The BootLoops toolkit

The BootLoops toolkit is the set of software packages behind the results on this site, grouped below by the job they do. Each page explains in plain terms what the tool does, works through examples, lists the tool's routines, and ends with its requirements and self-test command. BootLoops' own packages, 50 in the 1.0 release, each live in a folder under tools/ of the bootloops-dev repository, named on the page. Entries marked (external) are public programs by other authors, and their pages say what BootLoops uses them for and what code, if any, it adds around them.

The BootLoops 1.0 code is published under the GitHub organization BootLoops-ai: the toolkit (the tools/ packages, the toolkit/ catalogs, BootLoops' own engines under upgrades/ and the self-test runner) is the bootloops-dev repository, while JaCK & Jill (jackandjill-dev), the three patched engine forks (amflow-cpp-dev, kira-dev, blade-dev) and the Claude Code skills (skills-dev) are repositories of their own under the same organization. BootLoops code is released under the MIT license and the forks keep their upstream licenses (third-party and vendored components keep their own). Code written by Claude Fable 5.1 (Anthropic) under the supervision of Matthew D. Schwartz. The packages are by Matthew D. Schwartz and Claude; JaCK & Jill is by Matthew D. Schwartz, Scott V. Edwards, Paul O. Lewis and Claude.

Symbolic algebra and reduction to master integrals

A multi-loop calculation starts with algebra that produces thousands of Feynman integrals; integration-by-parts (IBP) reduction then writes all of them as exact combinations of a few master integrals. These tools do that step, size it, rescue it when it stops, and check the result.

• FORM (external) — Symbolic manipulation program for expressions too large for memory; BootLoops uses it for Dirac and color algebra, diagram generation and multiple zeta values.
• Formglue — Python wrappers around FORM: reads its output, exposes its multiple-zeta-value evaluator to Python, assigns loop momenta to generated diagrams, and drives HyperFORM.
• Kira, FireFly and Fermat (external) — IBP reduction: writes each integral of a family as an exact combination of master integrals; BootLoops adds a patched build and job-splitting scripts.
• Blade (external) — Block-triangular reduction: one expensive search per family, then cheap reductions at every kinematic point; BootLoops adds a Mathematica-free port and Python drivers.
• Seedling — Prepares and sizes a Kira reduction: finds the seed range the targets need, writes the job file, predicts memory, checks results, audits family symmetries.
• Dogtag — Identity checks on a family-definition file before anything runs on it: same family as a known one up to loop-momentum relabeling, planar or crossed in the physical leg order, cut signature, drawn graph versus propagator list, and the Yangian/loom applicability screen for conformal integrals.
• Winnow — Library for the elimination step of IBP reduction modulo a prime at one kinematic point, with a checkable certificate for each returned row.
• Trust — Checks a finished reduction table by three routes that share no core code and compares it coefficient by coefficient with an independently computed reduction; its receipt module writes and checks per-row certificates showing that each row follows exactly from the linear system it was reduced from.
• FFCapital — Recovers the exact rational functions already finished inside the saved state of an interrupted FireFly reconstruction, without resuming the run.
• Numkin — Unsticks two-scale Kira reductions by fixing one invariant to several rational values, reducing at each, and reconstructing the exact dependence afterward.

Singularities and geometry

Where an integral can become singular, and what geometric object sits on its maximal cut, together decide which class of functions the answer belongs to. These tools settle that before anything expensive is computed.

• Landau Alphabet — From a Feynman graph to the polynomials whose zeros locate its singularities (the symbol letters), sorted into classes and flagged as proven complete or not.
• PLD and SOFIA (external) — Two public programs that list where a diagram's integral can be singular; BootLoops adds a Mathematica-free SOFIA.jl and a bridge script for PLD.
• Leviathan — BootLoops' own Julia implementation of the Euler-characteristic-drop test for Landau singularities, with master-integral counts sector by sector.
• GeoTriage — Reads a graph and reports the geometry on its maximal cut (rational, elliptic, K3, Calabi–Yau, higher genus) with a recommended solution route and its assumptions.
• Maxcut — Maximal-cut package: classifies the cut's geometry from polynomial algebra, builds a canonical-basis rotation, assembles cut differential equations, and supplies sparse finite-field linear algebra.
• Dipstick — Cheap probes to run before an expensive calculation: the order of an integral's differential equation, the size of its family, Galois-group evidence and GKZ resonance, plus a check that a list of regions for an expansion by regions is complete (regions).
• Coalescer — Given a linear differential equation with polynomial coefficients and a series solution, both in a short JSON file, computes the coefficients with which that solution continues onto the fractional-power branches at a singular point, from the local monodromy; the threshold of a banana integral is the worked example.

Differential equations

Master integrals satisfy linear differential equations in the kinematic variables. These tools put the equations in canonical form, solve them along a path from a point where the values are known, and pin down the constants that remain.

