The manuscripts

This page presents the first science manuscripts produced or inspired by BootLoops. These projects were all done during the period July - September 2026. All of these manuscripts have been read by humans for accuracy. Many were drafted first by LLMs and some have not been fully checked yet by humans; those are watermarked as preliminary. For some projects the draft is not ready yet; those are marked as [in preparation]. As manuscripts get finalized and published, published versions or links to them will appear here.

— Matthew Schwartz (Sep 2026)

The BootLoops overview

  1. BootLoops: an LLM-driven living harness for precision science
    Matthew D. SchwartzProposes an organizational framework for AI in science: formally cleave the harness from the LLM — the harness grows into a substantial, testable, validated toolkit through human and AI contributions, while the LLM wields it to great effect. BootLoops began by porting code from the S-matrix bootstrap program into a common framework; Claude improved the ported tools and wrote new routines from papers with ideas but no public implementation, and each new tool or upgrade suggested new problems to try which needed new tools, feeding the exponential. The science expanded from particle physics to cosmology, earth science, evolutionary biology, genomics, ecology, economics, linguistics and public health — the methodology summarized here, with more than thirty detailed application papers in supplementary material.

Ecology and evolutionary biology

  1. Neutral biodiversity theory solved and tested against three decades of Barro Colorado censuses  ·  [in preparation]  ·  web summary  ·  video
    James P. O’Dwyer, Matthew D. SchwartzNeutral theory gives every tree in a forest the same chances of dying, reproducing and immigrating, and ecologists keep it as a null model whose failures show what a better model needs. This paper solves the spatially implicit theory exactly, reduces the spatially explicit theory to one exact equation, and compares both with the 1982–2015 censuses of the 50-hectare Barro Colorado plot in Panama, stating the theory’s known discrepancies precisely enough to say what a successor model must remedy. In preparation; not yet on the site.
  2. A minimal model of a forest after neutral theory  ·  web summary  ·  video
    James P. O’Dwyer, Matthew D. SchwartzNeutral theory predicts the overall shape of rarity and abundance in a forest but not which species gain or lose over time, and it underestimates the pace of change. This paper joins the two remedies proposed for those failings, environmental variation and species’ measured life histories (how fast they grow, survive and reproduce), with immigration into one minimal model, and finds that it predicts the dynamics and abundances of the species on the Barro Colorado plot in Panama: environmental variation enters as a single common effect on growth rates shared across all species, and regional commonness and rarity together with a simple competition–colonization trade-off predict relative abundances. What the censuses require is much simpler than the model’s ingredients would allow; whether a comparably simple model works in other forests is open.
  3. Predicting the life history of forests  ·  [in preparation]
    James P. O’Dwyer, Matthew D. SchwartzThe rates at which trees die, grow and reproduce govern everything a forest will become, yet they are measured well at only a handful of intensively studied sites. This paper fits one hierarchical model of survival, growth and recruitment jointly to remeasured forest inventories on four continents and tests it by prediction, scoring each new region before any of its data enters the fit. In the current draft, mortality totals fell inside their pre-stated bands in all forty-nine regions tested and predicted decade-scale changes in species composition correlate 0.70 to 0.95 with what the forests then did; the numbers may move before release.
  4. Exact Bayesian evidence for phylogenetic trees  ·  web summary
    Scott V. Edwards, Paul O. Lewis, Matthew D. SchwartzBayesian phylogenetics compares trees and models of DNA change through marginal likelihoods, the probability of the data under each hypothesis averaged over its parameters, normally estimated with uncontrolled error. This paper shows that for four- and five-species trees under any time-reversible substitution model the marginal likelihood is an exact ratio of two whole numbers, computable with methods borrowed from Feynman integrals. Among the results: two of the three resolutions of the hominid transfer-RNA quartet are exactly tied, and the third, nominally preferred though no site supports it, ranks last once a fifth species is added; the 1997 separation of Neandertal from modern human sequences is recovered as an exact Bayes factor; one mitochondrial locus does not by itself confirm the published placement of the extinct San Cristóbal tortoise; and in a rebuilt malaria-mosquito genome scan thousands of windows turn out to be exact ties, which ordinary arithmetic breaks only by rounding error.
  5. JaCK & Jill: exact marginal likelihoods for small phylogenetic trees  ·  web summary
    Scott V. Edwards, Paul O. Lewis, Matthew D. SchwartzMarginal likelihoods, the numbers used to choose between substitution models and between small trees, are normally estimated by sampling, with no bound on the error. JaCK & Jill (the Python package phyloexact) computes them for four species with guarantees. JaCK, the exact Jukes–Cantor and Kimura computations, returns the marginal likelihood as a ratio of two integers, for alignments of transfer-RNA length (shorter under Kimura’s two-parameter model), and proves ties between trees from symmetries of the data. Jill, the interval and sampling routines, gives proven intervals under HKY85 and GTR+Γ and, under all four models, an estimate with measured error that, at equal run time, is twenty to two hundred times more accurate than MrBayes, RevBayes and LoRaD, reaching a given accuracy in seconds where those samplers need minutes to hours. Further routines choose a model across loci without fixing a tree, scan genomes window by window, accept user-written models, and certify every number.

