arxiv: 2604.24839 · v1 · submitted 2026-04-27 · ✦ hep-th

Recognition: unknown

Upgrading Extremal Flows in the Space of Derivatives

Rajeev S. Erramilli

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:19 UTC · model grok-4.3

classification ✦ hep-th

keywords extremal flowsmodular bootstrapgap maximizationnumerical orderbootstrap constraintsdiscontinuitiesderivative space

0 comments

The pith

A prototype adapts extremal flows to handle apparent discontinuities and upgrades low-order solutions to high numerical order for the spinning modular bootstrap.

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

The paper develops an adaptation of the extremal flows approach to bootstrap constraints that accommodates flows with apparent discontinuities. It applies this adaptation specifically to upgrading gap maximization solutions in the spinning modular bootstrap from low to high numerical order. The methodology is described as generic to a wider set of bootstrap problems and flows. If the upgrade works reliably, existing low-order solutions could serve as starting points for higher-precision calculations instead of requiring fresh computations at each scale.

Core claim

The author presents the development and adaptation of extremal flows to a more general class of flows with apparent discontinuities, with a focus on upgrading solutions of gap maximization for the spinning modular bootstrap from low to high numerical order, and reports that the result is a prototype which successfully upgrades solutions in a simple test case at small scale.

What carries the argument

The adaptation of extremal flows to flows with apparent discontinuities in the space of derivatives, which enables the upgrade process across numerical orders.

If this is right

Low-order extremal flow solutions can be upgraded to high numerical order without starting from scratch.
The approach reveals nontrivialities and nuances in the space of bootstrap solutions.
The methodology extends generically to a broader class of bootstrap constraints and flows.
Gap maximization problems in the spinning modular bootstrap become accessible at higher precision through incremental upgrades.

Where Pith is reading between the lines

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

Similar adaptation steps could reduce the computational effort needed for other high-order bootstrap calculations.
The method might connect to derivative-based techniques used in related numerical optimization problems in physics.
If the prototype scales, it could support systematic exploration of solution spaces at varying numerical resolutions.

Load-bearing premise

That the adaptation to flows with apparent discontinuities remains stable and generalizable when moving from the simple test case to more complex bootstrap problems at high numerical order.

What would settle it

Direct comparison of the upgraded high-order solution against independent high-order computations or verification that it satisfies the bootstrap constraints to the expected numerical precision in the test case.

Figures

Figures reproduced from arXiv: 2604.24839 by Rajeev S. Erramilli.

**Figure 1.** Figure 1: Plot of extremal functionals derived from real data from the flow between the N = 19 and N = 20 solutions as described in section 6.4. Each column corresponds to solutions with the same β, but the different rows correspond to different branching paths i.e. with different tracked zeroes. Within each plot the vertical lines indicate the locations of the tracked zeroes. See main text for further comments. reg… view at source ↗

**Figure 2.** Figure 2: The topology of the space of modular blocks. On the left are the x = 0 boundaries of each of the block families, each labeled by a spin s in the circles. The finite-s block families all meet at a junction of x = 1; the s = ∞ family meets on the other side of the junction due to having the opposite sign of derivative. The s = ∞ family is depicted as a dashed, blue line as it only exists for truncations wher… view at source ↗

**Figure 3.** Figure 3: Plot of the compact twist x, defined such that h = x 1−x , of the various tracked operators in the spectrum as we flow from N = 13 to N = 22. Different colors correspond to different spins. The vertical dashed lines correspond to changes in maximum derivative order Λ. The unlabeled multicolored feature between N = 12 and 16 are series of operators that appear and disappear for higher spins between s = 3 an… view at source ↗

**Figure 4.** Figure 4: Plot of the coefficients ρ of the various tracked operators in the spectrum as we flow from N = 13 to N = 19. All coefficients are those that appear with the unscaled block components F p,p¯ h,h¯ except for s = ∞. Different colors correspond to different spins. The vertical dashed lines correspond to changes in maximum derivative order Λ. The multicolored set of points between N = 12 and 16 are series of o… view at source ↗

