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arxiv: 2605.29789 · v1 · pith:Z3QHBSNFnew · submitted 2026-05-28 · ✦ hep-th · hep-ph· math-ph· math.MP

Magic Relations and Critical Varieties of Feynman Integrals

Giulio Crisanti , Hjalte Frellesvig , Andrzej Pokraka , Sid Smith This is my paper

Pith reviewed 2026-06-29 06:38 UTC · model grok-4.3

classification ✦ hep-th hep-phmath-phmath.MP
keywords magic relationsFeynman integralsintegration-by-parts identitiescritical varietiesmaster integralsIBP reductionssymmetriescuts
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The pith

Magic relations in Feynman integrals always coincide with higher-dimensional critical varieties in the generating sector.

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

Magic relations are integration-by-parts identities where every integral in the generating sector drops out, causing standard reduction methods to fail. The paper observes and argues that these relations occur exactly when higher-dimensional critical varieties appear in that sector. This geometric coincidence supplies a direct computational test for the presence of magic relations and a way to locate them. The authors also show how to count master integrals in such cases and classify the relations under symmetries and cuts.

Core claim

We observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

What carries the argument

The coincidence of magic relations (IBP identities that nullify the entire generating sector) with higher-dimensional critical varieties in that sector, used as a diagnostic criterion.

Load-bearing premise

The observed coincidence between magic relations and higher-dimensional critical varieties holds for Feynman integral families in general rather than only the specific cases examined.

What would settle it

A Feynman integral family that exhibits magic relations but lacks higher-dimensional critical varieties in the generating sector, or the converse.

read the original abstract

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

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 manuscript observes and argues that magic relations—IBP identities emptying the generating sector—always coincide with the presence of higher-dimensional critical varieties in that sector for Feynman integral families. It supplies a practical detection test implemented in the ancillary Magic-Test.m, discusses counting master integrals when such varieties are present, classifies the behavior of magic relations under symmetries, and examines their interplay with cuts.

Significance. If the claimed coincidence is general, the work supplies a new diagnostic linking an important class of IBP breakdowns to geometric features of the integral families, together with a concrete computational test and guidance on master-integral counting. These elements would be directly useful to practitioners who encounter magic relations in multi-loop calculations.

major comments (1)
  1. [Abstract] Abstract: the central claim that magic relations 'always' coincide with higher-dimensional critical varieties is presented as an observation and argument; the provided test detects varieties but does not establish necessity or sufficiency for arbitrary families, so the universality statement requires either a general derivation or explicit verification beyond the examined cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that magic relations 'always' coincide with higher-dimensional critical varieties is presented as an observation and argument; the provided test detects varieties but does not establish necessity or sufficiency for arbitrary families, so the universality statement requires either a general derivation or explicit verification beyond the examined cases.

    Authors: We thank the referee for this comment. The manuscript deliberately phrases the central statement as an observation and argument (see abstract and Section 2), supported by explicit verification across a range of non-trivial families rather than a general theorem. The test implemented in Magic-Test.m is a practical diagnostic that identifies the higher-dimensional critical varieties whose presence we have found to coincide with magic relations in every case examined; it is not claimed to constitute a proof of necessity or sufficiency. Additional supporting examples and counting procedures appear in Sections 3 and 4. A general derivation establishing the coincidence for arbitrary families remains an open question outside the scope of the present work, which instead supplies a usable computational tool and guidance on master-integral counting when such varieties occur. The current wording already reflects the observational character of the claim, so we do not propose a change to the abstract. revision: no

Circularity Check

0 steps flagged

No circularity; central claim is observational coincidence without reduction to inputs or self-citations

full rationale

The paper states it 'observe[s] and argue[s]' that magic relations coincide with higher-dimensional critical varieties and supplies an external test (Magic-Test.m). No equations, definitions, or derivations in the provided text reduce the claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The argument is framed as empirical observation across examined families rather than a tautological or internally forced result, satisfying the requirement for independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, invented entities, or ad-hoc axioms; the work relies on standard domain concepts from Feynman integral reduction and algebraic geometry.

axioms (1)
  • domain assumption Standard definitions and properties of critical varieties apply to the parameter space of Feynman integral sectors
    Invoked implicitly when linking varieties to the occurrence of magic relations.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the spanning cuts consistency problem in the IBP reductions of Feynman integrals

    hep-ph 2026-06 unverdicted novelty 6.0

    Inconsistency in spanning cuts for IBP reductions arises because cuts can make hidden terms in IBP relations finite via pinch singularities that cancel vanishing parameters, linked to hidden linear relations between p...

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