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Algebraic versus physical uniqueness of MHV gravity numerators

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read At seven and eight points, MHV gravity numerators are underdetermined by degree and pair-zero conditions alone; Bose symmetry and one normalized boundary condition restore uniqueness.

desk verdict Refutes the Ref. [2] uniqueness conjecture with exact seven- and eight-point calculations; the all-multiplicity theorem has a fixable presentation gap. read the letter →

arxiv 2608.11792 v1 pith:73ALTXY3 submitted 2026-08-12 hep-th

Lin Mai , Yaobo Zhang This is my paper · ORCID
classification hep-th
keywords scatteringamplitudesMHVgravitynumeratorbootstrappair-idealintersectionflagvarietiessemistandardtableauxBosesymmetryBCFWscaling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Tree-level maximally-helicity-violating (MHV) gravity amplitudes are represented as a polynomial numerator divided by a fixed product of angle brackets, and this paper asks whether the numerator's degree together with its vanishing on every pairwise locus $\langle ij\rangle = [ij] = 0$ determines the amplitude. The answer is no in general: at seven points the algebraic solutions form a seven-dimensional space $S^{(2,1^5)}\oplus S^{(1^7)}$, consisting of the Hodges numerator line plus a six-dimensional hook representation. Requiring Bose symmetry (an alternating numerator) removes the hook and leaves the Hodges line. At eight points Bose symmetry leaves a two-dimensional alternating plane, and three physical conditions—the large-$z$ falloff, normalized collinear factorization, and the leading soft coefficient—impose the same linear condition and select the Hodges line. Within the fixed-common-denominator ansatz, the paper proves that one complete normalized marked-collinear boundary determines the numerator up to scale at any multiplicity.

What carries the argument

The load-bearing construction is the flag-variety realization of the spinor-helicity quotient: angle brackets become Plücker columns of height two, square brackets become complementary columns of height $n-2$, and the pair ideals $\langle ij\rangle,[ij]$ become column-restriction conditions in the coordinate ring of $\mathrm{Fl}(2,n-2)$. A standard-monomial basis indexed by semistandard tableaux of shape $\Lambda_n=(A_n+B_n,A_n+B_n,B_n,\ldots,B_n)$ makes the ansatz finite-dimensional, and Young projectors decompose the simultaneous restriction map into irreducible $S_n$ blocks, shrinking the seven-point calculation from 65,870 tableaux to fifteen blocks whose largest multiplicity space has dimension 475. The same machinery supports the eight-point character calculation and the marked collinear ideals $K_{ab}(\alpha)=\langle\langle ab\rangle,\langle ia\rangle-\alpha\langle ib\rangle\rangle$, whose joint injectivity—proved by successively extracting distinct angle-bracket divisors—is the engine of the all-multiplicity uniqueness theorem.

What would settle it

Under the $[1,2\rangle$ shift at the rational eight-point kinematics recorded in section G.1, compute the $z^6$ coefficient of $N/D_8$ for $N=7A-6B$: the paper predicts a nonzero value proportional to $7c_A-6c_B$, so a vanishing coefficient would overturn the large-$z$ selection. Independently, at $n=9$, search for a nonzero alternating fixed-degree numerator that lies in every pair ideal and in one marked collinear ideal $K_{ab}(2)$; theorem 6.2 predicts no such element exists.

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Extended reading notes

Core claim

The central discovery is that algebraic and physical uniqueness of MHV gravity numerators separate at higher multiplicity. Over $\mathbb{Q}$, the seven-point pair-ideal intersection at the target degree is $W_{7,\mathbb{Q}}\simeq S^{(2,1^5)}\oplus S^{(1^7)}$, so the degree-and-zero conditions admit six hook directions in addition to the Hodges line and the all-multiplicity conjecture that held at five and six points fails. Bose symmetry is an independent constraint that retains only the sign component, giving $W^{\mathrm{Bose}}_{7,\mathbb{Q}}=\mathbb{Q}\,N^{\mathrm{Hodges}}_7$. At eight points the Bose-compatible sector is a two-dimensional plane $QA\oplus QB$, and the conditions of $O(z^{-2})$ large-$z$ behavior, normalized $++$ collinear factorization, and the leading positive-helicity soft coefficient each reduce to the same linear condition $7c_A-6c_B=0$, whose solution is the Hodges line $Q(6A+7B)=QN^{\mathrm{Hodges}}_8$. For arbitrary $n$, an alternating fixed-degree numerator is injectively determined by one marked collinear restriction, so together with universal $++$ splitting and the five-point seed it is fixed up to normalization inside the common-denominator ansatz.

