BootLoops packages particle-physics integral reduction, experimental-mathematics integer-relation fitting, and ball arithmetic into an LLM-operable exact-computation toolkit that turns multi-loop Feynman integrals, Bayesian evidence integrals, and floating-point error re-runs into exact results
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Synopsis
BootLoops gathers integral reduction, differential equations, and high-precision evaluation from particle physics, integer-relation fitting from experimental mathematics, exact enumeration, and ball arithmetic into one toolkit that any agentic large language model can operate and extend, computing multi-loop Feynman integrals by bootstrap methods rather than direct integration across function classes including polylogarithms, elliptic functions, and periods of K3 surfaces and Calabi-Yau manifolds, and also evaluating Bayesian evidence integrals as exact rational numbers, enumerating finite configuration spaces with proof of completeness, showing when a sought closed form cannot exist, and redoing floating-point calculations with guaranteed digits.
Figure 4: At 16 and 50 digits of working precision, fixed-precision arithmetic returns a wrong value of x 100 x_{100} in Muller’s recurrence, Eq. ( 14 ), with no indication that it is wrong, whereas ball arithmetic returns either certified digits or an infinite radius. Top: the recurrence with its seeds, and the exact x 100 x_{100} , a ratio of a 79-digit integer to a 78-digit integer, approximately 6.000000016099565 6.000000016099565 . Middle: x 30 x_{30} and x 100 x_{100} computed in fixed precision at 16, 50 and 300 digits. At 16 and 50 digits x 100 x_{100} prints as exactly 100, because rounding error excites a second solution of the recurrence that grows about seventeen-fold per step; at 300 digits the value is correct. The notes at right give the step at which each fixed-precision iterate leaves the neighborhood of 6. Bottom: the same three computations in ball arithmetic. Where a divisor ball contains zero the radius is infinite and no digit is asserted; at 300 digits every printed digit of x 100 x_{100} is guaranteed. The midpoints in the bottom rows track the fixed-precision values because a ball’s midpoint is computed by the same floating-point operations; only the radius indicates which of its digits are established.
arXivInterpretation
The work presents BootLoops, a pool of cross-domain exact-computation programs in one place, including integral reduction, differential equations, high-precision evaluation, integer-relation fitting, exact enumeration, and ball arithmetic, designed so that any agentic large language model can operate and extend it. Previously these exact methods were scattered across distant fields, for example collider-physics reduction methods applying to Bayesian evidence and interval arithmetic from numerical analysis settling a regulatory threshold, while few people knew both fields; the work moves the carrier of the method from human cross-domain knowledge to a program collection a model can call. Based on the toolkit composition and design intent stated in the abstract; a system- and tool-level description with no benchmarks, sample sizes, or performance numbers.
On the physics side, the tools can compute multi-loop Feynman integrals by bootstrap methods rather than direct integration, across the function classes that arise, including polylogarithms, elliptic functions, and periods of K3 surfaces and Calabi-Yau manifolds. Shifts the solution path for multi-loop Feynman integrals from direct integration to bootstrap methods and explicitly names the covered function classes. A capability statement at the abstract level, with no specific integral cases, loop counts, or precision metrics.
Beyond physics, the tools can evaluate Bayesian evidence integrals as exact rational numbers, enumerate finite configuration spaces with proof of completeness, show when a sought closed form cannot exist, and redo floating-point calculations with guaranteed digits. Moves exactness from approximate estimation to provable outcomes: rational results, completeness proofs, non-existence determinations, and re-runs with guaranteed digits, corresponding to the Monte Carlo estimation, heuristic search, and rounding-error-untracked floating-point computation the abstract cites as the status quo. A capability statement at the abstract level, with no specific cases, error-bound numbers, or validation experiments.
The work argues this toolkit can bring the tools of mathematics, computer science, and physics to problems in genomics, statistics, ecology, and other fields. Extends the scope of cross-domain method transfer from within physics to other quantitative disciplines. An applicability claim phrased as 'In practice' in the abstract, with no listed application cases or results.
Perspective
The toolkit targets quantitative computations that need exact rather than approximate results: multi-loop Feynman integrals, Bayesian evidence integrals, enumeration of finite configuration spaces, existence determination for closed forms, and floating-point re-runs with guaranteed digits. The intended user is an agentic large language model that can operate and extend the pool, and the beneficiary fields named in the abstract include genomics, statistics, and ecology. The toolkit and its documentation are provided at the link given in the abstract.
The loaded text is an abstract, lacking figures, benchmarks, and concrete cases, so the accuracy, coverage, and runtime cost of each tool on real problems cannot be assessed; the cross-domain applicability claim phrased as 'In practice' also lacks supporting application results. These are open questions answerable only by consulting the toolkit and documentation.