theorem-search

name	theorem-search
description	Use TheoremSearch to find, verify, and cite mathematical theorem statements, named results, proof routes, and literature dependencies.

TheoremSearch Manual

Use this skill when a proof needs:

• a named theorem, lemma, proposition, corollary, definition, or remark;
• a literature dependency for a proof step;
• confirmation that a remembered theorem statement has the right hypotheses;
• leads for an external proof route;
• paper or theorem provenance for later human review.

TheoremSearch is supporting evidence only. It does not prove that a dependency is valid for the current proof. Always compare the returned statement against the exact proof obligation.

Official docs:

• Web docs: https://www.theoremsearch.com/docs
• API base: https://api.theoremsearch.com
• Main search endpoint: POST https://api.theoremsearch.com/search
• MCP endpoint: https://api.theoremsearch.com/mcp

Default to the REST API in this workspace: it is directly callable with curl, requires no agent-runtime configuration, and returns the search payload without protocol wrapping. Use MCP when the current agent runtime already has the TheoremSearch MCP server configured and tested.

Mental Model

TheoremSearch indexes theorem-level statements, not just whole papers. A query is embedded, matched against theorem slogans, and reranked. The best hit may be:

• the exact result;
• a nearby theorem with missing or extra hypotheses;
• a related result in the right paper;
• a statement whose generated slogan is more relevant than its full body.

Therefore, trust body and paper metadata more than slogan, name, or rank.

Basic Search

Use natural language, LaTeX, or a mix. Start broad enough to find the theorem, then narrow with filters or a sharper query.

curl -sS https://api.theoremsearch.com/search \
  -H 'Content-Type: application/json' \
  -d '{
    "query": "every projective module over a local ring is free",
    "n_results": 5
  }'

Expected top-level shape:

{
  "theorems": [
    {
      "slogan_id": 7725846,
      "theorem_id": 22812802,
      "name": "Theorem 5.11",
      "body": "If $P$ is a projective module over a local ring $R$, then $P$ is free.",
      "slogan": "A projective module over a local ring is free.",
      "theorem_type": "theorem",
      "label": null,
      "link": null,
      "paper": {
        "paper_id": "1011.0038v1",
        "source": "arXiv",
        "title": "Faithfully flat descent for projectivity of modules",
        "authors": ["Alexander Perry"],
        "link": "http://arxiv.org/abs/1011.0038v1",
        "primary_category": "math.AC",
        "categories": ["math.AC"],
        "citations": 0,
        "year": 2010,
        "journal_published": false
      },
      "similarity": 0.9086642212279158,
      "score": 0.9086642212279158,
      "has_metadata": true
    }
  ]
}

Search Request Fields

Required:

• query: natural-language or LaTeX description of the result.

Common optional fields:

• n_results: number of theorem results to return. Default is 10.
• sources: source filter. Officially documented examples include arXiv, Stacks Project, ProofWiki, CRing Project, HoTT Book, Open Logic Project, and An Infinitely Large Napkin. The live API may expose additional sources over time.
• authors: partial author-name filters.
• types: theorem-like object filters, for example theorem, lemma, proposition, corollary, definition, or remark. Results may preserve source casing inconsistently, so if a type filter misses, retry without it.
• tags: subject tags, especially arXiv categories such as math.AG, math.AC, math.NT, math.LO.
• paper_filter: substring filter over paper titles.
• year_range: inclusive [min_year, max_year].
• citation_range: inclusive [min_citations, max_citations].
• include_unknown_citations: when using citation_range, keep papers with no citation data. Default is true.
• citation_weight: blend citation count into ranking. Use 0.0 for pure semantic search. Increase only when standard/high-citation sources matter.
• prompt: optional instruction prepended before embedding. Use sparingly.
• db_top_k: candidate-pool size before reranking. Defaults to 2 * n_results. Increase when recall matters more than latency.

Filtered example:

curl -sS https://api.theoremsearch.com/search \
  -H 'Content-Type: application/json' \
  -d '{
    "query": "smooth Deligne Mumford stack has a dense open subscheme",
    "n_results": 8,
    "sources": ["Stacks Project", "arXiv"],
    "tags": ["math.AG"],
    "year_range": [2000, 2026],
    "include_unknown_citations": true,
    "db_top_k": 40
  }'

Query Strategy

1. Write the proof obligation as a theorem statement: "If [hypotheses], then [conclusion]."
2. Run an unfiltered search with n_results between 5 and 10.
3. Read body for every plausible hit. Ignore hits whose assumptions do not match, even if the slogan looks perfect.
4. If results are too broad, add one constraint at a time: sources, tags, authors, paper_filter, or types.
5. If results are too narrow or empty, remove filters, simplify notation, and search for equivalent phrases.
6. If a result is close but not exact, search again using distinctive words from the returned body, paper title, or named theorem.
7. Keep at least one exact or near-exact provenance record for every external dependency used in the final proof.

Good query patterns:

• Include the mathematical objects and the conclusion: "finite etale morphism is open and closed"
• Include key hypotheses: "normal noetherian domain height one localization discrete valuation ring"
• Use standard aliases: "Nakayama lemma finitely generated module local ring"
• Search by paper title plus theorem content when you know the source: "EGA flat morphism openness theorem"

Poor query patterns:

• "prove this" or "main theorem"
• overly local notation without context, such as "X is dense in U"
• long proof paragraphs with irrelevant narrative;
• filter-heavy first attempts that hide useful hits.

