{"schema_version":1,"research_id":"eoa_inverse_innovation_exp06_external_evaluation_20260803","source_assessment_id":"representation_independent_interface_contract__mathematics:P2:v0","cell_id":"representation_independent_interface_contract__mathematics","search_queries":["metric completion universal property dense isometry extension uniformly continuous formalization Lean mathlib","site:leanprover-community.github.io mathlib completion extension uniform continuous","site:github.com/leanprover-community/mathlib4 metric completion quotient Cauchy universal property issue","Isabelle metric completion universal property formalization","site:leanprover-community.github.io/mathlib4_docs \"Abstract theory of Hausdorff completions\"","site:github.com/leanprover-community/mathlib4 \"structure AbstractCompletion\"","site:github.com/leanprover-community/mathlib4 \"AbstractCompletion\" completion uniform","site:leanprover-community.github.io/mathlib4_docs UniformSpace AbstractCompletion extension","site:leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Completion.html \"AbstractCompletion\"","site:github.com/leanprover-community/mathlib4 \"Generalise the results here from the concrete\"","site:github.com/leanprover-community/mathlib4 AbstractCompletion TODO generalise completion","site:leanprover-community.github.io/contribute mathlib contribution process pull request maintainers review","site:leanprover-community.github.io/mathlib4_docs contribution guidelines maintainers review mathlib","mathlib contribution guide pull request review official","site:bls.gov/ooh software developers median pay 2025 BLS","site:bls.gov/oes mathematicians median wage May 2025"],"sources":[{"source_id":"S1","title":"Mathlib.Topology.UniformSpace.AbstractCompletion","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/AbstractCompletion.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Mathlib already defines an AbstractCompletion with a carrier, embedding, completeness, separation, uniform-inducing, and density obligations.","The existing API proves extension agreement and uniqueness, identity and composition laws, comparison maps, inverse comparison, and equivalences between completions.","This is a close implementation-level match to the proposed universal-property contract."]},{"source_id":"S2","title":"Mathlib.Topology.UniformSpace.Completion","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Completion.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Mathlib's concrete completion is built from Cauchy filters and packaged as an AbstractCompletion.","The concrete API exposes extension, uniqueness, map, identity, composition, and equivalence results.","The current extension function is total and returns an arbitrary constant when uniform continuity is absent, contrasting with the proposal's premise-carrying failure semantics."]},{"source_id":"S3","title":"Mathlib.Topology.Category.UniformSpace","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/UniformSpace.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Mathlib already packages completion categorically using a completion functor and extension morphisms.","The categorical layer is adjacent prior art for identity, composition, and universal-property coherence."]},{"source_id":"S4","title":"Mathlib.Analysis.Normed.Module.Completion","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Completion.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["The module defines structures and maps specifically for the concrete UniformSpace.Completion.","Its official TODO expressly asks to generalize these results to any AbstractCompletion, visibly documenting the coupling problem and desired adopter-side direction."]},{"source_id":"S5","title":"Mathlib.Topology.Algebra.UniformRing","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformRing.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Ring operations and morphism extensions are currently defined on the concrete Completion.","The official TODO independently repeats the request to generalize from concrete Completion to AbstractCompletion."]},{"source_id":"S6","title":"Mathlib.Topology.UniformSpace.CompareReals","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/CompareReals.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Mathlib already applies AbstractCompletion to compare Cauchy-sequence and uniform-space constructions of the reals.","The file describes this comparison as a cross-sanity check and says AbstractCompletion supplies the abstract characterization and isomorphisms.","This is a concrete precedent for the proposal's independent-construction differential comparison."]},{"source_id":"S7","title":"How to contribute to mathlib","publisher":"Mathlib Community","url":"https://leanprover-community.github.io/contribute/how-to-contribute.html","source_class":"OFFICIAL_GUIDANCE","publication_date":"n.d.","accessed_at":"2026-08-03","claims_supported":["Experimental work can occur in a fork and separate branch before a pull request.","Changes to production require successful CI and final approval from mathlib maintainers.","Small, self-contained pull requests and prior community discussion are encouraged."]