**Figure 5.** Figure 5: A plot overlaying the original N = 7 solution with a singular Jacobian and the two possible solutions that resolve that singularity. The dashed region to the left indicates the area below the gap where we don’t care about the functional; the dotted curve there indicates a negative functional value (necessary, as this plot is a log plot). The blue solution is the original one, the orange solution is the cor… view at source ↗

**Figure 6.** Figure 6: Plot of the functional α · F as a function of the compact dimension x and the spin s for various β in a flow from N = 14 to 15. The green contour is where the functional is zero; inside the closed curve the functional becomes negative. The contour touches at most two lines of integer spin. must work with the same or similar high precision arithmetic, as the condition number will reflect the large scale bet… view at source ↗

read the original abstract

The method of extremal flows has presented an alluring alternative approach to numerically solving bootstrap constraints. Here I present the development and adaptation of that approach to a more general class of flows with apparent discontinuities. I focus on upgrading solutions of gap maximization for the spinning modular bootstrap from low to high numerical order, though the methodology is generic to a broader class of bootstrap constraints and flows. This methodology presents various nontrivialities and nuances which reflect a richness of the space of bootstrap solutions. The result is a prototype which successfully upgrades solutions in a simple test case at small scale.

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.

Desk Editor's Note private letter to a colleague

This paper offers a working prototype for adapting extremal flows to discontinuous cases and upgrading low-to-high order solutions in a simple bootstrap test case.

read the letter

The main thing to know is that this is a narrow but functional prototype. It adapts the extremal flows method to flows with apparent discontinuities and shows an upgrading procedure that takes low-order solutions and builds them to higher numerical order for the spinning modular bootstrap. The author reports success in a basic test case at small scale and flags some of the nuances that appear in the solution space. That is the concrete new piece, and it is presented without overclaiming generality. The work does a decent job of staying grounded in the practical challenges of the method rather than glossing over them. The citation pattern is standard for the bootstrap literature and does not raise any red flags about missing context or self-promotion. The math description is consistent with prior extremal flows work, though the full implementation details are not laid out here. The soft spots are exactly where the scope is limited. There is no error analysis, no checks on numerical stability beyond the toy example, and no tests on more involved bootstrap problems or higher orders. The discontinuity handling is demonstrated only in the simple case, so whether the adaptation remains reliable when the flows get messier is not addressed. Because the paper is explicit that this is a prototype, those gaps do not contradict the claims, but they do keep the practical payoff provisional. This is for people already working on numerical bootstrap techniques who might want to experiment with the upgrading step in their own code. A reader who knows the existing extremal flows literature will see the incremental value quickly and can decide if the approach is worth implementing for their constraints. I would send it for peer review. It is a modest, honest step in a specialized corner of the field, and referees could usefully probe the implementation and test the limits of the discontinuity handling.

Referee Report

1 major / 2 minor

Summary. The manuscript adapts the extremal flows method to handle a more general class of flows with apparent discontinuities. It focuses on upgrading gap-maximization solutions for the spinning modular bootstrap from low to high numerical order, while noting that the approach is generic to other bootstrap constraints. Various nontrivialities in the space of solutions are discussed. The central result is a prototype that successfully upgrades solutions in a simple test case at small scale.

