Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erdős asked whether there exists a constant $δ>0$ such that $f_2(n)>δ$ for all $n$, offering \$100 for a proof and \$25 for a disproof. Recently, Bradač and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge3$. In particular, for every fixed integer $k\ge 3$, we prove that $\liminf_{n\to\infty} f_k(n) \ge 2/9$.

In the presence of a magnetic field, hyperfine clock qubits can acquire a vector differential light shift that can be tuned via polarization to suppress the total differential light shift of high-power, off-resonant laser light. We experimentally measure this "magic" polarization condition, suppressing differential light shifts in both the ${}^2\mathrm{S}_{1/2}$ ground and ${}^2\mathrm{F}_{7/2}^o$ metastable clock qubits of $^{171}\mathrm{Yb}^+$. We present calculations of the minimum bias magnetic fields required to suppress differential light shifts in the ground state clock qubits of commonly trapped ion species, finding that they are below the strengths of fields already typically present in experiments. We further present methods for metastable clock-qubit control in $^{171}\mathrm{Yb}^+$, demonstrating a state preparation and measurement infidelity of $2.9^{+3.0}_{-1.5}\times10^{-4}$ ($-35 \pm 4 \, \mathrm{dB}$).

The ZX-calculus is a powerful graphical language for manipulating quantum circuits, which has recently found many applications in quantum error correction. We extend this language to handle Floquet and other dynamical stabilizer codes via the connection between measurement-based code switching and gauge fixing (arXiv:1810.10037). We combine gauge-fixing steps to implement a closed loop in the space of stabilizer codes, returning to the original codespace up to a logical Clifford gate. These measurement-based paths in the space of stabilizer codes can be viewed as shortcuts, or "chutes and ladders", relative to single-qubit Clifford operations and qubit permutations. This yields a machine-interpretable method for constructing dynamical automorphisms and facilitates the search for implementations of desired logical gates. As an example, we implement a logical phase gate via distance-preserving code switching for the seven-qubit code bare code (arXiv:1702.01155), which has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations (arXiv:2409.18175).

For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers.

The nature of the ``Little Red Dots'' (LRDs) is one of the most profound mysteries posed by the JWST data. One promising class of models that can reproduce the observed LRDs spectra and morphology are quasi-stars: massive envelopes surrounding accreting black holes formed via the collapse of supermassive stars (SMSs). However, the canonical SMS pathway relies on a highly restricted set of environmental and structural conditions: strong Lyman--Werner (LW) backgrounds to suppress H$_2$ cooling, high and sustained gas inflow rates to enforce entropy stratified envelopes, and assume non-zero rotational support in order to prevent GR instability collapse before $\sim 10^6 M_{\odot}$. Here we show that supermassive dark stars (SMDSs), powered by dark matter (DM) annihilation rather than nuclear burning, naturally satisfy the key structural and energetic requirements for quasi-star (QS) formation while relaxing {\it all} of those restrictive conditions listed above. Moreover, quasi-stars formed through the SMDS pathway are born with prompt BH masses ($\gtrsim 10\%$) of the progenitor mass. They therefore enter directly into a late-stage quasi-star regime; subsequently the envelope expands and cools until its photosphere reaches the zero-metallicity opacity limit $(T_{\rm eff}\sim3000$-$6000\,{\rm K}$). Those cool, optically thick, unresolved photospheres can reproduce key features of many JWST LRDs.

Recently, a fit to $B \to PP$ decays ($P \in \{π, K, η, η'\}$) was performed (arXiv:2604.19612, "QCD-factorization amplitudes from flavour symmetries: beyond the $SU(3)$ symmetric case''}) using a formalism that combines topological diagrams with QCD factorization, and a good fit was found. We also recently performed such a fit, under the assumption that the $B \to PP$ amplitudes are related by flavour SU(3) symmetry, but we found a very poor fit. The two results therefore disagree with one another. The source of this disagreement is that we applied EWP-tree relations (ETRs). These were derived $\sim 30$ years ago, and relate different topological diagrams or reduced matrix elements, thus reducing the number of unknown parameters in the fit. In their paper, it is asserted that ETRs are invalid, so that analyses that use them are unreliable. We are writing this Comment to explain why this assertion is incorrect. The key point is that ETRs are mathematically rigorous, group theoretically. If SU(3) is unbroken, and the small Wilson coefficients $c_{7,8}$ in the weak effective Hamiltonian are neglected, ETRs follow automatically and are exact. That is, this is a group theory result -- no hadronic calculations are involved. In this Comment, we also point out several weaknesses of their formalism.

