The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on $ε$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality, and a curvature-driven pre-compactness result. The final part of the paper establishes several applications: we show that Chruściel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representations of persistence diagrams are typically constructed extrinsically through embeddings into auxiliary spaces. For filtrations with finite indexing sets, the associated virtual persistence diagram group obtained by Grothendieck completion of the persistence diagram monoid is a finitely generated lattice. We define a phase map sending each persistence interval to a circular coordinate and a character map aggregating the phases of intervals in a virtual persistence diagram. We introduce heat damping on characters of virtual persistence diagram groups to suppress the unstable frequencies. We derive Lipschitz bounds for the resulting kernels and apply them in a synthetic segmentation experiment.

The ``Pre-train, then fine-tune'' paradigm has revolutionized Natural Language Processing (NLP). In this context, transferable backdoors pose a severe threat to the Pre-trained Language Models (PLMs) supply chain, yet defensive research remains nascent, primarily relying on detecting anomalies in the output feature space. We identify a critical flaw that fine-tuning on downstream tasks inevitably modifies model parameters, shifting the output distribution and rendering pre-computed defense ineffective. To address this, we propose Patronus, a novel defense framework that shifts the defensive focus from output features to input-side invariance, exploiting the fact that adversarial triggers remain constant even as model weights change. To overcome the convergence challenges of discrete text optimization, Patronus introduces a multi-trigger contrastive search algorithm that effectively bridges gradient-based optimization with contrastive learning objectives. Furthermore, we employ a dual-stage mitigation strategy combining real-time input monitoring with model purification via adversarial training. Extensive experiments across 15 PLMs and nine tasks demonstrate that Patronus achieves $\geq98.3\%$ backdoor detection recall and reduces attack success rates to clean settings, significantly outperforming all state-of-the-art baselines in all settings. Code is available at https://github.com/zth855/Patronus.

We propose a hybrid variational quantum algorithm that has variational parameters used by both the quantum circuit and the subsequent classical optimization. Similar to the Variational Quantum Eigensolver (VQE), this algorithm applies a parameterized unitary operator to the qubit register. We generate this operator using diabatic state preparation. The quantum measurement results then inform the classical optimization procedure used by the Cascaded Variational Quantum Eigensolver (CVQE). We demonstrate the algorithm on a system of interacting electrons and show how it can be used on long-term error-corrected as well as short-term intermediate-scale quantum computers. Our simulations performed on IBM Brisbane produced energies well within chemical accuracy.

The Dark Energy Spectroscopic Instrument (DESI) in its first Data Release (DR1) already provides more than 100,000 galaxies with relative distance measurements. The primary purpose of this paper is to perform the calibration of the zero-point for the DESI Fundamental Plane and Tully-Fisher relations, which allows us to measure the Hubble constant, $H_0$. This sample has a lower statistical uncertainty than any previously used to measure $H_0$, and we investigate the systematic uncertainties in absolute calibration that could limit the accuracy of that measurement. We improve upon the DESI Early Data Release Fundamental Plane $H_0$ measurement by a) using a group catalog to increase the number of calibrator galaxies and b) investigating alternative calibrators in the nearby Universe. Our baseline measurement calibrates to the SH0ES/Pantheon+ type Ia supernovae, and finds $H_0=73.7\pm 0.06\;(\text{stat.})\pm 1.1\;(\text{syst.})$ km s$^{-1}$ Mpc$^{-1}$. Calibrating to surface brightness fluctuation (SBF) distances yields a similar $H_0$. We explore measurements using other calibrators, but these are currently less precise since the overlap with DESI peculiar velocity tracers is much smaller. In future data releases with an even larger peculiar velocity sample, we plan to calibrate directly to Cepheids and the tip of the red giant branch, which will enable the uncertainty to decrease towards a percent-level measurement of $H_0$. This will provide an alternative to supernovae as the Hubble flow sample for $H_0$ measurements.

