docs/PHASE3_COUNTEREXAMPLES.md

Phase 3 Design: Counterexamples and Semantic Divergence

Phase 3 will extend proof search from “no rule closed the claim” to a structured diagnosis of why the claim failed. It will distinguish a false claim, an under-specified claim, and a claim that diverges semantically from a competing formulation.

Counterexample pipeline

1. Normalize the claim. Parse relational expressions into a typed predicate form and retain the source fingerprint.
2. Select a model domain. Start with finite integers and rationals for order/equality claims; later add finite sets, Boolean algebras, matrices, and bounded real intervals.
3. Search bounded assignments. Enumerate assignments to symbols appearing in the premises and conclusion, recording the smallest domain and assignment that satisfies the premises while violating the conclusion.
4. Minimize the witness. Remove irrelevant variables and premises, then return a minimal counterexample certificate.
5. Classify the failure. Return refuted when a witness exists, under-specified when premises are satisfiable but no conclusion follows, and unsupported when the language is outside the current model theory.

Every witness will contain the model, variable assignment, satisfied premises, violated conclusion, search bound, and deterministic fingerprint.

Semantic divergence tracking

A divergence record compares two claims after normalization rather than comparing their raw strings. It records the changed symbols, relation operators, premises added or removed, source/status changes, and whether the difference changes the proof result. The system will expose a directed graph of revisions:

claim@hash₁ --[conclusion_changed]--> claim@hash₂
claim@hash₂ --[premise_added]-------> claim@hash₃

Divergence classes will be lexical (formatting only), structural (AST changed), logical (proof obligations changed), epistemic (status or assumptions changed), and provenance (source changed). A failed proof can then say not only “open,” but “open because revision 2 removed the bridge premise that made revision 1 transitive.”

API and UI contract

Phase 3 will add POST /api/counterexample, POST /api/divergence, and GET /api/divergence/<fingerprint>. The dashboard will show a minimized witness beside failed proof output and overlay divergence edges on the argument graph. No numerical witness will ever be promoted to a Lean proof; it is evidence for refutation or model exploration only.

Soundness boundary

The bounded search engine is a falsifier, not a complete theorem prover. A found witness is decisive for the declared model, while failure to find a witness is inconclusive. Lean remains the authority for formal proof, and all Phase 3 results will carry the model class and search bound explicitly.