{"schema_version":1,"research_id":"eoa_inverse_innovation_exp06_external_evaluation_20260803","source_assessment_id":"catalytic_pathway_enablement__mathematics:P1:v0","cell_id":"catalytic_pathway_enablement__mathematics","search_queries":["Gröbner basis ideal membership certificate original generators computer algebra reusable fixed ideal normal form documentation","formal verified Groebner basis ideal membership certificate Lean Coq research","Macaulay2 Gröbner basis lift original generators remainder documentation","Singular manual groebner basis reduce lift ideal membership","site:doc.sagemath.org ideal reduce groebner_basis polynomial ideal official documentation","site:oscar-system.github.io ideal groebner basis reduce membership documentation","Mayr Meyer ideal membership complexity Groebner bases primary paper","site:bls.gov mathematicians wage May 2025 software developers occupational employment wages","SageMath reference polynomial ideals groebner_basis reduce official","OSCAR documentation ideal membership groebner basis normal form official","Singular manual lift reduce std groebner official","proof logging Gröbner basis ideal membership certificate independent checker","certifying ideal membership proof logging Gröbner basis checker paper","certificate based verification Groebner basis computations ideal membership paper","incremental ideal membership queries fixed ideal reuse Gröbner basis","computer algebra independent verification Groebner basis certificate"],"sources":[{"source_id":"S1","title":"gb — compute a Gröbner basis","publisher":"Macaulay2 Project","url":"https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/_gb.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"Undated live documentation","accessed_at":"2026-08-03","claims_supported":["Macaulay2 computes reusable Gröbner-basis objects for ideals and modules.","ChangeMatrix and syzygy options retain relationships to original generators.","Reduction by the computed basis determines membership, substantially overlapping the proposed computational core."]},{"source_id":"S2","title":"Ideals · Singular.jl","publisher":"Singular.jl / OSCAR project","url":"https://oscar-system.github.io/Singular.jl/stable/ideal/","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"Undated live documentation","accessed_at":"2026-08-03","claims_supported":["Singular.jl supplies standard-basis computation, polynomial reduction, and generator-to-basis transformation matrices.","lift_std computes a Gröbner basis and transformation matrix satisfying G=M*T, enabling generator-linked evidence.","Existing APIs already separate basis computation from subsequent reductions and expose important domain restrictions."]},{"source_id":"S3","title":"Ideals of commutative rings — Sage Reference Manual","publisher":"SageMath Project","url":"https://sagemath.gitlab.io/documentation/html/en/reference/rings/sage/rings/ideal.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"Undated live documentation","accessed_at":"2026-08-03","claims_supported":["SageMath exposes ideal objects, Gröbner-basis computation, and reduction modulo an ideal.","The same broad reusable-basis workflow exists in another independently maintained computer-algebra system.","Examples demonstrate that basis size can be nontrivial even for standard benchmark ideals."]},{"source_id":"S4","title":"Short proofs of ideal membership","publisher":"Clemens Hofstadler and Thibaut Verron via arXiv; subsequently Journal of Symbolic Computation","url":"https://arxiv.org/abs/2302.02832","source_class":"PRIMARY_RESEARCH","publication_date":"2023-02-06; revised 2024-04-09","accessed_at":"2026-08-03","claims_supported":["A cofactor representation in the original generators is an ideal-membership certificate.","Certificates can vary greatly in complexity.","Finding bounded sparse representations is NP-complete, while the authors provide and experimentally evaluate a practical sparse-certificate method."]},{"source_id":"S5","title":"Formalizing Gröbner Basis Theory in Lean","publisher":"Junyu Guo, Hao Shen, Junqi Liu, and Lihong Zhi via arXiv","url":"https://arxiv.org/abs/2602.12772","source_class":"PRIMARY_RESEARCH","publication_date":"2026-02-13","accessed_at":"2026-08-03","claims_supported":["Lean formalization covers polynomial division, Buchberger's criterion, and existence and uniqueness of reduced Gröbner bases.","The formalized theorem states that zero remainder under a valid Gröbner basis is equivalent to ideal membership.","The paper reports that SageMath-generated certificates are already checked inside Lean by polyrith, that some new infrastructure has entered Mathlib, and that certification of external computation remains future work."]},{"source_id":"S6","title":"Certifying Algorithms for Automated Reasoning","publisher":"Schloss Dagstuhl – Leibniz Center for Informatics","url":"https://drops.dagstuhl.de/storage/04dagstuhl-reports/volume15/issue06/25231/DagRep.15.6.1/DagRep.15.6.1.pdf","source_class":"PRIMARY_RESEARCH","publication_date":"2026","accessed_at":"2026-08-03","claims_supported":["The report identifies ideal membership as central to computer algebra and relevant to geometry, verification, and symbolic computation.","It explicitly states that Gröbner-basis outputs can be complex and difficult to certify.","It presents a practical algebraic-calculus framework for tracking polynomial manipulations at different proof granularities."]