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SDO-FM is a foundation model using data from NASA's Solar Dynamics Observatory (SDO) spacecraft; integrating three separate instruments to encapsulate the Sun's complex physical interactions into a multi-modal embedding space. This model can be used to streamline scientific investigations involving SDO by making the enormous datasets more computationally accessible for heliophysics research and enable investigations that require instrument fusion. We discuss four key components: an ingestion pipeline to create machine learning ready datasets, the model architecture and training approach, resultant embeddings and fine-tunable models, and finally downstream fine-tuned applications. A key component of this effort has been to include subject matter specialists at each stage of development; reviewing the scientific value and providing guidance for model architecture, dataset, and training paradigm decisions. This paper marks release of our pretrained models and embedding datasets, available to the community on Hugging Face and sdofm.org.

For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.

In Outline of a Theory of Truth, Kripke introduces some of the central concepts of the logical study of truth and paradox. He informally defines some of these -- such as groundedness and paradoxicality -- using modal locutions. We introduce a modal language for regimenting these informal definitions. Though groundedness and paradoxicality are expressible in the modal language, we prove that intrinsicality -- which Kripke emphasizes but does not define modally -- is not. We characterize the modally definable relations and completely axiomatize the modal semantics., Comment: Some typos have been corrected. Some discussion compressed or excised

It is widely claimed that the natural axiom systems$\unicode{x2013}$including the large cardinal axioms$\unicode{x2013}$form a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by means of \emph{ad hoc} constructions. In this paper we formulate notions of proof-theoretic strength based on set-theoretic reflection principles. We prove that they coincide with orderings on theories given by the generalized ordinal analysis of Pohlers. Accordingly, these notions of proof-theoretic strength engender genuinely well-ordered hierarchies. The reflection principles considered in this paper are formulated relative to G\"odel's constructible universe; we conclude with generalizations to other inner models., Comment: 36 pages

Accurate estimation of thermospheric density is critical for precise modeling of satellite drag forces in low Earth orbit (LEO). Improving this estimation is crucial to tasks such as state estimation, collision avoidance, and re-entry calculations. The largest source of uncertainty in determining thermospheric density is modeling the effects of space weather driven by solar and geomagnetic activity. Current operational models rely on ground-based proxy indices which imperfectly correlate with the complexity of solar outputs and geomagnetic responses. In this work, we directly incorporate NASA's Solar Dynamics Observatory (SDO) extreme ultraviolet (EUV) spectral images into a neural thermospheric density model to determine whether the predictive performance of the model is increased by using space-based EUV imagery data instead of, or in addition to, the ground-based proxy indices. We demonstrate that EUV imagery can enable predictions with much higher temporal resolution and replace ground-based proxies while significantly increasing performance relative to current operational models. Our method paves the way for assimilating EUV image data into operational thermospheric density forecasting models for use in LEO satellite navigation processes., Comment: Accepted at the Machine Learning and the Physical Sciences workshop, NeurIPS 2023

We answer a question of Pakhomov by showing that there is a consistent, c.e. theory $T$ such that no theory which is definitionally equivalent to $T$ has a computable model. A key tool in our proof is the model-theoretic notion of mutual algebraicity., Comment: 21 pages

The ability to control magnetic properties of materials is crucial for fundamental research and underpins many information technologies. In this context, two-dimensional materials are a particularly exciting platform due to their high degree of tunability and ease of implementation into nanoscale devices. Here we report two approaches for manipulating the A-type antiferromagnetic properties of the layered semiconductor CrSBr through hydrostatic pressure and ligand substitution. Hydrostatic pressure compresses the unit cell, increasing the interlayer exchange energy while lowering the N\'eel temperature. Ligand substitution, realized synthetically through Cl alloying, anisotropically compresses the unit cell and suppresses the Cr-halogen covalency, reducing the magnetocrystalline anisotropy energy and decreasing the N\'eel temperature. A detailed structural analysis combined with first-principles calculations reveal that alterations in the magnetic properties are intricately related to changes in direct Cr-Cr exchange interactions and the Cr-anion superexchange pathways. Further, we demonstrate that Cl alloying enables chemical tuning of the interlayer coupling from antiferromagnetic to ferromagnetic, which is unique amongst known two-dimensional magnets. The magnetic tunability, combined with a high ordering temperature, chemical stability, and functional semiconducting properties, make CrSBr an ideal candidate for pre- and post-synthetic design of magnetism in two-dimensional materials., Comment: Main text: 17 pages, 4 figures. Supporting Information: 34 pages, 32 figures, 4 tables

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. First, we characterize ordinal analysis as a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $\Pi^1_1$-sound theories, namely, the ordering wherein $T< U$ if the proof-theoretic ordinal of $T$ is less than the proof-theoretic ordinal of $U$. The standard ways of measuring the strength of theories are consistency strength and inclusion of $\Pi^0_1$ theorems. We introduce analogues of these notions -- $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis., Comment: This work merges arXiv:2112.04980 and arXiv:2201.05284

Let's fix a reasonable subsystem $T$ of arithmetic; why are natural extensions of $T$ pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator in the limit. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally as strong as $\mathsf{Con}^\alpha_T$ yet cofinally as weak as $\mathsf{Con}_T$.