Genetics

  1. An exact solution for the site-frequency spectrum of a selected allele  ·  web summary  ·  video
    Michael M. Desai, Matthew D. SchwartzNatural selection removes the most damaging mutations fastest, so they are the rarest variants, and the site-frequency spectrum — how many variants appear on one, two or more sampled chromosomes — carries that signature. Its theory, the Sawyer–Hartl Poisson random field, underlies three decades of fitness-effect inference but could not be evaluated exactly at strong selection and large samples, where floating-point arithmetic fails. Using exact rational arithmetic, this paper evaluates the spectrum exactly at any selection strength for classical sample sizes, and with controlled error at biobank scale and for arbitrary dominance. A fit to roughly 730,000 human exomes in gnomAD places most gene-disabling mutations in the strongest-selection range the data can resolve. Checked against exact values, the standard packages polyDFE, dadi and fastDFE are mostly reliable at their intended sample sizes, and the paper maps where each loses accuracy.
  2. Signatures of gene conversion in the human two-site frequency spectrum  ·  web summary  ·  video
    Michael M. Desai, Matthew D. SchwartzMost methods that infer a population’s history from genomes assume the standard model of shared ancestry, the Kingman coalescent; recent work used the joint variant frequencies at pairs of nearby sites to reject it in fruit flies. This paper measures that two-site spectrum in humans from 5.7 billion pairs of nearby sites in 113 Gambian genomes and finds closely linked pairs inconsistent with the Kingman coalescent under any population history. Neither multiple-merger genealogies nor, within their measured rates, recombination-map and sequencing errors, population structure or linked selection explain the departure. A Kingman model with gene conversion (short stretches copied from one chromosome copy onto its partner) explains it best, though some discrepancies remain — the reverse of the fly result, where multiple mergers are favored. The fitted human gene-conversion rates are of the same order as pedigree and sperm-typing estimates, though lower for short tracts.
  3. Controlled inference of transcriptional bursting from single-cell RNA counts  ·  web summary  ·  video
    Martin Hemberg, Matthew D. SchwartzGenes make RNA in bursts: silent most of the time, briefly on. Fitting burst rates and sizes to single-cell RNA counts multiplies hundreds of delicate functions, and standard tools lose control of the errors; ball arithmetic — a tool from computer-assisted proofs — keeps the error bars honest. Applied to the large allelic dataset of Larsson et al., allelic differences in bursting turn out to be common, in roughly half of the genes tested, and the differences that can be resolved are in burst frequency. No gene in the dataset was measured in enough cells to reveal whether its promoter needs recovery time between bursts.
  4. Two kinds of time in mammalian promoters: a refractory class that detects change faster per burst cycle  ·  web summary  ·  video
    Martin Hemberg, Matthew D. SchwartzGenes in mammalian cells are transcribed in bursts separated by silences, and the standard model treats the promoter as a two-state switch with no memory: the chance of turning back on does not depend on how long the gene has been off. This paper fits exact likelihoods for that model and a multi-step alternative to time-resolved single-cell RNA counts from mouse fibroblasts. A subset of genes, enriched five-fold for secreted and matrix proteins, builds its silences from several sequential steps. These refractory promoters are poor in CpG sites, share no distinctive transcription-factor motif, and are methylated yet carry reduced levels of two active chromatin marks and more of a repressive one than memoryless promoters. A model suggests the benefit: a multi-step promoter can detect a change in its input several-fold sooner, counted in its own burst cycles, than a memoryless switch.