},{"source_id":"S8","title":"Table 1. National employment and wage data from the Occupational Employment and Wage Statistics survey by occupation, May 2025","publisher":"U.S. Bureau of Labor Statistics","url":"https://www.bls.gov/news.release/ocwage.t01.htm?mod=article_inline","source_class":"GOVERNMENT_OR_REGULATOR","publication_date":"2026-05-15","accessed_at":"2026-08-03","claims_supported":["May 2025 mean wages were $148,100 annually for software developers and $129,260 for mathematicians.","The wage data provide a public anchor for resource-equivalent labor-cost estimates, but not a quote for specialized Lean formalization."]}],"problem_evidence":{"support":"STRONG","rationale":"The problem is directly visible in current Mathlib: normed-module and uniform-ring completion files bind results to the concrete Completion and each contains an explicit TODO to generalize to AbstractCompletion. The concrete completion documentation also exposes construction-specific Cauchy-filter machinery. This establishes existence and relevance inside at least two library areas, although no dependency census or measured maintenance burden was found.","source_ids":["S2","S4","S5"]},"stakeholder_evidence":{"support":"STRONG","rationale":"Mathlib is an identifiable adopter; its own module TODOs express the need, while its contribution guidance identifies maintainers as final authorizers and specifies branch, CI, review, and merge workflow. No committed maintainer, grant, budget, deadline, or named downstream theorem owner was found.","source_ids":["S4","S5","S7"]},"prior_art":{"proximity":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"Mathlib AbstractCompletion","similarity":"Substantially implements the proposed representation-independent package: carrier, embedding, completeness, density, extension, uniqueness, map laws, comparison, and equivalence between completions.","remaining_difference":"The proposal would make extension premises unavailable at the type/API level, record an explicit assumption ledger, and test a frozen 12-theorem corpus; Mathlib's existing extension is derived and totalized with arbitrary behavior outside its valid premises.","source_ids":["S1","S2"]},{"name":"Mathlib completion functor and categorical packaging","similarity":"Packages completion as a functor with extension morphisms, covering the proposal's functorial identity/composition direction.","remaining_difference":"It does not by itself demonstrate migration of the proposed 12 downstream metric-analysis lemmas or the proposed leakage audit.","source_ids":["S3"]},{"name":"Mathlib CompareReals","similarity":"Uses AbstractCompletion to compare independently presented Cauchy-real and uniform-space completion constructions and obtain isomorphisms.","remaining_difference":"It is a specific comparison of real-number constructions, not the proposed closure-based fixed-space adapter and 12-lemma replay.","source_ids":["S6"]},{"name":"Existing requests to generalize concrete completion APIs","similarity":"Two official TODOs request exactly the diffusion move from concrete Completion to AbstractCompletion.","remaining_difference":"They do not specify the proposal's frozen corpus, assumption ledger, premise-carrying extension wrapper, or quantitative comparator.","source_ids":["S4","S5"]}],"distinctive_claim_remaining":"The remaining falsifiable claim is narrow and implementation-specific: for a frozen set of 12 currently concrete completion lemmas, reusing Mathlib's existing AbstractCompletion—optionally with a premise-carrying extension wrapper and explicit assumption ledger—will reproduce the exact statements and assumptions without concrete Completion access, while requiring less theorem-specific transport work than the baseline. The universal-property architecture and induced equivalence are not distinctive.","confidence":"HIGH"},"implementation_evidence":{"support":"STRONG","rationale":"Technical feasibility is demonstrated by an extant AbstractCompletion implementation, a concrete completion adapter, proved extension/uniqueness/coherence laws, a categorical completion layer, and a deployed cross-construction comparison. Workflow and authority are bounded by fork, branch, CI, review, and maintainer approval. The pilot manipulates pure formal artifacts and no personal data. Remaining uncertainties require actual Lean work: whether the chosen closure construction fits available instances, whether all 12 statements retain their assumptions, whether a safer partial extension API integrates cleanly, and the review burden of generalizing downstream algebraic structures.","source_ids":["S1","S2","S3","S6","S7"]},"scores":{"meaningful_impact":{"score":3,"rationale":"Official TODOs show real reuse value in multiple modules, but