jurgendn 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:
- Does it satisfy all the stated assumptions? (If not, it's not a counterexample — it's irrelevant.)
- 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.