Load-bearing premise

The all-multiplicity result rests on assuming that at every point count $n$ the ring of spinor bracket polynomials has no zero divisors, so a numerator that vanishes wherever one angle bracket vanishes must contain that bracket as a factor; the paper verifies this property only at seven points, and the physical selection statements also take the standard large-$z$ falloff, the universal $++$ splitting law, and the leading soft-graviton theorem as inputs rather than deriving them.

Editorial extensions

If this is right

  • The original conjecture that degree plus pair-zero conditions fix the numerator at every multiplicity is false; five- and six-point uniqueness was a low-multiplicity accident.
  • Bose symmetry is not implied by the pair-ideal conditions, but at seven points it restores uniqueness by killing the six hook directions.
  • At eight points the large-$z$, collinear, and soft conditions are interchangeable selectors within the Bose-compatible plane: any one normalized boundary condition fixes the amplitude.
  • At arbitrary multiplicity, reconstructing the numerator reduces to evaluating one complete marked collinear boundary, provided the common-denominator ansatz and the standard splitting law hold.
  • Higher-multiplicity calculations can be organized by permutation sectors, so the full $S_n$ decomposition, not just the sign sector, is the natural computational target.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is $n=9$: computing the full $S_n$-resolved pair-ideal kernel would show whether hook-type algebraic directions grow with multiplicity, and whether the physical selectors still agree on the Bose-compatible sector.
  • The all-multiplicity theorem assumes the bracket ring has no zero divisors at every $n$; locating the first multiplicity where this fails, or exhibiting a nonzero alternating numerator in the kernel of one marked restriction, would mark the boundary of the method.
  • Because the large-$z$, collinear, and soft selectors coincide exactly on the eight-point plane, a plausible extension is that every normalized boundary condition fixing the leading soft or collinear coefficient defines the same linear functional on the Bose-compatible sector, a statement not proved beyond eight points.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies whether a tree-level MHV gravity numerator is determined by its degree and by vanishing on the simultaneous pair loci <ij>=[ij]=0 for every pair, in the reduced form A_n=N_n/D_n with D_n=∏<ij>. The authors replace the original quotient-ring calculation by a flag-variety standard-monomial basis and an S_n-isotypic decomposition. They prove at seven points that W_7,ℚ ≅ S^{(2,1^5)}⊕S^{(1^7)}, so the pair-ideal conditions leave a six-dimensional hook in addition to the alternating Hodges line; this disproves the original one-dimensional algebraic uniqueness conjecture. Bose symmetry selects the sign component and gives the seven-point Hodges line. At eight points they compute (W_8,ℚ)_sgn = ℚA⊕ℚB, a two-dimensional plane, and show that same-helicity BCFW O(z^{-2}) falloff, normalized ++ collinear factorization, and the leading soft coefficient each impose the same linear condition 7c_A-6c_B=0, selecting the Hodges line. They also prove an all-multiplicity marked-collinear injectivity theorem and a recursive physical uniqueness theorem within the fixed-common-denominator ansatz, conditional on the standard factorization and soft theorems as external inputs. The finite-dimensional calculations are supported by exact integer/rational arithmetic, modular minors under two primes, explicit ideal-reduction lifts, and a Lean-verified consequence layer.

Significance. If the results hold, they cleanly separate algebraic, Bose-compatible, and physical uniqueness for MHV gravity numerators: the pair-ideal conditions are not sufficient, Bose symmetry removes the hook at seven points but not at eight, and one normalized boundary condition restores uniqueness. The paper's computational architecture is a genuine strength: the flag-variety tableau basis, S_n-resolved restriction blocks, exact rank bounds from modular minors, explicit 60-coefficient hook polynomial, global ideal-membership proofs with generator lifts, and the Lean formalization of the finite-dimensional consequence layer are all unusually well documented. The eight-point demonstration that BCFW scaling, normalized collinear factorization, and the leading soft coefficient reduce to the same linear condition is a valuable cross-check. The main all-multiplicity theorem, however, is conditional on a geometric domain/prime-divisor property that is stated only for n=7; filling that gap is necessary before the advertised all-n uniqueness statement is fully supported.

