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A single shift rule generates every strong-coupling correction in N=4 SYM

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2026-08-02 21:54 UTC pith:543C4MX7

load-bearing objection A practical and likely correct reorganization of the strong-coupling transseries for matrix Bessel determinants, but the load-bearing selection rule is still a conjecture. the 3 major comments →

arxiv 2602.18557 v3 pith:543C4MX7 submitted 2026-02-20 hep-th math-phmath.MP

Strong coupling structure of mathcal{N}=4 SYM observables with matrix Bessel kernel

Bercel Boldis This is my paper
classification hep-th math-phmath.MP
keywords strong coupling expansionmatrix Bessel kernelN=4 SYMtransseriesresurgencecusp anomalous dimensionStokes constantsalien calculus
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the strong-coupling expansion of a class of N=4 supersymmetric Yang-Mills observables, encoded in a determinant with a matrix Bessel kernel, has a hidden structure: every exponentially suppressed correction is labeled by finite sets of zeros of the modified symbol, each exponential scale appears at most once, and the 1/g series of every correction is obtained from the perturbative series by a simple shift of the mixing angle a→a−Δ plus a replacement of the moments. The Stokes constants that multiply these corrections are generated by two recurrence relations. If correct, this means the full transseries of the cusp anomalous dimension, multi-gluon amplitudes and the octagon form factor can be generated from the perturbative sector alone, and the resurgence structure of the strong-coupling expansion becomes transparent.

Core claim

The central discovery is that the transseries of the determinant Z_ℓ(g) can be reorganized so that each exponential correction corresponds to a finite pair of sets (δ+, δ−) of zeros of the modified symbol χ_α(x)=cosh(x/2+iα)/sinh(x/2), with each factor e^{−8πg x^±_j} appearing at most to first order. In this reorganization, the 1/g expansion of every nonperturbative coefficient D^{(δ+,δ−)}(g) is obtained from the perturbative coefficient D[I_n](g) by the rule D^{(δ+,δ−)}(g)=D[I_n^{(δ+,δ−)}](g)|_{a→a−Δ}, where Δ=|δ+|−|δ−| and the modified moments promote the selected zeros to poles. The Stokes constants S^{(δ+,δ−)} are generated recursively by two relations that multiply the previous constant

What carries the argument

The engine of the argument is the modified symbol χ_α(x)=cosh(x/2+iα)/sinh(x/2), whose zeros x^±_j = j+1/2∓a set the exponential scales e^{−8πg x^±_j}. The paper reorganizes the transseries of the determinant into a sum over finite sets (δ+, δ−) of these zeros, the 'one-per-zero' hierarchy. The core identity is the map (3.26), which states that each nonperturbative sector's 1/g series is the perturbative series with moments I_n → I_n^{(δ+,δ−)} and the explicit mixing angle shifted a → a−Δ; the Stokes constants are then fixed by the two recurrences (3.42)–(3.43). This machinery converts the problem of generating nonperturbative corrections into a purely perturbative calculation plus combinato

Load-bearing premise

The entire hierarchy rests on the assumption that every term in the strong-coupling transseries contains each exponential scale e^{−8πg x^±_j} at most once; if any sector with a squared exponential factor or higher multiplicity exists, the reorganized expansion and all its consequences miss it.

What would settle it

Extract the exponential term e^{−16πg x^+_0} (which would require a squared factor of the first zero) from high-precision numerical evaluation of the determinant or from the underlying integro-differential equations: the paper predicts it vanishes exactly, so any nonzero value for that sector would falsify the 'at most first order' ansatz.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The full strong-coupling transseries of the cusp anomalous dimension, multi-gluon scattering amplitudes and the octagon form factor can be generated to arbitrary exponential and 1/g order from the perturbative series alone.
  • Every previously observed cancellation in the old (n,m) transseries follows automatically: sectors not labeled by finite sets of zeros simply do not exist.
  • The resurgence structure is fully described by an alien algebra in which ∆^±_j adds the zero j to the sector, squares to zero, and all alien derivatives commute; the Stokes automorphism factorizes into first-order factors.
  • The coefficients of the alien derivatives are universal, equal to ∓2i sin(πa), independent of the sector, so the lateral resummation ambiguity is exactly cancelled by the complex phases e^{iπa∆}.
  • The median resummation of the reorganized transseries is real and supplies the physical strong-coupling answer.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the 'one-per-zero' hierarchy holds, any observable governed by a Bessel-type determinant with a meromorphic symbol should exhibit the same shift rule; one could test this against the O(6) model or generalized energy densities whose strong-coupling expansions are known.
  • The recurrence relations for the Stokes constants resemble a transfer-matrix product over zeros; they may admit a closed form as a product of Gamma functions, which would make the whole transseries a completely explicit analytic object.
  • The universality of the alien coefficients ∓2i sin(πa) suggests the Stokes automorphism in this regime is a fixed rotation in the space of sectors; it would be interesting to see whether the same constant appears in the weak-coupling resurgence of the same observables.
  • One could invert the shift rule and use numerical data from the first few nonperturbative sectors to bootstrap high-order perturbative coefficients, giving a new way to extract perturbative data from strong-coupling information.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the strong-coupling expansion of the determinant (1.1) with the matrix Bessel kernel and symbol (1.5), which encodes several planar N=4 SYM observables. Building on the prior transseries (1.7) from [24] and the α=0 structure of [47], the author proposes a reorganization (3.9) in which every exponentially suppressed sector is labelled by finite sets δ+,δ− of zeros of the modified symbol (1.9), with no factor e^{-8πg x^±_j} appearing more than once. The 1/g expansion of each sector is claimed to follow from the perturbative series by the replacement rule (3.26), and the Stokes constants are claimed to obey the recurrences (3.42)–(3.43). Section 4 uses the Bridge equations to derive an alien calculus (4.9)–(4.18), verifies the leading alien coefficients numerically, and constructs a median resummation. The paper closes with explicit applications to the cusp anomalous dimension at a=1/4.