• Counterweight — Finds an exact rational change of basis that puts a family's differential equation in canonical ε-form, verifies it, or reports why none was found.
• Canonify — Counterweight's Python front end: computes each master's leading singularity on the maximal cut and the rescaling that removes it, and calls the Julia engine.
• Wayfinder — Integrates a family's differential equation from known values to a target point and extracts the ε-expansion; also reconstructs the equation's matrices from numerical values, extrapolates another program's results on a grid of ε to the ε→0 limit (epslimit), and reduces large expressions to exact rational functions with FLINT (flintexport).
• Famhar — Evaluates a whole family from one configuration file: differential equation, starting values and branch points in, target values and derivatives along any path out.
• Vopclose — Solves a family's differential equations along a path as explicit closed-form functions of the path variable, order by order in ε, with symbolic checks.
• Annihilator — Finds the lowest-order recurrence satisfied by a sequence's opening terms and the differential (Picard–Fuchs) operator that annihilates its generating function.
• Frobenius — Finds the non-integer power behavior of a family's solutions at infinite mass and which exponents the physical solution allows, cutting down the unknown boundary constants.
• Holonomic — Driver around the public ore_algebra package: continues a solution of a linear differential equation to another point as a proven interval, as an independent check.

Evaluating integrals numerically and symbolically

Every exact formula on this site is tested against numerical values computed independently. These tools produce such values, several of them by methods that share no code with one another, and two of them integrate in closed form.

• AMFlow (external) — Evaluates Feynman integrals numerically to as many digits as requested by the auxiliary-mass method; BootLoops runs a patched amflow-cpp and adds the amflow-kit package: output checks, a predictor of the reduction-cache key a job will use, and a memory guard for launched jobs.
• PMflow — Supplies the boundary values AMFlow cannot compute on its own for a class of gravitational-scattering integrals, so that a normal AMFlow run can finish.
• Membound — Small-frequency boundary constants of frequency-space integrals with Bessel-function kernels, as they arise in worldline and post-Minkowskian expansions: the integrand goes in as a short JSON file and its Laurent series in ε comes out; the memory-region integrals of black-hole scattering are the worked example.
• Cosmoflow — Integrands and candidate symbol letters of wavefunction and correlator integrals in an expanding universe for any graph of sites (chains, rings, stars, multi-loop polygons), plus the Baikov polynomial of any one-loop polygon; the one-loop triangle's numerical master integrals and periods are the worked example.
• pySecDec and FIESTA (external) — Sector-decomposition programs used unmodified as a numerical check sharing no code with other methods; BootLoops adds two packages of its own: an arbitrary-precision evaluator for finite Euclidean integrals (hiprec-sectordecomp) and a staged runner that checks the error estimate of one long pySecDec evaluation (pysecdec-point).
• Nestor — High-precision numerical integration that reports only the digits two refinements agree on; nests for multiple integrals and spreads over many cores, and its dispersion module computes dispersion integrals of a spectral density with known threshold singularities.
• Tropical sampler — Monte Carlo integration of positive polynomial-power integrands sampled from their tropical approximation, returning an unbiased estimate with an error bar.
• SubTropica — Julia driver around the public HyperFLINT engine: exact ε-expansions of parametric integrals in zeta values and polylogarithms, with divergent integrals handled.
• Surd — Exact integration of a three-fold rational parametric integral on a one-parameter family into hyperlogarithms with algebraic arguments, plus evaluation and structure queries.

Periods, special functions and arithmetic geometry

When an integral is controlled by an elliptic curve, a K3 surface or a Calabi–Yau space, polylogarithms are not enough. These libraries evaluate the functions and periods that appear, most with proven error bounds, and two of them apply the same machinery to abelian varieties and string-theory vacua.

• Eichler — Periods, iterated integrals of modular forms, L-values and transported solutions attached to elliptic, K3, Calabi–Yau and genus-two geometries, returned as proven intervals, plus identification of a genus-two curve from its invariants (genus2).
• GPLEval — Julia evaluator for multiple, harmonic and classical polylogarithms to any precision, each value returned with a guaranteed error radius.
• GiNaC (external) — C++ symbolic library whose polylogarithm and modular-form iterated-integral evaluators BootLoops uses unmodified as independent reference values.
• PentagonFunctions-cpp (external) — Public evaluator of pentagon functions for two-loop five-point kinematics, built unchanged and used only as an independent reference value.
• Galois — Exact arithmetic with multiple zeta values: reduction to a canonical basis through weight 23, and Galois derivations turned into exact constraints on ansatz coefficients.
• Abacus — Point counts, Frobenius polynomials and local L-factors of an abelian fourfold given only by a certified period lattice, exactly, or a message saying why not.
• Terrier — Exact and certified computations about string-theory flux vacua: period values with proven radii, certified critical points, complete lattice-class lists, deduplicated vacuum counts.

