Results for “Complexity bounding” — Page 18

DOE OSTI2020

Automated Construction of Potential Energy Surfaces Suitable to Describe van der Waals Complexes with Highly Excited Nascent Molecules: The Rotational Spectra of Ar–CS( v ) and Ar–SiS( v )

Some reactions produce extremely hot nascent products which nevertheless can form sufficiently long-lived van der Waals (vdW) complexes—with atoms or molecules from a bath gas—as to be observed via microwave spectroscopy. Theoretical calculations of such unbound resonance states can be much more challenging than ordinary bound-state calculations depending on the approach employed. One encounters not just the floppy, and perhaps multiwelled potential energy surface (PES) characteristic of vdWs complexes, but in addition, one must contend with excitation of the intramolecular modes and its corresponding influence on the PES. Straightforward computation of the (resonance) rovibrational levels of interest, involves the added complication of the unbound nature of the wave function, often treated with techniques such as introducing a complex absorbing potential. Here, we have demonstrated that a simplified approach of making a series of vibrationally effective PESs for the intermolecular coordinates—one for each reaction product vibrational quantum number of interest—can produce vdW levels for the complex with spectroscopic accuracy. This requires constructing a series of appropriately weighted lower-dimensional PESs for which we use our freely available PES-fitting code AUTOSURF. The applications of this study are the Ar–CS and Ar–SiS complexes, which are isovalent to Ar–CO and Ar–SiO, the latter of which we considered in a previously reported study. Using a series of vibrationally effective PESs, rovibrational levels and predicted microwave transition frequencies for both complexes were computed variationally. A series of shifting rotational transition frequencies were also computed as a function of the diatom vibrational quantum number. The predicted transitions were used to guide and inform an experimental effort to make complementary observations. Comparisons are given for the transitions that are within the range of the spectrometer and were successfully recorded. Calculations of the rovibrational level pattern agree to within 0.2% with experimental measurements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Correlated Response of Protein Side-Chain Fluctuations and Conformational Entropy to Ligand Binding

The heterogeneous fast side-chain dynamics of proteins plays crucial roles in molecular recognition and binding. Site-specific NMR experiments quantify these motions by measuring the model-free order parameter (O axis 2 ) on a scale of 0 (most flexible) to 1 (least flexible) for each methyl-containing residue of proteins. In this work, we have examined ligand-induced variations in the fast side-chain dynamics and conformational entropy of calmodulin (CaM) using five different CaM–peptide complexes. O axis 2 of CaM in the ligand-free (O axis,U 2 ) and ligand-bound (O axis,B 2 ) states are calculated from molecular dynamics trajectories and conformational energy surfaces obtained using the adaptive biasing force (ABF) method. ΔO axis 2 = O axis,B 2 – O axis,U 2 follows a Gaussian-like unimodal distribution whose second moment is a potential indicator of the binding affinity of these complexes. The probability for the binding-induced O axis,U 2 → O axis,B 2 transition decreases with increasing magnitude of ΔO axis 2 , indicating that large flexibility changes are improbable for side chains of CaM after ligand binding. A linear correlation established between ΔOaxis2 and the conformational entropy change of the protein makes possible the determination of the conformational entropy of binding of protein–ligand complexes. The results not only underscore the functional importance of fast side-chain fluctuations but also highlight key motional and thermodynamic correlates of protein–ligand binding.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Direct Exciton Harvesting from a Bound Triplet Pair

Abstract Singlet fission is commonly defined as the generation of two triplet excitons from a single absorbed photon. However, ambiguities within this definition arise due to the complexity of the various double triplet states that exist in SF chromophores and the corresponding interconversion processes. To clarify this process, singlet fission is frequently depicted as sequential two‐step conversion in which a singlet exciton decays into a bound triplet‐pair biexciton state that dissociates into two “free” triplet excitons. However, this model discounts the potential for direct harvesting from the coupled biexciton state. Here, it is demonstrated that individual triplet excitons can be extracted directly from a bound triplet pair. It is demonstrated that due to the requirement for geminate triplet–triplet annihilation in intramolecular singlet fission compounds, unique spectral and kinetic signatures can be used to quantify triplet‐pair harvesting yields. An internal quantum efficiency for triplet exciton transfer from the triplet pair of >50%, limited only by the solubility of the compounds is achieved. The harvesting process is not dependent on the net multiplicity of the triplet‐pair state, suggesting that an explicit, independent dissociation step is not a requirement for using triplet pairs to do chemical or electrical work.

