Proof Exegesis

Treat an existing argument as mathematical evidence to interpret. The aim is to understand why it works, which ideas can be reused, and what broader structure it suggests. A short final theorem or a successful formal check does not by itself provide that understanding.

For a campaign assignment, honest-conjecture-resolution selects the question and integrates the interpretation; return mechanisms, verified claims, and candidate extensions to that driver. A standalone explanation needs no campaign.

Recover What Was Actually Proved

Read the exact statement, definitions, assumptions, and substantive dependencies. For formal proofs, inspect the encoded statement and its axioms; for computational proofs, identify the enumeration, certificates, coverage, and checking procedure. Keep a claimed result, a verified derivation under assumptions, and a theorem about the intended objects distinct. Use the relevant formalization or review workflow when checking those details is part of the task.

Recover prior interpretations and corrections from the campaign, sibling repositories, and shared literature. Preserve original proof and source links. If a gap prevents treating the argument as established, explain the gap and keep any useful conditional mechanism explicit.

Read The Argument At Three Levels

Choose the levels that illuminate the proof; not every argument has a useful description at all three.

• Elementary mechanism. Seek the cancellation, bijection, extremal choice, convexity argument, or other small mechanism doing the work. Unfold wrappers and choose a toy example that retains it. Separate notation and automation overhead from essential mathematical prerequisites. A simpler example can explain the mechanism without proving the full theorem.
• Reusable methods. Isolate constructions and transformations with explicit inputs, hypotheses, outputs, and losses. Locate where methods interact and why their assumptions are compatible. Test whether a proposed reusable lemma genuinely covers different instances. Use alon-method for importing a known technique and proof-calculus if the operations themselves deserve a theory.
• Structural explanation. Ask which definitions or laws account for the mechanism across a larger class. Formulate the proposed generalization and check that the original argument realizes its assumptions. Universal properties, dualities, or cohomology can explain something only after their relation to the proof is established. Use grothendieck-rising-sea when this becomes a sustained theory programme.

Move between these views. A small example may reveal a missing hypothesis in a generalization; a structural theorem may explain an initially mysterious local identity. Avoid forcing a categorical explanation or elementary proof that the evidence does not support.

Check The Distillation

Map each decisive step of the new account to the original argument or supply a new proof. Preserve quantifiers, constants, conventions, and exceptional cases. If hypotheses are weakened or the domain enlarged, that is a new mathematical claim requiring proof. Separately state what is established, what is explanatory analogy, and what is a candidate extension.

Count imported foundations and the work of checking assumptions when assessing simplicity. Fewer lines can hide the whole difficulty in a definition or theorem call. Conversely, substantial infrastructure can be valuable through reuse across many arguments. Do not equate minimum length with understanding or significance. An exhaustive proof may remain the available justification even when only part of its conceptual structure has been recovered; say so without inventing one.

Produce An Account Others Can Use

Provide the exact theorem and its status, a readable explanation of the central mechanism, a representative example, and links to the full justification. Add reusable lemmas or proposed generalizations only with their hypotheses and status. Keep the original proof accessible so the explanation can be checked against it.

Interpretation or a new presentation is not automatically a new theorem. Record what the account adds: simpler dependencies, a new proof, a wider scope, a useful operation, or improved understanding. Independently interesting methods and structural consequences can justify a separate paper or exploration branch; distinguish proved contributions from the work such a proposal still needs.

Keep research-specific findings in the campaign's writeups and result records. Contribute a new generic skill only when the extracted method has a distinct, reusable workflow, rather than turning every local proof trick into another skill.