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The enclosed data release consists of a subset of the calibration data from the Majorana Demonstrator experiment. Each Majorana event is accompanied by raw Germanium detector waveforms, pulse shape discrimination cuts, and calibrated final energies, all shared in an HDF5 file format along with relevant metadata. This release is specifically designed to support the training and testing of Artificial Intelligence (AI) and Machine Learning (ML) algorithms upon our data. This document is structured as follows. Section I provides an overview of the dataset's content and format; Section II outlines the location of this dataset and the method for accessing it; Section III presents the NPML Machine Learning Challenge associated with this dataset; Section IV contains a disclaimer from the Majorana collaboration regarding the use of this dataset; Appendix A contains technical details of this data release. Please direct questions about the material provided within this release to liaobo77@ucsd.edu (A. Li)., Comment: DataPlanet Access: https://dataplanet.ucsd.edu/dataset.xhtml?persistentId=perma:83.ucsddata/UQWQAV

The framework of templicial objects was put forth in arXiv:2302.02484v1 in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of "weak dg-categories (concentrated in homologically positive degrees)" according to arXiv:2005.04778v3. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation., Comment: 22 pages, no figures

We present a new method, called "forced-spectrum fitting", for physically-based spectral modelling of radio sources during deconvolution. This improves upon current common deconvolution fitting methods, which often produce inaccurate spectra. Our method uses any pre-existing spectral index map to assign spectral indices to each model component cleaned during the multi-frequency deconvolution of WSClean, where the pre-determined spectrum is fitted. The component magnitude is evaluated by performing a modified weighted linear least-squares fit. We test this method on a simulated LOFAR-HBA observation of the 3C196 QSO and a real LOFAR-HBA observation of the 4C+55.16 FRI galaxy. We compare the results from the forced-spectrum fitting with traditional joined-channel deconvolution using polynomial fitting. Because no prior spectral information was available for 4C+55.16, we demonstrate a method for extracting spectral indices in the observed frequency band using "clustering". The models generated by the forced-spectrum fitting are used to improve the calibration of the datasets. The final residuals are comparable to existing multi-frequency deconvolution methods, but the output model agrees with the provided spectral index map, embedding correct spectral information. While forced-spectrum fitting does not solve the determination of the spectral information itself, it enables the construction of accurate multi-frequency models that can be used for wide-band calibration and subtraction., Comment: 17 pages, 9 figures, 5 tables. Accepted for publication in MNRAS

Direct detection of the Cosmic Dawn and Epoch of Reionization via the redshifted 21-cm line of neutral Hydrogen will have unprecedented implications for studying structure formation in the early Universe. This exciting goal is challenged by the difficulty of extracting the faint 21-cm signal buried beneath bright astrophysical foregrounds and contaminated by numerous systematics. Here, we focus on improving the Gaussian Process Regression (GPR) signal separation method originally developed for LOFAR observations. We address a key limitation of the current approach by incorporating covariance prior models learnt from 21-cm signal simulations using Variational Autoencoder (VAE) and Interpolatory Autoencoder (IAE). Extensive tests are conducted to evaluate GPR, VAE-GPR, and IAE-GPR in different scenarios. Our findings reveal that the new method outperforms standard GPR in component separation tasks. Moreover, the improved method demonstrates robustness when applied to signals not represented in the training set. It also presents a certain degree of resilience to data systematics, highlighting its ability to effectively mitigate their impact on the signal recovery process. However, our findings also underscore the importance of accurately characterizing and understanding these systematics to achieve successful detection. Our generative approaches provide good results even with limited training data, offering a valuable advantage when a large training set is not feasible. Comparing the two algorithms, IAE-GPR shows slightly higher fidelity in recovering power spectra compared to VAE-GPR. These advancements highlight the strength of generative approaches and optimise the analysis techniques for future 21-cm signal detection at high redshifts., Comment: 14 pages, 9 figues, submitted to MNRAS, any comments welcome