It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of $\Pi^0_1$ theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories., Comment: This article has been merged with 2112.04980 to form 2209.09765

Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make., Comment: This article has been merged with 2201.05284 to form 2209.09765

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

Plasma, the fourth and most pervasive state of matter in the visible universe, is a fascinating medium that is connected to the beginning of our universe itself. Man-made plasmas are at the core of many technological advances that include the fabrication of semiconductor devices, which enabled the modern computer and communication revolutions. The introduction of low temperature, atmospheric pressure plasmas to the biomedical field has ushered a new revolution in the healthcare arena that promises to introduce plasma-based therapies to combat some thorny and long-standing medical challenges. This paper presents an overview of where research is at today and discusses innovative concepts and approaches to overcome present challenges and take the field to the next level. It is written by a team of experts who took an in-depth look at the various biomedical applications, made critical analysis, and proposed ideas and concepts that should help the research community focus their efforts on clear and practical steps necessary to keep the field advancing for decades to come.

There is no infinite sequence of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$ each of which proves $\Pi^1_1$-reflection of the next. This engenders a well-founded ``reflection ranking'' of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$. For any $\Pi^1_1$-sound theory $T$ extending $\mathsf{ACA}^+_0$, the reflection rank of $T$ equals the proof-theoretic ordinal of $T$. This provides an alternative characterization of the notion of ``proof-theoretic ordinal,'' which is one of the central concepts of proof theory. In this note we provide an alternative proof of this theorem using cut-elimination for infinitary derivations., Comment: This replaces an earlier paper containing joint work with Fedor Pakhomov

It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research.

In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, $\omega$-model reflection is equivalent to the claim that arbitrary iterations of uniform $\Pi^1_1$ reflection along countable well-orderings are $\Pi^1_1$-sound. This result yields uniform ordinal analyses of theories with strength between $\mathsf{ACA}_0$ and $\mathsf{ATR}$. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory $T$, which is the operator on countable ordinals that maps the order-type of $\prec$ to the proof-theoretic ordinal of $T+\mathsf{WO}(\prec)$. We obtain precise results about the growth of proof-theoretic dilators as a function of provable $\omega$-model reflection. This approach enables us to simultaneously obtain not only $\Pi^0_1$, $\Pi^0_2$, and $\Pi^1_1$ ordinals but also reverse-mathematical theorems for well-ordering principles., Comment: The proof of Lemma 5.2. in V1 contains a gap that is now fixed. Lemma 2.9 from V1 has been split into multiple claims

We present a new conceptual model of the Earth's glacial-interglacial cycles, one leading to governing equations for which the vector field has a hyperplane of discontinuities. This work extends the classic Budyko- and Sellers-type conceptual energy balance models of temperature-albedo feedback by removing the standard assumption of planetary symmetry about the equator. The dynamics of separate Northern and Southern Hemisphere ice caps are coupled to an equation representing the annual global mean surface temperature. The system has a discontinuous switching mechanism based on mass balance principles for the Northern Hemisphere ice sheet. We show the associated Filippov system admits a unique nonsmooth and attracting limit cycle that represents the cycling between glacial and interglacial states. Due to the vastly different time scales involved, the model presents a nonsmooth geometric perturbation problem, for which we use ad hoc mathematical techniques to produce the periodic orbit. We find climatic changes in the Northern Hemisphere drive synchronous changes in the Southern Hemisphere, as is observed for the Earth on orbital time scales., Comment: 21 pages, 5 figures

One of the Greater London Authority's (GLA) response to the COVID-19 pandemic brings together multiple large-scale and heterogeneous datasets capturing mobility, transportation and traffic activity over the city of London to better understand 'busyness' and enable targeted interventions and effective policy-making. As part of Project Odysseus we describe an early-warning system and introduce an expectation-based scan statistic for networks to help the GLA and Transport for London, understand the extent to which populations are following government COVID-19 guidelines. We explicitly treat the case of geographically fixed time-series data located on a (road) network and primarily focus on monitoring the dynamics across large regions of the capital. Additionally, we also focus on the detection and reporting of significant spatio-temporal regions. Our approach is extending the Network Based Scan Statistic (NBSS) by making it expectation-based (EBP) and by using stochastic processes for time-series forecasting, which enables us to quantify metric uncertainty in both the EBP and NBSS frameworks. We introduce a variant of the metric used in the EBP model which focuses on identifying space-time regions in which activity is quieter than expected.

During the COVID-19 pandemic, policy makers at the Greater London Authority, the regional governance body of London, UK, are reliant upon prompt and accurate data sources. Large well-defined heterogeneous compositions of activity throughout the city are sometimes difficult to acquire, yet are a necessity in order to learn 'busyness' and consequently make safe policy decisions. One component of our project within this space is to utilise existing infrastructure to estimate social distancing adherence by the general public. Our method enables near immediate sampling and contextualisation of activity and physical distancing on the streets of London via live traffic camera feeds. We introduce a framework for inspecting and improving upon existing methods, whilst also describing its active deployment on over 900 real-time feeds., Comment: Version accepted by The Computer Journal

We report a pressure-induced phase transition in the frustrated kagom\'e material jarosite at ~45 GPa, which leads to the disappearance of magnetic order. Using a suite of experimental techniques, we characterize the structural, electronic, and magnetic changes in jarosite through this phase transition. Synchrotron powder X-ray diffraction and Fourier transform infrared spectroscopy experiments, analyzed in aggregate with the results from density functional theory calculations, indicate that the material changes from a R-3m structure to a structure with a R-3c space group. The resulting phase features a rare twisted kagom\'e lattice in which the integrity of the equilateral Fe3+ triangles persists. Based on symmetry arguments we hypothesize that the resulting structural changes alter the magnetic interactions to favor a possible quantum paramagnetic phase at high pressure., Comment: Manuscript and Supplement included

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$., Comment: 11 pages. Corrects a mistake in the statements of two results