The research team, spearheaded by Gina Pizzo, a lecturer in Rice University’s Department of Statistics, and Aaron McCright, professor of sociology at Michigan State University, developed and validated a new way to measure what they call disciplinarity. The concept refers to the extent to which scientists remain closely connected to the core of the discipline in which they trained, or instead move toward its boundaries and beyond over the course of their careers. Rather than treating disciplinary affiliation as a fixed attribute, the new approach treats it as a continuum that can shift as research questions, collaborators and professional communities evolve. The measure was tested using survey responses from 4,113 tenure-system science faculty members at 83 U.S. research universities, and the co-author team also included Chad Gonnerman of the University of Southern Indiana, Troy Hall of Oregon State University, Kathryn Plaisance of the University of Waterloo and Brian Robinson of Texas A&M University-Kingsville.

Scientists don’t necessarily stay in the same intellectual box throughout their careers, Pizzo explained. Their research questions, collaborators and professional communities can change considerably, and the team wanted a way to capture that evolution rather than relying on a single disciplinary label. The motivation for the work grew out of a persistent methodological problem in the social studies of science. Researchers who survey scientists have long grouped them by characteristics such as the field of their highest degree, their academic department or the discipline with which they identify. But Pizzo and McCright found that such approaches often miss important differences among scientists who technically belong to the same field. Their review of previous surveys revealed that disciplinary affiliation has frequently been measured using only one or two indicators, a practice that can obscure the ways scientists’ careers diverge as they collaborate across fields or shift the focus of their research.

To close that gap, the researchers created a five-dimensional Disciplinarity Scale. The instrument is anchored to a scientist’s doctoral discipline and measures how closely five aspects of the scientist’s current professional life still align with that original field. Those dimensions include the scientist’s current department, primary professional association, publishing outlets, how their peers perceive their discipline and how they identify themselves. Taken together, the five measures place scientists along a continuum that runs from strong connection to a discipline’s core to broader association with its frontier or even beyond it. The design reflects a conceptual stance that McCright summarized succinctly: academic disciplines are useful categories, but they aren’t impermeable containers. The scale does not discard disciplines; instead, it quantifies the degree to which an individual scientist’s working life remains tethered to one.

The empirical findings lend strong support to the idea that disciplinary affiliation is plastic rather than permanent. The researchers found that scientists tended to have broader disciplinary affiliations the longer they had been out of graduate school, suggesting that careers naturally pull scholars away from the intellectual center of their training over time. The association was even stronger among scientists with more experience conducting cross-disciplinary research. Cross-disciplinary research experience was especially strongly associated with broader disciplinarity, Pizzo noted. When scientists repeatedly work with people who ask different questions, use different methods or publish in different places, those experiences can gradually reshape where they fit within science. In other words, interdisciplinarity is not merely a style of work; it appears to leave a measurable imprint on how scientists locate themselves within the broader landscape of academic fields.

The scale also proved capable of identifying scientists who had formally moved into a different discipline altogether. Among the survey respondents, about 16.6 percent reported a current discipline different from that of their doctorate, while roughly 83 percent remained within their original nominal discipline. Yet even among the scientists who had not formally changed fields, the new measure revealed meaningful differences in how closely their current work remained tied to their original discipline. Two faculty members might both be counted as physicists in a conventional survey, but one could be working near the traditional core of physics while the other operates largely at the boundary with climate science or materials science. Under older measurement approaches, those two scientists would be treated as equivalent. The new scale makes the distinction visible, and that distinction could matter for researchers who study how scientists’ backgrounds influence their views, behaviors or approaches to research.

The implications extend well beyond the sociology of science. Universities and funding agencies increasingly encourage interdisciplinary research aimed at problems that do not fit neatly within a single field, from climate change to emerging technologies to public health. If administrators, funders and scholars rely solely on doctoral discipline to characterize the scientific workforce, they risk drawing inaccurate pictures of who is actually equipped to tackle such problems and how expertise is distributed. Pizzo and McCright argue that future surveys could pair conventional disciplinary categories with the new scale, allowing researchers to capture both where scientists began their careers and how their disciplinary affiliation has evolved since. Such a paired approach would preserve the organizational utility of disciplines while adding a layer of nuance that reflects the increasingly fluid reality of scientific work.

Technically, the study contributes a validated instrument to a field that has often relied on ad hoc classification. By grounding the scale in the doctoral discipline and then systematically comparing five independent markers of current affiliation, the researchers created a measure that can detect gradations of disciplinarity rather than forcing scientists into binary categories such as disciplinary versus interdisciplinary. The validation across more than four thousand faculty members at a large number of research universities gives the instrument empirical weight, and the consistency of its central findings, that time since graduate school and cross-disciplinary experience both predict broader disciplinarity, suggests the construct captures something real about career trajectories in academic science. The work also demonstrates how survey-based measurement can be refined when existing indicators prove too coarse for the phenomena they are meant to describe.

The study’s authors are careful to frame their contribution as a refinement rather than a rejection of traditional categories. We’re not arguing that disciplines no longer matter, McCright said. They remain fundamental to how science is organized. But if we want to understand scientists accurately, we also need to account for the ways their relationship to those disciplines can change over time. In an era when the most pressing scientific questions increasingly demand collaboration across fields, the ability to measure precisely how far a scientist has traveled from the core of her training may become an essential tool for understanding, and supporting, the modern scientific workforce. The physicist who became a climate scientist is no longer an anomaly; with the new Disciplinarity Scale, she is finally a measurable and well-understood feature of the scientific landscape.

Subject of Research: A validated five-dimensional measure of how scientists' disciplinary affiliation changes relative to their doctoral discipline over the course of their careers.

Article Title: New measure captures how scientists’ identities shift across academic disciplines

Article References: New measure captures how scientists’ identities shift across academic disciplines. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: disciplinarity, interdisciplinary research, scientists, doctoral discipline, Rice University, Michigan State University, PLOS One, academic careers, survey research, scientific workforce, disciplinary affiliation, sociology of science

The finding matters because logarithmic distributions are not the kind of pattern one expects from random events spread evenly across a city. Instead, they describe situations in which a small number of categories, or in this case districts, account for a disproportionately large share of outcomes, while a long tail of districts experiences relatively few. This is the same family of heavy-tailed behavior that statisticians have documented in contexts as varied as word frequencies in language, species abundance in ecology, and the severity of terrorist attacks. Lane situates his result within this broader tradition of complexity studies, drawing explicit connections to the work of Lewis Fry Richardson, whose pioneering statistics of deadly quarrels in the mid-twentieth century anticipated modern quantitative approaches to violence.

The technical core of the study is a goodness-of-fit analysis. Lane assembled homicide counts for each of Chicago’s police districts using a series of official reports spanning more than half a century, from the Chicago police statistical report of 1965 through the department’s annual reports and, ultimately, data maintained by the Bureau of Detectives. For each year in the 1964 to 2018 window, the per-district homicide counts were compared against the theoretical probabilities predicted by a logarithmic distribution, and the coefficient of determination was used to assess how much of the observed variation the distribution could explain. The fits, according to the study, are consistently strong across a period that encompasses dramatic swings in the city’s overall homicide rate, from the violent peak of the early 1990s to the historic lows of the mid-2010s.

That consistency is itself the headline result. Chicago changed enormously between 1964 and 2018: neighborhoods transformed, policing strategies shifted, gang structures evolved, and the crack cocaine era drove homicide counts to levels far above what the city sees today. Yet through all of that turbulence, the relative distribution of murders across districts retained the same logarithmic shape. In statistical terms, the parameter governing the distribution may fluctuate with the overall level of violence, but the functional form remains stable. This kind of invariance is what physicists and complexity researchers look for when they suspect that a system is governed by underlying scaling laws rather than by the details of any particular moment.

The study draws on a rich interdisciplinary literature to interpret the result. In cosmology, Gerard de Vaucouleurs described the large-scale distribution of galaxies using similar statistical reasoning. In urban science, Michael Batty has documented scaling laws governing cities, neighborhoods, and buildings, arguing that competition in the built environment produces regular mathematical patterns. In ecology and paleontology, David Raup’s work on extinction highlighted how bad luck, rather than bad genes, often shapes the distribution of survival and loss. And in the study of conflict, Aaron Clauset and colleagues demonstrated that the severity of terrorist events follows a heavy-tailed distribution. Levy and Solomon have argued that power laws can be understood as logarithmic Boltzmann laws, a theoretical framing that Lane invokes to give the homicide finding a firmer mathematical footing.

