# Theory Counterexample Hunter > Search for edge cases, pathological constructions, and violated assumptions that break a conjecture or weaken a theorem. Use when a claim feels too broad, an assumption seems suspiciously weak, or a proof step needs stress-testing. Trigger on phrases like "is this always true", "does this hold in general", "check this claim", "try to break this", "stress-test this theorem", "find a counterexample", "is this assumption necessary", or whenever the user states a result and asks whether it can be trusted. Also trigger when reviewing a proof and a step feels hand-wavy or a quantifier feels wrong. - Skill: `jurgendn/theory-counterexample-hunter` (Agent Skill) - Install (CLI): `npx skillmds@latest add jurgendn/theory-counterexample-hunter` - Raw SKILL.md: https://api.skillmd.com/api/skills/jurgendn/theory-counterexample-hunter/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Research & Search - Author: jurgendn (https://skillmd.com/u/jurgendn) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/jurgendn/theory-counterexample-hunter --- # Counterexample Hunter Try to break the claim before trusting it. The two failure modes to avoid: - **Fake counterexample**: something that looks like it breaks the claim but actually doesn't violate any assumption. Always check the assumptions explicitly. - **Empty survival**: concluding "no counterexample found" without saying what was actually tested. A claim that survived three degenerate cases is not a claim that has been stress-tested. --- ## Step 1: Restate the claim precisely Write out: - The exact statement (universal quantifiers, domains, topology, finiteness — don't paraphrase) - Every assumption, explicit and implicit - What "breaks" the claim means (counterexample to existence? to uniqueness? to a bound?) If the claim is ambiguous, name the ambiguity and ask which reading to test. --- ## Step 2: Find the load-bearing assumption Ask: **which single assumption, if dropped, would make the claim obviously false?** That assumption is where to attack first. Weakening or violating it is the most efficient path to a counterexample. --- ## Step 3: Generate candidates systematically Work through each category. Don't skip a category just because it seems unlikely — the point is coverage. **Degenerate cases** Empty set, singleton, zero, identity, trivial structure. Does the claim hold vacuously? Does it break? **Boundary / extremal cases** What happens at the limit of each parameter? As n → ∞, as ε → 0, as the graph becomes a path or a clique? **Discontinuous or singular cases** Non-continuous functions where continuity was implicitly assumed. Singular matrices. Graphs with isolated nodes. Distributions with no mean. Measures with atoms. **High-asymmetry cases** One very large component and one very small one. A graph with one high-degree node and all others degree 1. A sequence that is Cauchy but not convergent in the given space. **Adversarial constructions** Build an object specifically designed to exploit the gap between what the assumptions say and what the proof needs. Ask: *what property does the proof silently use that isn't in the assumptions?* Construct something that has the stated assumptions but lacks that silent property. --- ## Step 4: Check each candidate honestly For each candidate, verify explicitly: 1. Does it satisfy all the stated assumptions? (If not, it's not a counterexample — it's irrelevant.) 2. Does it violate the conclusion? If both: **counterexample found**. State it, verify it, and stop. If only (1): the claim survived this case. Record it. If neither: the construction was malformed. Discard and move on. --- ## Step 5: Report the outcome **If a counterexample is found:** - State the counterexample concretely (give the object, not just the idea of it) - Verify it satisfies the assumptions - Verify it violates the conclusion - Identify which assumption, if strengthened, would block this counterexample **If the claim survived:** - List what was tested (be specific — "I tested disconnected graphs, the empty graph, singleton sets, and the limit as n → ∞") - Identify the narrowest version of the claim that the tests actually support - Name which adversarial construction felt closest to working — that's the assumption most worth examining **If the claim is unprovable in either direction from here:** - Say so. Sometimes the right output is "this needs a proof attempt, not a counterexample search." --- ## Principles A toy counterexample on ℤ/2ℤ or a two-node graph is enough if it genuinely breaks the claim. Simplicity is a virtue. Distinguish: *the claim is false* (counterexample exists) vs. *the claim needs stronger assumptions* (a family of near-misses that all get blocked by one missing condition) vs. *the claim is true but the proof is wrong* (no counterexample found, but the argument has a gap). Never report a counterexample without checking the assumptions. The most common error is constructing something that violates the conclusion but doesn't satisfy the hypotheses.