Homophily, the overrepresentation of interactions among similar individuals, and heterophily, the elevated prevalence of interactions among dissimilar ones, are frequently observed mixing patterns in social networks. As hypergraphs are increasingly used to represent social systems, a higher-order perspective on homophily and heterophily becomes ever more relevant. Here, we provide two complementary perspectives on this problem: First, we survey measures that can be used to quantify homophily (or heterophily) in hypergraphs -- emphasizing conceptual differences to existing pairwise measures -- and explain each measure through in-depth examples. Second, we provide an overview of hypergraph models for higher-order mixing patterns, distinguishing several model families with distinct use cases. By providing a guide to existing methods and synthesizing the current body of knowledge on higher-order homophily and heterophily, we lay the basis for informed methodological choices and future developments.

Space-filling designs are commonly used in deterministic computer experiments. However, they are ineffective for factor screening, which makes them inefficient when only a small subset of input factors is influential to the output. Recently developed screening designs, such as MOFAT designs, are effective at identifying important factors but lack space-filling properties, limiting their usefulness for surrogate modeling. In this article, we propose a new class of screening designs that improves the space-fillingness while retaining their screening capability. Through several numerical examples, we demonstrate that the proposed designs offer clear advantages over existing designs.

Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.

Incentive design studies how a central authority can influence strategic agents through payments, subsidies, or taxes, so that individual objectives align with collective welfare. This paper introduces a No-Regret Adaptive Incentive Design (RAID) framework for nonlinear games with continuous action spaces and private agent costs. In this framework, the authority (planner) designs incentives that regulate the Nash equilibrium toward a socially optimal action profile, while simultaneously learning agents' unknown preferences from repeated strategic responses. We formulate the RAID problem and construct a least-squares estimator whose strong consistency requires only diminishing excitation. Leveraging this weak excitation requirement, we propose a switching incentive policy that alternates between probing (exploration) and estimate-based (exploitation) incentives. The resulting policy achieves an $O(t^{-0.5})$ parameter estimation rate and accumulates $O(t^{0.5}\log t)$ squared social-cost regret, almost surely. We further extend the framework to an endogenous-noise response model, where standard least-squares estimation is biased due to an error-in-variables correlation between the noise and agent responses. We utilize a repeated-sampling estimator and corresponding switching policy that retain the same almost-sure convergence and regret rates. Numerical experiments validate the effectiveness and predicted convergence rates of the method.

Recent experiments have demonstrated the successful generation and detection of moderately squeezed vacuum states with integrated photonics. However, in order to benefit from the reduced noise of highly squeezed light, many different noise sources must be mitigated. Here, we quantify the fundamental limits these noise sources impose on squeezing measurements and find surprising generality across different platforms and designs. We combine these different limitations into a simple model that provides practical guidance for the design and benchmarking of next-generation integrated squeezed-light systems.

Subduction zones on the surface of the Earth, where abrupt sliding leads to earthquakes, are generally curved and localized. How does the geometry of these zones influence the occurrence of megathrust earthquakes? Here we use a combination of simple scaling arguments and data analysis using the differential geometry of surfaces to examine the relationship between the earthquake productivity of subduction zones and their shape. A scaling argument suggests how interface curvature changes both the accumulation and release of stress relative to planar interfaces; conformable sliding along relatively flat subduction zones should lead to rare but large events, while curved subduction zones should lead to frequent smaller events. To test this, we leverage global geometry datasets and analyze the correlation between the surface curvatures of the subduction zones and the frequency and magnitude of earthquakes therein. Our analysis shows that weakly curved slab geometries are associated with rarer larger magnitude events, while slab geometries with a larger relative dispersion in curvature are associated with frequent but smaller magnitude events. Using different scale-dependent shape metrics of the subduction zones, we show that the earthquake productivity is influenced by the conformability of the overriding and downgoing plates. More broadly, our results suggest the need to incorporate the large-scale geometry of subduction zones in computational models and predictive frameworks for earthquake risk.

We study the flow of an isothermal compressible Newtonian fluid around a body that performs a (time-independent) rigid motion. We derive a weak-strong uniqueness principle, and show that in the low Mach number limit, the governing equation is well approximated by the Navier-Stokes equations for incompressible rotating flow. Both results are based on the derivation of a relative energy inequality for weak solutions to this exterior-domain problem.