no prevalence count, maintenance-time measurement, or realized downstream impact is available.","source_ids":["S4","S5"]},"stakeholder_pull":{"score":4,"rationale":"The adopter and authorizer are identifiable and the desired generalization is explicitly stated twice; commitment, funding, and scheduling remain absent.","source_ids":["S4","S5","S7"]},"incremental_advantage":{"score":1,"rationale":"The central universal-property package, extension/uniqueness laws, functoriality, and comparison equivalences already exist. Only a bounded migration, safer premise interface, and assumption audit remain.","source_ids":["S1","S2","S3","S6"]},"distinctiveness_plausibility":{"score":1,"rationale":"External evidence establishes an implementation-level collision with the main proposal. Distinctiveness survives only at the selected-corpus and API-hardening level.","source_ids":["S1","S2","S6"]},"technical_implementability":{"score":5,"rationale":"Current Mathlib code demonstrates all principal mathematical mechanisms and one cross-construction comparison.","source_ids":["S1","S2","S3","S6"]},"adoption_authority_feasibility":{"score":4,"rationale":"A reversible fork/branch pilot is open to contributors; production adoption has a clear CI, review, and maintainer-approval path. No maintainer has pre-approved this specific work.","source_ids":["S7"]},"evidence_readiness":{"score":4,"rationale":"Official executable documentation supplies unusually direct prior-art and feasibility evidence, while the frozen dependency inventory and comparative effort data do not yet exist.","source_ids":["S1","S2","S4","S5","S6"]},"safety_net_benefit":{"score":4,"rationale":"An isolated branch, unchanged originals, CI, and maintainer review provide strong rollback and error containment. Formal equivalence does not eliminate automation or definitional-equality regressions.","source_ids":["S2","S7"]},"scalability":{"score":4,"rationale":"AbstractCompletion is generic and already reused, suggesting broad technical scalability; each downstream algebraic hierarchy may still require specialized adapters and review.","source_ids":["S1","S3","S4","S5"]}},"score_confidence":"HIGH","costs":{"first_evidence":{"band_2026_usd":"10K_TO_50K","scope":"Freeze 12 lemmas; inventory concrete dependencies and axioms; implement one AbstractCompletion-based replay and baseline comparator in a private branch.","confidence":"MODERATE","assumptions":["Approximately 4-8 specialist person-weeks.","Resource-equivalent rates are based on May 2025 U.S. mean wages of $148,100 for software developers and $129,260 for mathematicians, with allowance for benefits, tooling, and specialist scarcity.","No production migration or extended maintainer review is included."],"source_ids":["S7","S8"]},"initial_deployment_startup":{"band_2026_usd":"50K_TO_250K","scope":"Generalize a coherent first module, add adapters and regression tests, split work into reviewable pull requests, and resolve CI and reviewer feedback.","confidence":"LOW","assumptions":["Approximately 3-9 specialist person-months depending on typeclass and definitional-equality breakage.","Existing AbstractCompletion code is reused rather than replaced.","Volunteer review delay is not priced as paid labor but may extend calendar time."],"source_ids":["S1","S4","S7","S8"]},"operational_launch":{"band_2026_usd":"50K_TO_250K","scope":"Upstream the accepted generalized API, migrate selected normed-module or uniform-ring consumers, document assumptions and deprecations, and complete maintainer-approved CI rollout.","confidence":"LOW","assumptions":["Approximately 4-12 specialist person-months distributed across authoring, review, integration, and migration.","The launch is incremental and does not replace the canonical Completion construction.","Costs can be much lower when contributed by existing volunteers, but resource-equivalent labor is still counted."],"source_ids":["S4","S5","S7","S8"]},"annual_recurring":{"band_2026_usd":"10K_TO_50K","scope":"Maintain generalized lemmas and adapters across Lean/Mathlib changes, triage regressions, and review new construction-specific leakage.","confidence":"LOW","assumptions":["Approximately 3-10 specialist person-weeks per year.","No dedicated infrastructure or hosted service is required.","Recurring effort depends on downstream adoption breadth, which has not been measured."],"source_ids":["S7","S8"]}},"verified_pipeline_gates":{"externally_supported_problem":{"status":"YES","reason":"Two current official modules explicitly identify concrete-Completion coupling and request generalization to AbstractCompletion.","source_ids":["S4","S5"]},"externally_credible_adopter_or_authorizer":{"status":"YES","reason":"Mathlib is the direct adopter; its maintainers are the documented final merge authority.","source_ids":["S4","S5","S7"]},"distinct_testable_incremental_claim":{"status":"YES","reason":"Although the architecture is established practice, exact replay of 12 frozen lemmas