ATLAS and CMS have recently reported enhancements in the top-antitop production rate near threshold, a region where non-perturbative QCD dynamics associated with toponium formation become relevant. We investigate how this behaviour is modified in the presence of a neutral pseudoscalar that couples to gluons and top quarks, using an effective description that consistently incorporates perturbative Standard Model and new physics contributions, their interference and non-perturbative threshold effects. We show that the combined effect of those ingredients markedly shapes the viable region of the pseudoscalar parameter space, particularly for narrow resonances with masses close to twice the top mass. While Standard Model threshold effects could explain a sizeable part of the measured enhancements, the current data remain compatible with additional contributions from pseudoscalar interactions.

Imperfect molecular detection in single-cell experiments introduces technical noise that obscures the true stochastic dynamics of gene regulatory networks. While binomial models of molecular capture provide a principled description of imperfect detection, they have so far been analyzed only for simple gene-expression models that do not explicitly account for regulation. Here, we extend binomial models of capture to general gene regulatory networks to understand how imperfect capture reshapes the observed time-dependent statistics of molecular counts. Our results reveal when capture effects correspond to a renormalization of a subset of the kinetic rates and when they cannot be absorbed into effective rates, providing a systematic basis for interpreting noisy single-cell measurements. In particular, we show that rate renormalization depends on the level of regulatory detail in the model. For implicit regulatory models based on promoter state transitions, it arises whenever gene product synthesis does not trigger a promoter state change, as in the absence of promoter-proximal pausing or when pausing is short-lived. For models with explicit transcription factor binding, the same condition holds, together with sufficiently high transcription factor abundance, which in practice requires only a few tens of molecules per cell. In these cases, technical noise reduces the apparent mean burst size of synthesized gene products and accelerates the apparent rates of transcription factor binding reactions. This acceleration becomes stronger as the number of protein species and/or molecules involved in promoter switching increases. These effects hold for gene regulatory networks of arbitrary connectivity and remain valid under time-dependent kinetic rates.

Partial Differential Equations (PDEs) are fundamental mathematical models for describing physical phenomena, yet most PDEs of practical interest require numerical approximations. The feasibility of such methods is constrained by existing computational models. Since digital computers are the primary realizations of numerical computations, and Turing machines define their theoretical limits, computability of PDE solutions is of fundamental significance. It provides a rigorous framework to distinguish equations that are effectively solvable from those that encode undecidable or non-computable behavior. Once computability is established, complexity theory quantifies the resources required to approximate PDE solutions. In this work, we present a novel framework based on least-squares variational formulations and associated gradient flows to analyze the computability and complexity of PDE solutions from an optimization perspective. Our approach approximates PDE solution operators via discrete gradient flows, linking PDE properties, such as coercivity, ellipticity, and convexity, to solution complexity. Within this setting, we characterize representation- and discretization-dependent sufficient conditions for regimes where PDEs admit polynomial-time approximations, as well as regimes exhibiting complexity blowup, where polynomial-time input data produce solutions with super-polynomial complexity. In summary, this paper develops a variational framework for analyzing computability and computational complexity of PDE solution classes. The results show how PDE structure and solution regularity influence their complexity, by establishing sufficient conditions for computability and complexity bounds. Beyond the theoretical characterization, the framework provides guidelines for effective numerical methods and contributes to understanding the limitations of digital computation for PDE problems.

The rapid iteration cycles of modern live-service games make regression testing indispensable for maintaining quality and stability. However, existing regression testing approaches face critical limitations, especially in common gray-box settings where full source code access is unavailable: they heavily rely on manual effort for test case construction, struggle to maintain growing suites plagued by redundancy, and lack efficient mechanisms for prioritizing relevant tests. These challenges result in excessive testing costs, limited automation, and insufficient bug detection. To address these issues, we propose SAGE, a semanticaware regression testing framework for gray-box game environments. SAGE systematically addresses the core challenges of test generation, maintenance, and selection. It employs LLM-guided reinforcement learning for efficient, goal-oriented exploration to automatically generate a diverse foundational test suite. Subsequently, it applies a semantic-based multi-objective optimization to refine this suite into a compact, high-value subset by balancing cost, coverage, and rarity. Finally, it leverages LLM-based semantic analysis of update logs to prioritize test cases most relevant to version changes, enabling efficient adaptation across iterations. We evaluate SAGE on two representative environments, Overcooked Plus and Minecraft, comparing against both automated baselines and human-recorded test cases. Across all environments, SAGE achieves superior bug detection with significantly lower execution cost, while demonstrating strong adaptability to version updates.