What could produce such a pattern in something as grimly concrete as murder? The logarithmic distribution tends to arise in processes of repeated chance events where the probability of accumulating additional events shrinks in a specific way, and where aggregation occurs across many independent units. In an urban context, this is consistent with a picture in which a combination of concentrated disadvantage, networked conflict, and policing feedback loops channels violence unevenly across space. A small number of districts generate a large share of homicides, while most districts contribute only marginally, and the way those contributions stack up over time follows the logarithmic curve. The stability of the shape across five decades suggests that whatever mechanisms generate the pattern are structural features of the city rather than transient conditions.

The practical implications are where Lane’s report becomes provocative. If homicides are distributed according to a lawlike statistical pattern at the district level, then the conventional practice of investigating homicides primarily within district boundaries may be mismatched to the underlying structure of the problem. Lane argues that the finding may have consequences for how police departments allocate resources, and in particular that homicide investigation might be better organized on a citywide basis rather than a district level. Under a citywide model, investigative capacity could flow toward the districts where the logarithmic concentration predicts murders will cluster, rather than being fixed in place according to administrative geography drawn for other purposes.

This argument touches a long-running debate in criminology about the appropriate spatial scale for violence reduction. Place-based policing research has repeatedly shown that crime concentrates at very small geographic units, such as specific street blocks, and that such concentration is remarkably stable over time. Lane’s district-level analysis operates at a coarser resolution, but it points in a compatible direction: violence is not smoothly distributed across the urban landscape, and administrative units that ignore the statistical grain of the phenomenon may dilute investigative effectiveness. Detectives, forensic resources, and community outreach efforts could, in principle, be deployed with the distribution itself as a guide. The study stops short of prescribing specific policy changes, but it frames the logarithmic fit as an empirical benchmark that any resourcing model should reckon with.

Caveats remain. The analysis relies on official police statistics, which are subject to reporting and classification practices that may themselves vary across districts and decades. A logarithmic fit describes the aggregate pattern without identifying the causal mechanisms that generate it, and correlation across five decades of data does not by itself settle questions about what drives concentration. Lane’s report is explicitly framed as a short report, a first quantitative demonstration rather than a comprehensive theory of urban homicide. Still, the durability of the pattern, holding steady from the civil rights era through the post-pandemic surge, gives the result unusual weight. Future work, Lane suggests, should test whether comparable logarithmic structure appears in other major cities, which would indicate a general property of urban violence rather than a peculiarity of Chicago.

If the finding generalizes, the implications extend beyond one department. Homicide clearance rates in many American cities have fallen to historic lows, and researchers and reformers alike have questioned whether the traditional district-based detective model is adequate to the task. A statistical law that describes where murders cluster over more than half a century offers a rare kind of leverage: a stable target for resource allocation that persists even as the overall level of violence rises and falls. Lane’s contribution is to show that such a target exists, etched into five decades of Chicago’s records, waiting in plain sight within the numbers the city has been collecting all along.

The mathematical lineage behind the logarithmic distribution is worth underscoring. The distribution belongs to a family of discrete probability models in which the probability of observing a count decreases roughly in proportion to that count, producing the characteristic heavy tail. Statisticians have long catalogued how such distributions relate to one another, and Lane draws on that literature of univariate distribution relationships to place his homicide data within a coherent probabilistic framework. The connection matters because a good fit to a named distribution is more than a curve-fitting exercise; it suggests that the process generating the data may share features with other well-understood systems, from the frequencies of rare species to the sizes of extinction events documented by paleontologists.

The data foundation of the study deserves particular attention. Reconstructing fifty-five years of district-level homicide counts required assembling a patchwork of official documents, including statistical summaries from the mid-1960s, annual reports from the 1970s through the 2000s, and contemporary materials from the Bureau of Detectives. Few American cities maintain records of this depth and consistency, which is part of why Chicago has served as a proving ground for quantitative criminology. The reliance on the coefficient of determination as the measure of fit is a conventional but transparent choice, allowing readers to judge for themselves how much of the year-to-year variation in district-level counts the logarithmic model captures.

It is also notable that the study appears in a criminal justice journal rather than a physics or statistics venue, signaling a growing openness among criminologists to complexity-based framing. The keywords attached to the article, including fat tail, complexity studies, and statistics of deadly quarrels, situate the work explicitly in the tradition of Richardson, whose interwar analyses of conflict anticipated modern heavy-tailed modeling. Whether the logarithmic pattern proves unique to Chicago or emerges elsewhere, the study demonstrates that long-run administrative data can support the kind of scaling analysis more often applied to cities, galaxies, and ecosystems than to police districts.

Subject of Research: The statistical distribution of homicides across Chicago police districts from 1964 to 2018

Article Title: A Short Report on the Distribution of Murder in Chicago

Article References: Lane, D. J. (2026). A Short Report on the Distribution of Murder in Chicago. American Journal of Criminal Justice. https://doi.org/10.1007/s12103-026-09943-x

Image Credits: AI Generated

DOI: 10.1007/s12103-026-09943-x

Keywords: homicide, Chicago, logarithmic distribution, quantitative criminology, police districts, statistics, complexity studies, Lewis Richardson, policing, fat tail, crime concentration, resource allocation

The question of cosmic topology goes beyond the familiar issue of curvature. General relativity determines how space bends, but it does not fix whether space is simply connected or multiply connected. A universe could be geometrically flat, open or closed, yet still wrap around on itself like the three-dimensional analogue of a torus, the doughnut-shaped surface familiar from video games where a spaceship leaving one edge reappears on the other. In such a multiply connected cosmos, the Universe could be finite even though its local geometry looks Euclidean to every measurement we can make.

If space does wrap around, light from distant objects could reach us along multiple paths, producing ghost images of the same galaxies at different points on the sky. More subtly, a non-trivial topology would imprint distinctive signatures on the cosmic microwave background, the relic radiation from the hot early Universe. Because the CMB is the oldest light we can observe, emitted roughly 380,000 years after the Big Bang, it offers a unique window onto scales that no galaxy survey can reach. Topology at the largest scales would break the statistical isotropy and potentially the homogeneity that underpin the standard cosmological model.

The most celebrated signature is the matched circle-pair. If the last scattering surface – the shell from which the CMB photons were emitted – intersects its own topological images, the sky should contain pairs of circles along which the temperature patterns match exactly. This ‘circles in the sky’ method, pioneered by Neil Cornish, David Spergel and Glenn Starkman in the late 1990s, offers a potentially model-independent way of detecting the wrapping scale. Alternatively, cosmologists can perform full Bayesian likelihood analyses using topology-dependent covariance matrices, computing how the pattern of temperature fluctuations changes when the fundamental domain of the Universe is smaller than the horizon.

Successive space missions have powered increasingly sophisticated searches. The Cosmic Background Explorer, or COBE, provided the first constraints in the 1990s. The Wilkinson Microwave Anisotropy Probe and then the Planck satellite delivered maps of exquisite precision, allowing teams to exclude a range of topologies, including the much-discussed Poincare dodecahedral space that had once been proposed as an explanation for the puzzling lack of large-angle temperature correlations in the CMB. Yet the searches have yielded no definitive evidence for non-trivial topology.

That absence of detection, however, is far from a proof of infinity. The review emphasises that current constraints exclude only some topologies, some parameter ranges, and some possible observer positions. Many candidate manifolds – particularly the compact hyperbolic spaces whose mathematics was classified by William Thurston – remain only partially constrained, and the detectability of a given topology depends delicately on where our galaxy happens to sit within the fundamental domain.

Recent theoretical work has delivered a genuine surprise: detectable signals may persist even when the topology scale exceeds the size of the observable Universe. Because topological identifications correlate fluctuation modes across the entire sky, the induced correlations do not simply vanish once the wrapping scale passes beyond the horizon. This shifts the goalposts of the field, implying that even the null results from WMAP and Planck leave substantial parameter space unexplored – space that future experiments could yet probe.

The prospects rest on the next generation of instrumentation. Planned CMB experiments, including the LiteBIRD satellite and the balloon-borne Taurus experiment, will measure CMB polarisation with unprecedented sensitivity, adding an entirely new channel of topological information beyond temperature alone. High-precision galaxy surveys and line-intensity-mapping experiments could exploit topology-induced correlations at all accessible redshifts, mapping the three-dimensional distribution of matter in ways that reveal whether the cosmic web repeats itself.

