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DOE OSTI2025

A Theoretical Operational Model for Complex Experiments and its Invariance Theorems

We develop and systematize the Theoretical–Operational Model (TOM), a framework that treats preparation and measurement —including their operational residues— as intrinsic structures of physical theory. The central contribution is a principled geometric–algebraic organization of admissible operational deformations, formulated using quantum channels, renormalization-style flows, and information-geometric tools. Within this structure, operational residues and background processes are represented as effective morphisms attached to these operational components, whose invariants yield constraints on how theoretical parameters vary under specified classes of deformations. Illustrations drawn from muon–electron conversion, long-baseline neutrino oscillations, and quark–gluon-plasma phenomenology show how TOM maps operational effects into inferences about theoretical parameters, enables systematic cross-experimental comparisons, and stabilizes parameter estimation against defined deformation families. By embedding the operational layer—together with its residues—within a structured theoretical setting, TOM supports both theory testing and theory development, clarifying the conceptual relation between experimental realization and the physical quantities represented by the theory.

Pronskikh, Vitaly [Fermilab] (ORCID:00000002518174↗
DOE OSTI2025

Reweighting configurations generated by transferable, machine learned models for protein sidechain backmapping

Multiscale modeling requires the linking of models at different levels of detail, with the goal of gaining accelerations from lower fidelity models while recovering fine details from higher resolution models. Communication across resolutions is particularly important in modeling soft matter, where tight couplings exist between molecular-level details and mesoscale structures. While multiscale modeling of biomolecules has become a critical component in exploring their structure and self-assembly, backmapping from coarse-grained to fine-grained, or atomistic, representations presents a challenge, despite recent advances through machine learning. A major hurdle, especially for strategies utilizing machine learning, is that backmappings can only approximately recover the atomistic ensemble of interest. We demonstrate conditions for which backmapped configurations may be reweighted to exactly recover the desired atomistic ensemble. By training separate decoding models for each sidechain type, we develop an algorithm based on normalizing flows and geometric algebra attention to autoregressively propose backmapped configurations for any protein sequence. Critical for reweighting with modern protein force fields, our trained models include all hydrogen atoms in the backmapping and make probabilities associated with atomistic configurations directly accessible. We also demonstrate, however, that reweighting is extremely challenging despite state-of-the-art performance on recently developed metrics and generation of configurations with low energies in atomistic protein force fields. Through detailed analysis of configurational weights, we show that machine-learned backmappings must not only generate configurations with reasonable energies, but also correctly assign relative probabilities under the generative model. These are broadly important considerations in generative modeling of atomistic molecular configurations.

Monroe, Jacob I. [Univ. of Arkansas, Fayetteville,↗
DOE OSTI2026

Geometric invariants of quantum metrology

Here, we establish a conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum of the QFIM is invariant under unitary dynamics generated by these same operators. Each Lie algebra therefore endows any quantum state with a fixed “budget” of metrological sensitivity—an intrinsic resource that we show, like optical squeezing in interferometry, cannot be amplified by symmetry-preserving operations. The Uhlmann curvature tensor naturally inherits the same symmetry group, and so quantum incompatibility is similarly fixed. As a result, a metrological analog to Liouville's theorem appears; statistical distances, volumes, and curvatures are invariant under the evolution generated by the Lie algebra. We discuss this as it relates to the quantum analogs of classical optimality criteria. This enables one to efficiently classify useful classes of quantum states at the level of Lie algebras through geometric invariants.

Wilson, Christopher [University of Colorado, Bould↗
DOE OSTI2023

Constraints on sequential discontinuities from the geometry of on-shell spaces

We present several classes of constraints on the discontinuities of Feynman integrals that go beyond the Steinmann relations. These constraints follow from a geometric formulation of the Landau equations that was advocated by Pham, in which the singularities of Feynman integrals correspond to critical points of maps between on-shell spaces. To establish our results, we review elements of Picard-Lefschetz theory, which connect the homotopy properties of the space of complexified external momenta to the homology of the combined space of on-shell internal and external momenta. An important concept that emerges from this analysis is the question of whether or not a pair of Landau singularities is compatible — namely, whether or not the Landau equations for the two singularities can be satisfied simultaneously. Under conditions we describe, sequential discontinuities with respect to non-compatible Landau singularities must vanish. Although we only rigorously prove results for Feynman integrals with generic masses in this paper, we expect the geometric and algebraic insights that we gain will also assist in the analysis of more general Feynman integrals.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

