quantumquirkxyz Contract
- Input: a mathematical statement to prove (or a set of statements to evaluate).
- Output: a Markdown proof artifact saved to the repo, with statement, strategy, steps, verification, and theorem citations.
- Side effects: none.
- Dependencies: none.
- Stop condition: all proof steps justified; verification passed; artifact saved.
- Risk: low — reasoning-only, no code execution.
- Boundary: generates reasoning only; no code execution, no file system writes beyond the Markdown artifact.
Pure-Math Proofs
Construct a proof — an airtight argument from accepted axioms to a stated statement — using the strategy that fits the shape of the claim.
When to use
- User asks to "prove", "demonstrate", "show rigorously" a mathematical statement.
- The statement lives in number theory, algebra, analysis, combinatorics, or topology.
- A claim from another skill (e.g.
cs-formal-methods) needs a mathematical underpinning.
Process
1. Diagnose the claim
State the claim in your own words; identify its shape — universal (∀), existential (∃), implication, equivalence — and the domain (ℕ, ℤ, ℝ, groups, rings, vector spaces, metric spaces). A wrong diagnosis wastes every step after.
Completion criterion: the claim is restated with shape, domain, and the smallest hypothesis set that makes it true.
2. Pick the proof strategy
Choose the strategy that matches the shape:
- Direct — chain of implications.
- Contradiction — for "no such object exists" / "P implies Q" when direct fails.
- Contrapositive — for implications whose converse is easier.
- Induction — for statements indexed by ℕ; check base and inductive step explicitly.
- Strong induction — when the inductive step needs more than one previous case.
- Construction — for existential claims; exhibit the witness and verify.
- Pigeonhole / extremal / counting — for combinatorial statements.
- Diagonalisation — for non-constructive existence.
Completion criterion: strategy is named and one sentence justifies it.
3. Build the argument
Write the proof as numbered steps. Each step carries one logical move. State the rule you used (e.g. "by the axiom of choice", "since G is a group, inverses exist"). No gaps; a reader with the prerequisites should follow without filling in.
Completion criterion: every step is justified; no step is hand-waved with "clearly" or "obviously" on a non-trivial claim.
4. Verify
Sanity-check the proof three ways:
- Substitute a small instance (n = 1, 2, 3) and confirm the claim holds and the proof still applies.
- Look for the contrapositive / converse — does the proof accidentally prove something stronger or weaker?
- Adversarial pass — try to break it: find an edge case the proof ignores, a quantifier flip, an off-by-one.
Completion criterion: at least one verification pass completed; any flaw found is fixed before delivery.
5. Deliver
Produce the proof in a Markdown file with: statement, strategy, steps, verification notes, and any assumptions called out. Cite definitions or theorems used by name.
Completion criterion: Markdown artifact exists; statement, strategy, steps, verification all present; no step is unjustified.