Mathematical Proof Construction

Proof required

Submit the precise statement of the mathematical claim you intend to prove (using standard mathematical notation with all terms defined), a description of the existing proof techniques for related results you have reviewed, your chosen proof strategy (direct proof, proof by contradiction, mathematical induction, construction, or other), and a justification for why that strategy is appropriate for this particular claim.

What gets checked

• Claim is stated precisely in formal mathematical notation — every variable is quantified, every set is defined, and the statement is unambiguous
• Proof strategy justification connects the chosen approach to the specific structure of the claim — not 'I will use induction' but 'I will use strong induction on n because the step from case n-1 to n requires knowing the result holds for all smaller values'
• Evidence of research into related proofs — at least one reference to a related result whose proof technique informed the strategy choice

Common mistakes

• Selecting a claim that reproduces a standard textbook proof verbatim — the goal is genuine proof construction, not recall; choose an extension, a variant, or a novel application
• Committing to a proof strategy before exploring alternatives — the first strategy that seems plausible often fails at a key step; investing time in understanding why alternative approaches do or do not work strengthens the final proof

Resources

What a verifier looks for

• Ask the submitter to explain what would happen if the proof strategy failed at the key step — tests whether they have thought through the potential obstacles in their approach.
• Ask why the chosen strategy is more natural for this claim than an alternative — confirms genuine understanding of what makes a proof strategy appropriate.
• Ask the submitter to state the claim in plain language for a non-mathematician — mathematical precision is non-negotiable, but inability to explain the claim informally suggests incomplete understanding.

Proof required

Submit the complete written proof in standard mathematical notation. The proof must include all definitions required to understand the claim, every logical step with a justification (citing the rule, lemma, or theorem applied), explicit handling of all cases or edge conditions, and a clear conclusion statement. Use standard proof layout conventions (formal style: lemma/proof/QED, or written prose proof with clear step separation).

What gets checked

• Every logical step is justified — not 'clearly ...' or 'obviously ...' without justification; each step must name what it follows from
• All cases are addressed — a proof by cases that omits a case, or an inductive proof where the base case is skipped, is incomplete
• The proof is self-contained — a reader should not need to consult other documents to follow every step

Common mistakes

• Relying on informal arguments for steps that require formal justification — 'it is intuitively obvious that...' is not a proof step; every non-trivial step must be made rigorous
• Not checking the proof by working through it with specific examples — testing with concrete cases often reveals gaps or incorrect steps that are invisible in the abstract

Resources

What a verifier looks for

• Identify one step in the submitted proof that relies on an unstated assumption or an implicit argument and ask the submitter to make it rigorous — this is the standard peer-review challenge for mathematical proofs.
• Ask the submitter to test the proof on a specific example that represents a potential edge case — verifying that the proof works for a concrete case confirms the logical structure.
• Ask whether the proof generalises — can the argument be adapted to prove a stronger result or a related claim?

Proof required

Submit the final version of the proof (corrected after M2 informal review if applicable) and a Q&A record from the oral defence showing: at least one step the reviewer challenged and the submitter's explanation, at least one novel variant or modification the reviewer posed and the submitter's analysis of whether it holds or fails and why, and the reviewer's assessment of whether the proof is correct. The reviewer must be named and their mathematics background (degree, research area, or professional role involving mathematical proof) stated.

What gets checked

• Q&A record documents a genuine mathematical challenge — not 'the reviewer confirmed the proof was correct' but a specific step or claim that was questioned and how the submitter responded
• Novel variant is addressed — the submitter must reason correctly about at least one modification the reviewer poses, not simply confirm that the original proof stands
• Final proof is submitted with any corrections identified during the defence incorporated

Common mistakes

• A reviewer who confirms the proof is correct without posing challenges — the oral defence must include adversarial probing, not endorsement; a reviewer who cannot identify any potential gaps may lack the domain expertise to challenge the proof effectively
• Submitting the M2 proof unchanged as the final proof without engaging with the review — even a correct proof benefits from the precision improvements that come from explaining each step to a sceptical expert

Resources

What a verifier looks for

• Pose a novel variant of the claim — change one parameter, relax one condition, or modify the domain — and ask the submitter whether the proof still holds and why or why not.
• Ask the submitter to explain the key insight in the proof — the single step or observation that makes the proof work; this distinguishes genuine comprehension from successful reproduction.
• Identify the step in the proof that relies most heavily on the specific conditions of the claim and ask what would break if that condition were removed.