major comments (2)
  1. [Lemma 6.1, Theorem 6.2; Section 3.1; Appendix D.3] The all-multiplicity injectivity chain rests on an unproved domain/prime-divisor property. The proof of Lemma 6.1 passes from vanishing on the marked loci to f=<ab>g with the words "because this is a reduced Cartier divisor," and Theorem 6.2 then says that the distinct angle-bracket factors may be extracted successively. Both steps require Q_{n,ℚ} to be an integral domain, each <ab> to be a prime divisor, and the successive quotient by several <ab>'s to remain a domain (or at least a regular-sequence-type property). The paper states the domain property only at n=7, in Appendix D.3 ("Since Q_7,ℚ is a domain"), and gives no proof or citation for general n. If the identification of Q_{n,k} with the multihomogeneous coordinate ring of Fl(2,n-2;k^n) in Section 3.1 is intended to supply this, then the authors should state it explicitly as a proposition — including that this ring is a domain and that the Schubert divisors V(<ab>) are prime with prime successive intersections — with a proof or a precise reference. As written, Theorem 6.2, Corollary 6.3, and Theorem 6.4 are unsupported for n>7. This is a presentation gap in a load-bearing step rather than a detected contradiction, but it must be fixed before the all-multiplicity claim is accepted.
  2. [Theorem 6.4 and Section 6.3] The all-multiplicity physical uniqueness theorem is conditional on the universal ++ splitting law and on the five-point Hodges seed, and the paper is honest about this. However, the main-text statement of Theorem 6.4 should explicitly list all hypotheses in the theorem environment: the fixed-common-denominator ansatz, the fixed multidegree, the alternating character, one complete normalized ++ marked-collinear boundary, and the external factorization law. Currently these assumptions are distributed between the surrounding prose and Appendix A.2, which makes it easy to over-read the theorem as an unconditional derivation of MHV gravity from pair zeros. This is a scope-of-claim issue rather than a mathematical error, but it matters for the paper's advertised conclusion.
minor comments (4)
  1. [Equation (2.7)] The multidegree d(n) is written with entries n-5,...,n-5 without specifying how many entries there are or how the ordering aligns with the Hodge convention; please state this explicitly at first use, even though Appendix A later clarifies it.
  2. [Table 1 and Appendix C.3] There are typographical artifacts such as "T able 1," and the 60-coefficient hook table in Appendix C.3 would be easier to verify if the row-major basis IDs were explained in one sentence right before the table.
  3. [Appendix G.1] The BCFW selection uses one displayed kinematic point; the argument is logically sufficient because the Hodges line is known to have O(z^{-2}) falloff, but the text should state explicitly that [z^6]N is a nonzero linear functional on U8 whose kernel is the Hodges line, so the reader does not wonder whether the single-point evaluation is meant to prove a global coefficient identity.
  4. [Appendix D.3 and Section 5] The sentence "Since Q_7,ℚ is a domain" appears without justification; even if this follows from the flag-variety realization, give a citation or a one-line argument so that the localization step at n=7 is self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: exact computations and independent physical inputs select the Hodges line; the only caveat is an unproved domain/prime-divisor step in Lemma 6.1, which is a gap, not circularity.

full rationale

The derivation chain is largely self-contained and non-circular. The seven-point decomposition W_{7,Q} ≅ S^{(2,1^5)} ⊕ S^{(1^7)} is obtained from an explicit flag-variety standard-monomial basis, exact Young-projector rank bounds, and exact ideal reductions (Appendices B-D); the hook and sign representatives are proved to lie in all pair ideals by exact reductions with lifts, not by evaluation. The eight-point sign sector is bounded by exact character and stacked-rank computations and matched by the two explicit alternants A and B with global membership identities (Appendix F). The physical selectors in Section 7 and Appendix G are genuine boundary data: the BCFW z^6 coefficient, the normalized collinear residue, and the leading Weinberg coefficient are computed from A and B by exact arithmetic and compared with independent physical inputs (O(z^{-2}) gravity falloff, universal ++ splitting, soft theorem); none of these inputs is fitted to the eight-point Hodges numerator, and all three yield the same primitive condition 7x−6y=0, which identifies Q(6A+7B) only at the end. The self-citations [7] and [8] are related-work references and carry no load in the uniqueness proofs. The only flagged weakness is an unproved geometric hypothesis in Lemma 6.1: the proof passes from vanishing on D_ab to divisibility by ⟨ab⟩ with "Because this is a reduced Cartier divisor, f=⟨ab⟩g", which requires Q_{n,Q} to be a domain and each ⟨ab⟩ to be a prime divisor; Appendix D.3 states the domain property only for n=7, while Section 3.1 identifies Q_{n,k} with the coordinate ring of Fl(2,n−2;k^n), which would supply it, but the paper does not spell out the general prime-divisor argument. This is a presentation/correctness gap, not a circular reduction; it does not make any prediction identical to its input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants or invented entities appear in the proof. The only adjustable quantities, the boundary ratio c and the coefficients x,y in the eight-point plane, are either arbitrary nonzero labels or the coordinates being constrained by the physical conditions. The axioms are standard algebraic-geometry facts about the spinor-helicity and flag quotient and standard cited physical theorems on factorization, BCFW scaling, and soft limits. The paper is unusually honest that the physical inputs are assumptions: Appendix I.2 states that a Lean build does not establish that the imported data and definitions express the intended physics, leaving that correspondence as the authors' responsibility.