Significance. If the conjectured structure (3.9) is correct, it gives an extremely efficient and conceptually clean way to generate the full strong-coupling transseries and resurgence data for the cusp anomalous dimension, multi-gluon amplitudes, and the octagon form factor, unifying them with the Tracy-Widom-type structure found in [47]. The numerical checks are substantive: 50 1/g terms at 50-digit precision for three parameter sets, recurrences checked up to Λ^8_-Λ^8_+, and reproduction of the known cusp-anomalous-dimension results [43,45,54]. The paper also makes concrete, falsifiable predictions for the alien algebra. However, the central expansion (3.9) is explicitly conjectural, and several coefficients — including the (−1)^Δ prefactors and the alien coefficients A±_l — are fitted numerically rather than derived. The paper's strength is its computational verification; its main weakness is that the load-bearing selection rule is not proven.

major comments (3)
  1. [Sec. 3, Eqs. (3.4)–(3.9), (2.19)] The central ansatz (3.9) is a conjecture, as the author states, but it is load-bearing: the recurrences (3.42)–(3.43) and the alien algebra (4.9)–(4.18) presuppose that the transseries contains no sector with a squared exponential factor e^{-16πg x^±_j} and no higher multiplicity. The text supports this only by matching exponents to the empirical condition (2.19) and by the α=0 case of [47]. This is not an analytic selection rule. The printed inequality (2.19), 0≤m−n≤p, is moreover inconsistent with the leading sector (n,m)=(1,0), for which m−n=−1, so the empirical rule itself is not stated correctly. The author should either prove the cancellation of all sectors outside (3.5) directly from the determinant/Fredholm structure, or clearly frame the paper as a conjectural framework with finite-order checks. As written, any missing multiplicity-2 sector would invalidate both the recurrence r
  2. [Sec. 3.1, Eqs. (3.22)–(3.26)] The rule (3.26), D^{(δ+,δ−)}(g)=D[I_n^{(δ+,δ−)}](g)|_{a→a−Δ}, is motivated by the observation that the leading sectors are obtained by promoting zeros to poles, as in (3.18)–(3.21). But for general δ+,δ− this promotion is asserted, not derived: the modified symbol (3.22) is an invented object, and the paper verifies (3.26) only for a handful of non-degenerate and low-lying degenerate sectors (three parameter sets, up to Λ^8_-Λ^8_+). Since the D^{(δ+,δ−)} are the building blocks of the entire construction, an independent derivation — for instance from the integro-differential system of [24] — is needed before the claim 'completely describes the large-g expansion' is justified.
  3. [Sec. 3.2 and Sec. 4, Eqs. (3.40)–(3.43), (4.15)–(4.18)] The (−1)^Δ factors in the Stokes recurrences (3.40)–(3.41) and the explicit formulas (3.42)–(3.43) are fitted to numerical data up to Λ^5_-Λ^5_+; the paper states that their origin is understood only after a later phase redefinition. Similarly, the alien coefficients A±_l in (4.15) are extracted numerically up to l=2 and then extrapolated to (4.17). These are essential ingredients of the claimed 'complete' resurgence description. The paper should either derive these constants from the original transseries (1.7) or from exact asymptotics, or clearly label the alien algebra as a numerically supported conjecture rather than a derived result.
minor comments (4)
  1. [Throughout]
  2. [Sec. 3.3, Eqs. (3.53)–(3.55)]
  3. [Sec. 4.2, Fig. 2]
  4. [References]

Circularity Check

0 steps flagged

No significant circularity: the new transseries reorganization is explicitly conjectural and is checked against the prior transseries and external benchmarks, though some constants are numerically fitted.