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies $E_{1}-E_{6}$, $E_{11}$, $E_{12}$, $E_{16}$, and $E_{18}$ that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys., Comment: 79 pages, 9 figures. v3: version published in JCAP

In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest sporadic simple Mathieu groups. To do this we first elucidate a connection between a special case of umbral moonshine and the imaginary quadratic class numbers. Then we use this connection to classify a distinguished set of modules for the smallest sporadic Mathieu group. Finally we establish consequences of the classification for the congruent number problem of antiquity., Comment: 36 pages, 5 tables

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight $2$ newforms under the $\xi$-operator. Our work uses a new non-analytic completion of the Kleinian $\zeta$-function from the theory of Abelian functions.

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU$_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU$_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity., Comment: 51 pages, v2: matches published version

We present the first empirical constraints on the polymer scale describing polymer quantized GWs propagating on a classical background. These constraints are determined from the polymer-induced deviation from the classically predicted propagation speed of GWs. We leverage posterior information on the propagation speed of GWs from two previously reported sources: 1) inter-detector arrival time delays for signals from the LIGO-Virgo Collaboration's first gravitational-wave transient catalog, GWTC1, and 2) from arrival time delays between GW signal GW170817 and its associated gamma-ray burst GRB170817A. For pure-GW constraints, we find relatively uninformative combined constraints of $\nu = 0.96\substack{+0.15 \\ -0.21} \times 10^{-53} \, \rm{kg}^{1/2}$ and $\mu = 0.94\substack{+0.75 \\ -0.20} \times 10^{-48} \, \rm{kg}^{1/2} \cdot s$ at the $90\%$ credible level for the two polymer quantization schemes, where $\nu$ and $\mu$ refer to polymer parameters associated to the polymer quantization schemes of propagating gravitational degrees of freedom. For constraints from GW170817/GRB170817A, we report much more stringent constraints of $\nu_{\mathrm{low}} =2.66\substack{+0.60 \\ -0.10}\times 10^{-56}$, $\nu_{\mathrm{high}} = 2.66\substack{+0.45 \\ -0.10}\times 10^{-56} $ and $\mu_{\mathrm{low}} = 2.84\substack{+0.64 \\ -0.11}\times 10^{-52}$, $\mu_{\mathrm{high}} = 2.76\substack{+0.46 \\ -0.11}\times 10^{-52}$ for both representations of polymer quantization and two choices of spin prior indicated by the subscript. Additionally, we explore the effect of varying the lag between emission of the GW and EM signals in the multimessenger case., Comment: Invited contribution to "Focus issue on Quantum Gravity Phenomenology in the Multi-Messenger Era: Challenges and Perspectives" in Classical and Quantum Gravity, 18 pages, 6 figures, Part of the COST Action CA18108: Quantum gravity phenomenology in the multi-messenger approach. V2: added comments on polymer quantization & statistical method used. Minor modification to plot colors. Published version

We construct a weight $1/2$ multiplier system for the group $\Gamma_0^+(p)$, the normalizer of the congruence subgroup $\Gamma_0(p)$ where $p$ is an odd prime, and we define an analogue of the eta function and Rademacher symbol and relate it to the geometry of edge paths in a triangulation of the upper half plane., Comment: 9 pages

Real-world data, such as administrative claims and electronic health records, are increasingly used for safety monitoring and to help guide regulatory decision-making. In these settings, it is important to document analytic decisions transparently and objectively to ensure that analyses meet their intended goals. The Causal Roadmap is an established framework that can guide and document analytic decisions through each step of the analytic pipeline, which will help investigators generate high-quality real-world evidence. In this paper, we illustrate the utility of the Causal Roadmap using two case studies previously led by workgroups sponsored by the Sentinel Initiative -- a program for actively monitoring the safety of regulated medical products. Each case example focuses on different aspects of the analytic pipeline for drug safety monitoring. The first case study shows how the Causal Roadmap encourages transparency, reproducibility, and objective decision-making for causal analyses. The second case study highlights how this framework can guide analytic decisions beyond inference on causal parameters, improving outcome ascertainment in clinical phenotyping. These examples provide a structured framework for implementing the Causal Roadmap in safety surveillance and guide transparent, reproducible, and objective analysis., Comment: 26 pages, 4 figures