Earth and planetary science

  1. The Great Oxidation from a minimal ocean--atmosphere model  ·  web summary  ·  video
    David T. Johnston, Matthew D. SchwartzFor two billion years Earth’s air carried almost no oxygen; the textbook story is a stepwise rise beginning with the Great Oxidation 2.4 billion years ago. The paper builds a minimal ocean–atmosphere model of the transit with every input fixed by published measurements, the present-day steady state, the dated onset of the transition and the glacial record. Its Great Oxidation is not a step but an oscillation entangled with four glaciations from 2.45 billion years ago, each returning the air to anoxia if oxygen made beneath the ice is consumed there; oxygen becomes permanent only at the last deglaciation, 2.22 billion years ago. The model’s third glaciation ends about 20 million years later than dated ash beds allow, and CO2 values computed from two paleosols inside the transit are consistent with it.
  2. Demography of sunspots: aging, and death by turbulent erosion?  ·  web summary  ·  video
    Cecilia Garraffo, Matthew D. SchwartzSunspot groups emerge, grow and dissolve over days to weeks, but the Sun’s rotation carries each out of view within about nine days, so lifetimes are hard to measure and proposed decay laws disagree fourfold. This paper applies survival analysis, the statistics of partly observed lifespans, to the daily registry of sunspot groups. Groups seem to age: at fixed size and hardiness the risk of dying rises about twenty percent per day in the baseline model, and aging speeds up near each solar-cycle maximum, fading over about two years. Every published decay law fails against the data. Of some seventy candidate mechanisms tested, none is established by the data; the one not eliminated is the hypothesis that turbulence erodes a group’s subsurface root to a breaking point, which gives the rising risk at a rate solar convection can supply.
  3. Starspot regions die without aging from G dwarfs to the fully convective boundary  ·  web summary  ·  video
    Cecilia Garraffo, Matthew D. SchwartzA companion paper finds that sunspot groups age. On other stars spots show up only as recurring dips in brightness as the star rotates. This paper follows 10,051 spot features from birth to death on 509 heavily spotted Kepler and TESS dwarf stars and finds that they die without aging: within a star, at fixed depth at birth, an old feature is no likelier to die than a young one, and on G and K dwarfs a rise in log death rate of 0.05 per rotation would have been detected. Median lifetimes are four to seven rotations (tracking cannot follow features much longer) and scale with rotation period as P0.828. No aging is seen on early-M dwarfs either. A model of active sites where spot groups keep emerging can reconcile this with the Sun, though no version tested matches every observation.

Collider physics

  1. Automated computation of Feynman integrals with BootLoops
    Matthew D. SchwartzHow the BootLoops harness works. A large language model plays physicist, mathematician, and programmer at once: it classifies each integral, picks tools from a toolkit ported into one uniform open framework, and runs the bootstrap — reduce to master integrals, fix the space of possible answers from the problem's geometry, pin the last unknowns with high-precision fits — accepting a result only when it reproduces independent values its derivation never used, to thirty digits or more.
  2. Bootstrapping elliptic and Calabi--Yau Feynman integrals
    Matthew D. SchwartzThe frontier of collider integrals lives on elliptic curves and Calabi–Yau manifolds, where the familiar function dictionary runs out. This paper shows the bootstrap carries over — the space of possible answers now comes from the geometry of the integral itself — and builds the missing toolkit, importing mathematics that had not yet arrived in physics. A worked example, the generic-mass ice-cream cone, shows the machinery end to end.
  3. Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap
    Matthew D. SchwartzMulti-loop Feynman integrals are a bottleneck of precision collider physics: many are polylogarithmic, but those with internal masses and more scales can involve elliptic curves and Calabi–Yau manifolds. This paper presents thirty integrals from all three classes, computed with a hybrid bootstrap by a large language model operating the BootLoops toolkit, each with a closed form or a differential equation with boundary data and a standalone evaluation code. Fifteen are known, recomputed as a test; fifteen are, to the author’s knowledge, new, including the nonplanar branch of two-loop W-pair production with exact top-quark mass on one unphysical kinematic line, the simplest two-mass nonplanar hexa-box for diboson-plus-jet production along two kinematic paths, two elliptic masters of $gg \to Z\gamma$, three-loop light-by-light integrals, among them a crossed box in which K3 periods appear, and threshold coefficients of the three- and four-loop banana periods when one mass equals the sum of the others.
  4. The hidden sunrise in the energy-energy correlator  ·  web summary
    Matthew D. Schwartz, Xiaoyuan ZhangThe energy–energy correlator (EEC) records how the energy from a particle collision is shared between pairs of directions, and is one of the few collider observables computable analytically to high order. In the much-studied model theory N = 4 super Yang–Mills, at next-to-next-to-leading order, one piece of the published answer had been left as an unevaluated two-fold integral. This paper completes the result. The missing piece is elliptic and lives on exactly the curve of the sunrise Feynman integral, the modular curve Γ1(6): the j-invariants of the two curves agree identically after a Möbius change of variable. Written in iterated Eisenstein integrals, the completed answer evaluates to high precision within seconds, and its closed form permits a first study of a Landau bootstrap for the EEC. Understanding its function space in N = 4 super Yang–Mills gives a concrete handle on the elliptic part of the EEC in QCD.
  5. New mathematics and new physics in the four-point energy correlator  ·  web summary
    Matthew D. Schwartz, Xiaoyuan ZhangEnergy correlators measure how energy lands across a particle detector, and the four-point correlator is where new mathematics enters: alongside two elliptic curves, a genus-2 — two-holed — surface appears, in real QCD and not just the supersymmetric toy theory. The paper evaluates the correlator at generic angles and compares it with jets from CMS Open Data and DELPHI events, with good agreement.