The theoretical stakes extend to quantum gravity and the origin of the Universe. Ideas ranging from the no-boundary wavefunction of Hartle and Hawking to string-theoretic arguments about allowed topologies suggest that the shape of space may be linked to the deepest laws of physics. Casimir-type quantum effects in compact spaces, chaotic mixing in the early Universe, and the quantum creation of compact inflationary universes all connect cosmic topology to fundamental theory. Whether cosmic topology is observable remains uncertain, but as this review makes clear, current and future data offer an unprecedented opportunity to determine the global structure of the Universe – and perhaps, at last, to answer whether space is finite.

The intellectual lineage of cosmic topology is older than the observational era it now confronts. As early as 1900, Karl Schwarzschild pondered the possibility that space might be multiply connected, asking in effect whether an astronomer could detect the curvature measure of the cosmos by looking for self-intersections of light paths. In the decades that followed, the mathematical foundations laid out by figures such as Duncan Sommerville, and the expanding solutions of de Sitter, Friedmann and Lemaître, opened a conceptual space in which global shape and local geometry could be treated as genuinely independent questions. By the second half of the twentieth century, theorists including George Ellis and later Yakov Zeldovich and his collaborators had begun to articulate how a finite universe with non-Euclidean identifications might be recognised observationally, setting the stage for the systematic searches that define the field today.

One of the more colourful consequences of a compact universe is its effect on timekeeping and simultaneity. In multiply connected spaces, the classic twin paradox acquires a topological twist: a traveller circumnavigating the universe can return younger than a stay-at-home twin without ever accelerating to superluminal speeds, because the global identification of space breaks the usual equivalence between inertial frames. Studies of this effect, along with analyses of how the Copernican principle must be reformulated in compact spacetimes, illustrate that topology is not merely a geometric curiosity but alters the operational meaning of fundamental relativistic concepts.

Between the direct imaging of ghost images and the statistical analysis of the microwave background lies a family of intermediate techniques. Cosmic crystallography, developed in the mid-1990s by Roland Lehoucq, Marc Lachieze-Rey and Jean-Pierre Luminet, exploits the idea that in a multiply connected space the set of distances between pairs of objects shows spikes at the characteristic lengths associated with topological translations. If a sufficiently deep catalogue of galaxy positions were available, a histogram of pair separations would reveal these spikes as fingerprints of the fundamental domain. Subsequent refinements showed both the promise and the practical limits of the method, which demands redshift surveys of a depth and precision that remain challenging even for present-day instruments.

The statistical approach to topology rests on the eigenmodes of the Laplacian in the candidate space. In a simply connected universe, the modes of the temperature and matter fluctuations take well-known forms; in a compact space, the allowed wavelengths are quantised by the finite volume, suppressing power on scales exceeding the topology scale. Computing these eigenmodes for flat, spherical and hyperbolic manifolds – including lens spaces, prism spaces and the more exotic horned topologies – has been a major technical undertaking, requiring numerical methods for spaces whose mode structure resists closed-form solution. The resulting covariance matrices encode how topology reshapes the correlation of fluctuations across the sky, and they form the basis of the likelihood analyses applied to COBE, WMAP and Planck data.

A notable recent development is the systematic treatment of non-orientable manifolds, spaces in which a journey around a closed loop can flip handedness. Work published through 2025 has extended the eigenmode and correlation-matrix machinery to these non-orientable Euclidean spaces, which had previously received far less attention than their orientable counterparts. Related calculations for spin-2 perturbations – the mathematical description of polarisation and gravitational waves in such spaces – have revealed that a multiply connected universe can generate apparent parity violation in the microwave background even when the underlying microphysics respects parity exactly. This offers a striking example of how global geometry can mimic signatures usually attributed to new particle physics, underscoring the need to treat topology as a systematic uncertainty in searches for parity-violating cosmological signals.

The scale of the modern effort is reflected in the emergence of dedicated collaborations. The COMPACT collaboration, coordinating researchers across Europe and North America, has produced a series of papers cataloguing the observable consequences of candidate manifolds, from orientable and non-orientable Euclidean spaces to the detectability prospects for future surveys. A 2024 analysis in Physical Review Letters made the case that future searches hold genuine promise, quantifying how much of the topology parameter space remains accessible to planned instruments. This programme transforms what was once a collection of individual case studies into a coherent research programme with well-defined targets.

For observers, the practical message is that temperature maps alone have largely exhausted their discriminating power for many candidate topologies. Polarisation of the microwave background, generated by Thomson scattering in the early Universe, responds to topological identifications in ways that differ from the temperature field, and spin-2 modes carry information about the global structure that scalar modes cannot. Combining temperature, polarisation and three-dimensional matter correlations across a wide range of redshifts therefore offers the most promising route forward – a multi-channel strategy that could either reveal the wrapping of space or push the limits on finite universes to scales of extraordinary magnitude.

Subject of Research: The observable signatures of non-trivial cosmic topology in the cosmic microwave background and large-scale structure.

Article Title: The topology of the Universe

Article References: Copi, C. J., Mihaylov, D. P., Negro, A., Samandar, A., Starkman, G. D., Akrami, Y., Alestas, G., Anselmi, S., Carrón Duque, J., Cornet-Gomez, F., Htat Lu, L., Jaffe, A. H., Kosowsky, A., Martin Barandiaran, M., Pereira, T. S., Petretti, C., & Tamosiunas, A. (2026). The topology of the Universe. Nature Astronomy. https://doi.org/10.1038/s41550-026-02930-6

Image Credits: AI Generated

DOI: 10.1038/s41550-026-02930-6

Keywords: cosmic topology, cosmic microwave background, matched circle-pairs, CMB anomalies, Planck, WMAP, LiteBIRD, compact spaces, Bayesian analysis, large-scale structure, Universe shape, polarization

The research, published in the Journal of Machine Learning Research, addresses a subtle but critically important deception that can occur within a family of AI models known as concept-based models. These systems were designed specifically with human oversight in mind. Rather than processing raw data, such as the pixel values of a medical scan, and emitting an inscrutable answer, concept-based models are constructed to reason through clinically meaningful intermediate variables: blood pressure readings, the presence or absence of fever, the size of a tumour, or the visibility of abnormalities in a medical image. The promise is seductive and intuitive. If a model must justify its conclusions in terms of concepts that a trained clinician can inspect, verify, and contest, then the clinician retains meaningful control over the decision-making process. The AI becomes less of an oracle and more of a colleague whose reasoning can be followed.

In principle, this is exactly what emerging AI regulations around the world demand. Regulators increasingly emphasise transparency and meaningful human oversight as prerequisites for deploying automated systems in high-stakes environments, and healthcare is the most consequential of these. Clinicians need to be able to identify when a system has made a mistake and to intervene in its decision-making before harm occurs. An AI that predicts patient outcomes or flags signs of disease with high accuracy is of limited value, and potentially considerable danger, if the experts responsible for patient care cannot audit the chain of reasoning that led to its conclusions.

But here lies the problem the King’s College team set out to expose. Concept-based models, for all their apparent clarity, can suffer from a phenomenon known as information leakage. This occurs when the concepts used by the model contain additional, unintended information that is not visible to the human reviewing the decision. The concept label might say “fever present” or “tumour size: moderate,” and the human reviewer sees exactly that. But hidden within the numerical representation of that concept may be a wealth of auxiliary information, correlations and signals inherited from the raw data, that the model exploits when forming its final prediction. The consequence is profound: the model appears interpretable while still relying on information that a human cannot see or assess. In effect, it behaves like a black box model disguised as an interpretable one. The transparency is an illusion, and the clinician’s sense of oversight is misplaced confidence rather than genuine understanding.

The scale of this problem becomes clear when one considers what is at stake. If a concept-based diagnostic system recommends an aggressive treatment pathway, and the apparent basis for that recommendation is a set of concepts the clinician has reviewed and found reasonable, the clinician may approve the recommendation believing they have verified its justification. If, however, the model’s actual decision was driven substantially by hidden information smuggled inside those concepts, the clinician has not verified anything of consequence. The audit has failed silently, and the regulatory requirement for meaningful human oversight has been satisfied only in appearance.

Distinguishing genuine transparency from its counterfeit has, until now, lacked a rigorous foundation. It is one thing to suspect that leakage might occur; it is quite another to define it precisely enough to measure it. This is the central contribution of Dr Enrico Parisini, Senior Research Fellow in Machine Learning and first author of the paper, and Dr Chris Banerji, AI+ Senior Fellow and senior author, who developed a mathematical framework that defines and quantifies information leakage in concept-based AI systems with exactitude. The framework consists of two complementary measures. The first, concepts-task leakage, abbreviated CTL, captures hidden information linked to the final prediction, that is, the extent to which the model’s output depends on information concealed within the concepts rather than on the concepts as the human perceives them. The second, interconcept leakage, or ICL, captures hidden information shared between concepts, revealing how the internal representations of different concepts may be entangled in ways invisible to the reviewer.