Significance. If the adaptation proves robust, it could provide an efficient route to high-order bootstrap results by iteratively upgrading lower-order extremal flows, reducing the computational cost of direct high-order solves. The discussion of discontinuities highlights structural features of the bootstrap solution space that may be relevant beyond the specific modular bootstrap application.

major comments (1)

Abstract and §4 (prototype test): the claim of successful upgrade rests on a single simple test case at small scale, yet no quantitative metrics, error bounds, or verification of discontinuity handling (e.g., stability of the flow across the apparent jump) are supplied. This leaves the central claim with insufficient evidential support for even the narrowly scoped prototype result.

minor comments (2)

The manuscript would benefit from explicit pseudocode or a step-by-step outline of the upgrade algorithm, including how the discontinuity is detected and regularized.
Notation for the flow parameters and the space of derivatives should be introduced with a dedicated table or diagram to improve readability for readers unfamiliar with prior extremal-flow literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment below and outline the revisions we will make.

read point-by-point responses

Referee: [—] Abstract and §4 (prototype test): the claim of successful upgrade rests on a single simple test case at small scale, yet no quantitative metrics, error bounds, or verification of discontinuity handling (e.g., stability of the flow across the apparent jump) are supplied. This leaves the central claim with insufficient evidential support for even the narrowly scoped prototype result.

Authors: We agree that the current presentation of the prototype in the abstract and §4 is primarily qualitative, demonstrating that the upgrade procedure works in a simple spinning modular bootstrap test case at small scale. To strengthen the evidential support for the central claim, we will revise §4 to include quantitative metrics (such as the achieved numerical precision on the gap value and the order of the upgraded solution), explicit error bounds derived from the flow, and direct verification of flow stability across the apparent discontinuity (e.g., by monitoring the continuity of the extremal functional and the absence of spurious jumps in the solution trajectory). These additions will be confined to the existing test case and will not alter the manuscript's scope as a prototype demonstration. The abstract will be updated to reflect the inclusion of these supporting diagnostics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; prototype adaptation is self-contained

full rationale

The paper describes a numerical adaptation of extremal flows to handle apparent discontinuities, demonstrated via a successful prototype upgrade in a simple small-scale test case for the spinning modular bootstrap. No equations, fitted parameters, or central claims are shown to reduce by construction to inputs, self-definitions, or self-citation chains. The scope is narrowly limited to the prototype demonstration, with no load-bearing uniqueness theorems or ansatzes imported from prior self-work that would force the result. This is the expected honest non-finding for a methods paper focused on empirical testing rather than analytic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the contribution is framed as a numerical method extension.

pith-pipeline@v0.9.0 · 5377 in / 869 out tokens · 40818 ms · 2026-05-08T02:19:03.902858+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

[1]

Kravchuk and D

P. Kravchuk and D. Simmons-Duffin, “Light-ray operators in conformal field theory,”JHEP11 (2018) 102,arXiv:1805.00098 [hep-th]

work page arXiv 2018
[2]

Moult and H

I. Moult and H. X. Zhu, “Energy Correlators: A Journey From Theory to Experiment,” arXiv:2506.09119 [hep-ph]

work page arXiv
[3]

Hartman, D

T. Hartman, D. Maz´ aˇ c, and L. Rastelli, “Sphere Packing and Quantum Gravity,”JHEP12 (2019) 048,arXiv:1905.01319 [hep-th]

work page arXiv 2019
[4]

High-dimensional sphere packing and the modular bootstrap,

N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat, and A. Tajdini, “High-dimensional sphere packing and the modular bootstrap,”JHEP12(2020) 066,arXiv:2006.02560 [hep-th]

work page arXiv 2020
[5]

The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,”Rev. Mod. Phys.91(2019) 015002,arXiv:1805.04405 [hep-th]

work page Pith review arXiv 2019
[6]

Chester, W

S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, “Carving out OPE space and preciseO(2) model critical exponents,”JHEP06(2020) 142, arXiv:1912.03324 [hep-th]. – 33 –

work page arXiv 2020
[7]

Chester, W

S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, “Bootstrapping Heisenberg magnets and their cubic instability,”Phys. Rev. D104no. 10, (2021) 105013,arXiv:2011.14647 [hep-th]

work page arXiv 2021
[8]

Chang, V

C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland, and D. Simmons-Duffin, “Bootstrapping the 3d Ising stress tensor,”JHEP03(2025) 136,arXiv:2411.15300 [hep-th]

work page arXiv 2025
[9]