assumptions (6)
  • domain assumption Q_{n,Q} is an integral domain for every n at least 5, so <ab> is a prime or reduced Cartier divisor and divisibility by <ab> can be read off from vanishing on the pair divisor.
    Used in Lemma 6.1 and Theorem 6.2 to write f=<ab>g and to extract all angle-bracket factors successively. The paper states the domain property only at n=7 in Appendix D.3 and does not prove it for general n.
  • domain assumption The spinor-helicity variety is irreducible and the pair divisor V(<ab>) is reduced and Cartier for all pairs, so a function vanishing on the dense marked loci vanishes on the full divisor.
    Basis of the density and Cartier argument in Lemma 6.1; used without an explicit all-n proof.
  • domain assumption The universal ++ gravity splitting law fixes the normalized collinear boundary line L_{n,ab,c} from the (n-1)-point Hodges amplitude.
    Physical input to Theorem 6.4 and to the collinear selection at eight points; cited to Ref. [17], not derived in the paper.
  • domain assumption Einstein gravity tree amplitudes have BCFW large-z falloff O(z^{-2}) under the [1,2> shift.
    External physical input used in Theorem 7.1; cited to Refs. [11,18,19].
  • domain assumption The leading positive-helicity soft-graviton theorem gives the normalized epsilon^{-3} coefficient as the Weinberg operator times A7.
    Input to the soft selection in Section 7.3 and Appendix G.3; cited to Ref. [20].
  • standard math Standard-monomial theory for the multihomogeneous coordinate ring of the two-step flag variety, including straightening laws and Kostka multiplicity dim V7 = 65,870.
    Computational representation used throughout Sections 3 and 4 and Appendix A; accepted as background from Refs. [13,14].

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Pith. "Pith review of Algebraic versus physical uniqueness of MHV gravity numerators." pith.science (2026). https://pith.science/paper/73ALTXY3

@misc{pith2026260811792,
  author       = {Pith},
  title        = {Pith review of: Algebraic versus physical uniqueness of MHV gravity numerators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73ALTXY3}},
  note         = {Machine review of arXiv:2608.11792}
}
abstract

We study whether a tree-level MHV gravity numerator is determined by its degree and by vanishing on $\langle ij\rangle=[ij]=0$ for every pair. A flag-variety standard-monomial basis and an $S_n$-resolved restriction map reduce the problem to exact finite-dimensional calculations. At seven points we find $W_{7,\mathbb{Q}}\simeq S^{(2,1^5)}\oplus S^{(1^7)}$. The Hodges numerator spans the sign summand, while the six-dimensional hook gives additional algebraic solutions. The pair-ideal conditions therefore do not determine a unique algebraic solution, but Bose symmetry selects the Hodges line. At eight points, pair-ideal conditions and Bose symmetry leave a two-dimensional alternating space. Same-helicity BCFW scaling, normalized collinear factorization, and the leading soft coefficient impose the same linear condition and select the Hodges line. We also prove that, at arbitrary multiplicity, an alternating fixed-degree numerator is determined by its full value on one collinear boundary with the marked legs and their spinor ratio fixed. Together with standard factorization, this determines the numerator up to normalization within the fixed-common-denominator ansatz. All rank and ideal-membership calculations use exact integer or rational arithmetic, and their finite-dimensional consequences are checked separately in Lean.

