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arxiv: 2605.25863 · v1 · pith:GEMDAPAUnew · submitted 2026-05-25 · ⚛️ physics.comp-ph · hep-ph· hep-th

Linac: linear algebra with CUDA over finite fields

Giuseppe De Laurentis , Jack Franklin This is my paper

Pith reviewed 2026-06-29 19:27 UTC · model grok-4.3

classification ⚛️ physics.comp-ph hep-phhep-th
keywords Gaussian eliminationfinite fieldsCUDAGPU computingquantum field theoryscattering amplitudeslinear systemsparallel algorithms
0
0 comments X

The pith

Linac provides a CUDA-based parallel implementation of Gaussian elimination over finite fields for quantum field theory amplitude reconstruction.

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

The paper presents Linac as an open-source tool that runs Gaussian elimination in parallel on GPUs, supporting both finite-field arithmetic and floating-point numbers. This addresses the cubic scaling bottleneck in solving polynomial linear systems that appear when reconstructing scattering amplitudes analytically. A sympathetic reader would care because the approach promises higher throughput than CPU methods while using finite fields to sidestep precision loss that grows with system size. The work focuses on making this capability available for quantum field theory calculations.

Core claim

With Linac, we present a high-performance, open-source, parallel implementation of Gaussian elimination over finite fields and floating-point arithmetic developed for applications to analytic reconstruction of scattering amplitudes in quantum field theory.

What carries the argument

Parallel Gaussian elimination executed on CUDA GPUs, using finite fields (integers modulo a prime) in place of floating-point numbers to maintain numerical stability.

If this is right

  • Large linear systems from polynomial equations can be solved at higher throughput by moving the elimination step to GPUs.
  • Finite-field arithmetic offers a precision-preserving alternative that scales without the rounding errors that appear in floating-point at large sizes.
  • Analytic reconstruction workflows in quantum field theory gain a new open-source component for the linear-algebra stage.
  • The same code base supports both finite-field and floating-point paths, allowing direct comparison within one framework.

Where Pith is reading between the lines

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

  • The same GPU finite-field approach could be tested on linear systems from other domains that currently rely on high-precision floating-point arithmetic.
  • Integration with existing amplitude generators would reveal whether the reported performance gain translates into end-to-end speedups for full calculations.
  • Extensions to other elimination variants or preconditioners could be explored once the core CUDA kernels are public.

Load-bearing premise

The parallelism in Gaussian elimination maps efficiently onto GPU hardware and finite-field arithmetic preserves the necessary information for the polynomial systems that arise in scattering amplitude reconstruction.

What would settle it

A direct timing comparison on representative QFT-derived linear systems showing that the GPU version runs slower than a standard CPU implementation, or a case where finite-field results deviate from known floating-point answers in a physically relevant way.

Figures

Figures reproduced from arXiv: 2605.25863 by Giuseppe De Laurentis, Jack Franklin.

Figure 1
Figure 1. Figure 1: Matrix layout for CUDA RowReduce kernel: thread blocks operate to the right of column (j). CudaRescaleRows, which divides each row by the corresponding maximum found in the previous call. Looping All following kernel calls happen in a loop, whereby this sequence of kernels is called as many times as the number of columns of the matrix. This is important to ensure a row reduced echelon form irrespective of … view at source ↗
Figure 2
Figure 2. Figure 2: Matrix layout for CUDA LoadMatrix kernel. ensuring that all the data has been fetched, each thread works on a different monomial, computing the multiplication reduction. If the number of columns exceeds 1024, the maximum number of threads in a block, then each thread works on multiple monomials, as many as necessary to complete the computation. 3.4 Optimisations Several low-level optimisations were impleme… view at source ↗
Figure 3
Figure 3. Figure 3: Cubic fit of the data points illustrating the performance of Gaussian [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quadratic fit of the data points illustrating the performance of constructing [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