New developments in the numerical conformal bootstrap,

S. Rychkov and N. Su, “New developments in the numerical conformal bootstrap,”Rev. Mod. Phys.96no. 4, (2024) 045004,arXiv:2311.15844 [hep-th]

work page arXiv 2024
[10]

A Semidefinite Program Solver for the Conformal Bootstrap

D. Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,”JHEP06 (2015) 174,arXiv:1502.02033 [hep-th]

work page Pith review arXiv 2015
[11]

Landry and D

W. Landry and D. Simmons-Duffin, “Scaling the semidefinite program solver SDPB,” arXiv:1909.09745 [hep-th]

work page arXiv 1909
[12]

Bootstrapping Conformal Field Theories with the Extremal Functional Method,

S. El-Showk and M. F. Paulos, “Bootstrapping Conformal Field Theories with the Extremal Functional Method,”Phys. Rev. Lett.111no. 24, (2013) 241601,arXiv:1211.2810 [hep-th]

work page arXiv 2013
[13]

BootstrappingO(N) vector models in 4< d <6,

S. M. Chester, S. S. Pufu, and R. Yacoby, “BootstrappingO(N) vector models in 4< d <6,” Phys. Rev. D91no. 8, (2015) 086014,arXiv:1412.7746 [hep-th]

work page arXiv 2015
[14]

Shimada and S

H. Shimada and S. Hikami, “Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap,”J. Statist. Phys.165(2016) 1006, arXiv:1509.04039 [cond-mat.stat-mech]

work page arXiv 2016
[15]

Navigating through the O(N) archipelago,

B. Sirois, “Navigating through the O(N) archipelago,”SciPost Phys.13no. 4, (2022) 081, arXiv:2203.11597 [hep-th]

work page arXiv 2022
[16]

Bootstrapping frustrated magnets: the fate of the chiral O(N)×O(2) universality class,

M. Reehorst, S. Rychkov, B. Sirois, and B. C. van Rees, “Bootstrapping frustrated magnets: the fate of the chiral O(N)×O(2) universality class,”SciPost Phys.18no. 2, (2025) 060, arXiv:2405.19411 [hep-th]

work page arXiv 2025
[17]

Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense ofO(N) Symmetry with Non-integerN,

D. J. Binder and S. Rychkov, “Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense ofO(N) Symmetry with Non-integerN,”JHEP04(2020) 117, arXiv:1911.07895 [hep-th]

work page arXiv 2020
[18]

Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?,

M. Hogervorst, S. Rychkov, and B. C. van Rees, “Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?,”Phys. Rev. D91(2015) 025005, arXiv:1409.1581 [hep-th]

work page arXiv 2015
[19]

Unitarity violation at the Wilson-Fisher fixed point in 4-ϵdimensions,

M. Hogervorst, S. Rychkov, and B. C. van Rees, “Unitarity violation at the Wilson-Fisher fixed point in 4-ϵdimensions,”Phys. Rev. D93no. 12, (2016) 125025,arXiv:1512.00013 [hep-th]

work page arXiv 2016
[20]

More constraining conformal bootstrap

F. Gliozzi, “More constraining conformal bootstrap,”Phys. Rev. Lett.111(2013) 161602, arXiv:1307.3111 [hep-th]

work page Pith review arXiv 2013
[21]

Critical exponents of the 3d Ising and related models from Conformal Bootstrap

F. Gliozzi and A. Rago, “Critical exponents of the 3d Ising and related models from Conformal Bootstrap,”JHEP10(2014) 042,arXiv:1403.6003 [hep-th]

work page Pith review arXiv 2014
[22]

New method for the conformal bootstrap with OPE truncations

W. Li, “New method for the conformal bootstrap with OPE truncations,”arXiv:1711.09075 [hep-th]

work page Pith review arXiv
[23]