Reading Results

For each plausible theorem, inspect:

• body: exact theorem statement. This is the primary evidence.
• name: local theorem label, for example Theorem 5.11.
• theorem_type: theorem, lemma, proposition, corollary, definition, remark.
• slogan: generated natural-language summary. Useful for search, not proof.
• link: statement-level link if available.
• paper.source: corpus/source, for example arXiv or Stacks Project.
• paper.title: paper or project title.
• paper.authors: authors, when available.
• paper.link: paper-level URL, often arXiv.
• paper.primary_category and paper.categories: subject categories.
• paper.year, paper.journal_ref, paper.journal_published.
• paper.citations: citation count if known.
• similarity: semantic similarity before citation weighting.
• score: final ranking score after optional weighting.
• has_metadata: whether richer metadata was available.

Do not treat score thresholds as correctness thresholds. A low-scoring exact statement can be useful; a high-scoring result can still have wrong hypotheses.

Validation Checklist

Before using a result in a proof, check:

• objects match: rings, schemes, stacks, categories, fields, modules, spaces;
• hypotheses match: finiteness, noetherian, separated, proper, smooth, local, algebraically closed, characteristic, completeness, boundedness;
• conclusion matches: existence, uniqueness, equivalence, isomorphism, vanishing, density, openness, freeness, exactness;
• variance and direction match: source/target, left/right, covariant/ contravariant, injection/surjection;
• base category and notation match the current proof;
• no hidden condition is buried in surrounding paper notation;
• the result is appropriate to cite as a theorem rather than as a lead.

If a dependency is only close, write it as unresolved or as a search lead. Do not silently strengthen or weaken theorem statements.

Reporting Provenance

When a theorem is used, record enough information for audit:

Dependency: projective modules over local rings are free
TheoremSearch hit: Theorem 5.11
Statement: If $P$ is a projective module over a local ring $R$, then $P$ is free.
Source: arXiv, "Faithfully flat descent for projectivity of modules"
Authors: Alexander Perry
Year: 2010
Link: http://arxiv.org/abs/1011.0038v1
Score: 0.9086642212279158
Assessment: exact match for the required algebra lemma.

If the proof only needs a standard result and the search hit is a pointer to a secondary paper, prefer citing the classical theorem name in the proof while keeping the search provenance in notes.

Paper Search

Use paper search when you know a title fragment or arXiv ID and need a canonical paper identifier.

curl -sS 'https://api.theoremsearch.com/paper-search?q=derived%20categories&limit=5'

Expected shape:

{
  "papers": [
    {
      "paper_id": "f23156d8-433b-4f66-8b7b-8b58faabea6d",
      "title": "A Course on Derived Categories",
      "external_id": "1206.6632",
      "source": "arXiv"
    }
  ]
}

This endpoint is autocomplete-like. It is useful for locating a paper but does not replace theorem-level search.

Graph and Detail Endpoints

The public website documents dependency graph exploration. The live OpenAPI schema exposes these graph/detail paths:

• GET /graph/paper?external_id=<id>&sources=<source>
• GET /graph/paper/{paper_id}
• GET /graph/statement/{statement_id}
• GET /graph/embedding?query=<query>
• GET /graph/pagerank?query=<query>
• GET /statement/{statement_id}
• GET /paper/{paper_id}
• POST /statements with {"ids": [...]}
• POST /papers with {"ids": [...]}

Use these only as auxiliary tools after search has identified a likely paper or statement. Availability can vary by corpus and deployment; a 404 or empty graph does not invalidate a theorem search hit.

Paper graph example:

curl -sS 'https://api.theoremsearch.com/graph/paper?external_id=1011.0038&sources=arXiv&mode=minimal'

Statement graph example:

curl -sS 'https://api.theoremsearch.com/graph/statement/<statement_id>?direction=src&formality=informal&mode=full'

Use graph output to discover:

• what a theorem depends on;
• what later statements depend on it;
• nearby restatements or equivalent results;
• whether a paper's internal lemmas form a usable proof route.

Do not cite graph edges as mathematical facts unless the corresponding statements are inspected.

MCP Reference

TheoremSearch also exposes an MCP server at:

https://api.theoremsearch.com/mcp

The MCP server has a single theorem_search tool with the same parameters as POST /search. MCP is useful when the agent runtime is already configured to call MCP tools; REST is simpler for shell-based proof work.

Example MCP JSON-RPC request, for reference only:

curl -sS -X POST https://api.theoremsearch.com/mcp \
  -H 'Content-Type: application/json' \
  -d '{
    "jsonrpc": "2.0",
    "id": "1",
    "method": "tools/call",
    "params": {
      "name": "theorem_search",
      "arguments": {
        "query": "every projective module over a local ring is free",
        "n_results": 5
      }
    }
  }'

Failure Policy

If search fails:

• Network/server/API error: report TheoremSearch unavailable and include the endpoint and error text.
• Empty results: retry once with fewer filters and simpler language.
• Low relevance: report unresolved, summarize the closest hit, and explain the mismatch.
• Conflicting hits: list the competing statements and the hypothesis conflict.
• Missing links or metadata: use available paper title/source/body, but say what is missing.
• Graph endpoint failure: continue with /search; graph is auxiliary.

Never fabricate:

• theorem names;
• statement bodies;
• authors;
• links;
• scores;
• years;
• dependency edges;
• paper metadata.

Final Answer Pattern

When reporting a search to a parent solver, use one of these concise outcomes:

TheoremSearch: RESOLVED
Need: <proof obligation>
Result: <name/type>
Statement: <exact body>
Source: <source>, <paper title>, <authors>, <year>, <link>
Score: <score>
Use: <why the hypotheses and conclusion match>

TheoremSearch: PARTIAL
Need: <proof obligation>
Closest result: <name/type and exact body>
Source: <metadata>
Mismatch: <specific missing/extra hypothesis or wrong conclusion>
Next query: <sharper query or filter>

TheoremSearch: UNRESOLVED
Need: <proof obligation>
Searches tried: <queries/filters>
Reason: <empty, low relevance, inconsistent, unavailable>