Chemistry↗

DOE OSTI2025

An unusual Co–S bond links B 12 chaperones in an interprotein complex

Chaperones in trafficking pathways ensure specificity of transition metal loading and protection against adventitious side reactions. In B 12 , an essential cofactor for humans, a cobalt ion is coordinated to a >1.3 kDa tetrapyrrolic scaffold, posing logistical challenges for its translocation. In this study, we report the 3.4 Å crystal structure of the human MMACHC and MMADHC B 12 chaperones, tethered via a rarely seen covalent cobalt–sulfur bond. B 12 is bound in the base-off state to MMACHC, with Cys-261 on MMADHC serving as the upper axial ligand; the lower-axial position is vacant. The propensity of thiolato-cobalamin derivatives bound to MMACHC to undergo spontaneous decomposition via general acid catalysis or reduction/oxidation chemistry is averted in the interprotein complex with MMADHC. An exposed face in the complex suggests an exit route for B 12 . No known clinical variants localize to the interprotein interface, consistent with the cobalt–sulfur bond being key to forming the high-affinity complex.

Science & Technology - Other Topics↗

DOE OSTI2026

Fyn–Saracatinib Complex Structure Reveals an Active State-like Conformation

Fyn is a Src-family tyrosine kinase implicated in synaptic dysfunction and neuroinflammation across multiple neurodegenerative disorders, including Alzheimer’s disease (AD) and Parkinson’s disease (PD). Saracatinib (AZD0530) is a potent Src-family inhibitor that has been explored as a repurposed therapeutic; however, its clinical utility is limited by poor kinase selectivity caused by high sequence conservation within Src-family ATP-binding sites. Here, we combine surface plasmon resonance (SPR) and X-ray crystallography to define saracatinib recognition by the Fyn kinase domain (KD). SPR single-cycle kinetics shows that saracatinib binds the isolated Fyn KD and full-length Fyn with low-nanomolar affinity, whereas dasatinib binds with subnanomolar affinity and markedly slower dissociation. We determined the crystal structure of the Fyn KD-saracatinib complex at 2.22 Å resolution. The kinase adopts an active-like conformation with the DFG motif and αC-helix in the ‘in’ state and a conserved β3 αC Lys-Glu salt bridge. Saracatinib occupies the adenine and ribose pockets, and engages the hinge through direct and water-mediated hydrogen bonding while complementing a hydrophobic back pocket by van der Waals contacts. Comparison with reported saracatinib-bound structures of other kinases suggests that the active-state geometry observed for Fyn creates a pocket not observed in inactive-like complexes, providing a structural handle for designing Fyn-selective inhibitors. Comparison with all saracatinib-bound kinase co-structures currently available in the PDB (ALK2 and PKMYT1) indicates a conserved monodentate hinge binding mode but kinase-dependent αC-helix conformations, providing a structural rationale for designing Fyn-selective analogues.

AZD0530↗

DOE OSTI2023

Solar water splitting Pt-nanoparticle photosystem I thylakoid systems: Catalyst identification, location and oligomeric structure

In this study, photosynthetic conversion of light energy into chemical energy occurs in sheet-like membrane-bound compartments called thylakoids and is mediated by large integral membrane protein-pigment complexes called reaction centers (RCs). Oxygenic photosynthesis of higher plants, cyanobacteria and algae requires the symbiotic linking of two RCs, photosystem II (PSII) and photosystem I (PSI), to split water and assimilate carbon dioxide. Worldwide there is a large research investment in developing RC-based hybrids that utilize the highly evolved solar energy conversion capabilities of RCs to power catalytic reactions for solar fuel generation. Of particular interest is the solar-powered production of H 2 , a clean and renewable energy source that can replace carbonbased fossil fuels and help provide for ever-increasing global energy demands. Recently, we developed thylakoid membrane hybrids with abiotic catalysts and demonstrated that photosynthetic Z-scheme electron flow from the light-driven water oxidation at PSII can drive H 2 production from PSI. One of these hybrid systems was created by self-assembling Pt-nanoparticles (PtNPs) with the stromal subunits of PSI that extend beyond the membrane plane in both spinach and cyanobacterial thylakoids. Using PtNPs as site-specific probe molecules, we report the electron microscopic (EM) imaging of oligomeric structure, location and organization of PSI in thylakoid membranes and provide the first direct visualization of photosynthetic Z-scheme solar water-splitting biohybrids for clean H 2 production.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Mechanisms of activation and desensitization of full-length glycine receptor in lipid nanodiscs