The spatial wavefunction of half-integer spins

Within a nonrelativistic framework, spin is generally included as an intrinsic angular momentum. It is proposed here that consistent results can be obtained with a spatial wavefunction of an oriented complex exponential ${e}^{\mp {\boldsymbol{i}}\hat{{\bf{a}}}\varphi }$, where the bivector ${\boldsymbol{i}}\hat{{\bf{a}}}$ is written in terms of the pseudoscalar i = e x e y e z and the axis of rotation given by the unit vector $\hat{{\bf{a}}}$, which is generally an (effective) magnetic field direction (this can also be written as ${e}^{\mp i{\boldsymbol{\sigma }}\cdot \hat{{\bf{a}}}\varphi }$ in terms of the Pauli vector σ). The wavefunction is a multivector and not a complex scalar. The signs in the exponential correspond to the two directions of rotation around the axis. The transformation properties of these wavefunctions are given by the Pauli spinors. Finally, spin can be viewed as a zero-point rotation arising from the noncommutativity of the momentum and the vector potential.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Quantum Algorithms for Representation-Theoretic Multiplicities

Kostka, Littlewood-Richardson, Plethysm, and Kronecker coefficients are the multiplicities of irreducible representations in the decomposition of representations of the symmetric group that play an important role in representation theory, geometric complexity, and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups. The conjecture about Kronecker coefficients was disproved by Panova [Polynomial time classical versus quantum algorithms for representation theoretic multiplicities, arXiv:2502.20253] with a classical algorithm which, if optimal, points to a 𝒪⁡(𝑛 4+2⁢𝑘 ) vs $\tilde{Ω}$⁡(𝑛 4⁢𝑘 2 +1 ) polynomial gap in quantum vs classical computational complexity for an integer parameter 𝑘.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Subregion-subalgebra duality: Emergence of space and time in holography

In holographic duality, a higher dimensional quantum gravity system emerges from a lower dimensional conformal field theory (CFT) with a large number of degrees of freedom. We propose a formulation of duality for a general causally complete bulk spacetime region, called subregion-subalgebra duality, which provides a framework to describe how geometric notions in the gravity system, such as spacetime subregions, different notions of times, and causal structure, emerge from the dual CFT. Subregion-subalgebra duality generalizes and brings new insights into subregion-subregion duality (or equivalently entanglement wedge reconstruction). It provides a mathematically precise definition of subregion-subregion duality and gives an independent definition of entanglement wedges without using entropy. Geometric properties of entanglement wedges, including those that play a crucial role in interpreting the bulk as a quantum error correcting code, can be understood from the duality as the geometrization of the superadditivity of certain algebras. Using general boundary subalgebras rather than those associated with geometric subregions makes it possible to find duals for general bulk spacetime regions, including those not touching the boundary. Applying subregion-subalgebra duality to a boundary state describing a single-sided black hole also provides a precise way to define mirror operators. Published by the American Physical Society 2025

Leutheusser, Sam (ORCID:0000000339228128)↗
DOE OSTI2026

Parametric reduced order models for graded lattice structures

Graded lattice structures, characterized by smoothly varying mechanical properties, hold significant promise for optimizing material distribution in advanced engineering applications. However, accurately modeling these structures poses substantial computational challenges due to the continuous geometric variations within their unit cells. Here, to address these challenges, this paper introduces a novel Efficient Reduced Order Model (EROM) that integrates the Matrix Discrete Empirical Interpolation Method (MDEIM) and Discrete Empirical Interpolation Method (DEIM) with polynomial regression to efficiently manage geometric parametrization in lattice structures. Unlike traditional reduced order models (ROMs) that require extensive precomputed libraries for each geometric configuration, our approach enables continuous geometric variations through a flexible algebraic formulation, significantly reducing computational costs while preserving high accuracy. The method constructs projection matrices for individual unit cells that can be efficiently assembled into global systems, leveraging the repetitive nature of lattice structures. Numerical studies demonstrate that our EROM achieves displacement errors below 1% and von Mises stress prediction errors below 4%, coupled with computational speedups exceeding two orders of magnitude compared to full-order simulations. The proposed method's modularity and scalability make it particularly suitable for design optimization and real-time simulation of functionally graded lattice structures, with applications spanning aerospace to biomedical engineering.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2026

