Problem 1
Use the precise definitions, state all hypotheses, apply revision memorandum, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Peer Review, Gap Repair and Exposition Revision
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Peer Review, Gap Repair and Exposition Revision.
Problem 4
The argument must address responding to criticism with documented mathematical changes with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in advanced coursework, independent study, undergraduate research, graduate preparation, and mathematical publication, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.