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Agent skill

Construct Counterexamples

by frenzymath in frenzymath/Danus

Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail.

Apache-2.0Auto-check passed

Install Construct Counterexamples

skills CLI
$ npx skills add frenzymath/Danus --skill construct-counterexamples -a claude-code

Project install by default; add -g for ~/.claude/skills/.

GitHub CLI
$ gh skill install frenzymath/Danus construct-counterexamples --agent claude-code

Project scope by default; add --scope user for a personal install. Needs GitHub CLI 2.90.0 or later (public preview).

Manual copy
$ git clone --depth 1 https://github.com/frenzymath/Danus.git skills-src && mkdir -p .claude/skills && cp -r skills-src/agents/skills/worker/construct-counterexamples .claude/skills/construct-counterexamples && rm -rf skills-src

Use ~/.claude/skills/ instead of .claude/skills for a personal install. The folder must contain SKILL.md.

Claude Code skills documentation · loads skills from .claude/skills/

Facts

Skill name
construct-counterexamples
GitHub stars
475
Token cost
~790 tokens
SKILL.md length
285 words
Files
2
Skills in repo
16
Repo updated
First seen
Licence
Apache-2.0

At a glance

Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail.

  • Works in 6 steps: Identify the assumptions that must hold… → Use reasoning, decomposition, and… → Decide status → …
  • A proposed conjecture/claim feels fragile
  • SKILL.md covers Input Contract, Procedure, Output Contract and Tools, plus 1 more section
  • Instructions only: no scripts, shell commands, URLs or credentials in SKILL.md

What it does

Construct Counterexamples is an agent skill from frenzymath/Danus. Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail. Use when a proposed conjecture/claim feels fragile or unproved, or when you are stuck in reasoning and want to see where the assumptions take effect and gain intuition.

Its SKILL.md is about 790 tokens, which your agent loads only when the skill is triggered. The skill folder holds 2 other files (for example `agents/openai.yaml`).

The repository describes itself as: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory. The licence is Apache-2.0.

When your agent uses it

  • A proposed conjecture/claim feels fragile
  • You are stuck in reasoning and want to see where the assumptions take effect and gain intuition

Example prompts

  • “/construct-counterexamples”

Workflow steps

6 steps, taken from the first numbered list in SKILL.md.

  1. Identify the assumptions that must hold and the conclusion to fail.
  2. Use reasoning, decomposition, and retrieval to search for standard obstructions, pathological constructions, or previously known…
  3. Decide status
  4. If the search produces a concrete example that is informative but is not actually a counterexample, save that example as well in…
  5. If refuted, store the counterexample for reuse against future claims and mark impacted branches/lemmas as invalid.
  6. If no counterexample is found, treat that only as evidence that the claim may be correct, not as a proof.

What it can do on your machine

Read from SKILL.md and the folder at commit 6d92e8d. It shows what the files ask for, not the result of running them.

  • Tool permissions

    Pre-approves nothing: there is no allowed-tools line, so your agent's usual permission prompts apply.

    From allowed-tools in the SKILL.md frontmatter.

  • Runs code

    No scripts in the folder and no shell commands in SKILL.md (its code samples are json).

    From the folder's file list and the shell code blocks in SKILL.md.

  • Network

    No URLs in SKILL.md.

    From URLs in SKILL.md, links to its own repository left out.

  • Credentials

    Names no API keys, tokens, secrets or passwords.

    From names ending in _API_KEY, _TOKEN, _SECRET, _KEY or _PASSWORD in SKILL.md.

Context cost

Construct Counterexamples loads about 790 tokens when it runs. Until then it costs about 91 tokens; SKILL.md has 285 words of instructions outside code blocks.

Always · name and description, kept in context so the agent knows when to use it
~91
When it runs · the whole SKILL.md, loaded when a task matches
~790

Estimates: characters ÷ 4, the usual rule of thumb; real counts depend on the model's tokenizer. Scripts and assets cost tokens only if the agent reads them.

Safety

Auto-check passed

The automated check found no risky patterns in SKILL.md.

Automated static check — not a guarantee. Review scripts before installing. It scans the text of SKILL.md for risky patterns (piping downloads into a shell, reading credential files, hidden Unicode, destructive commands); files beside SKILL.md are not scanned.

SKILL.md

The full file from frenzymath/Danus at commit 6d92e8d, republished under its Apache-2.0 licence (© frenzymath). 285 words, ~790 tokens.

