Literature Reviewer: Comprehensive Mathematical Literature Search

Systematically search for and review mathematical literature related to the current research. Find references, discover connections, and identify results that strengthen the paper's context or yield new consequences.

Local results before web search

Every prior-work search starts with the common extracted literature, even when web browsing will also be needed. Search succinct result statements and project notes first with python3 scripts/research_library.py search 'terms'; add --sources to search full extracted text. Read merged-results.md when present, then the underlying extraction and its provenance. These local statements may match the current question better than an abstract or web snippet. Browse for gaps, primary-source verification, or updates; finding a paper online is not a reason to download or convert it again when an adequate local version exists.

How to Use This Skill

The user wants to find related work for their paper or a specific result. Follow the workflow below.

Step 0: Scope the Search

Determine the search scope by asking (if not specified):

Focus: Are we searching broadly for the whole paper, or for a specific definition/theorem/technique?
Goals: What are we looking for?
- Missing references for the bibliography
- Prior results that overlap with or imply our results
- Connections to other areas that could yield new results
- Standard terminology and notation to align with
- All of the above
Known references: What does the user already know about? (to avoid redundant results)

Step 1: Identify Search Terms

1.1 Extract Key Concepts

From the paper or the user's description, identify:

Core mathematical objects (e.g., simplicial complexes, Yang-Baxter equation, Betti numbers)
Techniques used (e.g., shellability, discrete Morse theory, spectral sequences)
Properties studied (e.g., acyclicity, Cohen-Macaulay, collapsibility)
Equivalent or related formulations in other areas

1.2 Generate Search Queries

For each concept, generate multiple query variants:

Standard mathematical terminology
Alternative names for the same concept (different communities may use different terms)
Broader and narrower terms
Combinations of key concepts that capture the paper's unique angle

Step 2: Search

2.1 Sources

Search across multiple sources:

arXiv (math sections: math.CO, math.AT, math.AC, etc.) — preprints and recent work
Google Scholar — broad coverage, citation counts
MathSciNet / zbMATH — reviewed publications with MSC classification
Specific journals relevant to the area
Textbooks and surveys — for foundational results and standard references

2.2 Search Strategy

Direct keyword search: search for the core terms.
Citation tracking: from known key papers, check what cites them (forward) and what they cite (backward).
Author search: identify the main contributors to the area and check their recent work.
Survey/textbook search: find surveys that cover the area — their bibliographies are goldmines.
Cross-area search: search for the same structures under different names in adjacent fields.

2.3 Record Findings

For each potentially relevant result, record:

Full citation (authors, title, year, venue/arXiv ID)
The specific result or concept that is relevant
How it connects to our work
Relevance level: essential / important / useful context / tangential

2.4 Durable Source Packets

Before downloading or converting a paper, use the shared-literature skill. Locate the common library through .research-library.json and search all subject areas by DOI, arXiv ID, title, authors, and aliases. Reuse an existing source and extraction; if absent, reserve a canonical entry with research_library.py. Do not create an independent repo-local literature cache or leave durable sources only in chat, a browser cache, or /tmp.

Recommended layout:

<library>/<domain>/<citation-key>/
  metadata.json          # citation, identifiers, version, access and extraction provenance
  source.txt             # durable converted text from PDF, TeX, or HTML
  source.tex             # source TeX when available and useful
  source-html.html       # official HTML capture when appropriate
  original.pdf           # when storage policy permits
  key-results.md         # precise reusable theorem/lemma extraction
  project-notes/<project>.md  # project notation and relevance
  variants/<origin>/     # alternative versions and preserved extractions

Use a stable citation key, but check identifiers and aliases before choosing a new one. Subject directories organize papers; they are not independent caches. Existing project literature paths are compatibility views, not locations for new downloads. Commit source packets in the shared library's owning repository.

Extraction priority:

Prefer arXiv source TeX when available because theorem environments and labels survive.
Prefer official HTML when it preserves math and section structure.
Otherwise convert the PDF to text, for example pdftotext -layout paper.pdf source.txt.

Record the exact source URL and extraction commands in metadata.json, including commands such as curl, arXiv source unpacking, pdftotext -layout, HTML-to-text conversion, or manual cleanup. Do not repeatedly download or reconvert the same paper if a packet already exists; inspect and update the packet instead.

Commit metadata.json, source.txt, key-results.md, and durable notes whenever they are used for research decisions. If the original PDF or source archive should not be committed, still store its URL, checksum when available, and the derived converted/extracted files that repo policy permits.

2.5 Key Result Extraction

For any paper that may be cited or used in a proof route, write a compact key-results.md. This is the quick-load artifact future agents should read before opening the raw conversion.

