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arxiv: 2604.20954 · v1 · submitted 2026-04-22 · ✦ hep-th · hep-ph

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SubTropica

Mathieu Giroux , Sebastian Mizera , Giulio Salvatori
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:37 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords tropical geometryFeynman integralssymbolic integrationmulti-polylogarithmsMathematica packageEuler integralshyperlogarithmslinearly-reducible integrals
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The pith

SubTropica is a Mathematica package for symbolic integration of multi-polylogarithmic integrals via tropical geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a software tool that performs symbolic integration on integrals appearing in particle physics by drawing on tropical geometry methods. It targets linearly-reducible Euler integrals, a class that includes many Feynman integrals, and expands them through a dedicated subtraction scheme. The underlying engine handles hyperlogarithm integration as a standalone component. A reader would care because these integrals arise routinely in precision calculations of scattering amplitudes and other observables where closed-form results are needed but manual methods become intractable.

Core claim

We present SubTropica, a Mathematica package that performs symbolic integration of multi-polylogarithmic integrals using recent advances in tropical geometry. It focuses on the class of linearly-reducible Euler integrals, such as Feynman integrals, and expands them using a tropical subtraction scheme. The engine behind it is HyperIntica, a native Mathematica package for hyperlogarithm integration that can be used independently. This paper documents both packages and illustrates their usage on examples from across different physics applications. Additionally, we introduce an AI-driven library of Feynman integrals, which catalogs diagrams discussed in the literature and serves as a database.

What carries the argument

The tropical subtraction scheme that expands linearly-reducible Euler integrals for subsequent evaluation by the HyperIntica hyperlogarithm integration engine.

Load-bearing premise

The tropical subtraction scheme and HyperIntica engine together handle every linearly-reducible Euler integral without hidden restrictions or failures on realistic cases.

What would settle it

A concrete linearly-reducible Euler integral taken from the literature for which the package either fails to produce a symbolic result or yields an answer that disagrees with an independently known closed form.

Figures

Figures reproduced from arXiv: 2604.20954 by Giulio Salvatori, Mathieu Giroux, Sebastian Mizera.

Figure 1
Figure 1. Figure 1: A four-loop Feynman diagram with nine propagators, one massive in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The crown diagram features a non-sign-definite Symanzik polynomial. [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Newton polytope describing the singularities of the Euler integral [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Correlation between the score Sorder given to a given LR order and its resulting timing Torder in seconds. Different colors represent data points associated with different Feynman integrals. The straight line is the power-law fit and the shaded areas denote the 95% confidence intervals. In particular, we find that the variance is approximately constant, which means that Torder follows an approximately log-… view at source ↗
Figure 5
Figure 5. Figure 5: Left: The e −e − → e −e − (Møller) scattering one-loop box. Right: a two-loop triangle-box topology with three distinct external masses M1, M2, M3. Wavy and dashed lines are massless. True, which instructs STFasterFubini to factor these quadratics into linear fac￾tors by introducing auxiliary square-root variables, restoring linear reducibility: result = STIntegrate[diag, "FindRoots" −> True] The integrati… view at source ↗
Figure 6
Figure 6. Figure 6: The algorithmic steps of STIntegrate described in Sec. 3. Numbered cir￾cles mark the two checkpoints where the pipeline can be paused via "StopAt" and resumed later via "StartAt". The single-headed dashed feedback arrow indicates that the integration step can be run multiple times with different "SelectFaces" selections, with completed faces reused via "ReuseExistingResults". 56 [PITH_FULL_IMAGE:figures/f… view at source ↗
read the original abstract

We present SubTropica, a Mathematica package that performs symbolic integration of multi-polylogarithmic integrals using recent advances in tropical geometry. It focuses on the class of linearly-reducible Euler integrals, such as Feynman integrals, and expands them using a tropical subtraction scheme. The engine behind it is HyperIntica, a native Mathematica package for hyperlogarithm integration that can be used independently. This paper documents both packages and illustrates their usage on examples from across different physics applications. Additionally, we introduce an AI-driven library of Feynman integrals, which catalogs diagrams discussed in the literature and serves as a database for computed results. Its online version is available at: https://subtropi.ca and features a graphical user interface for diagram input and retrieval of records.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper presents SubTropica, a Mathematica package for symbolic integration of multi-polylogarithmic integrals via tropical geometry, targeting linearly-reducible Euler integrals (e.g., Feynman integrals) through a tropical subtraction scheme. It documents the standalone HyperIntica engine for hyperlogarithm integration and introduces an AI-driven online database of Feynman integrals with a GUI at https://subtropi.ca, illustrated by usage examples from physics applications.

Significance. If the described tropical subtraction scheme and HyperIntica engine correctly and efficiently handle the stated class of integrals, the work would deliver a practical computational resource for perturbative calculations in high-energy physics. The accompanying database could aid literature synthesis and result reuse. The manuscript's descriptive focus on a software tool and database, rather than new derivations, means its impact hinges on demonstrated reliability rather than theoretical novelty.

major comments (1)
  1. Abstract: The central claim that SubTropica performs correct symbolic integration of the full class of linearly-reducible Euler integrals is presented without any test cases, benchmarks against known results, error analysis, or implementation details, preventing assessment of whether the tropical subtraction scheme works as asserted on realistic examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the major comment below and will revise the abstract to better highlight the illustrative examples already present in the paper.

read point-by-point responses

  1. Referee: Abstract: The central claim that SubTropica performs correct symbolic integration of the full class of linearly-reducible Euler integrals is presented without any test cases, benchmarks against known results, error analysis, or implementation details, preventing assessment of whether the tropical subtraction scheme works as asserted on realistic examples.

    Authors: The abstract provides a concise summary of the package's purpose and scope. The full manuscript documents the tropical subtraction scheme, the HyperIntica engine, and illustrates usage on concrete examples drawn from physics applications, which serve as demonstrations on realistic linearly-reducible Euler integrals. Implementation details appear in the subsequent sections. We agree that a brief reference to these examples would strengthen the abstract and aid assessment; we will revise the abstract accordingly. Comprehensive benchmarks and formal error analysis for the entire class lie beyond the current scope but can be expanded in follow-up work. revision: yes

Circularity Check

0 steps flagged

Software documentation paper with no derivation chain

full rationale

This is a tool and database paper documenting the SubTropica Mathematica package and its HyperIntica engine for symbolic integration of linearly-reducible Euler integrals via a tropical subtraction scheme. It contains no mathematical derivations, theorems, predictions, or load-bearing equations that could reduce to self-referential inputs, fitted parameters, or self-citations. The text is purely descriptive, consisting of package documentation, usage examples across physics applications, and an AI-driven integral library catalog. No steps in the presentation rely on circular reductions, self-definitional claims, or imported uniqueness results; the work is self-contained as software documentation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software and database presentation paper with no mathematical derivations or theoretical claims that introduce free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5415 in / 1120 out tokens · 27697 ms · 2026-05-09T23:37:01.994890+00:00 · methodology

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