with unchanged statements and axioms, no concrete access, and lower transport effort is a bounded contextual claim.","source_ids":["S1","S4","S5"]},"bounded_next_evidence_step":{"status":"YES","reason":"A 12-lemma isolated-branch replay with named baseline, treatment, comparators, and predeclared falsifiers is bounded and reversible.","source_ids":["S1","S2","S7"]},"no_unresolved_safety_or_authority_stop":{"status":"YES","reason":"The pilot changes no production definitions, handles no personal data, and production integration remains subject to CI and maintainer approval. No unresolved legal or physical-safety issue was found.","source_ids":["S7"]},"credible_cost_scope_and_range":{"status":"YES","reason":"The work units are bounded and public 2025 occupation wages anchor 2026 resource-equivalent bands, though specialized Lean productivity and review costs remain uncertain.","source_ids":["S7","S8"]}},"next_evidence_step":"Run a diffusion pilot, not a new-package invention study. In a forked branch, freeze 12 named concrete-Completion lemmas across extension, uniqueness, density, convergence, functoriality, and one algebraic structure. Record their exact statements, imported axioms, proof dependencies, compile time, and proof size. Treatment: restate and prove them using existing Mathlib AbstractCompletion, adding only a separately named premise-carrying extension wrapper if required. Comparator A: the unchanged concrete Completion proofs; comparator B: pairwise equivalence/transport without generalized lemmas. Add a second adapter using the existing CompareReals pattern or a genuinely independent closure construction. Success requires all 12 statements and assumptions to remain byte-for-byte or propositionally equivalent, zero access to concrete Cauchy-filter representation, successful identity/composition and induced-equivalence checks, and no increase in undeclared axioms. Falsify the incremental claim if AbstractCompletion cannot express any frozen theorem without stronger assumptions, concrete access, or changed mathematical content; if the premise-carrying wrapper adds no detectable safety or proof benefit; or if migration effort is not lower than comparator B. Before any upstreaming, discuss scope with topology/analysis maintainers and divide accepted changes into small CI-tested pull requests.","blocking_evidence":["No frozen 12-lemma corpus or construction-specific dependency inventory has been produced.","No measured comparison of proof effort, compile time, maintenance burden, or theorem-transport count exists.","No maintainer has committed to sponsor or review the specific pilot.","It is unknown whether a closure-based adapter can be made genuinely independent of the incumbent foundational definitions.","It is unknown whether replacing totalized extension with a premise-carrying wrapper improves usability enough to justify an additional API.","World novelty, patentability, freedom to operate, market size, and realized impact were not evaluated."],"research_disposition":"KNOWN_PRACTICE_DIFFUSION","world_novelty_boundary":"World novelty remains unmeasured. The bounded search establishes only that the proposal's central construct is already implemented in current Mathlib as AbstractCompletion, including extension, uniqueness, identity/composition, comparison maps, induced equivalences, categorical packaging, and a concrete cross-construction application. The only uncollided boundary found is contextual diffusion to a frozen theorem corpus, explicit assumption tracking, and possibly a premise-carrying extension API. No patent, freedom-to-operate, market-size, or realized-impact conclusion is made.","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_version":0,"controller_recommendation":{"action":"STOP_DIFFUSION","repairable":false,"material_progress_observed":true,"progress_targets":["Replace the proposed new CompletionPackage with reuse or a minimal refinement of existing Mathlib AbstractCompletion.","Choose one official generalization TODO and freeze 12 concrete declarations as the diffusion target.","Obtain topology/analysis maintainer feedback on scope and on whether a premise-carrying extension wrapper belongs in Mathlib.","Produce the dependency-and-assumption ledger and measure treatment versus concrete and pairwise-transport comparators.","Demonstrate a genuinely independent second adapter and preserve exact theorem statements, axioms, and CI behavior.","Document any irreducible need for definitional equality or construction-specific namespaces instead of weakening the abstract contract."],"reason":"The evaluation found strong problem and adopter evidence but also an established, implementation-level prior-art collision: Mathlib already has the proposed universal-property abstraction and comparison machinery. The worthwhile remaining activity is diffusion into modules whose own TODOs request generalization, not continued treatment as a distinct innovation."},"proposal_index":2}