Spicules and propagating coronal disturbances (PCDs) are ubiquitous dynamic features of the solar atmosphere, yet their physical connection remains an open question of paramount importance to the mass and energy transport in the solar atmosphere. Using concurrent multiwavelength high-resolution observations from the Swedish 1-m Solar Telescope and the Solar Dynamics Observatory, supported with two-dimensional radiative magnetohydrodynamic (MHD) simulations, we find that i) shock waves in the chromosphere generated from non-linear wave steepening drive some spicules, ii) in the corona, these shock waves may transition into large amplitude non-linear compressive MHD waves depending on the magnetic field strength and the ambient coronal conditions. In either case, the shocks or the large-amplitude compressive waves in the corona, also transport upward mass flux and produce intensity variations in the form of PCDs in coronal passbands. Further a multi-height wavelet analysis shows dominant $\sim$5 minute periods in the lower chromosphere that evolve into longer periods ($\ge$10 minutes) at higher atmospheric layers, consistent with dispersive propagation in a stratified medium. The observational characteristics together with the numerical simulations, demonstrate that a shock-driven MHD mechanism links spicule formation to coronal disturbances. Finally, mass flux estimates from both the observations and the simulations indicate that these PCDs can also aid in supplying mass to the solar wind.

We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly $d$-sphere with no more than $d+4$ vertices is combinatorially equivalent to a Murai sphere for any $d\geq 1$.

Geometrical frustration, a central paradigm in condensed matter physics, provides a unifying language for systems in which locally preferred interactions cannot be made globally compatible. Here, we use this language to formulate a minimal theory of frustrated neural timing, mapping repulsively coupled rhythmic units onto antiferromagnetic XY models. Within this framework, the condensed-matter concepts of local constraints, degenerate ground-state manifolds, metastability, and quench dynamics become a concrete diagnostic framework for structured neural phase dynamics. We analyze a hierarchy of geometries: a triangle as the minimal frustrated motif with two chiral 120° timing states, a tetrahedron whose reduced ground-state manifold consists of intersecting continuous branches associated with antipodal pairings, and a kagome lattice on which local constraints define a constrained three-coloring manifold. The kagome lattice reveals the central dynamical result: zero-temperature relaxation suppresses global synchrony but typically selects low-energy metastable torque-balanced states rather than exact ground states. Finally, we show how the phase theory can be carried back towards biophysical neural models by treating it as an effective-interaction target, where geometrical timing frustration is realized through preferred phase lags that become incompatible around closed motifs. This perspective suggests that weak global coherence in neural systems does not necessarily signal disordered activity, but can reflect structured local timing order shaped by a frustrated dynamical landscape.

In this paper we consider quark star solutions to Liu et al.'s \cite{Liu_2019} quasi-topological electromagnetism (QTEM), a recently proposed form of dark energy. Since the QTEM contribution is trivial for pure electric/magnetic charge, we consider the dyonic case in pure QTEM which does induce (dark) non-trivial dynamics from the non-linear theory. Besides the introduction of a dyonic charge distribution generally pushing the characteristic quark star `hook' shape to larger masses and radii, it also induces a second branch at very large mass and radius for stars with a small dyonic charge ratio. This second set of solutions have a negative pressure envelope surrounding a positive pressure core. As we explore the parameter space these features interact and evolve in interesting ways, with the two branches eventually merging in $M/R$ space before settling into a characteristic `paperclip' shape as the dyonic charge ratio becomes large.

In the last few years, AI-based models have become the centre of attention in weather forecasting due to their increasing accuracy and efficiency. Pioneering among weather services, ECMWF has developed its Artificial Intelligence Forecasting System (AIFS) model, which was first to provide data-driven ensemble forecasts in June 2024. Since July 2025, the AIFS ensemble model has been operational and runs in parallel with ECMWF's physics-based Integrated Forecasting System (IFS), which is considered the gold standard in weather prediction. The new AIFS model can generate forecasts ten times faster than the classical numerical weather prediction model, while consuming approximately a thousand times less energy. We present the results of our systematic assessment of the performance of the IFS and AIFS models by comparing the accuracy of raw and post-processed medium-range 10-m wind-speed ensemble forecasts generated operationally by the two models for the period between July and November 2025 for more than 9000 synoptic observation stations across the globe. The post-processed case involves the parametric ensemble model output statistics (EMOS) as well as the non-parametric quantile regression (QR) approach to correct any systematic inaccuracies in the raw forecasts. The predictive performance of raw IFS ensemble forecasts proves to be substantially superior to the skill of the raw AIFS predictions for all investigated forecast horizons. As expected, post-processing significantly improves the skill of both IFS and AIFS predictions, and, across most verification metrics, EMOS is superior to QR, especially for short lead times. Compared to the raw ensemble, the differences in skill between the matching IFS and AIFS predictions are substantially decreased by post-processing and are mostly significant at short lead times, when the IFS forecasts outperform their AIFS counterparts.