Anyonic Membranes and Pontryagin Statistics

Anyons, unique to two spatial dimensions, underlie extraordinary phenomena such as the fractional quantum Hall effect, but their generalization to higher dimensions has remained elusive. The topology of Eilenberg-MacLane spaces constrains the loop statistics to be only bosonic or fermionic in any dimension. In this work, we introduce the novel anyonic statistics for membrane excitations in four dimensions. Analogous to the $\mathbb{Z}_N$-particle exhibiting $\mathbb{Z}_{N\times \gcd(2,N)}$ anyonic statistics in two dimensions, we show that the $\mathbb{Z}_N$-membrane possesses $\mathbb{Z}_{N\times \gcd(3,N)}$ anyonic statistics in four dimensions. Given unitary volume operators that create membrane excitations on the boundary, we propose an explicit 56-step unitary sequence that detects the membrane statistics. We further analyze the boundary theory of $(5{+}1)$D 1-form $\mathbb{Z}_N$ symmetry-protected topological phases and demonstrate that their domain walls realize all possible anyonic membrane statistics. We then show that the $\mathbb{Z}_3$ subgroup persists in all higher dimensions. In addition to the standard fermionic $\mathbb{Z}_2$ membrane statistics arising from Stiefel-Whitney classes, membranes also exhibit $\mathbb{Z}_3$ statistics associated with Pontryagin classes. We explicitly verify that the 56-step process detects the nontrivial $\mathbb{Z}_3$ statistics in 5, 6, and 7 spatial dimensions. Furthermore, in 7 and higher dimensions, the statistics of membrane excitations stabilize to $\mathbb{Z}_{2} \times \mathbb{Z}_{3}$, with the $\mathbb{Z}_3$ sector consistently captured by this process.

Abstract algebra↗
DOE OSTI2025

Foams and KZ-equations in Rozansky-Witten theories

In this paper, we present a geometric description of foams, which are prevalent in topological quantum field theories (TQFTs) based on quantum algebra, and reciprocally explore the geometry of Rozansky-Witten (RW) theory from an algebraic perspective. This approach illuminates various aspects of decorated TQFTs via geometry of the target space X of RW theory. Through the formulation of the Knizhnik-Zamolodchikov (KZ) equation within this geometric framework, we derive the corresponding braiding and associator morphisms. We discuss applications where the target space of RW theory emerges as the Coulomb branch of a compactified 6d SCFT or Little String Theory, with the latter being particularly intriguing as it results in a compact X.

Gukov, Sergei [California Institute of Technology,↗
DOE OSTI2024

Local Poincaré algebra from quantum chaos

The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Commutative Algebra Modeling in Materials Science – A Case Study on Metal–Organic Frameworks (MOFs)

Metal-organic frameworks (MOFs) are a class of important crystalline and highly porous materials whose hierarchical geometry and chemistry hinder interpretable predictions in materials properties. Commutative algebra is a branch of abstract algebra that has been rarely applied in data and material sciences. We introduce the first ever commutative algebra modeling and prediction in materials science. Specifically, category-specific commutative algebra (CSCA) is proposed as a new framework for MOF representation and learning. It integrates element-based categorization with multiscale algebraic invariants to encode both local coordination motifs and global network organization of MOFs. These algebraically consistent, chemically aware representations enable compact, interpretable, and data efficient modeling of MOF properties such as Henry’s constants and uptake capacities for common gases. Compared to traditional geometric and graph-based approaches, CSCA achieves comparable or superior predictive accuracy while substantially improving interpretability and stability across data sets. By aligning commutative algebra with the chemical hierarchy, the CSCA establishes a rigorous and generalizable paradigm for understanding structure and property relationships in porous materials and provides a nonlinear algebra-based framework for data-driven material discovery.

Khaemba, Caleb S.↗
DOE OSTI2025

Geometric entropies and their Hamiltonian flows

In holographic theories, the Hubeny-Rangamani-Takayanagi (HRT) area operator plays a key role in our understanding of the emergence of semiclassical Einstein-Hilbert gravity. When higher derivative corrections are included, the role of the area is instead played by a more general functional known as the geometric entropy. It is thus of interest to understand the flow generated by the geometric entropy on the classical phase space. In particular, the fact that the associated flow in Einstein-Hilbert or Jackiw-Teitelboim (JT) gravity induces a relative boost between the left and right entanglement wedges is deeply related to the fact that gravitational dressing promotes the von Neumann algebra of local fields in each wedge to type II. This relative boost is known as a boundary-condition-preserving (BCP) kink-transformation. In a general theory of gravity (with arbitrary higher-derivative terms), it is straightforward to show that the flow continues to take the above geometric form when acting on a spacetime where the HRT surface is the bifurcation surface of a Killing horizon. However, the form of the flow on other spacetimes is less clear. In this paper, we use the manifestly-covariant Peierls bracket to explore such flows in two-dimensional theories of JT gravity coupled to matter fields with higher derivative interactions. The results no longer take a purely geometric form and, instead, demonstrate new features that should be expected of such flows in general higher derivative theories. We also show how to obtain the above flows using Poisson brackets.