Planning trajectories for automated vehicles in urban environments requires methods with high generality, long planning horizons, and fast update rates. Using a path-velocity decomposition, we contribute a novel planning framework, which generates foresighted trajectories and can handle a wide variety of state and control constraints effectively. In contrast to related work, the proposed optimal control problems are formulated over space rather than time. This spatial formulation decouples environmental constraints from the optimization variables, which allows the application of simple, yet efficient shooting methods. To this end, we present a tailored solution strategy based on ILQR, in the Augmented Lagrangian framework, to rapidly minimize the trajectory objective costs, even under infeasible initial solutions. Evaluations in simulation and on a full-sized automated vehicle in real-world urban traffic show the real-time capability and versatility of the proposed approach.

In this paper, we construct Hecke eigenforms for two families of quotient spaces of meromorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. We show that each quotient space in the first (resp. second family) is isomorphic as a Hecke module to the space $S_{2k}$ (resp. $M_{2k}$) of cusp forms (resp. holomorphic modular forms) of the same weight on $\mathrm{SL}_2(\mathbb{Z})$., Comment: 20 pages, comments welcome

Artificial intelligence (AI) has demonstrated the ability to extract insights from data, but the issue of fairness remains a concern in high-stakes fields such as healthcare. Despite extensive discussion and efforts in algorithm development, AI fairness and clinical concerns have not been adequately addressed. In this paper, we discuss the misalignment between technical and clinical perspectives of AI fairness, highlight the barriers to AI fairness' translation to healthcare, advocate multidisciplinary collaboration to bridge the knowledge gap, and provide possible solutions to address the clinical concerns pertaining to AI fairness.

The upcoming Square Kilometre Array Observatory (SKAO) will produce images of neutral hydrogen distribution during the epoch of reionization by observing the corresponding 21-cm signal. However, the 21-cm signal will be subject to instrumental limitations such as noise and galactic foreground contamination which pose a challenge for accurate detection. In this study, we present the SegU-Net v2 framework, an enhanced version of our convolutional neural network, built to identify neutral and ionized regions in the 21-cm signal contaminated with foreground emission. We trained our neural network on 21-cm image data processed by a foreground removal method based on Principal Component Analysis achieving an average classification accuracy of 71 per cent between redshift $z=7$ to $11$. We tested SegU-Net v2 against various foreground removal methods, including Gaussian Process Regression, Polynomial Fitting, and Foreground-Wedge Removal. Results show comparable performance, highlighting SegU-Net v2's independence on these pre-processing methods. Statistical analysis shows that a perfect classification score with $AUC=95\%$ is possible for $8(10\, {\rm cMpc})^3$ at $z>9$, for follow-up studies with infrared/optical telescopes to detect these sources., Comment: 19 pages, 10 figures, 3 tables, 2 appendixes

Libraries are increasingly relying on computational methods, including methods from Artificial Intelligence (AI). This increasing usage raises concerns about the risks of AI that are currently broadly discussed in scientific literature, the media and law-making. In this article we investigate the risks surrounding bias and unfairness in AI usage in classification and automated text analysis within the context of library applications. We describe examples that show how the library community has been aware of such risks for a long time, and how it has developed and deployed countermeasures. We take a closer look at the notion of '(un)fairness' in relation to the notion of 'diversity', and we investigate a formalisation of diversity that models both inclusion and distribution. We argue that many of the unfairness problems of automated content analysis can also be regarded through the lens of diversity and the countermeasures taken to enhance diversity., Comment: 14 pages