Together, these two measures provide a diagnostic instrument of genuine practical power. The researchers tested their framework across several datasets and found that it could reliably detect leakage when it was present. More impressively, the framework could predict how models would respond when their concepts were deliberately changed. This is a crucial validation: a measure of leakage is only useful if it reflects real causal structure within the model, and the ability to anticipate model behaviour under intervention demonstrates that the framework captures something true about the system’s internal workings, not merely a statistical artefact.

The implications reach well beyond diagnostics of existing systems. Because the framework quantifies leakage rather than merely detecting it, it provides practical guidance for designing concept-based models that minimise leakage from the outset. Developers can, in effect, use the measures as design targets, iteratively refining their architectures until the concepts presented to the human user genuinely carry the information that drives the decision. This transforms transparency from a vague aspiration into an engineering specification with a measurable standard, something that has been conspicuously absent from the discourse around explainable AI.

The work also arrives at a moment of acute relevance for the regulation of medical AI. As health systems around the world begin to deploy machine learning tools for triage, diagnosis, and prognostication, regulators are grappling with how to certify that these systems are safe and auditable. A framework that can quantify whether a model’s claimed interpretability is genuine offers regulators and developers a common language for assessing transparency claims. It moves the conversation from reassurance to measurement, from assurances that a model “uses clinically meaningful concepts” to a demonstrable quantity describing exactly how much hidden information those concepts conceal.

The researchers themselves frame the contribution as a deliberate act of caution in a field defined by speed. Dr Banerji, the senior author, described the motivation bluntly: rather than putting the cart before the horse, while the field of AI is moving quickly to apply models to real-world problems, the team stepped back to consider what is needed to make these systems safe and reliable. The focus, he explained, has been on understanding limitations and addressing them, so that the deployment of these models in clinical practice can proceed in a safer way. It is a philosophy that stands in quiet contrast to the prevailing urgency of the AI race, and one that may prove prescient as the first generation of clinical AI deployments encounters the messy realities of medical practice.

Dr Parisini, the paper’s first author, emphasised the dual nature of the finding. Concept-based AI has genuine potential to make AI systems more transparent, he noted, but the research shows that models can appear interpretable while still relying on information hidden from the person using them. By identifying and measuring this hidden information, he argued, the field can take concrete steps toward AI systems that are transparent in substance rather than in appearance. It is a distinction that may come to define the next phase of the explainable AI movement: the difference between systems that look accountable and systems that are accountable.

The team is now working to apply these approaches to real clinical problems, a step that will test whether the framework’s mathematical guarantees translate into safer deployments in hospitals and clinics. If they do, the work could become a standard component of the AI development pipeline, a verification layer through which every concept-based clinical model must pass before it is trusted with patients. The research was supported by the Turing-Roche Strategic Partnership, the King’s College London AI+ Fellowship, and PharosAI.

For a field that has long promised transparency as the antidote to the black box, this study delivers an uncomfortable but necessary message: interpretability itself can be faked, sometimes without anyone intending it, and only mathematics can tell the difference. In making that difference measurable, the King’s College London researchers have given clinicians, regulators, and developers something they have lacked until now, a way to know whether the window they are looking through into an AI’s mind is truly open, or merely a mirror.

The study was led by corresponding author Bernard Le Foll, MD, PhD, of the Centre for Addiction and Mental Health in Toronto, Canada, a researcher whose career has focused on the pharmacology and treatment of substance use disorders. By selecting a randomized crossover design — widely regarded as the gold standard for examining the acute behavioral effects of psychoactive substances — the research team was able to compare each participant’s driving performance across different conditions while that individual served as his or her own control. This methodological choice matters enormously in a field cluttered with confounding variables: cannabis users differ widely in tolerance, metabolism, body composition, and baseline driving skill. By randomizing the order of exposure and washing out between sessions, the investigators could isolate the causal contribution of THC dose itself, rather than merely observing correlations between cannabis use and poor driving outcomes in observational data.

The pharmacological subtlety at the heart of the research concerns edibles specifically. Unlike inhaled cannabis, which delivers THC rapidly to the bloodstream through the lungs and produces peak effects within minutes, edible cannabis undergoes first-pass metabolism in the liver, where THC is converted to 11-hydroxy-THC, a metabolite that is itself powerfully psychoactive and may cross the blood-brain barrier even more readily than the parent compound. This transformation produces a delayed onset of intoxication — often thirty minutes to two hours — a longer duration of effect, and a notoriously unpredictable relationship between the dose consumed and the degree of impairment experienced. The delayed onset is also what makes edibles uniquely hazardous in the driving context: consumers who do not immediately feel “high” may ingest additional doses, misjudge their state, and then get behind the wheel while their impairment is still climbing.

What makes the new findings so legally significant is the disconnect they expose between blood THC concentration and functional impairment. Several jurisdictions — including a number of U.S. states that have legalized recreational cannabis, as well as Canada under its nationwide framework — have adopted per se laws that mirror drunk-driving statutes: if a driver’s blood contains a specified nanogram-per-milliliter quantity of THC, that driver is presumed impaired by law, regardless of observed behavior. These thresholds were adopted largely for administrative convenience, because blood THC levels offer an objective, enforceable number. But THC pharmacokinetics frustrate this logic. Unlike alcohol, whose blood concentration tracks intoxication reasonably well across individuals, THC is highly lipophilic, sequestering rapidly in fatty tissues and releasing slowly over days or weeks in regular users. A chronic cannabis consumer may register blood THC levels far above any legal per se threshold while being functionally unimpaired, while a naive user may exhibit profound cognitive and psychomotor deficits at concentrations that would be legally permissible.

Against this backdrop, the crossover trial’s central result acquires its sting: measurable, dose-dependent degradation of simulated driving performance occurred at blood THC concentrations below the very thresholds that define legal impairment. In practical terms, this means a driver could pass a roadside per se screening and still be operating a vehicle with compromised abilities — slower reaction times, degraded lane-keeping, impaired divided attention, and reduced capacity to respond to unexpected hazards. Simulated driving paradigms, the standard instrument of this research field, quantify such deficits with precision, capturing metrics such as standard deviation of lateral position, speed variability, response latency to unexpected events, and collision rates, all while eliminating the ethical impossibility of testing truly impaired drivers on public roads.

The implications ripple outward in several directions. For policymakers, the study suggests that per se THC thresholds may be simultaneously over-inclusive — penalizing tolerant regular users who are not impaired — and under-inclusive, missing impairment in less tolerant individuals whose blood levels have already fallen below the cutoff. The alternative approaches that toxicologists have long advocated, such as combining blood or oral fluid testing with standardized field sobriety testing, or developing functional impairment assessments, gain empirical support from these results. For public health communicators, the study supplies a clear and urgent message that legal limits do not function as a “safe to drive” certificate the way blood alcohol limits roughly do. For consumers, particularly the young adults who represent the demographic most likely to consume edibles and also most likely to be involved in motor vehicle crashes, the message is that the absence of a legally detectable blood concentration offers no protection against the pharmacological reality of impairment.

The publication also arrives amid a broader re-evaluation of cannabis and road safety driven by the rapid normalization of the drug. Legalization in Canada in 2018 and in a growing roster of American states has been accompanied by increases in self-reported cannabis-impaired driving and by stubborn uncertainty among law enforcement agencies about how to detect it. Unlike alcohol, for which the breathalyzer provides a cheap, instant, and legally robust measurement, cannabis detection requires blood or oral fluid sampling, laboratory analysis, and interpretation against thresholds whose scientific validity this study now directly challenges. The research community has warned for years that THC blood levels are a poor proxy for impairment; this trial elevates that warning from pharmacokinetic theory to controlled experimental evidence.

It is worth emphasizing what the crossover design contributes to the credibility of this conclusion. Placebo-controlled cannabis administration studies face substantial regulatory and ethical hurdles, and few research centers in the world possess the licenses, facilities, and expertise to conduct them. The fact that impairment tracked dose systematically — rather than appearing as a scattered or inconsistent pattern across participants — strengthens the inference that the relationship is causal and pharmacological rather than driven by expectancy effects. Participants in such trials are typically aware they may receive active drug, which can bias performance; the dose-dependency observed here suggests that expectancy alone cannot account for the results, since expectation would not scale neatly with the amount of THC actually administered and absorbed.