},{"source_id":"S7","title":"The first Mayr-Meyer ideal","publisher":"Irena Swanson via arXiv","url":"https://arxiv.org/abs/math/0209154","source_class":"PRIMARY_RESEARCH","publication_date":"2002-09-12","accessed_at":"2026-08-03","claims_supported":["Polynomial ideal-membership instances can exhibit doubly exponential complexity.","Worst-case resource growth remains a substantive feasibility and scalability constraint despite basis reuse."]},{"source_id":"S8","title":"National employment and wage data by occupation, May 2025","publisher":"U.S. Bureau of Labor Statistics","url":"https://www.bls.gov/news.release/ocwage.t01.htm","source_class":"GOVERNMENT_OR_REGULATOR","publication_date":"2026-06; May 2025 survey data","accessed_at":"2026-08-03","claims_supported":["Mean wages were approximately $62.15 per hour for mathematicians, $71.20 for software developers, and $53.60 for software-quality-assurance analysts.","These wage anchors support labor-based 2026 resource-equivalent cost estimates, subject to benefits, overhead, seniority, and inflation assumptions."]}],"problem_evidence":{"support":"MODERATE","rationale":"The underlying problem visibly exists: ideal membership is central, certificates may be complicated, certification remains difficult, and worst-case computations can grow doubly exponentially. However, no external source verifies the candidate's specific premise that a particular collaborative mathematics project repeatedly reconstructs one unchanged ring, ideal, and monomial order or that this is its dominant delay.","source_ids":["S4","S5","S6","S7"]},"stakeholder_evidence":{"support":"WEAK","rationale":"Mathlib/Lean contributors and the Dagstuhl certification community are identifiable stakeholders actively developing reusable formal infrastructure and external-computation certification. Existing CAS projects also maintain the required primitives. No named research collaboration, algebra lead, funder, or theorem-review authority has expressed intent to adopt this exact governed lane, supplied a case queue, or committed staff for a probe.","source_ids":["S1","S2","S3","S5","S6"]},"prior_art":{"proximity":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"Macaulay2 Gröbner-basis object with ChangeMatrix","similarity":"Computes a basis once, preserves its relationship to original generators, and supports subsequent membership reductions.","remaining_difference":"Does not itself provide the proposed immutable service version, independent organizational checker, intake governance, queue dashboard, or controlled withdrawal process.","source_ids":["S1"]},{"name":"Singular.jl std/lift_std/reduce workflow","similarity":"Directly implements reusable basis computation, transformation matrices, and reduction of later polynomials.","remaining_difference":"The proposal adds project-level eligibility, independent checking, monitoring, stewardship, and comparative workflow measurement rather than a new algebraic method.","source_ids":["S2"]},{"name":"SageMath ideal and reduction APIs","similarity":"Packages an ideal as a reusable computational object and exposes basis computation and modular reduction.","remaining_difference":"Certificate provenance, separate checking, immutable deployment versions, and authority boundaries are not established by this documentation.","source_ids":["S3"]},{"name":"Certificate-producing ideal-membership methods","similarity":"Treats original-generator cofactor representations as checkable membership certificates and explicitly optimizes certificate complexity.","remaining_difference":"The proposal combines certificates with repeated fixed-context service operation; it does not propose a new certificate algorithm.","source_ids":["S4","S6"]},{"name":"Lean/Mathlib Gröbner formalization and checked external certificates","similarity":"Separates external certificate computation from checking in a trusted formal environment and formalizes the membership criterion.","remaining_difference":"End-to-end certification of external Gröbner computations is still described as future work, and the proposal's workflow metrics and governance are outside the formalization.","source_ids":["S5"]}],"distinctive_claim_remaining":"Only a local operational claim remains: for a real stream of eligible queries sharing exactly one ring, ideal, generators, and monomial order, an immutable precomputed basis plus independently checked generator-linked certificates and governed intake will reduce total setup-plus-computation-plus-checking resource use relative to per-case recomputation and best available theorem-specific handling, without any wrong classification, increased exception burden, or downstream-review bottleneck. This is contrastive and falsifiable but is not a world-novel algebraic claim.","confidence":"HIGH"},"implementation_evidence":{"support":"STRONG","rationale":"Multiple mature systems already supply the core basis, reduction, transformation-matrix, and ideal-object primitives; research demonstrates generator-linked certificates and formal checking. A shadow-mode wrapper with immutable context hashes, resource limits, logging, exception routing, and a checker using an independently implemented validation path is technically credible. Residual risks are expression explosion, certificate size, shared-bug correlation, incorrect handling of nonzero remainders, stale context, and checker or downstream-review saturation. No special legal authority is needed for a non-live probe, but production theorem use must remain under ordinary mathematical review and applicable software licenses must be checked.","source_ids":["S1","S2","S3","S4","S5","S6","S7"]},"scores":{"meaningful_impact":{"score":3,"rationale":"Potentially