arxiv: 2603.10627 · v2 · submitted 2026-03-11 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Efficient Conformal Block Evaluation with GoBlocks

James Chryssanthacopoulos , Vasilis Niarchos , Constantinos Papageorgakis , Alexander G. Stapleton

Authors on Pith no claims yet

Pith reviewed 2026-05-15 13:31 UTC · model grok-4.3

classification ✦ hep-th

keywords conformal blocksconformal bootstraprecursive relationsIsing modelOPE coefficientsnumerical methodsGo programmingCFT data

0 comments

The pith

GoBlocks enables fast on-the-fly conformal block evaluation using recursive relations implemented in Go.

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

The paper introduces GoBlocks, a package written in the Go programming language that generates conformal blocks through recursive relations for rapid parallel computation. It targets the conformal bootstrap problem where no closed analytic form exists for these functions in odd spacetime dimensions. Benchmarks against scalar_blocks demonstrate meaningful speed gains when moderate accuracy suffices rather than ultra-high precision. The authors apply the tool to the mixed-correlator bootstrap of the three-dimensional Ising model, treating it as a non-convex optimization that simultaneously varies external scaling dimensions and OPE data. They further examine how performance scales when the number of mixed correlators grows in general O(N) vector models.

Core claim

GoBlocks is a novel conformal-block generator in Go that uses recursive relations to support rapid, on-the-fly, parallel evaluation. The implementation handles both multi-point and derivative-based bootstrap approaches with tunable accuracy and performance settings. Direct comparisons show substantial speed improvements over the scalar_blocks package in regimes where computational speed matters more than extreme precision. As a concrete demonstration, the package is used to formulate and solve the mixed-correlator bootstrap of the three-dimensional Ising model as a non-convex optimization problem, optimizing external scaling dimensions together with OPE CFT data while also exploring scaling,

What carries the argument

Recursive relations for conformal blocks, coded in Go to enable parallel on-the-fly evaluation with controllable accuracy.

If this is right

Bootstrap calculations for the 3D Ising model and similar theories become feasible at larger truncation orders or with more operators.
Simultaneous optimization over external scaling dimensions and OPE coefficients is practical within a non-convex framework.
The method extends to mixed-correlator studies of O(N) vector models and shows how runtime grows with added correlators.
Applications that prioritize speed over ultra-high precision gain a practical new option for repeated block evaluations.

Where Pith is reading between the lines

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

The parallel recursive design could support iterative or real-time bootstrap workflows that adjust truncation schemes dynamically.
Similar recursive implementations might be ported to languages optimized for GPUs or distributed clusters to tackle even larger operator sets.
The speed advantage may open exploration of bootstrap problems in higher odd dimensions where closed forms remain unavailable.
Integration with gradient-based or stochastic optimizers could further reduce the cost of global searches over CFT data.

Load-bearing premise

The recursive relations are implemented correctly and remain numerically stable at the precision levels needed for the Ising bootstrap without hidden instabilities or truncation effects.

What would settle it

A direct numerical comparison of GoBlocks output for a known low-lying conformal block against an independent high-precision reference calculation at the same derivative order would confirm or refute the accuracy and stability claims.

Figures

Figures reproduced from arXiv: 2603.10627 by Alexander G. Stapleton, Constantinos Papageorgakis, James Chryssanthacopoulos, Vasilis Niarchos.

**Figure 1.** Figure 1: Runtime of scalar blocks as order varies, with and without post-processing to construct block derivatives and convert to derivatives of F±. Without post-processing, scalar blocks runs in under a second for most order values. With post-processing, the runtime can increase to two seconds. for systematic comparison, enabling both scalar blocks and the GoBlocks derivative approach to be evaluated across a rang… view at source ↗

**Figure 2.** Figure 2: Accuracy and runtime of GoBlocks with increasing ℓmax. Fig. (a) shows block accuracy varying from ∼ 20% to ∼ 4 × 10−5% as ℓmax increases from 12 to 30 for fixed k max 1,2 = 20. Fig. (b) shows runtime as a function of ℓmax for fixed k max 1,2 = 20 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of average runtimes between scalar blocks and GoBlocks for given accuracies with k max 1,2 = 10. Overall, GoBlocks is approximately 4.6 times faster than scalar blocks between the 0.1% and 10% accuracy levels. 3. Applications We will next demonstrate the utility of GoBlocks in conformal bootstrap studies with multiple correlators, starting with the 3D Ising model. We will also remark on applying… view at source ↗