The study is accompanied by a commentary in JAMA Network Open, a signal that the journal and its editors view the findings as consequential enough to warrant explicit scholarly interpretation. Commentaries attached to clinical trials often serve to translate technical results into clinical and policy guidance, and in this case the pairing underscores the tension between the scientific evidence and the enforcement frameworks currently in place. Researchers, toxicologists, and legal scholars will now face intensified pressure to develop impairment-detection tools that reflect functional capacity rather than chemical concentration — a shift comparable to what would be required if blood alcohol levels were discovered to diverge substantially from actual driving deficit.

The work also carries lessons for the evolving edible marketplace. As legalized products migrate toward higher-potency edibles, gummies, beverages, and novel formulations, the dose-response relationship documented here becomes a matter of product labeling, dosing guidance, and consumer education. Many jurisdictions mandate standard serving sizes of a few milligrams of THC, yet studies of edible consumption repeatedly show that users frequently exceed recommended doses, misjudge onset, and combine edibles with alcohol. Each of these behaviors amplifies the risks that the trial quantifies, and none of them is captured by a blood threshold measured hours after consumption.

For the researchers at the Centre for Addiction and Mental Health and their collaborators, the trial represents a milestone in translating cannabinoid pharmacology into road-safety policy. The rigorous demonstration that impairment occurs below legal thresholds does not by itself rewrite any statute, but it hands regulators, prosecutors, and public health authorities a scientific mandate to reconsider how cannabis-impaired driving is defined, detected, and deterred. As edible cannabis continues its expansion into mainstream markets, the gap between what the law can measure and what the brain can no longer do has now been documented under the most controlled conditions science can provide — and closing that gap has become an urgent task for legislators and scientists alike.

The findings, published as “Dose-dependent effects of cannabis edibles on simulated driving performance” in JAMA Network Open, are available to the public through an access-token link provided by the journal, with an accompanying commentary offering further interpretation of the results and their policy significance.

News Publication Date: 31-Aug-2026

Web References: EurekAlert! news release, JAMA Network Media Center

The sessions will be led by Derek Haseltine, Director of the Hertz Fellowship, who will be joined by current Hertz Fellows. Rather than a conventional webinar format, the sessions have been structured around interaction: after a short introduction covering the fellowship and the mechanics of the application, participants will move into breakout rooms where Hertz Fellows will facilitate open question-and-answer discussions. This format reflects a deliberate philosophy at the foundation, one that treats the application process not as a bureaucratic hurdle but as the first stage of a mentoring relationship. Prospective applicants will have the rare opportunity to speak directly with people who have already navigated the process, earned the award, and gone on to careers at the highest levels of research, entrepreneurship, and public service.

The topics scheduled for the breakout rooms reveal a great deal about the fellowship’s character and the kinds of questions that most often occupy applicants’ minds. One group will focus on the timing of the application, addressing how the fellowship fits the circumstances of college seniors, students taking gap years, and first-year graduate students, each of whom occupies a different position along the pipeline toward doctoral study. Another session, offered on September 10 only, will address the particular challenges faced by dual-degree MD/PhD applicants, a population whose extended training timelines and dual commitments raise questions that standard fellowship guidance often leaves unanswered. Choosing a graduate school that aligns with one’s research interests, and choosing a research project once enrolled, will also be explored, reflecting the foundation’s view that the fellowship is not merely a funding instrument but a framework for shaping an entire scientific career.

Additional breakout rooms will tackle questions of personal sustainability and identity in research careers. A session on building resiliency in graduate school acknowledges what many experienced researchers know well: that doctoral training can be psychologically grueling, and that the ability to withstand setbacks, failed experiments, rejected papers, and long stretches of uncertainty is as essential to scientific success as intellectual brilliance. A dedicated session for those underrepresented in STEM reflects the foundation’s ongoing effort to widen the demographic and disciplinary reach of its fellowship community. Finally, a session on coordination with other fellowships will address the practical realities of stacking or sequencing awards, a topic of considerable importance given that many strong applicants will also be weighing offers from the National Science Foundation Graduate Research Fellowship program, the Department of Defense’s National Defense Science and Engineering Graduate fellowships, and institutional awards.

The Hertz Fellowship itself occupies a unique position in the American research landscape. Established more than sixty years ago by the Fannie and John Hertz Foundation, the fellowship is explicitly committed to advancing American scientific and technological leadership by identifying the nation’s most promising young technical minds and giving them the freedom to pursue their most ambitious ideas. Unlike many fellowships that specify fields or constrain research direction, the Hertz Fellowship is famously open-ended: recipients may pursue doctoral studies in any of the applied physical, biological, or engineering sciences at any institution in the United States. That freedom extends to the money itself. Hertz Fellowships provide five years of support, structured so that Fellows can follow their intellectual curiosity across disciplines and institutions without the financial pressures that often push early-career researchers toward safer, more incremental projects.

The selection process has become legendary for its rigor. Beyond the standard review of academic records, essays, and letters of recommendation, Hertz finalists are subjected to a multi-day examination process that includes extended technical interviews probing not just what candidates know, but how they think. Interviewers frequently pose open-ended problems drawn from real research frontiers, watching for creativity, logical precision, the ability to recover from dead ends, and a certain playful courage in the face of the unfamiliar. The foundation has long argued that this method identifies a quality that transcripts cannot measure: the capacity for genuine scientific invention. The argument is supported by outcomes. Over its history, the fellowship has produced a remarkable roster of alumni, including Nobel laureates, founders of transformative technology companies, leaders of national laboratories, and researchers whose work has shaped fields from artificial intelligence and materials science to molecular biology and quantum information.

For the foundation, the upcoming sessions are also part of a broader effort to demystify the application and, implicitly, to broaden the pool of people who consider applying. The Hertz Foundation’s announcement states plainly that it wants every prospective applicant to fully understand the fellowship and feel confident in assembling a strong application. That framing matters. Selective fellowships have historically drawn disproportionate numbers of applicants from a narrow set of elite institutions, and outreach efforts such as these are increasingly recognized as important tools for ensuring that extraordinary talent at less prominent universities has the information and encouragement needed to compete. The inclusion of breakout rooms specifically devoted to underrepresented students and to non-traditional pathways, such as gap years and first-year graduate applications, signals a deliberate push in this direction.

The timing of the sessions, roughly seven weeks before the October 30 application deadline, is strategic as well. Information gathered in September can meaningfully shape the essays, recommendation strategies, and school-selection decisions that applicants must finalize in the following weeks. Fellowship advisors at universities often note that one of the most common mistakes in applications like the Hertz is generic writing, essays that describe ambition in abstract terms without connecting it to concrete technical problems the applicant has grappled with. Hearing directly from Fellows about what distinguished successful applications, and about how the foundation evaluates intellectual independence rather than conventional prestige, can help candidates calibrate their materials in ways that generic advice cannot.

For students still deciding whether doctoral study in the applied sciences is right for them, the sessions also serve a wider informational purpose. Discussions about how to choose a research project, how to align graduate school choice with research interests, and how to balance fellowship obligations with other funding sources offer guidance that extends well beyond the Hertz application itself. In an era when the economics of doctoral education are under scrutiny, when many PhD students face stipend pressures and uncertain career pipelines, fellowships that grant genuine financial independence have acquired an outsized significance. The Hertz model, which frees recipients to take intellectual risks and discourages narrowly defined, grant-driven research at the earliest career stage, is often cited as a counterexample to prevailing trends in graduate training.

Prospective applicants who wish to attend either of the two sessions can register online through the Hertz Foundation’s fellowship website, with separate registration links for the September 10 evening session and the September 14 midday session. The foundation has also opened a direct channel for other questions about the fellowship or the events, directing inquiries to fellowshipinfo@hertzfoundation.org. With the application window now open through the end of October 2026, and with the fellowship’s reputation for identifying the architects of America’s future technical leadership intact after more than six decades, the two September sessions represent a low-cost, high-value opportunity for anyone contemplating one of the most consequential applications in American graduate education. For students whose ambitions reach beyond incremental science, an evening or midday spent in conversation with Hertz Fellows may well be the first step in a career measured in decades of impact.

Conjectures are the lifeblood of number theory. Statements such as the Riemann hypothesis or the prime number theorem began as observed regularities in numerical data before mathematicians set out to prove or disprove them. Historically, formulating such conjectures has depended on the pattern-recognition abilities of exceptional human minds, supported by increasing amounts of computer-assisted experimentation. The new work aims to formalize that experimentation stage. Rather than treating conjecture generation as a black-box or a purely heuristic exercise, the researchers define an explicit sequential pipeline that runs from data acquisition through representation learning to hypothesis construction, with statistical validation woven into every stage.