Mathematical physics and string theory

  1. The anisotropic Watson integral  ·  [in preparation]  ·  web summary
    Noam Elkies, Thomas W. Grimm, Matthew D. SchwartzA random walker on a line or a plane always comes home; in three dimensions it can escape, and Watson computed the return probability for the symmetric cubic lattice in 1939. With a different hopping rate along each axis the return integral had resisted every method. This paper identifies it with a period of a K3 surface, shows that its moments form a Lauricella series, and proves that for general rates its fifth-order differential equation is irreducible with monodromy group SO(5), which no product of elliptic integrals can have; on the walls where two rates coincide an extra algebraic cycle appears and the classical product formulas of Watson, Joyce and Delves–Joyce are recovered. A pair of genus-two curves, explicit in the rates, expresses the integral as a two-by-two determinant of hyperelliptic integrals, and at one point with complex multiplication its value in Gamma functions is proved with its absolute normalization.
  2. Factorization without clustering and the six-point Grassmannian string integral  ·  web summary
    Matthew D. SchwartzGrassmannian string integrals extend the Veneziano amplitude, an integral over points on a line, to other configurations. The first case not known to be a string theory is K(3,6), an amplitude-like integral over six points in the projective plane. This paper shows that its residue at every pole is a finite sum of string integrals at degenerate configurations, so it factorizes onto its boundaries as a string amplitude does. What it lacks is the further split of each residue into two smaller amplitudes summed over states, which in field theory comes from cluster decomposition and cannot be formulated for its kinematics. The double copy, the relation that builds closed-string amplitudes from pairs of open-string ones, holds for K(3,6) though no worldsheet is known; and the low-energy coefficients are proved to be multiple zeta values to all orders, only ordinary ζ(n) appearing through weight eight.
  3. Exact methods for string flux vacua  ·  web summary
    Matthew D. SchwartzThe string landscape holds infinitely many candidate flux vacua, usually explored one by one, though the deepest questions concern them all at once. This paper puts two tools to work on such questions: the genus theory of quadratic forms, which enumerates flux lattices and proves the list complete, and interval arithmetic, which proves critical points exist. On K3×K3 it proves, for every flux whose support has rank at most three, the bound indicated by the searches of Bena, Blåbäck, Graña and Lüst, that full stabilization costs at least 25 of the 24 available flux units, and at higher rank constructs a fully stabilizing flux inside the budget. In the symmetric sector of a Hulek–Verrill fourfold it proves the first vacuum appears at charge 4 over a stated range of fluxes and moduli. For the benchmark KKLT vacuum it proves the critical point of the superpotential exists and computes its value with a proven error bound.
  4. Irreducible multiple zeta values of depth three in four-edge modular graph functions  ·  web summary
    Matthew D. SchwartzString theory’s concrete predictions are corrections to Einstein’s equations whose one-loop coefficients are integrals of modular graph functions over the shape of the torus a string sweeps out. For a thin torus these functions reduce to Laurent polynomials whose coefficients are conjectured to be single-valued multiple zeta values that map into one another under the cosmic Galois group; the melon and three-edge families computed earlier met only odd zeta values, for which both properties hold automatically. This paper computes in closed form the Laurent polynomials of all forty-nine two-vertex four-edge functions at weights seven through twelve. Multiple zeta values of depth three occur, and the paper proves they can appear in only one Laurent coefficient, fixed by the weight; they span a space of the full dimension the Broadhurst–Kreimer conjecture allows, the Galois generators can send them only to products of two odd zeta values, every image lies in the span of the family’s own lower-weight coefficients, and through weight twenty-one every coefficient is single-valued.