Glycinergic synapses play a central role in motor control and pain processing in the central nervous system. Glycine receptors (GlyRs) are key players in mediating fast inhibitory neurotransmission at these synapses. While previous high-resolution structures have provided insights into the molecular architecture of GlyR, several mechanistic questions pertaining to channel function are still unanswered. Here, we present Cryo-EM structures of the full-length GlyR protein complex reconstituted into lipid nanodiscs that are captured in the unliganded (closed), glycine-bound (open and desensitized), and allosteric modulator-bound conformations. A comparison of these states reveals global conformational changes underlying GlyR channel gating and modulation. The functional state assignments were validated by molecular dynamics simulations, and the observed permeation events are in agreement with the anion selectivity and conductance of GlyR. These studies provide the structural basis for gating, ion selectivity, and single-channel conductance properties of GlyR in a lipid environment.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2020

Computing Bottleneck Structures at Scale for High-Precision Network Performance Analysis

The Theory of Bottleneck Structures is a recently-developed framework for studying the performance of data networks. It describes how local perturbations in one part of the network propagate and interact with others. This framework is a powerful analytical tool that allows network operators to make accurate predictions about network behavior and thereby optimize performance. Previous work implemented a software package for bottleneck structure analysis, but applied it only to toy examples. In this work, we introduce the first software package capable of scaling bottleneck structure analysis to production-size networks. Here, we benchmark our system using logs from ESnet, the Department of Energy's high-performance data network that connects research institutions in the U.S. Using the previously published tool as a baseline, we demonstrate that our system achieves vastly improved performance, constructing the bottleneck structure graphs in 0.21 s and calculating link derivatives in 0.09 s on average. We also study the asymptotic complexity of our core algorithms, demonstrating good scaling properties and strong agreement with theoretical bounds. These results indicate that our new software package can maintain its fast performance when applied to even larger networks. They also show that our software is efficient enough to analyze rapidly changing networks in real time. Overall, we demonstrate the feasibility of applying bottleneck structure analysis to solve practical problems in large, real-world data networks.

benchmark↗

DOE OSTI2025

Spectroscopic and Theoretical Studies of Ruthenium Complexes with a Noninnocent N 2 S 2 Ligand in Different Redox States

Herein we report an electronic structure investigation of neutral and oxidized Ru complexes containing a redox noninnocent N 2 S 2 ligand derived from o-phenylenediamide (L1). UV–vis spectroelectrochemistry (SEC) studies were conducted on the square pyramidal complex [Ru I I(L1)(PPh 3 )] (1) and the six-coordinate complexes [Ru I I(μ-BH 3 )(L1)(PPh 3 )] (2) – which has BH 3 bound in a metal–ligand cooperative (MLC) fashion across Ru and L1 – and [Ru II (L1)(PPh 3 )(MeCN)] (3). The SEC results yielded spectra assigned to singly and doubly oxidized 1 and 3, revealing electronic structure changes as a function of oxidation state and in response to the presence and absence of bound MeCN. By contrast, the SEC results of 2 showed that it rapidly loses MLC-bound BH 3 upon oxidation. The SEC results for 1 and 3 were compared to single-crystal XRD data and UV–vis, EPR, and P K-edge, S K-edge, and Ru L 3 -edge X-ray absorption spectroscopy (XAS) data collected on isolated samples of chemically oxidized 3. The data revealed that the first two oxidations are primarily localized on the ligand, which was supported by DFT and TDDFT calculations. DFT calculations for the doubly oxidized species revealed a singlet ground state with a singlet–triplet gap of 8.9 kcal/mol. CASPT2 calculations corroborated the DFT calculations and further revealed that the singlet ground state is multiconfigurational with 21% radical character. Collectively, the results establish redox formalisms and the underlying electronic structure of Ru complexes containing a noninnocent tetradentate ligand in different oxidation states.

electronic structure↗

DOE OSTI2023

Operator Relaxation and the Optimal Depth of Classical Shadows

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via “shallow shadows,” a recently proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth t. Here we show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (“twirling”) circuit—namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially contiguous Pauli operators of weight k, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth t~log⁡(k), guarantees an exponential gain in sample complexity over the t=0 protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our Letter connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly optimized protocols for learning different properties of quantum states.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity

We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector q exhibit almost no relaxation until times O ( exp ( 1 / q ) ) , at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions. Published by the American Physical Society 2024

Physics↗

DOE OSTI2023

Tetracopper σ‐Bound μ ‐Acetylide and ‐Diyne Units Stabilized by a Naphthyridine‐based Dinucleating Ligand