At the heart of the approach is the encoding of number-theoretic objects as graphs. Prime numbers, for example, are organized into what the authors call a prime number graph, in which individual primes become nodes and the relationships between them, defined by modular and structural properties, become edges. Similarly, the coefficients of the Ramanujan tau function, a central object in the theory of modular forms, are represented in a modular form graph. Once the arithmetic objects are cast in graphical form, graph neural networks can be applied in earnest. These networks operate by passing messages between nodes, allowing each node to build up a representation, or embedding, that captures not only its own properties but also the structure of its neighborhood. Crucially, the embedding stage is designed to preserve arithmetic and structural invariants, ensuring that the learned representations do not discard the very mathematical properties that matter.

The team trained and evaluated their models on curated datasets drawn from established mathematical repositories, including the L-functions and modular forms database (LMFDB) and the Online Encyclopedia of Integer Sequences (OEIS). In practice, the experiments covered 15,000 prime numbers and 2,000 Ramanujan tau coefficients. The neural architectures were implemented using PyTorch Geometric, the widely used graph deep learning library, with graph autoencoders employed to learn compact representations in an unsupervised manner and ReLU activations and Adam optimization standardizing the training procedure. The results were striking: the models achieved area-under-the-curve scores of 0.91 on the prime data and 0.88 on the tau coefficient data, indicating strong discriminative power, while clustering quality in the learned embedding spaces remained robust, with silhouette scores of at least 0.52.

Once stable embeddings are obtained, the pipeline turns to the extraction of latent statistical patterns. The researchers apply clustering algorithms to group similar objects in the high-dimensional representation space, and then deploy symbolic regression to search for closed-form mathematical relationships that describe the regularities the network has uncovered. Symbolic regression is particularly significant here because it produces human-readable formulas rather than opaque numerical predictions, allowing mathematicians to inspect, test, and potentially prove the relationships the machine has identified. In this sense, the system acts less like an oracle and more like a highly disciplined research assistant, surfacing candidate patterns that would otherwise require months of manual computation to notice.

The candidate conjectures that emerge are then subjected to rigorous evaluation. The experimental design partitions datasets into training, validation, and testing subsets, and multiple random seeds are used across runs to ensure that results are not artifacts of a particular initialization or data split. Statistical significance is assessed through controlled hypothesis testing, with false discovery rate correction applied to account for the multiple comparisons problem, a well-known pitfall in which testing many hypotheses simultaneously inflates the chance of spurious findings. Quantitative assessment relies on goodness-of-fit measures such as the coefficient of determination and out-of-sample validation, while qualitative assessment checks the extracted patterns for consistency with known theoretical results, including classical results such as the prime number theorem.

The outcomes are notable both for their quality and their diversity. According to the study, the pipeline’s GNN-based models were able to predict trends in prime gaps, the irregular intervals between consecutive primes, and properties of tau coefficients, and 75 percent of the generated conjectures were independently rated as either plausible or novel by the evaluation framework. Visual interpretability played a supporting role throughout: t-distributed stochastic neighbor embedding (t-SNE) plots, scatter plots, and silhouette diagrams were used to visualize how number-theoretic objects organize themselves in the learned embedding space, giving researchers an intuitive window into why the model groups certain primes or tau coefficients together. The authors are candid, however, that scalability remains a challenge; extending the framework to vastly larger datasets and more complex arithmetic structures will require further engineering and theoretical work.

What distinguishes this study from earlier experiments at the intersection of machine learning and mathematics, such as the well-known work on using neural networks to study knots and representation theory, is its emphasis on the full conjecture-generation pipeline rather than a single predictive task. The framework formally defines each transformation, from raw numerical data to embedding, from embedding to extracted pattern, and from pattern to formulated hypothesis, and it builds bias-reduction into the process by replacing intuition with statistically validated regularities. The authors argue that this reduces the subjective bias associated with purely human-driven conjecture formation while establishing a bridge between deep learning-based pattern discovery and formal mathematical reasoning.

The implications reach beyond number theory. A reproducible, interpretable, and scalable pipeline for conjecture generation could in principle be adapted to other fields of pure mathematics where large structured datasets exist, from combinatorics to representation theory. The work resonates with a broader movement in the mathematical community toward machine-assisted discovery, exemplified in recent years by collaborative projects that pair mathematicians with large language models and automated reasoning tools. By providing an open, documented pipeline built on established libraries and public databases, the Jilin team has contributed a template that other groups can replicate, extend, and stress-test. The study’s openness is reinforced by its publication under a Creative Commons license, allowing unrestricted non-commercial reuse with attribution.

For mathematicians, the message is not that machines will soon replace the profound creative leaps of human conjecture-making, but that the earliest, most laborious stage of discovery, combing through oceans of numerical data for hints of structure, can now be systematically automated. For computer scientists, the study demonstrates that graph neural networks, already proven on molecules and social graphs, can carry genuine mathematical weight when paired with careful statistical hygiene. And for the growing community of researchers exploring AI for mathematics, the result offers a concrete answer to a pressing question: how do you know a machine-generated conjecture is worth a mathematician’s time? The answer emerging from Jilin is that you test it the way you would test any scientific hypothesis, with controlled experiments, corrected statistics, and honest out-of-sample validation. If 75 percent of the output passes that bar today, the framework’s authors and their successors will be working to raise that figure as the architecture scales to the vast, unexplored territories of the mathematical universe.

For decades, the sector label has been the first thing an analyst reaches for when sizing up a company: technology versus energy, healthcare versus financials, utilities versus consumer discretionary. A new study of every one of the 500 firms in the S&P 500 suggests that habit captures far less than investors assume. When researchers in Spain trained artificial intelligence models to recognize a company’s sector using nothing but its accounting numbers, the best-performing model — an algorithm known as K-nearest neighbors — succeeded only 49.3 percent of the time: well above blind guessing, but wrong more often than right. And when the team let an unsupervised algorithm group the firms purely by financial similarity, ignoring official labels altogether, it uncovered nine distinct financial families, every one of which cut across sector boundaries. The findings, published in The Journal of Finance and Data Science, stop short of declaring sector classifications obsolete. But they make a striking case that the labels organizing the world’s most closely watched stock index describe only part of each company’s financial anatomy — and that data-driven peer groups can reveal the rest.

Comparing companies within a sector is one of the oldest rituals in finance. Analysts benchmark profit margins against industry rivals, screen stocks by sector to diversify portfolios, and build risk models on the premise that firms facing similar markets should resemble one another on the balance sheet. Entire index families and research disciplines are organized around that assumption. Yet it is, at bottom, an empirical claim — one that had rarely been tested at the scale of an entire flagship index. A research team at the Catholic University of Ávila in Spain, working within its Dekis Research Group and led by corresponding author Ricardo Reier Forradellas, decided to put the claim to a formal test. Their question was deceptively simple: if you strip away a company’s name, its industry narrative and its stock chart, and hand a machine only its financial ratios, how often does the machine land on the same sector label that humans assigned? The answer, according to the new paper, is not nearly often enough for sectors to be treated as complete financial descriptions.

The study began with a comprehensive snapshot of the U.S. large-cap market: the fiscal year 2022 financial statements of all 500 constituents of the S&P 500. From each statement, the researchers distilled a battery of accounting ratios spanning the dimensions that fundamental analysts track most closely — profitability, which measures how efficiently a firm converts sales and assets into earnings; leverage, which captures its reliance on borrowed money; liquidity, which gauges its ability to meet short-term obligations; efficiency, which reflects how productively it deploys its resources; and cash generation, which reveals whether reported earnings are backed by real cash flow. Ratios like these compress thousands of line items into comparable numbers, making them the raw material of fundamental analysis. The team’s first step was classical rather than computational: a statistical examination of how strongly those ratios actually differed from sector to sector. The verdict was nuanced rather than clean. The ratios did vary across sectors — the labels are not arbitrary — but those differences accounted for only part of the financial variation among the 500 firms.

Then came the artificial intelligence. The researchers trained seven different supervised machine-learning models on a single task: given a company’s accounting ratios, predict its sector. Supervised learning of this kind works by letting an algorithm study examples whose answers are known — here, firms carrying official sector labels — and internalize the patterns connecting inputs to outputs. Among the seven contenders, the strongest performer was K-nearest neighbors, a deceptively simple method that makes predictions by analogy. Rather than deriving an explicit formula, the algorithm stores the training companies as points in a multidimensional space of financial ratios and classifies each new company by finding its closest neighbors and adopting whatever label dominates among them. In effect, the model asks: which established sector residents does this firm most resemble on paper? Performance was measured on validation data withheld from training — a safeguard that prevents the algorithm from simply memorizing answers it has already seen — and K-nearest neighbors reached a validation accuracy of 49.3 percent, the highest figure any of the seven approaches achieved.