Download SKILL.mdSave it as .claude/skills/construct-counterexamples/SKILL.md (or your agent's skills folder). This skill also uses 1 other file; get the full folder from GitHub.
name
construct-counterexamples
description
Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail. Use when a proposed conjecture/claim feels fragile or unproved, or when you are stuck in reasoning and want to see where the assumptions take effect and gain intuition.

Construct Counterexamples

Actively falsify proposed conjectures or intermediate claims by finding examples that satisfy the assumptions but violate the claimed conclusion.

Input Contract

Read:

  • the specific conjecture/claim to test
  • active branch assumptions
  • candidate lemmas/proof steps
  • current immediate_conclusions and toy_examples
  • previously found counterexamples that can be reused against new claims

Procedure

  1. Identify the assumptions that must hold and the conclusion to fail.
  2. Use reasoning, decomposition, and retrieval to search for standard obstructions, pathological constructions, or previously known counterexamples.
  3. Decide status:
    • refuted: assumptions hold and the claim fails
    • not_refuted: no counterexample found yet
    • inconclusive: search space unclear or partially explored
  4. If the search produces a concrete example that is informative but is not actually a counterexample, save that example as well in toy_examples.
  5. If refuted, store the counterexample for reuse against future claims and mark impacted branches/lemmas as invalid.
  6. If no counterexample is found, treat that only as evidence that the claim may be correct, not as a proof.

Output Contract

Publish to global memory with gm_add (kind counterexample): claim = what is refuted/tested, evidence = the candidate construction, plus these fields:

json
{
  "target_claim": "...",
  "candidate_counterexample": "...",
  "status": "refuted|not_refuted|inconclusive",
  "assumptions_satisfied": ["..."],
  "failed_conclusion": "...",
  "impact": "...",
  "branch_id": "optional",
  "subgoal_id": "optional"
}

If status="refuted" and it kills a branch, also publish a dead_end finding (gm_add, kind dead_end) so siblings skip that branch.

If the search produced a concrete non-refuting example, also publish an example finding (gm_add, kind example):

json
{
  "example": "...",
  "why_relevant": "constructed while testing the claim ...",
  "assumptions_satisfied": ["..."],
  "conclusion_verified": true,
  "where_assumptions_take_effect": "...",
  "observed_pattern": "...",
  "supports_branch_ids": ["optional"],
  "subgoal_id": "optional"
}

Do this whenever the constructed example is useful enough to test future claims or clarify the current branch, even if it did not refute the target claim.

Tools

  • gm_add (publish counterexample / dead_end / example findings)
  • gm_search (recall stored counterexamples to reuse against new claims)
  • Codex built-in web search and search_arxiv_theorems to find standard counterexample patterns

Failure Logging

If no meaningful counterexample space is identified, append:

  • events.event_type="counterexample_space_unclear"

© frenzymath, Apache-2.0. Rendered from Markdown: HTML in the file is shown as text, images as links, and headings moved down two levels. Raw file

Files

SKILL.md and 1 other file in agents/skills/worker/construct-counterexamples of frenzymath/Danus.

  • SKILL.md
  • agents/openai.yaml

Open the folder on GitHubat commit 6d92e8d

Compare with similar skills

Construct Counterexamples next to the 5 skills that share the most tags, products or categories with it. Stars are the repository's; “used in” counts other GitHub owners with a copy.

Construct Counterexamples compared with similar skills
SkillStarsUsed inTokensAuto-checkLicenceRepo updated
Construct Counterexamples this skillfrenzymath/Danus475—~790Automated safety check: PassApache-2.0
Geo Proposalsickn33/agentic-awesome-skills47k1 repos~3.2kAutomated safety check: NotesMIT
Better Proposals AutomationComposioHQ/awesome-claude-skills77k3 repos~764Automated safety check: PassNone
Contract And Proposal Writeralirezarezvani/claude-skills28k2 repos~3.4kAutomated safety check: PassMIT
ProposalChorus-AIDLC/Chorus1.2k—~5.5kAutomated safety check: PassAGPL-3.0
Proposal Writerholaboss-ai/holaOS11k—~574Automated safety check: PassCustom licence

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More from frenzymath/Danus

All 16 skills in this repo

  • Check Referenced Statements

    frenzymath/Danus

    Validate externally referenced theorems by querying arXiv theorem search first and Codex's built-in web search second.