**Figure 4.** Figure 4: Sampled points in the complex z plane. This subset of points results in a scalar crossing violation of order 10−2 when evaluated on the 18 stable operators of the 3D Ising model with dimensions ∆ ≤ 8, as reported in [24]. vector norms of Eqs. (3.1)–(3.4), while the ground-truth values against which we compare the optimisation results are taken from [24]. Constraints were also imposed to ensure the scaling … view at source ↗

**Figure 5.** Figure 5: Schematic overview of multi-point block evaluation steps. 23 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗

**Figure 6.** Figure 6: Schematic overview of block derivative evaluation steps. 24 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

**Figure 7.** Figure 7: Runtime versus accuracy for scalar blocks and GoBlocks with a selection of hyperparameters, with the 1% accuracy level depicted by a dashed horizontal line. Appendix E. Tuning GoBlocks for Optimal Performance As discussed in Section 2.4, the performance of GoBlocks is predominantly controlled by k max 1,2 and ℓmax. In the absence of a canonical strategy for configuring these parameters, it is useful to tu… view at source ↗

read the original abstract

Conformal blocks in odd spacetime dimensions are not known in closed analytic form. To facilitate efficient computations in the conformal bootstrap, we introduce $\texttt{GoBlocks}$: a novel conformal-block generator implemented in the Go programming language, designed for rapid, on-the-fly, parallel evaluation using recursive relations. The package supports both multi-point and derivative-based bootstrap approaches and allows flexible control over accuracy and performance. We benchmark $\texttt{GoBlocks}$ against the $\texttt{scalar_blocks}$ package, finding significant speed improvements in applications where computational speed and moderate accuracy are critical, but ultra-high precision is not essential. As an illustration, we apply $\texttt{GoBlocks}$ to the mixed-correlator bootstrap of the three-dimensional Ising model, formulated as a non-convex optimisation problem in a suitable truncation scheme. We simultaneously optimise over external scaling dimensions and OPE CFT data. In addition, we discuss how the approach scales as we increase the number of mixed correlators in more general $O(N)$ vector models.

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

GoBlocks is a practical new Go tool for faster recursive conformal block evaluation that could help bootstrap calculations, but it needs explicit accuracy checks to confirm the speedups don't come with hidden errors.

read the letter

GoBlocks is a new Go package that computes conformal blocks recursively with parallel evaluation and user controls for accuracy. The authors benchmark it against scalar_blocks, report speed gains for moderate-precision work, and apply it to a mixed-correlator Ising bootstrap cast as a non-convex optimization over external dimensions and OPE data. They also sketch how the approach extends to larger sets of correlators in O(N) models. That is the core contribution: a fresh implementation in a language suited to concurrency, aimed at on-the-fly use rather than precomputed tables. It is a useful software step for people running many blocks inside optimization loops where ultra-high precision is not the priority. The code choice and the parallel design are the parts that stand out as genuinely new relative to existing scalar_blocks and similar packages. The Ising example shows they can actually run the full pipeline, which gives the work some grounding. The soft spot is the thin numerical validation. The manuscript describes benchmarks and the application but does not include error tables, convergence plots under increasing recursion depth, or block-by-block comparisons at the precise truncation (spins, derivatives, external dimensions) used in the optimization. Without those, the central claim that the speed improvements preserve the quality of the minima rests on unshown implementation details. Truncation artifacts or floating-point accumulation in the recursion could shift results even if the raw timing numbers look good. This paper is for bootstrap practitioners who need quicker block generation for 3d mixed-correlator studies and are willing to trade some precision for runtime. A reader who wants an alternative tool to experiment with would get concrete value from the reported speedups and the scaling discussion. It deserves peer review because the software contribution is substantive and the application is real, even though revisions should add the missing validation data to make the efficiency claims fully convincing.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces GoBlocks, a conformal-block generator implemented in the Go programming language that evaluates blocks on-the-fly via recursive relations for use in conformal bootstrap. It reports benchmarks against scalar_blocks showing speed gains at moderate accuracy, and demonstrates the package on the mixed-correlator bootstrap of the 3D Ising model cast as a non-convex optimization over external dimensions and OPE coefficients; scaling to larger O(N) models is also discussed.