High-precision and certified arithmetic

Ball arithmetic stores every number as a midpoint plus a radius guaranteed to contain the true value. These are the libraries underneath every proven error bound on the site; Baller also carries the checkers that catch precision mistakes in code that uses them.

• Arb and mpmath (external) — Arbitrary-precision arithmetic: Arb with proven error bounds, mpmath without; used as released underneath every many-digit computation here.
• Baller — Ball-arithmetic drivers on Arb: proven enclosures for expressions, iterations, integrals, differential-equation continuation and solution certificates, plus consistency and source checks such as dps_lint, which finds mpmath numbers created before the working precision is set.
• ERAS — Rigorous range bounds for a function over a small interval of a shared parameter when ordinary interval arithmetic is uselessly wide, with a verification program.

Exact reconstruction and integer relations

The last step of many calculations turns a number known to many digits, or a function sampled exactly, into an exact formula. These tools plan the fit, run it with checks that make a negative answer meaningful, and recover exact rational functions and linear-system structure.

• PSLQ and LLL (external) — Integer-relation and lattice-reduction algorithms from mpmath, fplll and Nemo that turn a many-digit number into an exact formula; BootLoops adds the surrounding checks.
• Lockpick — Integer-relation finding with checks: a PSLQ driver, the multi-point lattice fitter mplll, a sample-point selector, and a ready-made library of high-precision constants.
• Ansatzer — Counts, weight by weight, how many unknown coefficients survive in a polylogarithmic integral's symbol ansatz, and so how many numerical values a fit needs.
• Gatekeeper — Checks the numerical values going into a fit for duplicates and bad data, then accepts a fitted formula only if every left-out value is reproduced.
• Rankscreen — Rank, pivots and contradictory equations of a large exact linear system, found by parallel elimination modulo several primes that must all agree.
• Ratfit — Recovers an exact one-variable rational function from exact samples and confirms it on samples not used in the fit; also chooses safe sample points and, when reconstruction modulo primes stalls, says whether to add a prime or enlarge the sampling grid (degree_alias).

Statistics and inference packages

Packages from the applications outside particle physics. The standard is the same: an answer is exact, or an interval with proven endpoints, or it states its measured error, and several carry certificates a separate routine can recheck.

• Mixalot — Exact Bayesian evidence of finite mixture models for count data, as a rational number, with the exact posterior over the number of components; its boxwalk module evaluates the underlying integrals of products of polynomial powers over a box exactly, by recurrences modulo primes.
• Popcorn — Expected site-frequency spectrum under natural selection to any requested precision, exact or as a guaranteed interval, with derivatives, DFE averaging and dominance.
• JaCK & Jill — Bayesian evidence of phylogenetic trees under simple substitution models: exact fractions, proven intervals or measured estimates, each with a recheckable certificate.
• POSQ — Bayesian evidence of a small Markov model on a four-leaf tree as an interval with both endpoints proven, by exact positive quadrature in ball arithmetic.
• Clinch — Proves a fitted optimum of a hierarchical model is the only stationary point in a small box, with positive-definite Hessian, or names the failed test.
• Qinvert — Works backward from exactly published summary statistics to the smallest sets of added or removed data rows that reproduce them, in exact arithmetic.

Records and repository utilities

One package keeps quoted numbers tied to the files they came from. The repository also carries, under ops/, a utility for running long jobs on a shared machine; it is not one of the 50 toolkit packages and nothing in the toolkit depends on it.

• Emitall — Recomputes every number quoted in a report from its result files and lists disagreements; a companion linter flags hand-typed digits in the generating scripts.
• Turnstile (repository utility, ops/turnstile/) — Bash wrapper that decides when a long, memory-hungry job may start on a shared Linux machine, by priority and by the memory and threads each job declares, and logs estimated versus measured memory and run time.

Renamed and retired pages

These addresses are kept so that old links still work; each one redirects to the page named after the arrow.

• Bigram-queue → Turnstile (the repository utility under ops/turnstile/)
• Collapse Predictor → Ansatzer
• Connection Sampler → Wayfinder (its sampler module)
• RingMaker → Lockpick (its ring15 constant library)
• Degree-alias → Ratfit (its degree_alias module)
• Dispersify → Nestor (its dispersion module)
• Dps-lint → Baller (its dps_lint checker)
• Family Audit → Seedling (its audit and identity commands)
• Flint-export → Wayfinder (its flintexport module)
• Geometry Classifier → GeoTriage
• GravityFlow → PMflow
• Kira → Kira, FireFly and Fermat
• Loomcheck → Dogtag (its loomcheck.py screen)
• Oracle Harness → Gatekeeper
• Receipt → Trust (its receipt module)
• series_pf → Annihilator
• Topology Audit → Dogtag
• Rowscripts is retired: each solved integral's standalone evaluator script is now on that integral's own page in the Feynman diagrams gallery.

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