To judge whether 49.3 percent is impressive or damning, one must consult the study’s baseline. Because the S&P 500’s sectors are unevenly populated, a lazy classifier that always guessed the most common sector — the majority class — would have been correct 14.8 percent of the time. Viewed against that yardstick, the machine-learning result is more than three times better, confirming that accounting ratios do carry a genuine sector signal: utilities genuinely do look different from banks on a balance sheet. But the same number carries a more provocative message. Even the best model misidentified a company’s sector more often than it identified it correctly. In practical terms, most of the index’s members behave as financial hybrids, their ratio profiles confusable with those of firms from entirely different industries. If sector membership were a full description of financial structure, a well-trained classifier should approach near-perfect accuracy. Instead, the evidence indicates that a company’s industry tells you something real about its finances — but far from everything.

Faced with that ceiling, the team changed tactics. Instead of asking the data to reproduce the human-made labels, they asked it to ignore the labels entirely. “Hence, we used unsupervised learning to group firms by financial similarity rather than by their existing labels,” explains corresponding author Ricardo Reier Forradellas of the Catholic University of Ávila. “This produced nine economically interpretable clusters.” Unsupervised learning is the branch of machine learning that finds structure without a teacher: the algorithm receives no answers, only measurements, and must discover on its own which companies naturally bunch together in ratio space. The result was not a mirror of the official taxonomy but an alternative map of the U.S. corporate economy, drawn exclusively in the currency of profitability, leverage, liquidity, efficiency and cash flow. Crucially, the clusters were not statistical noise. Each of the nine could be described in plain financial language, giving the researchers confidence that the algorithm had surfaced economically meaningful structure rather than accidental groupings.

The clearest evidence for that meaningfulness lay in how tightly knit the new groups were. Every one of the nine clusters contained companies drawn from more than one official sector, confirming that financial similarity respects no industry border. Yet the clusters were generally more internally coherent than the sectors themselves: across most of the accounting ratios, firms inside a data-driven group showed lower internal dispersion — a smaller statistical spread around the group’s typical value — than firms sharing a sector label. In other words, a company’s closest financial peers were more likely to be found inside its algorithmic cluster than inside its sector. Some familiar signatures did survive the analysis. Utilities, real estate companies and financial firms proved more readily identifiable than several other sectors, a reflection of business models that imprint themselves unmistakably on the accounts: capital-intensive networks, property-heavy balance sheets and debt-fueled intermediation leave deep accounting fingerprints. Other kinds of firms, by contrast, proved harder to pin down, slipping quietly across sector lines on the machine’s map.

Forradellas is careful to frame the result as an addition to financial practice rather than a demolition of it. “Our findings do not mean that sector classifications are obsolete,” he says. “They show that sectors tell only part of the story. When the aim is to compare companies by financial structure, accounting-based peer groups can provide a useful additional perspective.” The distinction matters for anyone who relies on comparisons professionally. For questions about regulation, industry competition or supply chains, sector membership remains the natural organizing principle. But for questions about valuation, credit risk or benchmarking financial performance — questions that turn on how a company actually funds itself, generates cash and manages its short-term obligations — the study suggests that peers defined by accounting similarity may be the more honest reference group. The two lenses answer different questions, and neither one alone captures the whole financial creature.

The research also carries a warning for anyone tempted to enshrine the new clusters as a permanent replacement taxonomy. When the team compared cluster assignments across later annual reporting periods, they found only moderate persistence: companies did not stay put in their financial families from one reporting period to the next. “This indicates that these peer groups should be updated rather than treated as fixed categories,” Forradellas adds. “Our approach complements sector taxonomies for benchmarking, peer comparison, and financial analysis.” That drift is not a flaw in the method so much as a feature of corporate life. Firms alter their capital structures, pivot their strategies, acquire rivals and ride macroeconomic cycles, and their ratio profiles shift accordingly. A company can migrate from a cash-rich cluster to a heavily leveraged one without ever changing its ticker symbol or its industry. Any financial map built from accounting data, the authors imply, must be redrawn periodically — a living taxonomy rather than a carved-in-stone one.

The findings arrive as machine learning steadily permeates quantitative finance, and they offer a template for how data-driven classification might sit alongside traditional taxonomies rather than clash with them. For index providers, the results hint at complementary ways to construct peer sets for benchmarking; for analysts and portfolio managers, they suggest that screening by algorithmic financial similarity could surface valuation signals and risks that sector screens miss. The study is also a sober reminder of the limits of AI: even the best of seven supervised models fell well short of the accuracy that would make sectors predictable from accounts alone, and unsupervised groupings still demand expert interpretation before they become economically meaningful. Published open access by KeAi, a publishing venture of Elsevier and China Science Publishing & Media Ltd, the paper — “Characterization of S&P 500 companies by sector using artificial intelligence: Statistical evidence and machine learning application” — reexamines a tool so familiar that few thought to question it. The sector, the study concludes in effect, is where a company works. The balance sheet is who it is. Investors reading only the first are seeing half the picture.

The timing matters. Mathematics curricula worldwide have pushed algebra into ever-earlier grades, on the strength of evidence that children who learn to notice and express generalizations of mathematical structure early make a smoother transition to formal algebra later. In Spain, where the study took place, the national primary curriculum explicitly introduces algebraic thinking from first grade, asking children to “recognise and describe regularities and patterns in numerical, geometric, and functional contexts.” Yet the empirical picture of autism and mathematics is far more complicated than the savant myth. Approximately 20 percent of students with autism spectrum disorder (ASD) show signs of a mathematics learning disability — nearly triple the 7 percent rate in the general population — while only about 4 percent display mathematical giftedness. Previous research has tied math outcomes in autistic learners to working memory, processing speed, language, and visuospatial skill, but almost exclusively through the lenses of arithmetic and word-problem solving. How any of these abilities support algebraic thinking, the recognized gateway to higher mathematics, had never been directly examined.

To close that gap, a team spanning the University of Cantabria in Spain and the University of Cyprus recruited 26 children with ASD and 26 without, all aged 6 to 12 and all with Full-Scale IQs of at least 70 on the Wechsler Intelligence Scale for Children–Fifth Edition (WISC-V). Every autistic participant was matched to a non-autistic peer of the same sex, age, school, grade, and even classroom, and each diagnosis was confirmed against DSM-5 criteria by a psychiatrist on the research team. Children with comorbid ADHD, dyslexia, or language disorders were excluded, sharpening the focus on autism-specific cognition. Recruitment ran from July 2019 to February 2021 through child psychiatry and pediatric clinics, family associations, and school counseling services in a Spanish region, with approval from the Cantabria Research Ethics Committee, and children needed a minimum raw score of 26 on the Test of Early Mathematics Ability (TEMA-3) — roughly a mathematical age of five and a half years — to guarantee a baseline of arithmetical knowledge. Across two to three videotaped sessions per child, psychologists measured six cognitive domains — working memory, processing speed, spatial reasoning, verbal reasoning, fluid reasoning, and theory of mind, the latter using the Theory of Mind subtest of the NEPSY-II neuropsychological battery — while mathematics educators assessed basic arithmetic with the TEMA-3, which yields a “mathematical age” expressing arithmetical skill relative to age-based norms.

The centerpiece was a figural pattern generalization task adapted from classic early-algebra research and built on the linear function f(x) = 2x + 2. Children saw square tables joined in a row with people seated around them and faced seven questions: how many people could sit around 3, 4, 5, 8, 18, and even 100 tables, and — the decisive generalization step — how to determine the head-count from any number of tables. Far-term questions like the 100-table case demand functional thinking: a shift from counting isolated cases to grasping the invariant relationship between two variables, the conceptual seed of a symbolic rule such as y = 2x + 2. Researchers describe learners as progressing through levels of generalization, from factual observations such as “add 4 each time,” to contextual verbal descriptions of structure, to fully symbolic rules — and children can arrive at the same correct answer through different strategies and representations. Each correct answer here earned one point, to a maximum of seven, and the test showed high internal consistency (Cronbach’s alpha = 0.853). Interviewers supplied reading help when needed and pressed children to explain the reasoning behind every response.