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Reference graph

Works this paper leans on

39 extracted references · 11 canonical work pages

  1. [2]

    Koefler, U

    J. Koefler, U. Oktem, S. Paranjape, J. Trnka and B. Zacovic,Uniqueness of MHV gravity amplitudes,Le Matematiche80(2025) 347 [2412.08713]

    arXiv 2025

  2. [1]

    Arkani-Hamed, L

    N. Arkani-Hamed, L. Rodina and J. Trnka,Locality and unitarity from singularities and gauge invariance,Phys. Rev. Lett.120(2018) 231602 [1612.02797]

    arXiv 2018

  3. [3]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons,JHEP10(2024) 231 [2312.16282]

    arXiv 2024

  4. [4]

    Rodina,Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes inTr(ϕ 3)theory,Phys

    L. Rodina,Hidden zeros are equivalent to enhanced ultraviolet scaling and lead to unique amplitudes inTr(ϕ 3)theory,Phys. Rev. Lett.134(2025) 031601 [2406.04234]

    arXiv 2025

  5. [5]

    Paranjape and J

    S. Paranjape and J. Trnka,Gravity amplitudes from double bonus relations,Phys. Rev. Lett. 131(2023) 251601 [2309.05710]. – 39 –

    arXiv 2023

  6. [6]

    Smooth Splitting and Zeros from On-Shell Recursion

    C.R.T. Jones and S. Paranjape,Smooth splitting and zeros from on-shell recursion,JHEP07 (2025) 251 [2505.02520]

    work page Pith review arXiv 2025

  7. [7]

    Li and Y

    S. Li and Y. Zhang,Inverse Cachazo–He–Yuan integrand construction from factorization channels with exact integer constraint propagation,Phys. Rev. D113(2026) 105006 [2508.02248]

    work page arXiv 2026

  8. [8]

    An adaptive inverse-problem framework for one-loop five-gluon BCJ numerators

    L. Mai and Y. Zhang,An adaptive inverse-problem framework for one-loop five-gluon BCJ numerators,2607.13412

    work page Pith review arXiv

Show all 39 references
  1. [9]

    Hodges,A simple formula for gravitational MHV amplitudes,1204.1930

    A. Hodges,A simple formula for gravitational MHV amplitudes,1204.1930

    1930 arXiv

  2. [10]

    Mason and D

    L. Mason and D. Skinner,Gravity, twistors and the MHV formalism,Commun. Math. Phys. 294(2010) 827 [0808.3907]

    2010 arXiv

  3. [11]

    Nguyen, M

    D. Nguyen, M. Spradlin, A. Volovich and C. Wen,The tree formula for MHV graviton amplitudes,JHEP07(2010) 045 [0907.2276]

    2010 arXiv

  4. [12]

    De Laurentis and B

    G. De Laurentis and B. Page,Ansätze for scattering amplitudes fromp-adic numbers and algebraic geometry,JHEP12(2022) 140 [2203.04269]

    2022 arXiv

  5. [13]

    El Maazouz, A

    Y. El Maazouz, A. Pfister and B. Sturmfels,Spinor-helicity varieties,2406.17331

    arXiv

  6. [14]

    Brion and V

    M. Brion and V. Lakshmibai,A geometric approach to standard monomial theory,Represent. Theory7(2003) 651 [math/0111054]

    2003 arXiv

  7. [15]

    Lee and L

    K. Lee and L. Li,Notes on a minimal set of generators for the radical ideal defining the diagonal locus of(C2)n,0901.1176

    arXiv

  8. [16]

    Lee and L

    K. Lee and L. Li,q,t-Catalan Numbers and generators for the radical ideal defining the diagonal locus of(C2)n,Electron. J. Combin.18(2011) [0909.1612]

    2011 arXiv

  9. [17]

    Bern, L.J

    Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky,Perturbative relationships between QCD and gravity and some implications,hep-th/9809163

    arXiv

  10. [18]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Kaplan,On tree amplitudes in gauge theory and gravity,JHEP04 (2008) 076 [0801.2385]

    2008 arXiv

  11. [19]

    McGady and L

    D.A. McGady and L. Rodina,Gravitons, bose symmetry, and bonus scaling,Phys. Rev. D91 (2015) 105010 [1408.5125]

    2015 arXiv

  12. [20]

    Cachazo and A

    F. Cachazo and A. Strominger,Evidence for a new soft graviton theorem,1404.4091

    arXiv

  13. [21]

    Fulton,Young Tableaux: With Applications to Representation Theory and Geometry, vol

