foundations-of-computational-mathematics - SKILL.md Agent Skill

name: foundations-of-computational-mathematics description: Use when targeting Foundations of Computational Mathematics (FoCM) or deciding whether a manuscript at the mathematics–computation interface fits this Springer journal. Encodes the journal's fit, framing, proof-and-rigor bar, house style, official-submission re-check, and desk-reject heuristics.

Foundations of Computational Mathematics (foundations-of-computational-mathematics)

Journal positioning

Foundations of Computational Mathematics, published by Springer for the Society for the Foundations of Computational Mathematics, is a leading venue for the theory at the interface of mathematics and computation. Its defining character is rigorous mathematical foundations for computation: numerical analysis, approximation and complexity theory, optimization, computational geometry and topology, information-based complexity, and the analysis of algorithms — always with theorems and proofs at the core rather than empirical performance. The journal rewards work that establishes the mathematical basis of computational methods: convergence and error analysis, complexity bounds, stability, and structural theory, addressed to researchers who treat computation as a mathematical subject. It is selective and emphasizes depth and lasting foundational value. This skill is a fit / venue-selection / re-framing tool. It does not replace the journal's current official submission guidelines or the editorial board's judgment. Before submitting, re-check the live author instructions on the FoCM / Springer site.

When to trigger

The author names FoCM as the target for a result that establishes rigorous mathematical foundations of a computational method or problem.
A manuscript proves convergence, error, complexity, or stability theory for numerical, optimization, or geometric/topological computation, and the author is choosing between FoCM and SIAM Review or Journal of the ACM.
A paper is theory-first at the math–computation interface and would be undersold by a venue that prioritizes empirical results.
The author needs FoCM's expectations on full rigor, exposition, and submission norms before committing.

Scope & topic fit

Numerical analysis with rigorous theory: convergence, error bounds, and stability of methods for differential equations, linear algebra, and approximation.
Approximation theory, sampling, and information-based complexity: optimal rates, lower bounds, and the mathematics of recovery.
Optimization theory: convergence and complexity of continuous and convex/nonconvex optimization, with rigorous guarantees.
Computational complexity in the real/algebraic and continuous settings, and the analysis of algorithms over continuous structures.
Computational geometry and topology: persistent homology, topological data analysis, and geometric computation with provable foundations.
Mathematical foundations of learning, signal recovery, and data analysis where the contribution is theorems and bounds.

Method & evidence bar

Results must be complete and fully proved: theorems established rigorously with explicit hypotheses, no gaps, and no reliance on empirical evidence in place of proof.
Foundational significance is required: the contribution must advance the mathematical understanding of a computational method or problem — sharper bounds, new convergence/complexity theory, or a structural insight — not merely a faster implementation.
Proofs must be referee-verifiable in full; constants, rates, and assumptions must be stated precisely, and lemmas proved or precisely cited.
Where numerical experiments appear, they illustrate or motivate the theory; they do not substitute for it, and the paper's claims rest on the proofs.
The Mathematics Subject Classification (MSC) should be assigned accurately; arXiv posting is standard and encouraged, with any computer-assisted proof documented for independent checking. There is no data/code-deposition norm, though making accompanying code available is welcomed.
Prior and concurrent results must be credited precisely, with the improvement over known rates/bounds stated exactly.

Structure & house style

Standard mathematical structure: abstract, introduction stating results and significance, preliminaries and setup, the body of theorems and proofs, optional illustrative experiments, and references.
The introduction must state the main theorems precisely, give the foundational significance, and position the rates/bounds against prior work.
Exposition must be rigorous and clear: precise definitions, explicit constants and assumptions, and proofs organized so an expert can verify each step.
Long arguments may be modularized into lemmas and propositions; technical estimates or auxiliary computations may go to appendices.
Illustrative numerical experiments, if included, must be clearly subordinate to the theory and described reproducibly.
LaTeX is expected; bibliographic and formatting conventions follow the journal's current style.

Official-submission checklist

Before giving submission-ready advice, read ../../resources/source-basis.md and ../../resources/official-source-map.md; start from the official source anchors for this journal family, then cite the current journal-specific page you checked.
Search the live site for "Foundations of Computational Mathematics submission guidelines" and follow the current Springer version.
Re-check the submission procedure, file-format and LaTeX requirements, and editor-handling conventions.
Re-check MSC classification requirements and the journal's expectations on exposition, length, and the role of any numerical experiments.
Re-check authorship, competing-interests, and AI-use disclosure requirements; confirm arXiv/preprint posting policy and documentation expectations for computer-assisted proofs.
If the live official instructions conflict with this skill, the official instructions win.

Pre-submission self-check

One sentence — the main theorem and the foundational advance it makes at the math–computation interface.
Every proof is complete and referee-verifiable, with explicit constants/rates/assumptions and lemmas proved or cited.
The contribution is theory-first; any numerical experiments illustrate rather than substitute for the proofs.
The introduction states results precisely and positions the rates/bounds honestly against prior work.
MSC classification is assigned; arXiv status and any computer-assisted-proof documentation are consistent with policy.
The result's foundational significance matches FoCM rather than an applications- or empirics-driven venue.

Common desk-reject triggers

An empirical or applied paper whose contribution is performance rather than proved foundations.
A result with incomplete proofs, unstated assumptions, or claims resting on experiments instead of theorems.
An incremental improvement of a known bound without a new method or structural insight of lasting value.
Overstated significance or imprecise positioning relative to existing rates and complexity results.
A manuscript whose exposition is too disorganized for a referee to verify the analysis.

Re-routing decision

A broad, expository, survey-style treatment of a computational-mathematics topic for a wide audience: siam-review.
A theoretical-computer-science result centered on algorithms and complexity rather than continuous/numerical foundations: journal-of-the-acm or siam-journal-on-computing.
A pure-mathematics result whose computational framing is incidental: a leading pure-math venue such as acta-mathematica.
An applied numerical-methods paper driven by application performance: a SIAM applied journal (e.g., SINUM, SISC).

Output format

[Fit] High / Medium / Low (one-line reason)
[Target] Foundations of Computational Mathematics
[Topic tags] <2–3 closest areas / MSC themes>
[Method/evidence] <are the proofs complete and referee-verifiable, and is the contribution a foundational advance at the math–computation interface?>
[Top risk] <the single most likely reason for rejection>
[Official items to re-check] <submission/LaTeX format / MSC classification / role of numerical experiments & length / arXiv & computer-proof documentation / disclosure>
[Re-route suggestion] <if not a fit, a better-matched venue>