We generalize Milnor link invariants to surface-links in 4-space, possibly with boundary. To this end, we introduce the notion of cut-diagram, which is a 2-dimensional analogue of Gauss diagrams. To each cut-diagram, we associate a group extending the fundamental group of the exterior of a surface-link, and we extract Milnor-type invariants from its successive nilpotent quotients. We show that this yields concordance invariants for surface-links, and that some even are link-homotopy invariants. We give several concrete applications, including realization and classification results. The theory of cut-diagrams is further investigated, heading towards a combinatorial approach to surfaces in 4-space.

Quantum networks and modular quantum computers require efficient spin-photon interfaces, often realized using optical resonators that enhance radiative decay on a desired transition. However, this requires small mode volumes and high quality factors, which limits multiplexing capacity and demands precise frequency tuning. Here, we demonstrate an alternative approach that circumvents these bottlenecks for upscaling. Using a W1 silicon photonic crystal waveguide with a tailored photonic bandgap, we selectively inhibit unwanted decay pathways, thereby redirecting emission to the desired transition. This enables efficient photon collection over a large frequency range, allowing the resolution and individual addressing of tens of erbium dopants. Their lifetimes are preserved, or even increased, compared to bulk material. The extended mode volume of the devices enables the use of lower dopant concentrations, thereby improving emitter coherence. Our approach can be combined with Purcell enhancement and applied to other spin-qubit platforms, opening intriguing perspectives for photonic quantum technologies.

We present a new method to estimate terminal bubble velocities during first-order phase transitions in a plasma in local equilibrium. The method relies on calculating the extrema of a modified potential function for the scalar field undergoing the transition. The shape of this function, which we refer to as the ``pseudopotential'', changes with the wall velocity, and if the dependence of the fluid temperature on scalar gradients is weak -- which is confirmed to hold with high accuracy in concrete examples -- the difference in pseudopotential between two appropriate extrema gives the net outward pressure acting on the bubble wall. It then follows that the correct terminal bubble velocities are those that lead to degenerate minima in the pseudopotential. This allows to compute bubble velocities without having to solve the equation of motion of the scalar field, and in contrast to other methods this can be done without relying on simplified equations of state for the plasma or without choosing a specific ansatz for the scalar field profile. We illustrate the method in a singlet extension of the Standard Model, computing the net outward pressure as a function of the wall velocity. We confirm the dip in outward pressure found in the literature for hybrid bubbles, which implies that stationary deflagrations are stable, while their detonation counterparts are unstable.

In gravitational theories containing matter fields and higher-derivative corrections, the standard Smarr formula often fails unless all dimensionful couplings are incorporated consistently. Traditionally, parameters such as the cosmological constant or the coefficients of higher-derivative terms are regarded as immutable features of the theory and therefore excluded from the thermodynamic phase space. In our recent work, we developed a fully general framework that promotes every such coupling to a dynamical, freely varying parameter of black hole solutions. This is accomplished by introducing, for each coupling, an auxiliary scalar and gauge field, through which the coupling appears as a conserved charge associated with the global sector of an emergent gauge symmetry. The corresponding conjugate variables naturally arise as electric potentials evaluated at the black hole horizon. As a result, the first law and the Smarr relation acquire additional, systematically determined contributions, yielding a consistent and universal extension of black hole thermodynamics. We illustrate the validity of this construction by revisiting several black hole examples in the literature where the Smarr formula remains inconsistent even after treating the cosmological constant as a thermodynamic variable. Our analysis shows that only by including all dimensionful couplings in this generalized manner can one obtain an internally consistent Smarr relation, thereby providing the foundation for a truly universal formulation of black hole thermodynamics.

We provide efficient algorithms to compute convergents of quadratic irrationals. We show that for square roots, in settings where Galois' refinement of Lagrange's theorem holds, certain decimations of the sequence of convergents are signed Chebyshev sequences, which can be also be generated by a Householder method.