The first surprise came from straight group comparisons. As predicted, the non-autistic children significantly outperformed their autistic peers in mathematical age (F = 8.34, p = .006), working memory (F = 10.20, p = .002), processing speed (F = 22.26, p < .001), verbal reasoning (F = 11.65, p = .001), and theory of mind (F = 19.26, p < .001) — mirroring the executive-function and language differences long documented in autism. But on three measures the two groups were statistically indistinguishable: spatial reasoning (p = .289), fluid reasoning (p = .582), and, critically, the early algebra test itself (p = .246). Despite measurable deficits in domains that conventional models treat as engines of mathematical abstraction, the autistic children generalized visual patterns just as well.

To uncover what was driving that performance, the researchers ran multiple linear regressions separately for each group, with the algebra score as the outcome and the seven cognitive and arithmetic measures as predictors. Because score variability ballooned in some age bands relative to others — Levene’s test for homogeneity of variances came back at p < .001 — the team used weighted least squares (WLS) regression, giving heavier statistical weight to age groups with consistent scores and lighter weight to the erratic ones, and verified that multicollinearity was negligible, with all variance inflation factors below 5. For the non-autistic children, the model explained 41.3 percent of the variance in algebra performance (R² = 0.413, a large effect, f² = 0.70) and singled out two significant predictors: fluid reasoning (standardized β = 0.508, p = .005) and mathematical age (β = 0.397, p = .024). For the autistic children the model was markedly stronger, accounting for 63.4 percent of the variance (R² = 0.634, a very large effect, f² = 1.73), and the predictors flipped entirely: theory of mind dominated (β = 0.681, p < .001), followed by spatial reasoning (β = 0.333, p = .021). Fluid reasoning and arithmetic maturity — the twin engines of early algebra in typical development — lost all predictive power in the autistic group.

The prominence of theory of mind is the study’s most provocative result. Although ToM is conventionally filed under social cognition, the authors argue it may also underwrite domain-general inference: interpreting functional relations, mapping relationships between different representational spaces, and adopting the kind of perspective that lets a child see a structure from outside a single example. On this reading, structural mathematical problems quietly recruit some of the same machinery that social reasoning uses, and that machinery becomes uniquely salient for autistic learners when the customary supports — arithmetic fluency and fluid reasoning — are not carrying the load. The parallel reliance on spatial reasoning fits a long line of findings documenting visuospatial strengths in autism, from superior visual search to block-design performance, and with earlier observations that autistic students lean heavily on drawings and visual elements during algebra tasks. Visual patterning, in other words, may open a gateway into algebraic understanding that plays directly to autistic children’s strengths.

Equally telling is what failed to predict anything. Working memory and processing speed — bedrock predictors of arithmetic in both populations, and areas where the autistic children lagged their peers — contributed no additional explanatory power in either group once the other abilities were in the model. The null result dovetails with prior work in typically developing children showing that algebraic thinking leans more on reasoning-based processes than on speed of processing, and it suggests that task format, not just diagnosis, determines which cognitive abilities matter. A task centered on identifying structural rules, rather than intensive calculation or symbolic manipulation, may simply not tax the capacities that dominate early arithmetic.

The educational stakes are considerable. If autistic children build algebraic ideas through visual-spatial structure and inference rather than through arithmetic drill, then instruction built exclusively on symbol manipulation and calculation fluency may systematically miss them. The authors argue for visual supports, patterning tasks, and opportunities for inference-based reasoning as more natural entry points, and for treating cognitive profiles as design constraints rather than obstacles in inclusive classrooms. They are also candid about the study’s limits: 26 children per group and an age span from 6 to 12 leave the regression models statistically fragile; a single tables-and-chairs task may privilege certain cognitive processes over others; and the TEMA-3 produced ceiling effects among older participants. Larger samples, a wider battery of algebraic tasks, and intervention studies that deliberately target each group’s cognitive strengths come next. But the headline finding is already hard to ignore: two children can produce the same correct rule for a hundred tables using minds that arrived there by entirely different roads — and teaching to the route each learner actually takes may be the future of inclusive mathematics education.

The (N)-body problem asks how a collection of mutually attracting objects moves under Newton’s law of gravity. For each body, the acceleration depends on the positions and masses of every other body, producing a system of coupled differential equations that becomes extraordinarily difficult to solve as (N) increases. Even when the governing equations are simple, the resulting trajectories can be chaotic, singular or impossible to express in elementary mathematical form. Special solutions—carefully organized motions in which the bodies preserve a recognizable geometric pattern—therefore provide rare windows into the structure of the equations.

A homothetic solution is one of the most rigid of these special motions. The bodies retain the same relative arrangement while the entire configuration expands or contracts by a common scale factor. Imagine a three-dimensional constellation whose points move directly away from, or toward, its center while preserving every angle and ratio of distances. The shape does not rotate or distort; only its overall size changes. In a gravitational setting, such behavior requires the geometry and mass distribution to balance in a highly constrained way, linking the spatial arrangement of bodies to the forces acting on each one.

The key geometric object behind these motions is a central configuration. In such a configuration, the gravitational acceleration of every body points toward—or away from—the common center of mass and is proportional to that body’s displacement from the center. In schematic form, the acceleration of body (i) must satisfy an equation of the type (mathbf{a}_i=-lambdamathbf{r}_i), where (mathbf{r}_i) is its position relative to the center of mass and (lambda) is the same proportionality factor for the entire system. This shared factor allows the spatial pattern to evolve through a single scale variable rather than through independent motion in every coordinate.

Shibayama’s result becomes “really perverse” because the same geometric configuration can satisfy the central-configuration condition for two distinct mass distributions. The distributions are not merely relabelings of identical bodies, nor are they equivalent descriptions of the same physical assignment. They differ while preserving two global quantities: the total mass and the location of the center of mass. Despite that difference, both produce the force pattern required for homothetic motion. The result reveals a surprising non-uniqueness in the inverse direction of the gravitational problem: knowing the shape and certain global properties does not necessarily determine how mass must be assigned to its points.

This is the reverse of a more familiar question. In the ordinary (N)-body problem, researchers specify masses and positions, then calculate the forces and future motion. The inverse problem asks whether a desired arrangement or motion can reveal the masses that produced it. For central configurations, intuition might suggest that the geometry should strongly constrain the mass distribution, perhaps determining it uniquely once the total mass and center of mass are fixed. Perverse solutions demonstrate that this expectation can fail. Two physically distinct allocations of mass can occupy the same framework and still generate compatible accelerations.

The new work moves this phenomenon from two dimensions into genuine three-dimensional space. Earlier examples of really perverse solutions were known in the planar (N)-body problem, but only for large values of (N). Shibayama proves existence in the spatial problem for every integer (N) from 27 through 55, according to the published result. These are not simply flat configurations viewed from an angle: the study concerns the spatial Newtonian problem, where bodies can occupy three-dimensional arrangements and the force-balance conditions include the additional freedom—and additional complexity—of motion out of the plane.

To establish explicit examples, the study uses interval arithmetic, a computer-assisted method designed to control numerical uncertainty. Ordinary floating-point calculations produce approximate values and can conceal small errors, especially when a solution depends on delicate cancellations among many gravitational forces. Interval arithmetic instead represents each quantity by a guaranteed range containing its true value, allowing calculations to propagate rigorous bounds through the equations. If the resulting intervals satisfy the required inequalities or contain a certified solution, researchers can establish existence without relying solely on a visually convincing numerical approximation. The article reports a construction rather than a data-generating experiment; no datasets were generated or analyzed.

The importance of the finding is not that it predicts a newly observed star cluster or a practical orbital arrangement for spacecraft. Real systems with dozens of bodies would be disturbed by imperfections, external gravitational fields and dynamical instabilities, and the article does not claim that these mathematical solutions are naturally realized in the cosmos. Their value is structural. They expose unexpected flexibility in Newtonian gravity and offer new test cases for theories of central configurations, bifurcations and the geometry of many-body dynamics. They may also help mathematicians understand how spatial solutions emerge from planar ones, a question connected to earlier work on the bifurcation of central configurations.

The word “perverse” in this context is technical rather than moral: it describes a solution that violates an anticipated uniqueness pattern in the equations. By constructing such solutions explicitly, Shibayama shows that the three-dimensional (N)-body problem contains hidden alternatives even under apparently restrictive conditions. The result broadens the known range of exceptional gravitational motions and suggests that the boundary between orderly geometry and many-body complexity is more intricate than conventional intuition implies. In a field famous for unstable trajectories and impossible general solutions, these carefully balanced exceptions provide a mathematical form of viral-worthy cosmic weirdness: different masses, the same center, and the same expanding or contracting shape.