We present entanglement measures between spatially separated regions and between two distinguishable or indistinguishable particles in one-dimensional two-particle continuous-time quantum walks governed by the Hubbard Hamiltonian. The left-right entanglement checks the entropy of coarse-grained states counting the numbers of particles on the left and right halves of the lattice while the particle-particle entanglement is based on the entropy of the singular values of the time-evolved Fock state. With separable, entangled, and doubly occupied initial states, we examine initial entanglement and the following growth in different entanglement measures. While the entanglement measures of the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. We also discuss possible implications for future research of entanglement in multi-particle quantum dynamics.

In previous work (Rave, 2008) it was proposed that closed loops should be treated as fundamental quantum entities, and such loops were presented in a quasi-probability framework. We demonstrate that the closed-loop decomposition of quantum probabilities is a direct consequence of unitarity, and that Bargmann invariants arise naturally as the phase-invariant quantities associated with these loops, rather than being introduced independently. This identifies interference not as mysterious cross terms, but as contributions from distinct classes of closed loops weighted by their associated Bargmann phases. Additionally, the Born rule is seen to reflect the fundamental quadratic structure arising from the product of forward and reverse amplitudes, which together define such loops.

Fractionalized excitations are among the most striking signatures of emergence in quantum matter. While widely sought in frustrated magnets, their detection and characterization remain challenging, motivating the exploration of new probes. Meanwhile, Spintronics offers versatile tools for probing spin-related phenomena. In particular, the spin Seebeck effect (SSE) converts thermally driven magnetic excitations into a voltage in an adjacent metal, providing electrical access to the underlying dynamics and transport properties. Here we employ the SSE to probe emergent magnetic monopoles in the non-collinear Ising magnet Dy$_2$Ti$_2$O$_7$, a rare instance of a three-dimensional fractionalized magnet. We observe an SSE signal featuring a pronounced peak at monopole proliferation, accompanied by characteristic frequency and angular dependence. Our results broaden the scope of spintronic methods for detecting exotic excitations, provide new insights into magnetic insulators generally and monopole physics specifically, and suggest the potential of quantum materials as functional interfaces.

Polar molecules are a promising platform for quantum-enhanced sensing and precision tests of fundamental physics, owing to their strong long-range dipolar interactions, broad sensitivity to electromagnetic fields, and sensitivity to potential physics beyond the Standard Model. However, the creation of metrologically useful entangled states in molecular systems has remained elusive. Here, we report the first observation of a class of metrologically useful entangled states - spin-squeezed states - in polar CaF molecules trapped in an optical tweezer array. The spin degree of freedom is encoded in rotational levels which are directly coupled by dipolar exchange interactions. By harnessing appropriate dynamical decoupling schemes we observe up to 3.0(3)dB of metrological gain, (2.2(3)dB without measurement correction) from direct exchange interactions. Using Floquet engineering, we further realize richer Hamiltonians that preserve spin squeezing while enabling the development of longer-range quantum correlations. Using site- and spin-resolved measurements we demonstrate that these entangled states enhance sensitivity to both homogeneous and spatially varying fields, and reveal strong non-classical correlations, including bipartite entanglement and Einstein-Podolsky-Rosen steering. Finally, we transfer the spin-squeezed states into long-lived and non-interacting hyperfine states, where the metrological enhancement persists for up to 100ms. Our results establish molecular optical tweezer arrays as a scalable platform for generating, controlling, characterizing, and storing entangled states of molecules, opening new opportunities for quantum-enhanced sensing and precision tests of fundamental physics.

18 HST parallaxes for Galactic classical Cepheids are unified to establish a Gaia-independent IC1613 distance and evaluate metallicity effect determinations, since the Gaia zeropoint is debated. Recently proposed classical Cepheid metallicity corrections of $γ(W_{VI}) \simeq 0, -0.25, -0.50$ mag dex$^{-1}$ are benchmarked, and yield $μ_{0,{W_{VI}}} \simeq 24.39, 24.16, 23.93$ ($\pm0.07$). Larger corrections are disfavored relative to a weighted mean of IC1613 TRGB and TRGB/JAGB distances of $24.39^{+0.07}_{-0.04}$ (EDD) and $24.36\pm0.06$/$24.45\pm0.11$ (CCHP). A more expansive metallicity baseline is desirable to scrutinize smaller corrections (e.g., $γ(W_{VI}) \lesssim -0.1$ mag dex$^{-1}$), while concurrently acquiring additional non-Gaia parallaxes since published concerns exist regarding DR3 (e.g., critical long-period Cepheids S Vul, SV Vul).