Include:

Citation and source packet path.
The exact theorem/lemma/proposition number or label from the paper.
Location: section/page/equation/source line when available.
Precise hypotheses and conclusion, paraphrased unless a short exact quote is essential.
Put notation translation into the project's own note under project-notes/.
How the result could be used: cite only, direct input, adaptable method, obstruction, or terminology alignment.
Any caveat: different definitions, missing hypothesis, unpublished status, proof gap, or dependence on another result.

Do not bury important theorems in a long narrative review. If a theorem is likely to be invoked again, promote it into key-results.md with enough precision that an agent can decide whether the raw source must be reopened.

2.6 Literature Index

After adding or changing packets, refresh the shared catalog and literature index:

python3 scripts/research_library.py index
python3 scripts/render_literature_index.py

Both commands use the configured common library. The renderer indexes canonical packets across subject directories without counting variants or compatibility views as additional papers. Commit the generated indexes with the packet changes in the library's owning repository. Link the common index or canonical packet from the project dashboard when it affects a research decision. The index is a table of contents; key-results.md carries the extracted results.

Step 3: Analyze Connections

3.1 Classify Found Results

Organize findings into categories:

Direct predecessors — Results our work builds on or generalizes.

How does our work extend or differ from these?
Are we citing them? Should we?

Parallel results — Results that prove similar things with different methods or in different settings.

Can their techniques be adapted to our setting?
Do their results imply any of ours (or vice versa)?
Should we discuss the relationship explicitly?

Connectable results — Results in adjacent areas that could combine with ours.

Can we derive new corollaries by combining their results with ours?
Do they provide alternative characterizations of our objects?
Do they suggest new conjectures?

Contextual references — Standard references for definitions, techniques, or background.

Are we using standard terminology consistently with these references?
Should we cite them for definitions we use?

Contradictions or overlaps — Results that conflict with or subsume our claims.

Does any found result already prove what we claim as new?
Does any result contradict our conjectures?
These must be addressed immediately.

3.2 Identify Gaps

After reviewing the literature:

Are there obvious related papers we haven't found?
Are there areas where the literature is thin (opportunity for our contribution)?
Are there open problems in found papers that our work addresses?

Step 4: Report

4.1 Literature Report Structure

# LITERATURE REVIEW: [Topic/Paper Title]
Date: [date]

## Search Summary
- Terms searched: [list]
- Sources checked: [list]
- Papers reviewed: [count]
- Durable packets created/updated: [`<library>/<domain>/<citation-key>/`, ...]
- Literature index refreshed: yes/no (shared `index.md` and `index.html`)

## Essential References (must cite)
1. [Citation] — [one-line relevance]. Connection: [how it relates to our work]
2. ...

## Key Results Extracted
1. [Citation, Theorem/Lemma X.Y] — [precise hypothesis/conclusion summary].
   Packet: `<library>/<domain>/<citation-key>/key-results.md`
   Use: [direct input / adaptable method / terminology / obstruction]

## Important Connections
1. [Citation] — [Result X] connects to our [Theorem Y] because [reason].
   Potential consequence: [what we could derive]
2. ...

## Suggested New Directions
1. [Citation] suggests that [idea]. This could lead to [conjecture/extension].
2. ...

## Terminology and Notation Alignment
- Our "[term A]" is called "[term B]" in [reference] — consider aligning.
- Standard notation for [concept] is [notation] per [reference].

## Potential Issues
- [Citation] may already prove [our result] — verify.
- [Citation] uses a different definition of [term] — clarify relationship.

## Missing Coverage
- No good reference found for [concept] — may need to be self-contained here.
- The area of [X] seems unexplored in connection with our work.

4.2 Bibliography Entries

For references to add, provide ready-to-use BibTeX entries:

@article{AuthorYear,
  author  = {Last, First and Last, First},
  title   = {Title},
  journal = {Journal},
  year    = {2024},
  volume  = {XX},
  pages   = {YY--ZZ},
  doi     = {10.xxxx/...}
}

4.3 Actionable Recommendations

Prioritize recommendations by impact:

Cite immediately — references that should be added to the paper now.
Investigate further — results that may yield new consequences and need closer reading.
Track — recent preprints or ongoing work to watch.

Guidelines

Search Quality

Thoroughness over speed: a missed key reference is worse than a long search.
Verify relevance: read abstracts and key results, don't just match keywords.
Check recency: prioritize recent work but don't ignore foundational older references.
Cross-community awareness: the same result may appear under different names in combinatorics, topology, algebra, or computer science.

Honesty

If a found result appears to subsume or contradict our work, report it immediately — do not downplay.
If the search is inconclusive for a topic, say so rather than claiming nothing exists.
Distinguish between "no results found" and "this area is unexplored."

When to Trigger This Skill

At the start of a project, to understand the landscape.
After proving a new result, to check novelty and find connections.
Before submission, for a final bibliography check.
When entering a new sub-topic or using a new technique.
When a reviewer or collaborator suggests "have you seen...?"

literature-reviewer

SKILL.md.claude/skills/literature-reviewer/SKILL.md