Significance. If the recursive implementation proves numerically stable and accurate at the truncation levels required for the Ising optimization, GoBlocks would provide a useful alternative implementation that leverages Go's concurrency for faster on-the-fly block generation, potentially enabling larger-scale or optimization-heavy bootstrap studies where ultra-high precision is not essential.

major comments (2)

[Benchmarks and Ising application sections] The central efficiency and reliability claims rest on the correctness of the recursive relations in GoBlocks, yet the manuscript provides no quantitative error tables, convergence tests under increasing recursion depth, or direct block-by-block comparisons against scalar_blocks (or known high-precision results) at the precise truncation (spins, derivatives, external dimensions) used in the Ising non-convex optimization. This validation is load-bearing for the application results.
[Ising application section] Without explicit checks for floating-point accumulation or truncation artifacts in the recursion at the moderate precision levels employed, it remains possible that the reported optimization minima are affected by implementation details not shown in the text.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a clearer statement of the target precision range (e.g., number of significant digits) at which the speed gains are claimed.
[Ising application section] Notation for the truncation parameters and the non-convex objective function should be defined more explicitly before the optimization results are presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address the major concerns point by point below and agree to enhance the validation sections in the revised version.

read point-by-point responses

Referee: [Benchmarks and Ising application sections] The central efficiency and reliability claims rest on the correctness of the recursive relations in GoBlocks, yet the manuscript provides no quantitative error tables, convergence tests under increasing recursion depth, or direct block-by-block comparisons against scalar_blocks (or known high-precision results) at the precise truncation (spins, derivatives, external dimensions) used in the Ising non-convex optimization. This validation is load-bearing for the application results.

Authors: We agree that additional quantitative validation would strengthen the claims. While the recursive relations are standard and our benchmarks indicate agreement with scalar_blocks within the moderate accuracy regime, we did not include detailed error tables at the exact truncation used for the Ising optimization. In the revised manuscript, we will add a dedicated subsection with convergence tests, error tables, and direct comparisons at the relevant truncation levels (including the spins, derivatives, and external dimensions employed in the optimization). revision: yes
Referee: [Ising application section] Without explicit checks for floating-point accumulation or truncation artifacts in the recursion at the moderate precision levels employed, it remains possible that the reported optimization minima are affected by implementation details not shown in the text.

Authors: We acknowledge the importance of verifying numerical stability. The implementation uses double-precision floating point, and the recursion depth is controlled to maintain accuracy, but explicit checks for accumulation errors were not presented. We will include in the revision numerical tests demonstrating the absence of significant floating-point artifacts at the precision levels used in the Ising bootstrap, such as comparisons with higher precision runs or stability under increased recursion depth. revision: yes

Circularity Check

0 steps flagged

No significant circularity; software implementation of known recursions with external benchmarks

full rationale

The manuscript introduces GoBlocks as an implementation of established recursive relations for conformal blocks in odd dimensions, benchmarked against the independent scalar_blocks package. No derivations are presented that reduce by construction to fitted inputs, self-definitions, or self-citation chains. Performance claims rest on direct timing measurements rather than any predicted quantities derived from the same data. The central contribution is the Go-language code and its measured speedups in the Ising bootstrap application; these are falsifiable by re-running the provided implementation against external references and do not rely on any load-bearing self-citation or ansatz smuggled from prior author work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on standard recursive relations for conformal blocks (domain knowledge) and introduces user-controlled accuracy parameters as the main tunable elements. No new physical entities are postulated.

free parameters (1)

accuracy and truncation controls
User-set parameters that trade speed for precision; their specific values are not fixed by the paper but chosen per run.

axioms (1)

domain assumption Recursive relations for conformal blocks in odd dimensions hold and are numerically stable when implemented in GoBlocks
Invoked throughout the description of the generator; correctness is assumed from prior literature on block recursions.