We introduce the notion of timelike ideal boundary of a Lorentzian length space as the set of asymptotic classes of future or past-directed timelike geodesic rays, a construction complementary to the causal boundary in the sense of Geroch-Kronheimer-Penrose and akin to the concept of ideal boundary of a metric space. We endow such a timelike ideal boundary with a natural cone topology and an angular metric, and establish upper curvature bounds for the resulting metric space. Finally, we consider generalized cones as a model and study the relation between the timelike ideal boundary and both the metric ideal boundary of the fiber and the asymptotic behaviour of the warping function.

Switched systems are a family of dynamical systems where a switching rule indicates which system is "switched on". This rule can be dependent on time and/or position in the state space. Stability of switched systems is a property that is often investigated using the existence of one (or multiple) Lyapunov function(s). We develop an algorithm using a scattered approximation method to construct a Lyapunov function for switched systems, accompanied with stability results. The construction adapts a previous method for autonomous ODEs that uses meshfree collocation and quadratic programming.

Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

Chern numbers are central to correlated and topological phenomena, yet most topological systems are associated with Chern numbers of order unity. Here we propose a scheme to achieve large Chern numbers in an optical Floquet quasicrystal with cold atoms, which can be directly measured via Thouless pumping. We study the quasienergy spectrum of Floquet quasicrystals and characterize the emergent Chern numbers using gap labeling theorem. We further investigate the Thouless pumping in the Floquet quasicrystal at different driving frequencies and amplitudes, revealing the connection between transport features and the quasienergy spectrum. Our findings open new avenues for exploring rich topological dynamics in Floquet quasicrystals and realizing fractional Chern insulating states.

The spanning cuts method is a powerful approach to reduce the cost of IBP reduction while computing Feynman integrals. However, its usage is limited due to the so-called consistency problem. It was unclear why the IBP reduction coefficients can be inconsistent with each other between different cuts. In this paper, we report a mechanism behind this inconsistency. We found that the IBP relations can be violated under the cuts, if we blindly erase the hidden terms that are proportional to the ``vanishing'' Feynman prescription parameters in the relations. In some cases, the cut introduces pinch singularities, which cancel the vanishing Feynman prescription parameters, making the hidden terms finite. In various cases, the error comes from omitting such finite hidden terms. We also claimed that the pinch singularity under the cuts are related to some hidden relations between the propagators. In this paper, we provide an algorithm and its implementation to find the linear hidden relations.

On-site disorder can be leveraged to induce a transition from a trivial to a topological insulator. However it is unclear if structural disorder in the absence of on-site disorder can aid a similar transition and, if so, which kind of structural disorder is more favourable. We numerically show that structural disorder can enhance and sustain a topological phase up to strong disorder in two dimensions provided that one penalises atomic sites from being close to one another. However, we find this effect is absent in three dimensions, where structural disorder appears generically detrimental to the phase. In our calculations we include disorder that can scramble the global spin-reference frame, an overlooked type of disorder expected to exist in strongly disordered solids. This disorder fatally scrambles the information necessary for the spin-Bott and the spin-Chern marker to correctly diagnose a topological phase. By using the spectral localizer, a local marker directly defined using the time-reversal symmetry operator rather than a spin-projection, we show how one can circumvent this limitation, providing a basis-indifferent theory for calculating Z2 invariants. Our work showcases that not all structural disorders are equally beneficial to topology, and highlights guiding principles to enhance and detect topological phases in both solid-state and metamaterial realisations.

We explore a reorganized framework for the Hartree Fock equations that allows varying patterns of locality to be imposed on the molecular orbitals while maintaining a highly efficient self-consistent field optimization algorithm. Rather than limiting orbitals' spread and then variationally minimizing the energy within those limits, our reorganization neatly pairs each local degree of freedom with a specific solution condition that itself has a naturally local interpretation. These pairs can each be turned on or off, and, regardless of the sparsity pattern used to make such choices, the overall method maintains a fast self-consistent field optimization. We use this structure to test reaction-matched schemes for imposing orbital locality and, through Fock builds that exploit this locality, achieve competitive timings even in modestly sized molecules. Our initial tests also suggest that this approach to imposed orbital locality can be arranged so as to do minimal damage to both Hartree Fock and MP2 reaction energy predictions.