meaningful where basis construction and contextual validation recur frequently, but no project-level prevalence, delay, or labor measurement establishes the attainable impact.","source_ids":["S6","S7"]},"stakeholder_pull":{"score":2,"rationale":"There is visible research and tool-maintainer interest in certification and reusable formalization, but no committed adopter for the proposed operational lane.","source_ids":["S5","S6"]},"incremental_advantage":{"score":2,"rationale":"Basis reuse, reduction, generator transformations, and checked certificates already exist; advantage depends on whether governance and amortization improve a particular workflow.","source_ids":["S1","S2","S3","S4"]},"distinctiveness_plausibility":{"score":1,"rationale":"The algebraic and software core is established practice. The remaining distinction is a locally testable service design, not demonstrated technical novelty.","source_ids":["S1","S2","S3","S4","S5","S6"]},"technical_implementability":{"score":4,"rationale":"Mature CAS APIs and emerging formal checking make a bounded implementation credible, although worst-case growth and checker independence require careful engineering.","source_ids":["S1","S2","S3","S4","S5","S7"]},"adoption_authority_feasibility":{"score":3,"rationale":"A project algebra lead could authorize shadow testing and ordinary theorem reviewers can retain final authority, but no actual lead or review body has been identified externally.","source_ids":["S5","S6"]},"evidence_readiness":{"score":4,"rationale":"The proposed twelve-case concealed comparison is bounded, reversible, and instrumentable with mature software; representative local cases and personnel are the missing inputs.","source_ids":["S1","S2","S4","S5"]},"safety_net_benefit":{"score":4,"rationale":"Immutable versions, independent checking, context rejection, withdrawal, and preservation of ordinary theorem review directly reduce stale-context and false-assurance risks if implemented independently.","source_ids":["S4","S5","S6"]},"scalability":{"score":3,"rationale":"Reuse can amortize basis construction across many queries, but coefficient growth, certificate size, exceptional cases, and doubly exponential worst cases limit predictable scaling.","source_ids":["S4","S7"]}},"score_confidence":"MODERATE","costs":{"first_evidence":{"band_2026_usd":"10K_TO_50K","scope":"Prepare one non-live immutable basis version, implement a minimal certificate/log wrapper, select and blind twelve adjudicated cases, execute the lane and two comparators, independently check outputs, and analyze results.","confidence":"MODERATE","assumptions":["Approximately 100-250 combined mathematician, developer, and checker hours.","Loaded labor is approximately $95-$150 per hour after applying overhead and modest 2026 escalation to BLS mean wages.","Existing open-source CAS and ordinary workstation or small cloud compute are adequate.","No difficult formal-verification development is required for the first probe."],"source_ids":["S1","S2","S3","S8"]},"initial_deployment_startup":{"band_2026_usd":"50K_TO_250K","scope":"Production-quality context hashing, immutable artifact storage, certificate schema, independently implemented checker, resource sandboxing, test corpus, incident and withdrawal tooling, documentation, and license review for one algebraic context.","confidence":"LOW","assumptions":["Approximately 0.5-1.5 combined technical FTE-years depending on checker assurance level.","Existing CAS engines are integrated rather than reimplemented.","Formal verification of the complete engine is excluded.","The deployment covers one project and a limited set of coefficient domains."],"source_ids":["S1","S2","S4","S5","S8"]},"operational_launch":{"band_2026_usd":"50K_TO_250K","scope":"Operate a limited live launch with dual review, monitoring, steward training, case migration, exception handling, audit sampling, and rollback readiness.","confidence":"LOW","assumptions":["A partial-time algebra steward, checker/reviewer, and software maintainer are required for six to twelve months.","Compute remains small relative to specialist labor for ordinary cases.","The lane supplements rather than replaces theorem review."],"source_ids":["S5","S6","S7","S8"]},"annual_recurring":{"band_2026_usd":"50K_TO_250K","scope":"Basis regeneration and revalidation, software maintenance, certificate audits, incident response, queue monitoring, exception review, compute, and succession coverage for one project.","confidence":"LOW","assumptions":["Approximately 0.4-1.2 total FTE across mathematics, software, and checking roles.","Context changes are occasional rather than per case.","Pathological cases are rerouted before consuming unbounded resources.","Costs can exceed this band if formal verification or high-performance computing becomes necessary."],"source_ids":["S4","S5","S7","S8"]}},"verified_pipeline_gates":{"externally_supported_problem":{"status":"YES","reason":"Primary research and an authoritative workshop report establish that ideal membership is central, can be computationally difficult, and has a real certification burden. The prevalence of repeated setup in the unnamed target project remains unverified.","source_ids":["S4","S6","S7"]},"externally_credible_adopter_or_authorizer":{"status":"UNCERTAIN","reason":"Mathlib/Lean and computer-algebra maintainers are credible adjacent adopters, but no named collaboration or algebra lead has expressed intent or authority to run this exact