Decompose Mathematical Solution Corpora

Extract reusable mathematical moves from a bounded solution corpus without turning the result into a file-by-file summary or a catalog of theorem wrappers. The unit of analysis is a method family: a recurring local mathematical move, its exact carrier, and the global reasoning that lifts it into a proof.

Use audit-mathematical-vocabulary for one bounded mathematical slice, learn-from-math-agent-trajectories for one completed or paused investigation, and audit-public-operation-contracts when a specific operation needs a deep contract review. This skill owns the repository- or corpus-level decomposition that precedes those focused audits.

Freeze the scope

Record the immutable revision of every source repository and the Jacobian revision and catalog used for comparison. Define the included directories, campaigns, or certificate families and any exclusions. Do not claim corpus closure from a partial clone, truncated artifact, generated summary, or unreadable dependency.

Inventory artifact types before reading deeply: papers and notes, Lean or other formal developments, exact Python or native programs, numerical experiments, solver encodings and proofs, prompts and trajectories, certificate bundles, test fixtures, and replay or publication metadata. Use the inventory to find method families, not to produce a chronological summary of every file.

Group by solution technique

Cluster sources by mathematical move rather than conjecture name or language. Examples include exact finite enumeration, dynamic programming, local-lemma witnesses, linear or semidefinite duality, algebraic elimination, interval enclosure, canonicalization, dependent rounding, coding-theoretic profiles, and finite-state transfer arguments.

Select representative sources from each family. Prefer sources that expose the local move, exact hypotheses, boundary behavior, and an independently replayable fixture. Continue sampling within a family until another representative no longer reveals a new carrier, postcondition, representation regime, or failure mode.

Decompose each representative

Describe the exact carrier, local move and stable postcondition, how the surrounding theorem uses it, and the evidence or fixture. Add representation, closure, and discovery details when they affect the proposed operation.

The local move is not automatically an operation. Reject boundaries that merely expose one loop iteration, solver control, callback, proof bookkeeping, or temporary data structure. Also reject the opposite boundary when it bundles the motivating theorem, search strategy, interpretation, and stopping rule. Look for one postcondition that remains meaningful if the surrounding paper or conjecture disappears.

Resolve the result's closure cases before calling it a complete finite value. Record the empty, zero, identity, repeated-root, singular, boundary, or continuum-locus case that applies to the proposed carrier. A result whose maximizers can be all points of a curve, for example, needs an exact locus variant rather than a fictitious complete finite witness list. This is part of the postcondition, not an implementation footnote.

For continuous or analytic sources, use the carrier checks to distinguish an exact local operation from a discretization that loses the source's decisive semantics.

Classify an established technique as a public-operation candidate, native-only function, private kernel, invariant or fixture, or caller reasoning. Technique names may be discovery vocabulary for a public operation without becoming separate operation IDs. Require an independently consumable postcondition before making an intermediate technique separately runnable.

Treat representation as mathematical execution evidence. State whether the carrier is materialized, succinct, generated, or oracle-backed; what expansion the implementation performs; whether that expansion is predictable before execution; and whether a compact representation changes the complexity class or output obligation.

Research the method

Trace the local move to primary literature or an authoritative formal/library source. Verify the exact hypotheses, conclusion, conventions, algorithmic regime, and representation-sensitive complexity. Distinguish neighboring methods that share vocabulary but prove different guarantees. Use secondary surveys only to discover sources or terminology, then verify the conclusion against the primary source.

Research maintained exact backends and standard algorithms in proportion to the candidate. The question is whether a bounded, typed Jacobian contract is feasible—not whether the corpus's handwritten implementation should be copied. Record uncertainty when the literature supports the theorem but not an admissible exact kernel at the required scale.

Compare with Jacobian

Inspect the current catalog, native API, canonical values, request and result models, tests, admission decisions, and narrowly related issues. Attempt the smallest exact composition before declaring a gap. A manual coordinate change, cheap projection, or theorem-specific assembly normally remains caller work; incompatible values, detached certificates, or hidden expansion may instead identify an interoperability or contract problem.

Give every method family one disposition:

• existing operation or exact composition;
• representation or interoperability repair;
• discovery repair;
• request/result contract repair;
• scale or backend improvement;
• new bounded operation candidate;
• defining, convention, adversarial, metamorphic, producer-consumer, or stress fixture;
• reasoning, theorem-specific workflow, or certificate infrastructure; or
• no supported Jacobian action.

For operation candidates, state the semantic domain, stable postcondition, source representation, controlling work and output quantities, reconstruction or defining invariant, typed incomplete states, and at least one discriminating fixture. Separate the existence of a reusable gap from public-catalog admission.

Route actions without overclaiming

Verify issue ownership narrowly before proposing a new issue. Reinforce the canonical owner when the operation, contract, or scale question is already in scope. Keep distinct semantics separate even when they share a backend. Do not file, comment, edit external systems, or make repository changes without user authorization.

Prefer compact in-thread findings and focused repository actions. Do not create large durable reports, copied source archives, or generated inventories unless the user requests them. Preserve only the small fixtures and evidence needed to replay a conclusion.

Establish closure

Stop when every inventoried method family has a disposition, every proposed operation has been compared with exact current composition and issue ownership, and additional representative sources yield no new local move, representation regime, postcondition, or fixture role. Report the frozen revisions, coverage, important exclusions, and unresolved uncertainties. “No gaps remain” means no unclassified reusable move within that declared scope, not that the corpus or mathematical literature contains nothing else.

Lead the final result with a compact technique-to-disposition matrix, followed by the few highest-value operation, contract, scale, and fixture actions. Keep the proof workflow separate from the atomic mathematical move throughout.