Representation Refinement

Find what a representation must remember for the mathematics it is meant to support. Work on a specific observation, operation, or failed law, rather than inventing a general abstraction without a concrete comparison question.

This is a focused process usable within theory building, calculus construction, or an ordinary proof attempt. It does not require a campaign or a previous failed proof. In a campaign, return the witness, repaired representation, proved laws, and remaining obligations to the existing state.

Diagnose Before Repairing

Recover the actual definitions, known results, and previous corrections. Specify concrete objects X, a representation alpha:X->A, intended observations Q, and the operations the representation should support. Record quantitative hypotheses and what success means: exact recovery, an upper bound, an approximation, or a usable composition law.

Classify the evidence and its scope:

Failure	Appropriate work
A concrete object violates the target	Correct or refute the target
A proposed auxiliary identity is false	Change the law; the target may survive
Equal retained data hide different required outcomes	Refine the representation or weaken its intended use
A natural construction leaves the object class	Enlarge the objects or presentations and prove comparisons
An identity holds but its quantitative control is missing	Prove the estimate with its actual assumptions

Several diagnoses can coexist. Do not infer falsity of a target from failure of one sufficient surrogate, or infer lost information merely from an unsuccessful proof search.

Test Sufficiency And Construct A Repair

For exact recovery of Q:X->Y, a map q:alpha(X)->Y with Q(x)=q(alpha(x)) exists exactly when Q is constant on every alpha-fibre. A witness pair with equal alpha and unequal Q rules out this exact use. Unequal Q values alone do not refute a common upper bound: an inequality witness must violate the proposed bound itself. For deterministic operations u, an induced operation on alpha(X) also requires alpha(x)=alpha(y) to imply alpha(u(x))=alpha(u(y)), on the appropriate domains. Track definedness as well as outputs for partial operations.

Choose candidate extra data from the failed operation: signed coefficients for cancellation, source provenance for dependence, transitions for gluing, or a remainder for an inexact identity. One repair is alpha'(x)=(alpha(x),r(x)). Show that it separates the witness, then prove the intended recovery/composition laws on the claimed class. Separating one pair does not prove sufficiency everywhere.

Compare alternative repairs, including changing the law, weakening the conclusion, retaining a latent presentation, or using a known richer representation. Added data should have mathematical meaning and an available construction; recording the unknown answer or a complete object can be sufficient yet offer no progress. Assess cost and usefulness after accounting for the work of constructing it.

Use Finite Refinement When Its Inputs Are Available

Read finite sufficient representations when X, its operations, and the observations are supplied explicitly. It gives a terminating partition procedure and a related observable-space/operator-algebra construction. The quotient is relative to those operations and observations; it may retain all of X. A supplied observable cannot be an oracle for the unresolved problem.

The finite construction uses standard automata partition-refinement ideas. Counterexample-guided abstraction refinement is a related verification precedent; its termination and correctness results do not automatically transfer to arbitrary mathematical research. Bounded countermodel search can find informative witnesses; not finding one does not prove an infinite-class law.

Integrate The Mathematics

Produce a concrete witness, a proved sufficiency/comparison result, a repaired law, or a precisely located remaining obligation. Keep examples, definitions, proofs, and the failed representation's scope in the existing project artifacts.

Use proof-calculus when repaired data support a language of arguments and a validity theorem; mathematical-framework-design when the setting or a substantial correspondence must be constructed; grothendieck-rising-sea when developing the surrounding theory. These processes can inform each other without fixed gates. Do not present a more elaborate representation as a solution until its concrete models, comparisons, and required estimates are proved.