    W. Fulton,Young Tableaux: With Applications to Representation Theory and Geometry, vol. 35 ofLondon Mathematical Society Student Texts, Cambridge University Press (1997), 10.1017/CBO9780511626241

    1997 doi

  14. [22]

    Peraro,Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP12(2016) 030 [1608.01902]

    T. Peraro,Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP12(2016) 030 [1608.01902]

    2016 arXiv

  15. [23]

    Cachazo and P

    F. Cachazo and P. Svrcek,Tree level recursion relations in general relativity,hep-th/0502160

    arXiv

  16. [24]

    Bedford, A

    J. Bedford, A. Brandhuber, B. Spence and G. Travaglini,A recursion relation for gravity amplitudes,Nucl. Phys. B721(2005) 98 [hep-th/0502146]

    2005 arXiv

  17. [25]

    Schwab and A

    B.U.W. Schwab and A. Volovich,Subleading soft theorem in arbitrary dimensions from scattering equations,Phys. Rev. Lett.113(2014) 101601 [1404.7749]

    2014 arXiv

  18. [26]

    Guevara, E

    A. Guevara, E. Himwich and N. Miller,Generating hodges’ graviton MHV formula with an lw1+∞ ward identity,2506.05460. – 40 –

    arXiv

  19. [27]

    Feng and Y

    B. Feng and Y. Zhang,Note on the labelled tree graphs,JHEP12(2020) 096 [2009.02394]

    2020 arXiv

  20. [28]

    B. Feng, C. Hu and Y. Zhang,Note on solutions of scattering equations,Phys. Rev. D105 (2022) 036015 [2105.03700]

    2022 arXiv

  21. [29]

    Arkani-Hamed, Y

    N. Arkani-Hamed, Y. Bai and T. Lam,Positive geometries and canonical forms,JHEP11 (2017) 039 [1703.04541]

    2017 arXiv

  22. [30]

    Tooby-Smith,HepLean: Digitalising high energy physics,Comput

    J. Tooby-Smith,HepLean: Digitalising high energy physics,Comput. Phys. Commun.308 (2025) 109457 [2405.08863]

    2025 arXiv

  23. [31]

    Tooby-Smith,Digitalizing Wick’s theorem,2505.07939

    J. Tooby-Smith,Digitalizing Wick’s theorem,2505.07939

    arXiv

  24. [32]

    Douglas, S

    M.R. Douglas, S. Hoback, A. Mei and R. Nissim,Formalization of QFT,2603.15770

  25. [33]

    de Moura and S

    L. de Moura and S. Ullrich,The Lean 4 theorem prover and programming language, in Automated Deduction – CADE 28, vol. 12699 ofLecture Notes in Computer Science, pp. 625–635, Springer, 2021, DOI

    2021

  26. [34]

    367–381, ACM, 2020, DOI [1910.09336]

    The mathlib Community,The Lean mathematical library, inProceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, pp. 367–381, ACM, 2020, DOI [1910.09336]

    2020 arXiv

  27. [35]

    Douglas,Axioms for physical reasoning: Codifying the Seiberg–Witten solution in Lean, 2607.06379

    M.R. Douglas,Axioms for physical reasoning: Codifying the Seiberg–Witten solution in Lean, 2607.06379

    arXiv

  28. [36]

    Cai, G.W

    T. Cai, G.W. Merz, F. Charton, N. Nolte, M. Wilhelm, K. Cranmer et al.,Transforming the bootstrap: Using transformers to compute scattering amplitudes in planarN= 4super Yang–Mills theory,Mach. Learn. Sci. Technol.5(2024) 035073 [2405.06107]

    2024 arXiv

  29. [37]

    Cheung, A

    C. Cheung, A. Dersy and M.D. Schwartz,Learning the simplicity of scattering amplitudes, SciPost Phys.18(2025) 040 [2408.04720]

    2025 arXiv

  30. [38]

    Shih,Learning to unscramble: Simplifying symbolic expressions via self-supervised oracle trajectories,2603.11164

    D. Shih,Learning to unscramble: Simplifying symbolic expressions via self-supervised oracle trajectories,2603.11164

    arXiv

  31. [39]

    Moynihan,Learning theS-matrix from data: Rediscovering gravity from gauge theory via symbolic regression,2602.15169

    N. Moynihan,Learning theS-matrix from data: Rediscovering gravity from gauge theory via symbolic regression,2602.15169. – 41 –

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