Final Validation Report: Roth 1982

Updated: 2026-08-04

1. Human Verdict

Formalized for finite equal sides with complete strict preferences. The checked results cover stable and side-optimal matching, proposing-side truthfulness, manipulation impossibilities, and the efficient truthful procedure. The broad extension to quotas is outside this named-result scope.

2. Closeout Status

• Completion status: formalized.
• Scope: ten named results, with their source definitions and model prerequisites.
• Human review: no annotations recorded.

3. Source and Scope

The audited source is Alvin E. Roth, The Economics of Matching: Stability and Incentives, Mathematics of Operations Research 7(4), 1982. The canonical local source artifact is Roth82StableMatching.txt, SHA-256 55d2c35a8077a74dc9953dcb6f838fcd6e91edc9e77079a2419f185835a79fd4. The source record is linked to https://doi.org/10.1287/moor.7.4.617.

Normal scope follows the repository’s named-theory policy. It includes the paper-facing definitions of the preference profile, matching procedure, marriage model, complete outcome, stability, individual quotas, stable procedure, dominant truthfulness, efficiency, side-optimality, and simple misrepresentation, together with every numbered theorem, corollary, and lemma. Algorithms, the Section 6 numerical example, proof-local notation, proof narration, and unnumbered extension prose are deep-only unless needed to prove one of those endpoints.

4. Researcher Summary of Checked Results

5. Remaining Boundaries and Gaps

No mathematical obligation remains in the configured normal named-theory scope. The broad unnumbered assertion that all arguments carry over virtually unchanged to arbitrary individual quotas is not used as source credit for a general many-to-many incentive theorem. The paper’s quota-outcome definition is formalized with explicit finiteness of every partner set and exact finite cardinality equal to the corresponding quota. The one-to-one deferred-acceptance machinery used by the named results is checked through the matching library. The exact many-to-one waiting-list runner is auxiliary quota support. A future deep-paper review could formulate and prove a precise responsive-preference theorem for the broad both-sided-quota model.

The proposal algorithms, termination narration, possible-partner notation, serial-procedure narration, successful-misrepresentation shorthand, and the Section 6 example remain support or deep-audit material under the normal-scope policy. They are not missing named results.

6. Additional Assumptions Beyond Paper

None.

7. Proof-Strategy Deviations

None.

8. Proof Tricks Worth Reusing

• Audit impossibility theorems against every true and reported profile used by the witness; a strictness condition on the true profile alone is not enough for a stable-procedure quantifier.
• When embedding a finite manipulation witness into a larger market, assign distinct positive fallback scores and force dummy self-matches through preference order. Do not use negative dummy scores when the source domain makes every pair feasible.
• State source definitions as closed equivalences and source theorems as closed propositions, then prove them separately. This exposes the full target even when the implementation uses many helper lemmas.
• Connect source algorithms to shared library implementations by a proved refinement theorem after establishing the source runner’s own semantics.
• When a finite-cardinality API totalizes infinite sets, make finiteness an explicit conjunct before using its numeric result as a source quota.
• For incentive statements, audit both the original deviation and every existential replacement against the legal action space; checking only the truthful profile is insufficient.

9. Generalizations, Conjectures, and Extensions

The direct batched waiting-list runner and its refinement to applicant deferred acceptance can support future many-to-one matching papers. A genuine many-to-many version of Roth’s broad quota sentence would require a precise outcome model, responsive preferences over partner sets, acceptability and outside-option conventions, and a separately proved stability and incentive theorem. It is not obtained merely by cloning seats or renaming agents.

10. Source Clarifications and Exact Readings

None.

11. Paper Issues or Caveats

None for the selected results. The unproved broad quota extension is identified in Section 5.

12. Detailed Formalization Evidence

The current direct review selects ten named-result specifications in PaperInterface.lean, with twelve separately reviewed paper prerequisites. MainTheorems.lean contains the three-by-three manipulation witness and its positive, strict-preference arbitrary-rank extension. GeneralMatching.lean proves refinement of the source waiting-list procedure to the shared deferred-acceptance runner. The claim map binds the source statements to their expanded specifications; the accepted graph and current ledgers are linked in Section 17.

13. Paper Assumption Provenance

The paper-facing conditions are the equal finite sides and complete strict preference profiles of the source marriage model, plus the exact first-ranked-partner condition used by the source’s simple-misrepresentation reduction. Every true profile and every reported profile used by a stability or incentive theorem is required to be legal. No stable outcome, side-optimality conclusion, manipulation witness, or runner invariant is accepted as a theorem premise.

The source-domain conditions are included in the current prerequisite review linked in Section 17.

14. Displayed Formula Provenance

The source has no separately selected displayed-equation item in the normal inventory. Preference comparisons and blocking-pair clauses are audited inside their complete definition or theorem proposition.

15. Library Lift Pass

The paper reuses the shared assignment, stability, proposer-optimality, and deferred-acceptance infrastructure. The exact paper-facing model bridges and finite counterexamples remain paper-local so their source semantics are inspectable. The source waiting-list runner is connected to the shared runner only by a proved refinement; shared declaration names supply no paper credit.

16. DAG Audit

The dependency-DAG source is docs/DependencyDAG.tex, with rendered artifact docs/DependencyDAG.pdf. It shows the 21 source items, the checked deferred-acceptance support route, and the broad quota prose as deep-only. It does not promote support algorithms or the unchecked deep-only general quota extension to normal source results.

Codex current-protocol DAG inspection (2026-09-03, 21:30–21:31 EDT). The rendered one-page PDF was visually inspected at 160 dpi. Its source-definition groups and ten named-result boxes were legible, the displayed arrows preserved the result dependencies, and the deep-only quota material remained visually separated from the normal result surface.

17. Validation Checks

The current source-to-Spec ledger records 10 matching judgments. The paper-prerequisite ledger has 12 entries and the library-prerequisite ledger has 0. The closure receipt points to the accepted obligation graph. These saved records distinguish semantic judgments from human annotations; this documentation edit does not reissue them.

18. Paper Definitions Checked

The general Assignment carrier, score-table aliases, operational stability, and deferred-acceptance state machinery remain visible support vocabulary. They do not create extra normal-scope source items.

19. Named Theorem Statements Checked

20. Paper-Facing Statement Validator Ledger

The current source-to-Spec ledger records 10 matching judgments. The paper-prerequisite ledger has 12 entries and the library-prerequisite ledger has 0. The closure receipt points to the accepted obligation graph. These saved records distinguish semantic judgments from human annotations; this documentation edit does not reissue them.

21. Source-Coverage Audit Ledger

The source map retains the full source inventory and each item’s disposition. The current selected direct results and their judgments are listed in Section 20; definitions and supporting source conditions are reviewed in the separate prerequisite ledger. Source inventory entries are not automatically counted as theorem proofs.