AdS-CFT correspondence↗
DOE OSTI2024

Modular flow in JT gravity and entanglement wedge reconstruction

It has been shown in recent works that JT gravity with matter with two boundaries has a type II ∞ algebra on each side. As the bulk spacetime between the two boundaries fluctuates in quantum nature, we can only define the entanglement wedge for each side in a pure algebraic sense. As we take the semiclassical limit, we will have a fixed long wormhole spacetime for a generic partially entangled thermal state (PETS), which is prepared by inserting heavy operators on the Euclidean path integral. Under this limit, with appropriate assumptions of the matter theory, geometric notions of the causal wedge and entanglement wedge emerge in this background. In particular, the causal wedge is manifestly nested in the entanglement wedge. Different PETS are orthogonal to each other, and thus the Hilbert space has a direct sum structure over sub-Hilbert spaces labeled by different Euclidean geometries. The full algebra for both sides is decomposed accordingly. From the algebra viewpoint, the causal wedge is dual to an emergent type III 1 subalgebra, which is generated by boundary light operators. To reconstruct the entanglement wedge, we consider the modular flow in a generic PETS for each boundary. We show that the modular flow acts locally and is the boost transformation around the global RT surface in the semiclassical limit. It follows that we can extend the causal wedge algebra to a larger type III 1 algebra corresponding to the entanglement wedge. Within each sub-Hilbert space, the original type II ∞ reduces to type III 1 .

2D Gravity↗
DOE OSTI2025

Absolute entropy and the observer’s no-boundary state

We investigate the no-boundary proposal for closed universes with an observer. We argue that the observer’s no-boundary state is the identity operator on the physical Hilbert space, i.e., the maximum entropy state and show this explicitly in Jackiw-Teitelboim gravity. Geometrically, the no-boundary state is a bra-ket wormhole. Expectation values in the no-boundary state provide a trace for the observer’s algebra, which allows one to define von Neumann entropy for observers in different universes as the relative entropy with respect to the no-boundary state. This result is consistent with all previously discussed cases of traces for invariantly defined regions.

Cosmological models↗
DOE OSTI2023

Parallel Algebraic Multigrid for Fusion and Higher-Order PDEs

Multigrid methods play a key role in large-scale scientific simulation because they are among the fastest and most scalable approaches for solving the underlying sparse linear systems of equations that arise from a wide array of Partial Differential Equation (PDE) discretizations. Algebraic multigrid (AMG) is a special type of multigrid method that depends only on the description of the linear system, giving it better portability and broader applicability than geometric multigrid, as it requires no explicit knowledge of the problem geometry. Even though these methods are widely used today, there are still applications where further development is needed. In this report, we focus on PDEs with higher-order terms (e.g., fourth order), concentrating on a PDE that arises in tokamak edge plasma simulations (a tokamak is a machine that confines a plasma using magnetic fields and is believed to be the leading plasma confinement concept for future fusion power plants). General multigrid relaxes a linear system on coarser grids and reverses this process with interpolation, but standard AMG methods struggle with the aforementioned higher-order PDEs. We investigate cyclic coarsening and interpolation heuristics, as well as new iterative approximation methods of refining the solution at each grid to improve the existing multigrid approach. To this end, we ensure that these techniques are transferable to a parallelized setting with LLNL’s supercomputers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A cluster of results on amplituhedron tiles

Abstract The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $$\mathcal {N}=4$$ N = 4 super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the $$m=4$$ m = 4 amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $$\text{ Gr}_{4,n}$$ Gr 4 , n . Secondly, we exhibit a tiling of the $$m=4$$ m = 4 amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for $$\text{ Gr}_{4,n}$$ Gr 4 , n . This paper is a companion to our previous paper “Cluster algebras and tilings for the $$m=4$$ m = 4 amplituhedron.”

Physics↗