We investigate a one-dimensional tight-binding model in which onsite potentials $\{\varepsilon_i\}$ exhibit power-law spatialcorrelations (with exponent $α$) and the hopping amplitudes decay as $t_{ij}\sim |i-j|^{-β}$. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the $(α,β)$ plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio $r$, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the $(α,β)$ parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.

Execution path reasoning is a key step towards program semantics understanding. It is crucial for generating test cases that cover certain branches/paths, or detecting bugs that are triggered by some paths without actually executing the program. Traditionally, execution path reasoning can be achieved by symbolic execution techniques, but existing SMT-based symbolic execution approaches struggle with complex data structures and external API calls. This challenge is even more pronounced in languages with highly flexible syntax, such as Python, resulting in a lack of widely adopted tools for reasoning on execution paths. Therefore, reasoning execution paths with AI-based approaches become a promising direction. In this paper, we investigate the feasibility of adopting large language models (LLMs) for execution path reasoning on Python, where traditional path-based symbolic execution tools are unavailable. We conduct an empirical study on two types of path reasoning tasks: generation tasks for test case generation and classification tasks for bug detection. We build new evaluation pipelines and benchmarks from both competition-level programs and real-world repositories. Our results show that state-of-the-art LLMs can perform correct reasoning on execution paths and improve test coverage on real-world software, though models with stronger reasoning abilities do not always outperform weaker ones. These findings highlight the potential of utilizing LLMs as a complementary heuristic for path-aware code reasoning, especially in program languages lacking mature symbolic execution tools. We have released our benchmark and evaluation scripts at https://github.com/jacobwwh/llm-path-study.

Achieving consensus among self-interested agents remains challenging in decentralized multi-agent systems, where agents often have conflicting preferences. Existing coordination methods enable agents to reach consensus without a centralized coordinator, but do not provide formal guarantees on system-level objectives such as efficiency or fairness. To address this limitation, we propose a regulated decentralized negotiation framework that augments a decentralized negotiation mechanism with limited regulatory oversight. The framework builds upon the trading auction for consensus, enabling self-interested agents with conflicting preferences to negotiate through asset trading while avoiding direct disclosure of private asset valuations. We introduce an oversight mechanism, which implements a taxation-like intervention that guides decentralized negotiation toward system-efficient and equitable outcomes while also regulating how fast the framework converges. We establish theoretical guarantees of finite-time termination and derive bounds linking system efficiency and convergence rate to the level of regulatory intervention. A case study based on the collaborative trajectory options program, a rerouting initiative in U.S. air traffic management, demonstrates that the framework can reliably achieve consensus among self-interested airspace sector managers, and reveals how the level of regulatory intervention regulates the relationship between system efficiency and convergence speed. Taken together, the theoretical and experimental results indicate that the proposed framework provides a mechanism for regulated decentralized coordination that preserves noncooperative final selection while safeguarding system-level objectives.

Recent high-precision parity-violating electron scattering (PVES) measurements on $^{208}$Pb (PREX-II) and $^{48}$Ca (CREX) reveal a tension in their simultaneous description within modern nuclear energy density functionals (EDFs). Analyses of these data suggest that an enhanced isovector spin-orbit interaction may help account for both measurements, but its relativistic origin in covariant density functional theory remains to be clarified. We show that, within the framework of a covariant density-dependent point-coupling EDF, an enhanced isovector tensor coupling can naturally induce such a strong isovector spin-orbit interaction. This mechanism provides a promising route toward a simultaneous description of the PREX-II and CREX results while preserving a reasonable description of finite nuclei and nuclear matter. PVES on $^{48}$Ca thus provides a sensitive probe of the covariant isovector tensor interaction.