lane.","source_ids":["S1","S2","S3","S5","S6"]},"distinct_testable_incremental_claim":{"status":"YES","reason":"The remaining claim compares total work, correctness, eligibility, exceptions, and downstream load under a governed reused basis against per-case recomputation and theorem-specific handling.","source_ids":["S1","S2","S4","S5"]},"bounded_next_evidence_step":{"status":"YES","reason":"A twelve-case concealed shadow trial with fixed comparators, resource ceilings, mismatch and withdrawal controls, and precommitted stop criteria is bounded and reversible.","source_ids":["S1","S2","S4","S5"]},"no_unresolved_safety_or_authority_stop":{"status":"YES","reason":"Shadow mode, independent certificate checking, immutable context versions, immediate withdrawal, exception routing, and preservation of ordinary theorem-review authority bound the first-step risks. Production authority is explicitly not granted.","source_ids":["S4","S5","S6"]},"credible_cost_scope_and_range":{"status":"YES","reason":"The work packages are bounded and labor-dominated; BLS occupational wages provide a defensible resource-equivalent anchor, although deployment estimates remain low-confidence until a stack and assurance level are selected.","source_ids":["S8"]}},"next_evidence_step":"Partner with one named commutative-algebra project and first audit its previous six months of candidate identities to verify that at least twelve cases share an identical ring, coefficient field, ideal version, and monomial order. If that prevalence condition holds, preregister a twelve-case concealed shadow experiment containing zero and nonzero normal forms, at least two high-expression-growth cases, one mismatched-context case, and one simulated withdrawn-version submission. Compare (A) the immutable shared-basis lane with independent checking, (B) per-case Gröbner-basis recomputation with equivalent checking, and (C) the project's best documented direct or theorem-specific method where available. Measure operator setup time, compute and memory, checker time, elapsed time, correct classifications, certificate size and completeness, exceptions, reroutes, and downstream review time. Require zero wrong or unverifiable classifications, correct rejection of both integrity controls, at least 25% lower median total resource-equivalent cost than comparator A, no material increase against comparator C, and no shift of the dominant queue to checking or theorem review. Halt on any wrong result, context leak, unverifiable certificate, or preset resource-ceiling breach. Failure of the prevalence threshold, the correctness conditions, or the resource advantage falsifies the proposed scope.","blocking_evidence":["No external audit establishes that a real project has a material recurring queue under one invariant algebraic context.","No named adopter, algebra lead, theorem-review authority, or funder has committed to the probe.","No head-to-head evidence shows that shared-basis operation reduces total human-plus-compute work rather than merely moving effort into certificate generation or checking.","The independence architecture for the checker is unspecified, leaving correlated reducer/checker defects unresolved for production use.","The eligible-case fraction and distribution of coefficient growth, memory demand, and certificate size are unknown.","Software-license, data-retention, and reproducibility requirements have not been assessed for a selected project and stack."],"research_disposition":"KNOWN_PRACTICE_DIFFUSION","world_novelty_boundary":"The search establishes substantial and mature prior art for computing a Gröbner basis once, reducing later polynomials, retaining transformations to original generators, producing membership certificates, and checking externally generated certificates. It does not measure world novelty, patentability, freedom to operate, market size, or realized impact. The only surviving claim is local workflow performance and governance under a particular repeated-query workload.","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_version":0,"controller_recommendation":{"action":"STOP_EMPIRICAL_RESEARCH_NEEDED","repairable":false,"material_progress_observed":true,"progress_targets":["Name a real adopting project, algebra lead, and theorem-review authority and document their willingness to supply cases and staff.","Audit historical work to quantify the number and share of queries that truly reuse an unchanged ring, field, ideal, generators, and monomial order.","Specify immutable context identifiers, certificate semantics for both membership and non-membership, checker independence, resource limits, and license obligations.","Run the preregistered twelve-case shadow comparison against per-case recomputation and theorem-specific handling.","Demonstrate zero wrong or unverifiable results, correct mismatch and withdrawal behavior, and at least a 25% median total resource-equivalent advantage without a checker or downstream bottleneck.","Revise the proposal as diffusion and governance of established computational-algebra practice rather than as a novel Gröbner-reduction technique."],"reason":"Bounded web research resolves the prior-art question: the computational core is established practice, while certification remains an active implementation area. The decisive remaining evidence—local prevalence, adopter commitment, comparative total-work advantage, checker independence in operation, and safe workflow integration—requires proprietary case records, fieldwork, and a live or shadow test and therefore cannot be repaired by additional web search."},"proposal_index":1}