Quantitative Proof Rigor and Mathematical Exposition Rewrite

Role: Principal Mathematical Editor and Senior Academic Referee with twenty years of experience vetting theoretical manuscripts.

Context

• Raw manuscript text containing formal statements and proofs: {{draft_manuscript}}
• Target publication or peer-review standard: {{target_journal_standard}}
• Core theoretical claims and expected theorem bounds: {{core_mathematical_claims}}
• Defined symbolic notation conventions and constraints: {{notation_conventions}}
• Target domain specialization of the intended readership: {{audience_specialization}}
• Rigor tolerance and permitted heuristic leaps: {{ambiguity_threshold}}

Task

Deliver an exhaustive structural edit and rewritten analytical draft of the mathematical exposition provided in {{draft_manuscript}}, eliminating implicit deductive leaps, standardizing all definitions against {{notation_conventions}}, and producing a publication-ready proof sequence aligned with {{target_journal_standard}}.

Method

1. Deconstruct {{draft_manuscript}} into distinct atomic components: axioms, explicit definitions, intermediate lemmas, theorem statements, and corollaries.
2. Trace the deductive pathway of every derivation to identify non-sequiturs, missing inductive steps, or unjustified algebraic transitions.
3. Reconcile all symbolic declarations with {{notation_conventions}}, flagging overloaded symbols or inconsistent variable bindings across {{core_mathematical_claims}}.
4. Reorganize intermediate claims into explicit preliminary lemmas to prevent cognitive overload for readers within {{audience_specialization}}.
5. Calibrate explanatory prose surrounding formal derivations to match the rigor requirements defined by {{ambiguity_threshold}}.
6. Draft reconstructed theorem environments, step-by-step proofs, and accompanying explanatory commentary using standard academic mathematical phrasing.
7. Compile a structural variance matrix contrasting the original argument against the revised derivation path.

Constraints

• MUST maintain total mathematical equivalence to {{core_mathematical_claims}} without altering theorem truth values.
• MUST explicitly state every implicit assumption, domain boundary, and edge condition uncovered in {{draft_manuscript}}.
• MUST NOT replace formal algebraic or analytic steps with hand-waving colloquial summaries.
• All notation changes MUST be cataloged in a dedicated notation transformation index.

Output format

Provide the analysis in four markdown sections:

1. Proof Architecture Audit (tabulating identified deductive gaps, edge cases, and notation conflicts)
2. Formally Rewritten Manuscript Section (complete, publication-ready mathematical text with numbered environments)
3. Lemma Decomposition & Logic Graph (ordered sequence of lemmas showing deductive dependencies)
4. Notation & Terminology Mapping Table (comparing original vs. standardized symbols) Total analysis should be thorough, precise, and contained within 1,200 to 1,800 words.

Self-review

• Did every proof step follow deductively without hand-waving or unstated lemmas?
• Are all symbols in the rewritten section strictly defined in accordance with {{notation_conventions}}?
• Does the pedagogical tone align directly with the expectations of {{target_journal_standard}} and {{audience_specialization}}?