We consider the optimal control of large-scale systems using distributed controllers whose network topology mirrors the coupling graph between subsystems. In this work, we introduce spatial regret, a graph-informed metric measuring the worst-case performance gap between a distributed controller and an oracle with access to additional sensor information. The oracle's graph is a user-specified augmentation of the information graph, yielding a benchmark policy that penalizes disturbances for which additional sensing would improve performance. Minimizing spatial regret yields distributed controllers - respecting the nominal information graph - that emulate the oracle's response to disturbances characteristic of large-scale networks, such as localized perturbations. We show that minimizing spatial regret admits a convex reformulation as an infinite program with a finite-dimensional approximation. To scale to large networks, we derive an upper bound on the spatial regret that can be efficiently minimized in a distributed way. Numerical experiments on power-system models show that the resulting controllers mitigate localized disturbances more effectively than those based on classical metrics.

We study double-well systems with strong magnetic fields and deep potential wells. We present lower bounds on tunneling rates for generic values of the coupling constant. This result was recently announced and complements our recent counter-example construction which exhibits vanishing tunneling for specially-constructed double-well potentials.

We study an inverse boundary value problem for the nonlinear wave equation in $2 + 1$ dimensions. The objective is to recover an unknown potential $q(x, t)$ from the associated Dirichlet-to-Neumann map using real-valued waves. We propose a direct numerical reconstruction method for the Radon transform of $q$, which can then be inverted using standard X-ray tomography techniques to determine $q$. Our implementation introduces a spectral regularization procedure to stabilize the numerical differentiation step required in the reconstruction, improving robustness with respect to noise in the boundary data. We give rigorous justification and optimal stability estimates for the regularized spectral differentiation of noisy measurements, which may be of independent interest. Numerical experiments demonstrate the feasibility of recovering potentials from boundary measurements of nonlinear waves and illustrate the advantages of the Radon-based reconstruction.

We consider nuclear gamma decays and $γ$-emitting reactions that can be an efficient source of hypothetical millicharged particles ($χ$). In particular, we revisit the production of millicharged particles in nuclear reactor environment, pointing out that $γ$ cascades from $^{239}$U is an overlooked yet a powerful source of $χ\barχ$ pairs. This leads to an increased flux compared to previous studies. We then apply new estimates of the flux to derive novel limits on the value of millicharge, $\varepsilon = Q_χ/e$, from the electron recoil searched for in a variety of experiments placed in proximity to the reactor cores. The derived limits on $\varepsilon$ are the strongest in the interval of masses $\sim 0.7-2$ MeV. We also derive the MCP flux from the Sun and point out potential sensitivity of the low-threshold dark matter search experiments.

Since dark matter particles have never been directly detected, we do not know how they move, and in particular we do not know how they fall inside gravitational potential wells. Usually it is assumed that dark matter only interacts gravitationally with itself and with particles of the standard model, and therefore that its motion is governed by Euler's equation. In this paper, we do test this assumption directly at cosmological scales, by combining measurements of galaxy velocities with measurements of gravitational potential wells, encoded in the Weyl potential. We find that current data are consistent with Euler's equation at redshifts $z\in [0.3,0.8]$, and we place constraints on the strength of a potential fifth force, which would alter the way dark matter particles fall. We find that a positive fifth force cannot exceed 7% of the gravitational interaction strength, while a negative fifth force is limited to 21%. The coming generation of surveys, including the Legacy Survey of Space and Time of the Vera C. Rubin Observatory and the Dark Energy Spectroscopic Instrument will drastically improve the constraints, allowing to constrain a departure from pure gravitational interaction at the level of 2%.

We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

We study the large-$N$ asymptotics of the central trace of the heat kernel on compact classical groups. For every classical family $G_N\subset \mathrm{GL}_N(\C)$, we prove a full large-$N$ asymptotic expansion, using a highest weights/partitions correspondence adapted to the large-rank regime, under which the eigenvalues of the Laplace--Beltrami operator stabilize as observables in the algebra of shifted symmetric functions. Then, we prove a random surface representation of the trace in terms of ramified coverings of the torus. We provide two independent applications: an explicit large-rank counting law for the Casimir spectrum, with exponential Hardy--Ramanujan-type growth in contrast with the polynomial behavior of Weyl's law at fixed rank, and a rigorous probabilistic formulation of the Yang--Mills/Hurwitz duality on a two-dimensional torus initiated by Gross and Taylor, completing a previous work of the authors. We also extend this duality to a Yang--Mills/Gromov--Witten duality by expressing the coefficients of the central heat trace as explicit functionals of the generating function of Gromov--Witten invariants.