Solving linear systems of polynomial equations is a ubiquitous problem in both mathematics and physics. The standard approach, Gaussian elimination, scales cubically with system size and often constitutes a computational bottleneck. The algorithm's inherent parallelism makes it well-suited for modern computing architectures, namely graphics processing units (GPUs), which offer significantly higher throughput than CPUs. Additionally, the use of finite fields -- integers modulo a prime -- in place of floating-point arithmetic offers a scalable solution to the issue of numerical precision loss, which becomes increasingly problematic at large system sizes. With Linac, we present a high-performance, open-source, parallel implementation of Gaussian elimination over finite fields and floating-point arithmetic. This tool has been developed for applications to analytic reconstruction of scattering amplitudes in quantum field theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents Linac, an open-source CUDA implementation of Gaussian elimination over finite fields and floating-point arithmetic. It targets the solution of linear systems of polynomial equations that arise in analytic reconstruction of scattering amplitudes in quantum field theory, claiming that GPU parallelism combined with finite-field arithmetic yields high performance while avoiding numerical precision loss.

Significance. If validated, a reliable GPU-accelerated finite-field linear algebra library could reduce a key computational bottleneck in QFT amplitude calculations. The approach of replacing floating-point with modular arithmetic to control precision is conceptually attractive for large sparse polynomial systems. However, the manuscript supplies no empirical evidence, so any significance assessment remains prospective rather than demonstrated.

major comments (2)
  1. [Abstract] Abstract: the central claims of 'high-performance' and a 'scalable solution to the issue of numerical precision loss' are asserted without any timing data, scaling curves, error analysis, or comparison against CPU baselines or existing packages (e.g., LinBox). This absence leaves the performance and accuracy assertions unverified.
  2. [Abstract] Abstract and application section: the claim that finite-field arithmetic provides an accuracy-preserving replacement for floating-point arithmetic is made for the specific sparse, high-degree polynomial matrices arising in QFT amplitude reconstruction, yet no tests on representative QFT-sized systems are reported. This is load-bearing for the stated use case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying the lack of empirical support for the performance and accuracy claims. We agree that these assertions require substantiation and will revise the manuscript to include the requested benchmarks, scaling studies, and application-specific tests.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of 'high-performance' and a 'scalable solution to the issue of numerical precision loss' are asserted without any timing data, scaling curves, error analysis, or comparison against CPU baselines or existing packages (e.g., LinBox). This absence leaves the performance and accuracy assertions unverified.

    Authors: We agree that the current manuscript asserts these properties on the basis of the algorithmic design and CUDA implementation without providing supporting measurements. The revised version will incorporate timing data on representative hardware, strong- and weak-scaling curves, floating-point versus finite-field error analysis, and direct comparisons against CPU baselines and libraries such as LinBox. revision: yes

  2. Referee: [Abstract] Abstract and application section: the claim that finite-field arithmetic provides an accuracy-preserving replacement for floating-point arithmetic is made for the specific sparse, high-degree polynomial matrices arising in QFT amplitude reconstruction, yet no tests on representative QFT-sized systems are reported. This is load-bearing for the stated use case.

    Authors: The referee correctly notes that no tests on QFT-scale systems are presented. Although the library was developed with these matrices in mind, the manuscript does not demonstrate the claimed accuracy preservation or performance on systems of the relevant size and sparsity. We will add such benchmarks in the revision, using polynomial systems drawn from actual amplitude-reconstruction workflows. revision: yes

Circularity Check

0 steps flagged

No circularity: software implementation paper with no derivation chain

full rationale

The manuscript presents an open-source CUDA implementation of Gaussian elimination over finite fields and floats, developed for QFT amplitude reconstruction. No equations, predictions, fitted parameters, or first-principles derivations are claimed. The abstract and description focus on algorithmic parallelism and numerical advantages without reducing any result to self-definition, self-citation load-bearing premises, or renaming of known patterns. The reader's assessment of 0.0 is confirmed; the work is a software artifact whose performance claims would require external benchmarks rather than internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no equations, data, or implementation sections are present from which free parameters, axioms, or invented entities can be extracted.

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discussion (0)

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Reference graph

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