pith-pipeline@v0.9.0 · 5483 in / 1240 out tokens · 45915 ms · 2026-05-15T13:31:06.841547+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

recursive relations for conformal blocks... order-by-order r-series expansion... F±,Δ,ℓ(u,v) construction
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

3D Ising mixed-correlator truncation... non-convex optimisation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 13 internal anchors

[1]

Bounding scalar operator dimensions in 4D CFT

R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi,Bounding scalar operator dimensions in 4D CFT,JHEP12(2008) 031 [0807.0004].✦D. Poland, S. Rychkov 32 and A. Vichi,The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,Rev. Mod. Phys.91(2019) 015002 [1805.04405].✦S. Rychkov and N. Su,New developments in the numerical conformal bootstrap,Re...

work page internal anchor Pith review Pith/arXiv arXiv 2008
[2]

Solving the 3D Ising Model with the Conformal Bootstrap

S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap,Phys. Rev. D86(2012) 025022 [1203.6064]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[3]

F. Kos, D. Poland and D. Simmons-Duffin,Bootstrapping Mixed Correlators in the 3D Ising Model,JHEP11(2014) 109 [1406.4858]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[4]

F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi,Precision Islands in the Ising and O(N)Models,JHEP08(2016) 036 [1603.04436]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[5]

Defects in conformal field theory

M. Bill` o, V. Gon¸ calves, E. Lauria and M. Meineri,Defects in conformal field theory, JHEP04(2016) 091 [ 1601.02883]. ✦ E. Lauria, M. Meineri and E. Trevisani,Radial coordinates for defect CFTs,JHEP11(2018) 148 [1712.07668]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[6]

Conformal Field Theories Near a Boundary in General Dimensions

D.M. McAvity and H. Osborn,Conformal field theories near a boundary in general dimensions,Nucl. Phys. B455(1995) 522 [cond-mat/9505127].✦P. Liendo, L. Rastelli and B.C. van Rees,The Bootstrap Program for Boundary CFT d,JHEP07 (2013) 113 [1210.4258]

work page internal anchor Pith review Pith/arXiv arXiv 1995
[7]

The Conformal Bootstrap at Finite Temperature

L. Iliesiu, M. Kolo˘ glu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin,The Conformal Bootstrap at Finite Temperature,JHEP10(2018) 070 [1802.10266]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[8]

Poland, V

D. Poland, V. Prilepina and P. Tadi´ c,The five-point bootstrap,JHEP10(2023) 153 [2305.08914].✦D. Poland, V. Prilepina and P. Tadi´ c,Improving the five-point bootstrap,JHEP05(2024) 299 [ 2312.13344]. ✦ D. Poland, V. Prilepina and P. Tadi´ c, Mixed five-point correlators in the 3d Ising model,JHEP10(2025) 237 [2507.01223]

work page arXiv 2023
[9]

More constraining conformal bootstrap

F. Gliozzi,More Constraining Conformal Bootstrap,Phys. Rev. Lett.111(2013) 161602 [1307.3111]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[10]

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 [1403.6003]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[11]

Gliozzi, P

F. Gliozzi, P. Liendo, M. Meineri and A. Rago,Boundary and Interface CFTs from the Conformal Bootstrap,JHEP05(2015) 036 [1502.07217].✦F. Gliozzi,Truncatable bootstrap equations in algebraic form and critical surface exponents,JHEP10(2016) 33 037 [1605.04175].✦W. Li,New method for the conformal bootstrap with OPE truncations,1711.09075

work page arXiv 2015
[12]

K´ antor, V

G. K´ antor, V. Niarchos and C. Papageorgakis,Solving Conformal Field Theories with Artificial Intelligence,Phys. Rev. Lett.128(2022) 041601 [2108.08859]

work page arXiv 2022
[13]

K´ antor, V

G. K´ antor, V. Niarchos and C. Papageorgakis,Conformal bootstrap with reinforcement learning,Phys. Rev. D105(2022) 025018 [2108.09330].✦G. K´ antor, V. Niarchos, C. Papageorgakis and P. Richmond,6D (2,0) bootstrap with the soft-actor-critic algorithm,Phys. Rev. D107(2023) 025005 [2209.02801]

work page arXiv 2022
[14]