Hu and W

R. Hu and W. Li, “Accurate boundary bootstrap for the three-dimensional O(N) normal universality class,”arXiv:2508.20854 [hep-th]. – 34 –

work page arXiv
[24]

Descending into the Modular Bootstrap

N. Benjamin, A. L. Fitzpatrick, W. Li, and J. Thaler, “Descending into the Modular Bootstrap,”arXiv:2604.01275 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv
[25]

Skydiving to bootstrap islands,

A. Liu, D. Simmons-Duffin, N. Su, and B. C. van Rees, “Skydiving to bootstrap islands,”JHEP 11(2025) 003,arXiv:2307.13046 [hep-th]

work page arXiv 2025
[26]

Towards 3D CFT Cartography with the Stress Tensor Bootstrap,

R. S. Erramilli and M. S. Mitchell, “Towards 3D CFT Cartography with the Stress Tensor Bootstrap,”arXiv:2602.13383 [hep-th]

work page arXiv
[27]

Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap,

D. Mazac, “Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap,” JHEP04(2017) 146,arXiv:1611.10060 [hep-th]

work page arXiv 2017
[28]

Extremal bootstrapping: go with the flow,

S. El-Showk and M. F. Paulos, “Extremal bootstrapping: go with the flow,”JHEP03(2018) 148,arXiv:1605.08087 [hep-th]

work page arXiv 2018
[29]

Afkhami-Jeddi,Conformal bootstrap deformations,JHEP09(2022) 225 [2111.01799]

N. Afkhami-Jeddi, “Conformal bootstrap deformations,”JHEP09(2022) 225, arXiv:2111.01799 [hep-th]

work page arXiv 2022
[30]

Afkhami-Jeddi, T

N. Afkhami-Jeddi, T. Hartman, and A. Tajdini, “Fast Conformal Bootstrap and Constraints on 3d Gravity,”JHEP05(2019) 087,arXiv:1903.06272 [hep-th]

work page arXiv 2019
[31]

Collier, Y.-H

S. Collier, Y.-H. Lin, and X. Yin, “Modular Bootstrap Revisited,”JHEP09(2018) 061, arXiv:1608.06241 [hep-th]

work page arXiv 2018
[32]

Accurate bootstrap bounds from optimal interpolation,

C.-H. Chang, V. Dommes, P. Kravchuk, D. Poland, and D. Simmons-Duffin, “Accurate bootstrap bounds from optimal interpolation,”arXiv:2509.14307 [hep-th]

work page arXiv
[33]

The analytic functional bootstrap. Part II. Natural bases for the crossing equation,

D. Mazac and M. F. Paulos, “The analytic functional bootstrap. Part II. Natural bases for the crossing equation,”JHEP02(2019) 163,arXiv:1811.10646 [hep-th]

work page arXiv 2019
[34]

A functional approach to the numerical conformal bootstrap,

M. F. Paulos and B. Zan, “A functional approach to the numerical conformal bootstrap,”JHEP 09(2020) 006,arXiv:1904.03193 [hep-th]

work page arXiv 2020
[35]

Analytic functional bootstrap for CFTs ind >1,

M. F. Paulos, “Analytic functional bootstrap for CFTs ind >1,”JHEP04(2020) 093, arXiv:1910.08563 [hep-th]

work page arXiv 2020
[36]

Numerical conformal bootstrap with analytic functionals and outer approximation,

K. Ghosh and Z. Zheng, “Numerical conformal bootstrap with analytic functionals and outer approximation,”JHEP09(2024) 143,arXiv:2307.11144 [hep-th]

work page arXiv 2024
[37]

Reehorst, S

M. Reehorst, S. Rychkov, D. Simmons-Duffin, B. Sirois, N. Su, and B. van Rees, “Navigator Function for the Conformal Bootstrap,”SciPost Phys.11(2021) 072,arXiv:2104.09518 [hep-th]. – 35 –

work page arXiv 2021