We present low-temperature magnetotransport measurements on selectively-grown Sb$_2$Te$_3$-based topological insulator ring structures. These topological insulator ring geometries display clear Aharonov-Bohm oscillations in the conductance originating from phase-coherent transport around the ring. The temperature dependence of the oscillation amplitude indicates that the Aharonov-Bohm oscillations originate from ballistic transport along the ring arms. The oscillations can therefore be attributed to topological surface states, which can maintain a quasi-ballistic transport regime in the presence of disorder. Further insight on the phase coherence is gained by comparing with similar Aharonov-Bohm-type oscillations in topological insulator nanoribbons exposed to an axial magnetic field. Here, quasi-ballistic phase-coherent transport is confirmed for closed-loop topological surface states in transverse direction enclosing the cross-section of the nanoribbon. In contrast, the appearance of universal conductance fluctuations indicates phase-coherent transport in the diffuse regime, which is attributed to bulk carrier transport. Thus, it appears that even in the presence of diffusive $p$-type charge carriers in Aharonov-Bohm ring structures, phase-coherent quasi-ballistic transport of topologically protected surface states is maintained over long distances., Comment: 11 pages (including Supplementary Material), 5 figures, 4 supplementary figures

We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $\xi$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight $2$ newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi--Weierstrass $\zeta$-function which is a generalization of the classical Weierstrass $\zeta$-function.

The argument of environment-assisted invariance (known as envariance) implying Born's rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can never give rise to Born's rule. Here, we address this apparent contradiction and point out an inconsistency in the assumptions underlying the arguments based on envariance. We use a construction in which the role of the measurement machine is made explicit and show that the presence of envariance does not imply every measurement will behave according to Born's rule. Rather, it implies that every quantum state allows a measurement machine to be constructed, which yields Born's rule when measuring that particular state. This resolves the paradox and is in agreement with the recent result of objective collapse models necessarily being non-linear.

Molecular hydrogen clouds are a key component of the interstellar medium because they are the birthplaces for stars. They are embedded in atomic gas that pervades the interstellar space. However, the details of how molecular clouds assemble from and interact with the atomic gas are still largely unknown. As a result of new observations of the 158~$\mu$m line of ionized carbon CII in the Cygnus region within the FEEDBACK program on SOFIA (Stratospheric Observatory for Infrared Astronomy), we present compelling evidence that CII unveils dynamic interactions between cloud ensembles. This process is neither a head-on collision of fully molecular clouds nor a gentle merging ofonly atomic clouds. Moreover, we demonstrate that the dense molecular clouds associated with the DR21 and W75N star-forming regions and a cloud at higher velocity are embedded in atomic gas and all components interact over a large range of velocities (20 km/s). The atomic gas has a density of 100 cm$^{-3}$ and a temperature of 100 K. We conclude that the CII 158 $\mu$m line is an excellent tracer to witness the processes involved in cloud interactions and anticipate further detections of this phenomenon in other regions, Comment: Nature Astronomy in press

Dynamic contrast-enhanced (DCE) magnetic resonance imaging (MRI) ideally requires a high spatial and high temporal resolution, but hardware limitations prevent acquisitions from simultaneously achieving both. Existing image reconstruction techniques can artificially create spatial resolution at a given temporal resolution by estimating data that is not acquired, but, ultimately, spatial details are sacrificed at very high acceleration rates. The purpose of this paper is to introduce the concept of spatial subspace reconstructions (SPARS) and demonstrate its ability to reconstruct high spatial resolution dynamic images from as few as one acquired radial spoke per dynamic frame. Briefly, a low-temporal-high-spatial resolution organization of the acquired raw data is used to estimate a spatial subspace in which the high-temporal-high-spatial ground truth data resides. This subspace is then used to estimate entire images from single k-space spokes. In both simulated and human in-vivo data, the proposed SPARS reconstruction method outperformed standard GRASP and GRASP-Pro reconstruction, providing a shorter reconstruction time and yielding higher accuracy from both a spatial and temporal perspective., Comment: commercialization