State-of-the-art magnetic devices rely on faster, more efficient memory elements. A major recent advance is the discovery of orbital torques, which use the orbital angular momentum of Bloch electrons to switch the magnetisation of an adjacent ferromagnet, motivating the search for orbitronic materials with strong orbital responses, exemplified by the orbital Hall effect (OHE). Here we propose $p$-type semiconductors, with a focus on Ge, as orbitronic platforms. We demonstrate that bulk holes in five common semiconductors exhibit a large orbital Hall conductivity of order $10^3 (\hbar/e)Ω^{-1}$cm$^{-1}$, exceeding the spin-Hall effect by 2-3 orders of magnitude. The calculation is performed within the framework of the modern theory of orbital magnetisation, while incorporating recently-discovered quantum corrections to the OHE. Moreover, we argue that bulk $p$-type Ge and Si serve as ideal testbeds for the orbital torque resulting from a charge current, since the spin- and orbital-Edelstein effects are forbidden by symmetry. Our results provide a blueprint for producing strong orbital torques in magnetic devices with $p$-type semiconductors, guiding experimental work in this direction.

Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids appear in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a finite meet-semilattice form a supersolvable convex geometry. As an application, we demonstrate that building sets and nested set complexes respect certain restrictions of finite meet-semilattices unifying and extending results of several authors.

Matrix-product state (MPS) skeletons are connected networks of Hamiltonians with exact MPS ground states that underlie a phase diagram. Such skeletons have previously been found in classes of free-fermion models. For the translation-invariant BDI and AIII free-fermion classes, it has been shown that the underlying skeleton is dense, giving an analytic approach to MPS approximation of ground states anywhere in the class. In this paper, we partially expose the skeleton in certain interacting spin chains: the $N$-state Onsager-integrable chiral clock families. We construct MPS that form a dense MPS skeleton in the gapped regions surrounding a sequence of fixed-point Hamiltonians (the generators of the Onsager algebra). Outside these gapped regions, these MPS remain eigenstates, but no longer give the many-body ground state. Rather, they are ground states in particular sectors of the spectrum. Our methods also allow us to find further MPS eigenstates; these correspond to low-lying excited states within the aforementioned gapped regions. This set of MPS excited states goes beyond the previous analysis of ground states on the $N=2$ free-fermion MPS skeleton. As an application of our results, we find a closed form for the disorder parameter in a family of interacting models. Finally, we remark that many of our results use only the Onsager algebra and are not specific to the chiral clock model representation.

Gamma-ray bursts (GRBs) are classified as long (LGRBs) and short (SGRBs), with collapsars and compact-object mergers (NS-NS or NS-Black Holes) as progenitors, respectively. LGRBs are expected to follow the cosmic star formation rate (SFR), while SGRBs follow a delayed version of the SFR. However, this division has come under question, most prominently by observational evidence of an excess of LGRBs at low redshifts by several investigations, summarized in \cite{Petrosian_2024}. Two recent observations of low-redshift LGRBs show associations with kilonovae. Both of these indicate compact mergers as a potential source of LGRBs as well. Most results showing this separation are based on analyses of small (less than 200) samples of LGRBs with measured redshifts. The aim of this paper is to use a larger sample of LGRBs. The number of LGRBs with measured redshifts has increased by more than a factor of 2 over the last decade. To this data set we add a sample of LGRBs whose redshifts are estimated using a machine learning (ML) method (\cite{Narendra_2025}). To account for the observational selection bias due to redshift measurements, we use the non-parametric, non-binning Efron-Petrosian method to establish the degree of correlation between luminosity and redshift, \textit{the luminosity evolution}, and then use the Lynden-Bell $C^-$ method to obtain the luminosity function. We find a low redshift excess for the larger sample with measured redshifts. Adding the sources with ML-estimated redshifts, which shows overabundance of the mid-range redshifts, the excess is reduced.