Economics and statistics

  1. An LLM Workflow That Reproduces, Improves, and Extends Published Economics Research  ·  NBER  ·  web summary
    Isaiah Andrews, Matthew Schwartz, Jesse M. Shapiro — NBER Working Paper 35782 (September 2026)Economics journals have long required replication packages, the data and code behind each article. This NBER working paper introduces an open-source workflow that lets an LLM take a replication package and do three things: try to reproduce every calculation not declared to need unavailable data, checking each against the printed value and running an automated sensitivity analysis; reimplement or reformulate calculations for speed and accuracy; and propose extensions within the article’s own goals and assumptions. Over 4,452 packages from five journals it flags a discrepancy in 3,460 articles or appendices (about a third of them at the level of rounding in the last printed digit), speeds a calculation up more than tenfold in 496 articles, and develops an extension in 923. The authors stress that the workflow does not aim to evaluate articles and that work is ongoing.
  2. Exact Bayesian evidence for mixture models: from two components to infinity  ·  web summary
    Matthew D. Schwartz, Subhabrata SenWhether data that look mixed come from one population or several is one of the oldest questions in statistics. The Bayesian answer compares marginal likelihoods, integrals treated as intractable for mixture models for decades. For mixtures of a discrete variable this paper makes them explicit: a closed form for two components, exact at every sample size; an exact finite sum for any number of components under uniform priors; and the infinite-component limit taken exactly. It also proves constants that the asymptotic theory leaves open: in a trial recording four yes-or-no outcomes per participant, strong evidence against a spurious second group takes 3.3 million participants at a fifty-fifty response rate, where the standard BIC criterion would declare it at 22. The exact values also benchmark common estimators; applications to capture–recapture, record linkage, environmental DNA and mutational signatures are in the supplement.

