, which unifies two independent generalizations of the classical supplemented module condition: $g$-radical supplements and $t$-sum terms. A module $M$ is $t$-$g$-radical supplemented if every submodule $N \leq M$ has a $g$-radical supplement that is simultaneously a $t$-sum term of $M$. We establish closure properties under $t$-sums, quotient modules, and homomorphic images, prove inheritance by $t$-sum terms, and classify the standard module classes (simple, semisimple, local, hollow, and Prüfer groups) within this framework. A key observation is that the class strictly contains the class of supplemented modules: both $\mathbb{Q}$ and $\mathbb{Z}_{p^{\infty}}$ are $t$-$g$-radical supplemented but not supplemented.

We study paired multiplication operators associated with orthogonal decompositions of $L^2(\mathbb{T})$ arising from Hardy spaces and Sz.-Nagy–Foias model spaces. For a non-constant inner function $\theta$, we characterize $\mathcal{K}_\theta$-paired and triply paired operators by finite-rank commutator identities with the bilateral shift $M_z$. We also apply these ideas to sums of truncated Toeplitz and truncated Hankel operators on $\mathcal{K}_\theta$. Using the displacement characterizations of Sarason and Gu–Ma, we derive a second-order finite-rank displacement formula for such sums and describe the ambiguity of the decomposition in terms of $\mathcal{I}_\theta=\mathcal{T}_\theta \cap \mathcal{H}_\theta$. In the finite-dimensional case $\theta(z)=z^n$, the framework recovers the classical Toeplitz-plus-Hankel matrix recurrence of Bevilacqua–Bonanni–Bozzo.

Tropical Pacific sea-surface-temperature (SST) variability spans interacting timescales, with the ENSO as its dominant interannual expression. Yet the dynamical structure organizing this variability and underpinning extended-range predictability remains difficult to extract from high-dimensional observations. Koopman operator learning offers spectral coordinates for nonlinear dynamics, yet finite geophysical records often produce dense, sampling-sensitive spectra whose physical content is ambiguous. We show that this apparent redundancy reflects coherent operator-level structure. Combining kernel Extended Dynamic Mode Decomposition with residual minimization and pseudospectral analysis, we use the Koopman eigenvalue relation as a physics-informed consistency test to organize learned spectra. Applied to ERA5 and HadISST tropical Pacific SST anomalies, the residual landscape identifies 19 robust residual-minimum frequencies with coherent spatial modes that persist across products and sampling realizations. Together, these modes define a compact Koopman backbone spanning low-frequency modulation through quasi-biennial components, including ENSO-band variability. The surrounding spectral cloud is structured by integer powers and nonlinear combinations of this backbone, forming a residual-ordered Koopman hierarchy. The backbone reconstructs substantial Nino3.4 variance and enables skillful out-of-sample forecasts, with greatest gains at 8-18-month leads. By embedding dynamical consistency into physics-informed operator learning, the framework turns opaque spectra into robust, interpretable and predictive representations of tropical Pacific variability.