Li,Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods,JHEP07(2024) 047 [2312.07866]

W. Li,Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods,JHEP07(2024) 047 [2312.07866]

work page arXiv 2024
[15]

Hu and W

R. Hu and W. Li,Accurate boundary bootstrap for the three-dimensional O( N) normal universality class,2508.20854.✦Y.-t. Huang, S.-C. Lee, H. Liao and J. Rumbutis, Bootstrapping non-unitary CFTs,2512.07706

work page arXiv
[16]

Niarchos, C

V. Niarchos, C. Papageorgakis, A. Stratoudakis and M. Woolley,Deep finite temperature bootstrap,Phys. Rev. D112(2025) 126012 [2508.08560]

work page arXiv 2025
[17]

A Semidefinite Program Solver for the Conformal Bootstrap

D. Simmons-Duffin,A Semidefinite Program Solver for the Conformal Bootstrap,JHEP 06(2015) 174 [1502.02033]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[18]

scalar blocks: Conformal Blocks for Scalar Correlators

The Bootstrap Collaboration, “scalar blocks: Conformal Blocks for Scalar Correlators.” https://gitlab.com/bootstrapcollaboration/scalar_blocks, 2023

work page 2023
[19]

Erramilli, L.V

R.S. Erramilli, L.V. Iliesiu, P. Kravchuk, W. Landry, D. Poland and D. Simmons-Duffin,blocks 3d: Software for General 3D Conformal Blocks,JHEP11 (2021) 006 [2011.01959]

work page arXiv 2021
[20]

Radial Coordinates for Conformal Blocks

M. Hogervorst and S. Rychkov,Radial Coordinates for Conformal Blocks,Phys. Rev. D 87(2013) 106004 [1303.1111]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[21]

The Effective Bootstrap

A. Castedo Echeverri, B. von Harling and M. Serone,The Effective Bootstrap,JHEP 09(2016) 097 [1606.02771]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[22]

Cavagli` a, N

A. Cavagli` a, N. Gromov, J. Julius and M. Preti,Integrability and conformal bootstrap: One dimensional defect conformal field theory,Phys. Rev. D105(2022) L021902 [2107.08510].✦A. Cavagli` a, N. Gromov, J. Julius and M. Preti,Bootstrability in defect CFT: integrated correlators and sharper bounds,JHEP05(2022) 164 [2203.09556]. 34

work page arXiv 2022
[23]

Niarchos, C

V. Niarchos, C. Papageorgakis, P. Richmond, A.G. Stapleton and M. Woolley, Bootstrability in line-defect CFTs with improved truncation methods,Phys. Rev. D108 (2023) 105027 [2306.15730]

work page arXiv 2023
[24]

The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

D. Simmons-Duffin,The Lightcone Bootstrap and the Spectrum of the 3D Ising CFT, JHEP2017(2017) [1612.08471]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[25]

Biscani and D

F. Biscani and D. Izzo,A parallel global multiobjective framework for optimization: pagmo,Journal of Open Source Software5(2020) 2338

work page 2020
[26]

T. King, S. Butcher and L. Zalewski,Apocrita - High Performance Computing Cluster for Queen Mary University of London, Mar., 2017. 10.5281/zenodo.438045

work page doi:10.5281/zenodo.438045 2017
[27]

Chester, W

S.M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su et al.,Carving out OPE space and preciseO(2)model critical exponents,JHEP06(2020) 142 [1912.03324]

work page arXiv 2020
[28]

Chester, W

S.M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su et al., Bootstrapping Heisenberg magnets and their cubic instability,Phys. Rev. D104(2021) 105013 [2011.14647]

work page arXiv 2021
[29]

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 [2104.09518]

work page arXiv 2021
[30]

Gon¸ calves, R

V. Gon¸ calves, R. Pereira and X. Zhou, 20 ′ Five-Point Function fromAdS 5 ×S 5 Supergravity,JHEP10(2019) 247 [1906.05305].✦D. Poland and V. Prilepina, Recursion relations for 5-point conformal blocks,JHEP10(2021) 160 [2103.12092]. 35

work page arXiv 2019