    475 GitHub stars~852 tokensUpdated 1 mo ago
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  • Construct Toy Examples

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    Generate and analyze simpler examples that satisfy both the assumptions and the conclusion of a theorem statement or subgoal.

    475 GitHub stars~541 tokensUpdated 1 mo ago
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  • Direct Proving

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    Screen a decomposition plan by first trying to prove all of its subgoals directly, then identifying the key stuck points if the plan does not fully go through.

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  • Identify Key Failures

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    Synthesize the common stuck points across failed decomposition plans.

    475 GitHub stars~586 tokensUpdated 1 mo ago
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  • Obtain Immediate Conclusions

    frenzymath/Danus

    Derive immediate mathematical consequences from a theorem statement or subgoal.

    475 GitHub stars~587 tokensUpdated 1 mo ago
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  • Propose Subgoal Decomposition Plans

    frenzymath/Danus

    Propose multiple subgoal decomposition plans for the current theorem using the information already gathered.

    475 GitHub stars~580 tokensUpdated 1 mo ago
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Questions about Construct Counterexamples

What does Construct Counterexamples do?

Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail. Construct Counterexamples is an agent skill from frenzymath/Danus. Construct candidate counterexamples to test a proposed conjecture, lemma, or intermediate claim by keeping the assumptions true while making the claimed conclusion fail.

When should I use Construct Counterexamples?

Construct Counterexamples fits situations like: A proposed conjecture/claim feels fragile; you are stuck in reasoning and want to see where the assumptions take effect and gain intuition.

How do I install Construct Counterexamples in Claude Code?

Run `npx skills add frenzymath/Danus --skill construct-counterexamples -a claude-code`. Or copy the skill folder (agents/skills/worker/construct-counterexamples in frenzymath/Danus) into .claude/skills/construct-counterexamples in your project. Claude Code loads it when a task matches its description.

How do I install Construct Counterexamples in Codex?

Run `npx skills add frenzymath/Danus --skill construct-counterexamples -a codex`. Or copy the skill folder (agents/skills/worker/construct-counterexamples in frenzymath/Danus) into .agents/skills/construct-counterexamples in your project. Codex loads it when a task matches its description.

Can I use Construct Counterexamples in Cursor, Gemini CLI or GitHub Copilot?

Cursor, Gemini CLI, GitHub Copilot and OpenCode also load SKILL.md folders. With the skills CLI, run `npx skills add frenzymath/Danus --skill construct-counterexamples -a cursor` (or -a gemini-cli, github-copilot or opencode for the others). To copy it by hand, put the folder in .cursor/skills/construct-counterexamples, .gemini/skills/construct-counterexamples, .github/skills/construct-counterexamples and .opencode/skills/construct-counterexamples in your project.

What does Construct Counterexamples need to run?

SKILL.md names no scripts, command-line tools or credentials: Construct Counterexamples is instructions for the agent only.

Does Construct Counterexamples access the network?

SKILL.md contains no URLs. Any network use would come from the scripts or tools the agent runs. This is read from the text; nothing was executed.

Is Construct Counterexamples safe to install?

Our automated static check of SKILL.md found no risky patterns, such as piping downloads into a shell, reading credential files or hidden Unicode. It is not a guarantee. Review the folder before installing.

What licence does Construct Counterexamples use?

Construct Counterexamples is published under the Apache-2.0 licence (the repository's licence). It allows redistribution, so the full SKILL.md is shown on this page.

How many tokens does Construct Counterexamples use?

About 790 tokens (SKILL.md is roughly 3.2k characters). Agents keep only the skill's name and description in context until a task matches; then they load SKILL.md in full.

What are the alternatives to Construct Counterexamples?

Skills that share tags, products or a category with Construct Counterexamples: Geo Proposal (sickn33/agentic-awesome-skills, 47k stars), Better Proposals Automation (ComposioHQ/awesome-claude-skills, 77k stars), Contract And Proposal Writer (alirezarezvani/claude-skills, 28k stars) and Proposal (Chorus-AIDLC/Chorus, 1.2k stars). The comparison table on this page puts their stars, adoption, token cost, safety result and licence side by side.

Who maintains Construct Counterexamples?

frenzymath (a GitHub organization) maintains it in frenzymath/Danus, which has 475 GitHub stars. The repository holds 16 skills in this directory. The repository was last updated on August 27, 2026.

Source: frenzymath/Danus on GitHub. Facts on this page come from the repository at the commit we read; the author's words are quoted as theirs.