full rationale

The central claim (3.9) is introduced explicitly as a conjecture ('I conjecture that the strong coupling expansion of Z_l(g) is given by...') and is verified numerically against the prior transseries (1.7) and against independent external results [43,45,54]. The rule (3.26), relating each non-perturbative D^{(δ+,δ−)} to the perturbative functional with shifted moments, is a nontrivial statement tested on 50 1/g-coefficients at 50-digit precision for several parameter values; it is not an identity by construction. The Stokes-constant recurrences (3.42)-(3.43) and the Alien-derivative coefficients A±_l are admittedly inferred from numerical data ('The prefactors (−1)^Δ ... were introduced by investigating numerical results', 'with a 10−6 relative error, I found...'), but the paper does not present these fitted values as first-principles predictions. Use of the author's earlier work [24,47] is substantial, but it is continuation rather than circularity: the old transseries is an independent input obtained from integro-differential equations, and no uniqueness theorem from the authors is invoked to exclude alternatives. The finite-set ansatz (3.5) is a real unproven assumption and a correctness risk, but it is openly labelled as inferred and conjectural, not disguised as a derivation. Overall, no load-bearing step reduces by construction to its own input.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 1 invented entities

The central structural claims rest on the prior transseries (1.7) of [24], on the α=0 universality of [47], and on a Wiener-Hopf factorization of the symbol. The genuinely new ingredients — the promotion-of-zeros rule, the Stokes recurrences and the universal alien constants — are conjectured or fitted against numerical data rather than derived. No new physical entities are postulated.

free parameters (3)
  • Alien-derivative coefficients A±_l = ∓2i sin(πa), conjectured from numerically extracted l=0,1,2 values
    In Sec. 4.2 the first few A±_l are extracted from large-order asymptotics of the numerically generated coefficients, then the general form (4.17) is conjectured. These constants control the whole Stokes automorphism (4.18).
  • Sign prefactors (−1)^Δ in the Stokes recurrences = (−1)^Δ
    In Sec. 3.2 the prefactors are introduced ad hoc after 'investigating numerical results with different values of a and ℓ up to Λ^5_-Λ^5_+'; they are matched to data rather than derived from first principles.
  • Widom-Dyson constant B_ℓ(a) = fitting expression from Eq. (5.18) of [24]
    The overall prefactor A_ℓ(g) in (1.11) uses B_ℓ(a) obtained by numerical fitting in the author's earlier paper [24]; it is not re-derived here.
axioms (6)
  • domain assumption The determinant representation (1.1)-(1.5) correctly describes the cusp anomalous dimension, multi-gluon amplitudes, and octagon form factor in planar N=4 SYM.
    Taken from prior literature [5-27]; if this correspondence is wrong, the physical interpretation of all transseries results fails.
  • domain assumption The strong-coupling transseries (1.7) derived in [24] is complete and correct, and can serve as the benchmark for verifying the new form (3.9).
    The paper's numerical checks compare (3.9)/(3.26) against (1.7) from the author's own [24]; any error or omitted sector in [24] propagates into this work.
  • domain assumption The Wiener-Hopf factorization (3.1)-(3.2) of the modified symbol χ_α and its zeros x^±_j determine all exponential scales of the transseries.
    The new finite-set parameterization and the promotion-of-zeros rule (3.22)-(3.24) rest on this analytic structure.
  • ad hoc to paper The construction (3.22)-(3.23) that promotes zeros to poles produces the correct nonperturbative series D^{(δ+,δ−)}.
    Rule (3.26) is motivated by analogy with [47] and verified numerically for a few sectors; it is not derived from the integral equations.
  • ad hoc to paper The Bridge equations and the resulting Alien algebra (4.9)-(4.10) apply to the two-sector transseries (3.9) with complex phases e^{iπaΔ}.
    Appendix A derives the algebra for a simpler one-sector transseries; the extension to two weight sectors and phases is assumed, with undetermined prefactors fitted numerically.
  • standard math Standard Fredholm determinant, Bessel kernel, Borel resummation and transseries techniques are valid as used.
    Background material from [4,23,28-42] is relied on throughout.
invented entities (1)
  • Modified symbol functions χ^{(δ+,δ−)}_α no independent evidence
    purpose: Generate the nonperturbative 1/g expansions D^{(δ+,δ−)}(g) by replacing the moment integrals I_n with I_n^{(δ+,δ−)}.
    These auxiliary functions are defined in (3.22)-(3.23) by promoting zeros to poles. They have no direct empirical handle outside the computed expansions, although the resulting coefficients match known cusp results.

pith-pipeline@v1.3.0-alltime-deepseek · 32848 in / 13067 out tokens · 113395 ms · 2026-08-02T21:54:16.556021+00:00 · methodology

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read the original abstract

In this paper I continue the program of studying the strong coupling expansion of certain observables in $\mathcal{N}=4$ supersymmetric Yang-Mills theory, which are given by a determinant with a matrix Bessel kernel. I show that, by reorganizing the transseries of the determinant at large values of the 't Hooft coupling, a simple underlying structure emerges, in which each exponentially suppressed correction is related to the perturbative series in a simple way. This new approach provides an efficient method to generate the full transseries for $\mathcal{N}=4$ SYM observables, such as the cusp anomalous dimension, multi-gluon scattering amplitudes, and the octagon form factor. Using high-precision numerical analysis, I verify the results and provide a complete description of the resurgence structure of the strong coupling expansion.

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

  1. From Fredholm Determinants to AdS/CFT Observables: A Universal Strong-Coupling Framework

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