Abstract Reactions of a dicopper(I) tert ‐butoxide complex with alkynes possessing boryl or silyl capping groups resulted in formation of unprecedented tetracopper(I) μ ‐acetylide/diyne complexes that were characterized by NMR and UV/Vis spectroscopy, mass spectrometry and single‐crystal X‐ray diffraction. These compounds possess an unusual μ 4 ‐ η 1 : η 1 : η 1 : η 1 coordination mode for the bridging organic fragment, enforced by the rigid and dinucleating nature of the ligand utilized. Thus, the central π system remains unperturbed and accessible for subsequent reactivity and modification. This has been corroborated by addition of a fifth copper atom, giving rise to a pentacopper acetylide complex. This work may provide a new approach by which metal‐metal cooperativity can be exploited in the transformation of acetylide and diyne groups to a variety of substrates, or as a starting point for the controlled synthesis of copper(I) alkyne‐containing clusters.

Ríos, Pablo↗

DOE OSTI2023

Tetracopper σ‐Bound μ ‐Acetylide and ‐Diyne Units Stabilized by a Naphthyridine‐based Dinucleating Ligand

Abstract Reactions of a dicopper(I) tert ‐butoxide complex with alkynes possessing boryl or silyl capping groups resulted in formation of unprecedented tetracopper(I) μ ‐acetylide/diyne complexes that were characterized by NMR and UV/Vis spectroscopy, mass spectrometry and single‐crystal X‐ray diffraction. These compounds possess an unusual μ 4 ‐ η 1 : η 1 : η 1 : η 1 coordination mode for the bridging organic fragment, enforced by the rigid and dinucleating nature of the ligand utilized. Thus, the central π system remains unperturbed and accessible for subsequent reactivity and modification. This has been corroborated by addition of a fifth copper atom, giving rise to a pentacopper acetylide complex. This work may provide a new approach by which metal‐metal cooperativity can be exploited in the transformation of acetylide and diyne groups to a variety of substrates, or as a starting point for the controlled synthesis of copper(I) alkyne‐containing clusters.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Investigation of Uranyl Perchlorate Anion Complexes in the Gas Phase via Infrared Multiphoton Dissociation and Collision‐Induced Dissociation

The infrared multiphoton dissociation spectrum of the gas-phase uranyl perchlorate anion ([UO 2 (ClO 4 ) 3 ] − ) was measured. Here, the asymmetric uranyl stretch lies somewhere between 930 and 1030 cm −1 , though it could not be deconvoluted from a closely located perchlorate stretching mode. The experimentally measured spectrum was in good agreement with spectra calculated using density functional theory with the B3LYP and TPSSh functionals and Grimme's D3 dispersion corrections with Becke–Johnson damping. The calculations suggest that the asymmetric uranyl stretch lies in the range of 970–980 cm −1 , about 20–30 cm −1 higher than for the analogous uranyl trinitrato complex and consistent with perchlorate as a weaker ligand than nitrate. Like the uranyl trinitrato anion, fragmentation of [UO 2 (ClO 4 ) 3 ] − by collision-induced dissociation resulted in loss of a ClO 3 • radical to form [UO 3 (ClO 4 ) 2 ] − . Infrared multiphoton dissociation measurements of [UO 3 (ClO 4 ) 2 ] − indicate that the structure of the product ion has an O − ligand bound to the uranyl in a T-arrangement and two perchlorate ligands, matching the findings from uranyl nitrato complexes. The uranyl stretch in this complex was slightly separated from the perchlorate modes and was measured at 906 cm −1 , slightly higher than for the analogous nitrate complex, again consistent with perchlorate as a weaker ligand than nitrate in the gas phase.

37 - INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL C↗

DOE OSTI2026

Trafficking of a nitrogenase FeMo-cofactor assembly intermediate

The maturation of the unique FeMo-cofactor of molybdenum nitrogenase is a multistep process requiring the sequential action of a series of maturase complexes. As a final step, the NifEN complex forms FeMo-cofactor from the precursor NifB-co, also called L-cluster, through replacement of an apical iron ion by molybdenum and the attachment of an organic homocitrate ligand. NifB-co is delivered by a small cofactor chaperone, NifX, and initially bound near the surface of the maturase NifEN. Here, we report high-resolution cryo-electron microscopy structures of NifEN in complex with NifX, showing NifB-co binding to NifEN in full detail, capturing both interacting partners in the act of cluster transfer. In a dynamic transfer complex, the large metal cluster is coordinated by single residues from both NifEN and NifX. In silico studies concur with these structures but suggest a third, internal conversion site where cluster maturation likely takes place.

metalloproteins↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

In-Situ Magnetic Field Reconstruction in the MAGIS-100 Experiment

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.; Ferm↗