We lay the foundations for a theory of quasi-categories in a monoidal category $\mathcal{V}$ replacing $\mathrm{Set}$, aimed at realising weak enrichment in the category $S\mathcal{V}$ of simplicial objects in $\mathcal{V}$. To accomodate non-cartesian monoidal products, we make use of an ambient category $S_{\otimes}\mathcal{V}$ of templicial - or 'tensor-simplicial' - objects in $\mathcal{V}$, which are certain colax monoidal functors following Leinster. Inspired by the description of the categorification functor due to Dugger and Spivak, we construct a templicial analogue of the homotopy coherent nerve functor which goes from $S\mathcal{V}$-enriched categories to templicial objects. We show that an $S\mathcal{V}$-enriched category whose underlying simplicial category is locally Kan, is turned into a quasi-category in $\mathcal{V}$ by this nerve functor., Comment: 45 pages, no figures. Added some references and acknowledgements

The light-cone (LC) anisotropy arises due to cosmic evolution of the cosmic dawn 21-cm signal along the line-of-sight (LoS) axis of the observation volume. The LC effect makes the signal statistically non-ergodic along the LoS axis. The multi-frequency angular power spectrum (MAPS) provides an unbiased alternative to the popular 3D power spectrum as it does not assume statistical ergodicity along every direction in the signal volume. Unlike the 3D power spectrum which mixes the cosmic evolution of the 21-cm signal along the LoS $k$ modes, MAPS keeps the evolution information disentangled. Here we first study the impact of different underlying physical processes during cosmic dawn on the behaviour of the 21-cm MAPS using simulations of various different scenarios and models. We also make error predictions in 21-cm MAPS measurements considering only the system noise and cosmic variance for mock observations of HERA, NenuFAR and SKA-Low. We find that $100~{\rm h}$ of HERA observations will be able to measure 21-cm MAPS at $\geq 3\sigma$ for $\ell \lesssim 1000$ with $0.1\,{\rm MHz}$ channel-width. The better sensitivity of SKA-Low allows reaching this sensitivity up to $\ell \lesssim 3000$. Note that due to the difference in the frequency coverage of the various experiements, the CD-EoR model considered for NenuFAR is different than those used for the HERA and SKA-Low predictions. Considering NenuFAR with the new model, measurements $\geq 2\sigma$ are possible only for $\ell \lesssim 600$ with $0.2\,{\rm MHz}$ channel-width and for a ten times longer observation time of $t_{\rm obs} = 1000~{\rm h}$. However, for the range $300 \lesssim \ell \lesssim 600$ and $t_{\rm obs}=1000~{\rm h}$ more than $3\sigma$ measurements are still possible for NenuFAR when combining consecutive frequency channels within a $5 ~{\rm MHz}$ band., Comment: 21 pages, 12 figures, 3 tables, accepted for publication in MNRAS

The measurement of the $21$ cm signal from the Cosmic Dawn is a major goal for several existing and upcoming radio interferometers such as NenuFAR and the SKA. During this era before the beginning of the Epoch of Reionization, the signal is more difficult to observe due to brighter foregrounds but reveals additional information on the underlying astrophysical processes encoded in the spatial fluctuations of the spin temperature of hydrogen. To interpret future measurements, controlling the level of accuracy of the Lyman-$\alpha$ flux modelling is mandatory. In this work, we evaluate the impact of various approximations that exist in the main fast modelling approach compared to the results of a costly full radiative transfer simulation. The fast SPINTER code, presented in this work, computes the Lyman-$\alpha$ flux including the effect of wing scatterings for an inhomogeneous emissivity field, but assuming an otherwise homogeneous expanding universe. The LICORICE code computes the full radiative transfer in the Lyman-$\alpha$ line without any substantial approximation. We find that the difference between homogeneous and inhomogeneous gas density and temperature is very small for the computed flux. On the contrary, neglecting the effect of gas velocities produces a significant change in the computed flux. We identify the causes (mainly Doppler shifts due to velocity gradients) and quantify the magnitude of the effect in both an idealised setup and a realistic cosmological situation. We find that the amplitude of the effect, up to a factor of $\sim 2$ on the $21$ cm signal power spectrum on some scales (depending on both other model parameters and the redshift), can be easily discriminated with an SKA-like survey and already be approached, particularly for exotic signals, by the ongoing NenuFAR Cosmic Dawn Key Science Program., Comment: 14 pages, accepted for publication in A&A