Astrophysics and cosmology

  1. Elliptic master integrals and the one-loop wavefunction of the universe  ·  web summary
    Amara McCune, Matthew D. SchwartzThe coefficients of the wavefunction of the universe encode the correlations left by the universe’s first moments. For the one-loop triangle coefficient of a conformally coupled scalar, an elliptic differential equation among the master integrals had suggested that cosmological loops leave the polylogarithms already at one loop. Computed in flat space, the seed for de Sitter and power-law backgrounds, the coefficient turns out to be polylogarithmic: the elliptic periods cancel because the integrand’s residue at the poles carrying the curve is a flat-space amplitude. With one leg per site the result is three dilogarithms, and for all site energies it is a combination of dilogarithms given in closed form. The one-loop box is not polylogarithmic even with a single leg at each site: its soft discontinuity is an elliptic function, also given in closed form, and the periods of a K3 surface enter as well, so the triangle is the last polygon of this theory that is polylogarithmic.
  2. Memory back-reaction in black-hole scattering at fifth post-Minkowskian order  ·  web summary
    Matthew D. SchwartzWhen two black holes swing past each other, their gravitational waves leave detectors permanently displaced, the memory effect. The memory also pushes back on the orbit, and in the post-Minkowskian expansion (a series in Newton’s constant) that back-reaction enters at fifth order and second order in the mass ratio, where radiation reaction has not yet been computed. There a spurious divergence must cancel, which fixes the back-reaction up to a constant set by the propagator prescription for the zero-frequency gravitons, and the integrals involve the periods of a K3 surface, a geometry one step beyond an elliptic curve. This paper rederives the K3 surface from the maximal cut of the integrals and shows that the divergence occurs at the leading Landau singularity of the cut. For the smallest integral family it computes boundary constants on three of the four leading top sectors, and it continues the K3 and Calabi–Yau periods to bound orbits, working from maximal cuts and boundary limits without an integration-by-parts reduction of the full four-loop integrand.
  3. The loop expansion of galaxy clustering in redshift space  ·  [in preparation]  ·  web summary
    Mikhail M. Ivanov, Siddharth Mishra-Sharma, Matthew D. SchwartzGalaxy surveys draw much of their cosmological information from mildly nonlinear scales, where clustering admits a loop expansion that current analyses truncate at one loop; the omitted two-loop term sets their scale cuts. This first paper of the series computes the two-loop power spectrum and the one-loop bispectrum and trispectrum of biased tracers in redshift space on one complete fifth-order operator basis, with redshift-space kernels derived recursively and checked against the known fourth-order kernels in exact arithmetic. The two-loop power spectrum decomposes into 149 diagrams whose integrands cancel strongly, so each is integrated directly in momentum space in double and extended precision with a stated error (474 angular components of 136 diagrams are tabulated so far); renormalization subtracts each integrand’s hard-momentum limits, and infrared resummation damps each term by its loop order. With the one-loop bispectrum (568 coefficient functions on 786 triangles) and the nine-type one-loop trispectrum (5209 coefficient functions, evaluated on the 2400 configurations of the power-spectrum covariance), the three statistics can be analyzed jointly at a consistent loop order. Preliminary version.
  4. KITE: a two-loop galaxy clustering pipeline at stated precision  ·  [in preparation]  ·  web summary
    Mikhail M. Ivanov, Siddharth Mishra-Sharma, Matthew D. SchwartzFull-shape analyses of galaxy clustering rely on public codes such as CLASS-PT, PyBird and velocileptors, which evaluate the one-loop power spectrum at each sampled cosmology; the two-loop integrals needed at smaller scales are too slow to recompute at every cosmology, and the bispectrum and trispectrum add many nuisance parameters. This second paper of the series presents the Kernel-Integral Tensor Engine (KITE), which evaluates the two-loop power spectrum, one-loop bispectrum and one-loop trispectrum in redshift space from cosmology-independent tables, each entry carrying an error bound propagated to parameter shifts. KITE integrates out every linearly entering parameter in one closed-form Gaussian marginal, images the stochastic terms through the survey window, and samples the exact posterior by delayed-acceptance Hamiltonian Monte Carlo with a Boltzmann code in the final acceptance test. All tests use simulations, one posterior is reproduced bitwise across processor vendors, and on a light-cone mock four of five parameters designated beforehand are recovered within one standard deviation, short of the tighter survey-use benchmark. Preliminary version.
  5. Paper III: fits to DESI survey data  ·  [in preparation]
    Mikhail M. Ivanov, Siddharth Mishra-Sharma, Matthew D. SchwartzThe third paper of the series applies the theory of the first and the KITE instrument of the second to galaxy-clustering data from the DESI survey. In preparation; not yet on the site.

Language and textual analysis

  1. ACCSTACK: word stress and tone in the world’s languages, with sources  ·  web summary
    Matthew K. Gordon, Joe Pater, Kevin M. Ryan, Matthew D. SchwartzACCSTACK, a catalog at accstack.org, records what published descriptions say about word stress and tone for 5,561 languages: each entry gives the stress type, the stressed syllables at each word length the source states, the exceptions and domain of the rule, and the tone system where the source describes one. A large language model compiled it, reading the sources under the authors’ written instructions, quoting the passage that decides each entry and coding it. A catalog of this size is beyond what a linguist or a small group could compile by hand in any reasonable time, so not every entry has been checked by hand; but every entry was made to the same standard and shows its citation, the quoted passage and the reason for the ruling, so that readers can judge the interpretation for themselves and corrections are easy to make.
  2. Measuring shared authorship in the Federalist Papers, Shakespeare and the Bible  ·  web summary
    Matthew D. SchwartzFunction-word statistics have mostly been used to assign a disputed text to one author; hypotheses of mixed or multiple authorship have seldom received probabilities of their own. This paper treats such a text as a statistical mixture, so each proposed division becomes a hypothesis with its own Bayesian evidence, computed in closed form, and applies the method to the Federalist, the Shakespeare canon and the Bible. The joint Federalist papers are Madison-dominant under every word list and model; six Shakespeare plays show a second hand, in Henry VIII closer to Middleton’s profile than Fletcher’s, and four others long enough to test show none; Isaiah divides after chapter 39 with no support for a third Isaiah; in the Pentateuch, Priestly material separates while J and E cannot be told apart; and the Bible’s books fall into roughly a dozen to thirty distinguishable profiles. Most results confirm the scholarly consensus; a few do not.
  3. How the Voynich Manuscript was written  ·  web summary
    Matthew D. SchwartzThe Voynich manuscript is an illustrated fifteenth-century book in an unread script, and text with many of its statistics can be produced mechanically. This paper writes readings and text generators alike as probability models, fitted on half of a scribe’s pages and judged on the rest. In six candidate languages, readings by substitution, consonantal writing, abbreviation, codebooks and the Naibbe cipher all fail on withheld pages, though such a failure excludes a reading only where the test is shown to detect disguised real text of the same kind, so far only for scripture. A keyless procedure in which the scribes reused their words, occasionally copied a recent one and coined others by letter habit reproduces nearly all the summary statistics of both principal hands; two weak traces of content remain. The low cost of the writing and the genres the pictures imitate suggest a book made to be looked at, not read.
  4. Sharpening language trees with exact statistics and AI-curated cognate sets  ·  [in preparation]  ·  web summary
    Matthew D. SchwartzFamily relationships among the world’s languages are decided by Bayes factors computed over expert-curated word lists. This paper computes those quantities exactly, with no Monte Carlo sampling, and rebuilds the word lists with AI language models. The new data agree with every previously established result tested, with stronger conclusions, and add new ones, including a resolution of the Dravidian subgrouping and an Indo-European tree that begins with a three-way split into Anatolian, Tocharian and the core.

Health policy and medicine

  1. Stars misaligned: Medicare star ratings and the exact optimum of their clustering criterion  ·  web summary
    Matthew D. SchwartzSix Medicare rating systems turn quality scores into one to five stars by binning them so that the total squared distance of the scores from their bin centers is as small as possible. The bins are found in practice by clustering algorithms not guaranteed to reach that minimum, but the scores are one-dimensional, and in one dimension the optimal bins can be computed exactly. The paper computes them, without judging the agency’s intent or interpreting the law. In Medicare Advantage, where stars set bonus payments, the clustering step misses the minimum in 104 of 123 cut-point computations; the optimal bins move 34 published ratings across the bonus threshold, some up and some down, and roughly a billion dollars in gross bonus payments depend on the choice. About one hospital in fifteen displays fewer stars than the optimal bins assign, and two patient-experience programs miss the minimum on most measures examined.
  2. How truncation and rounding affect attainment of the U.S. ozone standard  ·  web summary
    Matthew D. SchwartzUnited States law limits ozone in outdoor air to 70 parts per billion, and whether a monitoring site meets the limit is judged by a three-year statistic of its readings, the design value, which the regulation truncates at its last digit for ozone but rounds for fine particles. Environmental groups have repeatedly asked the EPA to round for ozone too. The paper notes that digits are discarded at three points in the chain of calculations behind the ozone statistic, and recomputes from public monitoring data every published 2020–2025 design value within one unit of a standard. For about one in three of these borderline values, attainment depends on the final digit alone: 61 of 190 borderline ozone values that attain as published would violate under rounding, and at least 148 if the eight-hour averages were rounded as well. The paper lists the affected values and suggests EPA flag them.
  3. Exact versus simulated operating characteristics of published adaptive and Bayesian trial designs  ·  web summary
    Matthew D. SchwartzBefore a Phase II trial begins, its design is judged on how often it would declare an ineffective drug effective and how often it would miss an effective one; for adaptive and Bayesian designs these operating characteristics are usually estimated from a few thousand simulated trials, each with Monte Carlo noise. Applying standard-error formulas to the tables of nine published designs, the paper finds that 351 of 977 reported error rates and powers lie within noise of their thresholds, from simulations too small to resolve them by a median factor of 18. Most need no simulation at all: where the decision rule is stated in full the quantity is a finite sum or low-dimensional integral, which the paper computes exactly, with exact rational arithmetic and cross-checked quadrature. Recomputing the borderline values wherever possible confirms the reported side in 87 cases and finds the other side in 30, about what noise alone predicts.