The inability of Schrodinger's unitary time evolution to describe measurement of a quantum state remains a central foundational problem. It was recently suggested that the unitarity of Schrodinger dynamics can be spontaneously broken, resulting in measurement as an emergent phenomenon in the thermodynamic limit. Here, we introduce a family of models for spontaneous unitarity violation that apply to generic initial superpositions over arbitrarily many states, using either single or multiple state-independent stochastic components. Crucially, we show that Born's probability rule emerges spontaneously in all cases., Comment: 12 pages, 5 figures, appendices added

Diversity Searcher is a tool originally developed to help analyse diversity in news media texts. It relies on a form of automated content analysis and thus rests on prior assumptions and depends on certain design choices related to diversity and fairness. One such design choice is the external knowledge source(s) used. In this article, we discuss implications that these sources can have on the results of content analysis. We compare two data sources that Diversity Searcher has worked with - DBpedia and Wikidata - with respect to their ontological coverage and diversity, and describe implications for the resulting analyses of text corpora. We describe a case study of the relative over- or under-representation of Belgian political parties between 1990 and 2020 in the English-language DBpedia, the Dutch-language DBpedia, and Wikidata, and highlight the many decisions needed with regard to the design of this data analysis and the assumptions behind it, as well as implications from the results. In particular, we came across a staggering over-representation of the political right in the English-language DBpedia.

This White Paper, prepared for the Fundamental Symmetries, Neutrons, and Neutrinos Town Meeting related to the 2023 Nuclear Physics Long Range Plan, makes the case for double beta decay as a critical component of the future nuclear physics program. The major experimental collaborations and many theorists have endorsed this white paper., Comment: white paper submitted for the Fundamental Symmetries, Neutrons, and Neutrinos Town Meeting in support of the US Nuclear Physics Long Range Planning Process

In this note, we propose a decomposition of the quantum matrix group SL$_q^+(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebra generators of Faddeev's modular double of $\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$. The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a $q$-BF formulation of the amplitudes of both of these gravitational models., Comment: 32 pages, v3: added material on regular representation and Hopf duality, typos fixed and references added, matches published version

Short-range interactions and long-range contacts drive the 3D folding of structured proteins. The proteins' structure has a direct impact on their biological function. However, nearly 40% of the eukaryotes proteome is composed of intrinsically disordered proteins (IDPs) and protein regions that fluctuate between ensembles of numerous conformations. Therefore, to understand their biological function, it is critical to depict how the structural ensemble statistics correlate to the IDPs' amino acid sequence. Here, using small-angle x-ray scattering (SAXS) and time-resolved F\"orster resonance energy transfer (trFRET), we study the intra-molecular structural heterogeneity of the neurofilament low intrinsically disordered tail domain (NFLt). Using theoretical results of polymer physics, we find that the Flory scaling exponent of NFLt sub-segments correlates linearly with their net charge, ranging from statistics of ideal to self-avoiding chains. Surprisingly, measuring the same segments in the context of the whole NFLt protein, we find that regardless of the peptide sequence, the segments' structural statistics are more expanded than when measured independently. Our findings show that while polymer physics can, to some level, relate the IDP's sequence to its ensemble conformations, long-range contacts between distant amino acids play a crucial role in determining intra-molecular structures. This emphasizes the necessity of advanced polymer theories to fully describe IDPs ensembles with the hope it will allow us to model their biological function.