{"schema_version":1,"research_id":"eoa_inverse_innovation_exp06_external_evaluation_20260803","source_assessment_id":"predictive_residual_processing__mathematics:P3:v0","cell_id":"predictive_residual_processing__mathematics","search_queries":["validated numerical continuation interval Newton parameterized equations branch singularity certified enclosures","rigorous continuation solution branches radii polynomial method parameter dependent equations","interval arithmetic validated numerics software official documentation continuation CAPD JuliaValidatedNumerics","validated continuation computational cost certificates many parameter values","site:dl.acm.org validated continuation interval arithmetic predictor corrector certificate","rigorous path tracking certified homotopy continuation adaptive precision predictor corrector","validated continuation parameter intervals branch proof interval Newton predictor corrector","IEEE 1788 interval arithmetic standard official reproducible decorations exception handling","alphaCertified software official certification numerical solutions polynomial systems documentation","CAPD rigorous numerics official validated continuation software","INTLAB verified numerical computations official interval toolbox applications","computer assisted proof validated numerics research center expressed need reliable numerical results"],"sources":[{"source_id":"S1","title":"IEEE 1788.1-2017: IEEE Standard for Interval Arithmetic (Simplified)","publisher":"IEEE Standards Association","url":"https://standards.ieee.org/ieee/1788.1/6074/","source_class":"STANDARD","publication_date":"2018-01-31","accessed_at":"2026-08-03","claims_supported":["An active standard specifies binary64 interval operations and decorations for propagating computation properties and exceptional conditions.","Standardized interval semantics provide part of the arithmetic-authority layer, but do not specify continuation certificates, residual messages, audits, or branch identity."]},{"source_id":"S2","title":"IntervalArithmetic.jl API","publisher":"JuliaIntervals","url":"https://juliaintervals.github.io/IntervalArithmetic.jl/dev/manual/api/","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"undated","accessed_at":"2026-08-03","claims_supported":["A maintained open-source implementation provides guaranteed enclosures, correct rounding, configurable precision, and IEEE 1788-compatible interval types.","Existing software reduces the implementation burden for certified arithmetic but does not supply the proposed residual-certificate workflow."]},{"source_id":"S3","title":"A General Framework for Validated Continuation of Periodic Orbits in Systems of Polynomial ODEs","publisher":"Vrije Universiteit Amsterdam","url":"https://www.math.vu.nl/~janbouwe/code/continuation/continuation.pdf","source_class":"PRIMARY_RESEARCH","publication_date":"2019-03-22","accessed_at":"2026-08-03","claims_supported":["Parameterized Newton-Kantorovich and radii-polynomial methods already rigorously validate numerically computed solution branches.","The implementation already combines predictor-corrector continuation, interval arithmetic, adaptive step prediction, and explicit concern for computational cost.","The method relies on general analytic estimates plus finite computer-checked inequalities and has an accompanying Matlab/INTLAB implementation."]},{"source_id":"S4","title":"Validated Continuation for Equilibria of PDEs","publisher":"Rutgers University","url":"https://sites.math.rutgers.edu/~mischaik/papers/continuation.pdf","source_class":"PRIMARY_RESEARCH","publication_date":"2005-11-22","accessed_at":"2026-08-03","claims_supported":["Researchers studying many parameter values face a real tradeoff between validation and computational cost.","The paper already integrates predictor-corrector continuation with validation by reusing continuation computations.","Its example reports validated continuation taking 56 seconds versus 44 seconds for continuation alone and estimates validation overhead below 1.5 times continuation cost.","Adaptive predictor step length can prove existence and uniqueness along continuous finite branches."]},{"source_id":"S5","title":"Certified Homotopy Tracking Using the Krawczyk Method","publisher":"arXiv / ISSAC 2024 authors","url":"https://arxiv.org/abs/2402.07053","source_class":"PRIMARY_RESEARCH","publication_date":"2024-05-29","accessed_at":"2026-08-03","claims_supported":["Parametric Krawczyk interval methods already certify approximate solution paths for general parameter homotopies.","The method has correctness and termination results, while preliminary experiments indicate competitiveness with specialized certified trackers."]},{"source_id":"S6","title":"A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation","publisher":"arXiv / paper authors","url":"https://arxiv.org/abs/1909.04984","source_class":"PRIMARY_RESEARCH","publication_date":"2020-09-10","accessed_at":"2026-08-03","claims_supported":["Path jumping is a documented failure mode in numerical continuation.","Adaptive Padé-based prediction and step-size selection already detect nearby singular behavior and path-jumping danger.","Published experiments show path jumping in several comparison implementations under difficult conditions, supporting the importance of protected transitions and fallback."]},{"source_id":"S7","title":"Pathtracking Routines","publisher":"HomotopyContinuation.jl","url":"https://www.juliahomotopycontinuation.org/HomotopyContinuation.jl/v0.1/pathtracker.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"undated","accessed_at":"2026-08-03","claims_supported":["Operational software already implements affine and spherical predictor-corrector tracking.","The product exposes correction tolerance, maximum corrector iterations, adaptive step increases and decreases, and high-precision operation.","This establishes workflow implementability but not rigorous residual-certificate reconstruction."]},{"source_id":"S8","title":"Validated Numerics for Computer-Assisted Proofs","publisher":"International Centre for Mathematical Sciences","url":"https://icms.ac.uk/activities/workshop/validated-numerics-for-computer-assisted-proofs/","source_class":"OFFICIAL_ORGANIZATION_DATA","publication_date":"2026-07-06","accessed_at":"2026-08-03","claims_supported":["An identifiable international validated-numerics community is actively reviewing software platforms, common standard features, open problems, and applications.","The organizers explicitly seek collaborative research projects, funding applications, and a continuing special-interest group, demonstrating credible stakeholder pull for infrastructure work.","The expressed agenda is broad and does not specifically request residual certificate serialization or cold-start audit protocols."]}],"problem_evidence":{"support":"STRONG","rationale":"S4 directly identifies the cost-versus-validity problem when researchers examine many parameter values, while S6 documents path-jumping risk and S3-S5 show that rigorous branch certification remains an active technical subject. The exact magnitude of duplicated certificate-transfer, storage, and human-review cost asserted by the candidate was not measured.","source_ids":["S3","S4","S5","S6"]},"stakeholder_evidence":{"support":"MODERATE","rationale":"ICMS identifies a current international expert community seeking common validated-numerics software features, collaborative projects, and funding. Researchers and maintainers behind S2, S3, S5, and S7 are plausible adopters or authorizers. No source expresses demand for this exact residual-certificate protocol or commits to adopting it.","source_ids":["S2","S3","S5","S7","S8"]},"prior_art":{"proximity":"SUBSTANTIAL_COLLISION","closest_analogues":[{"name":"Validated continuation for PDE equilibria","similarity":"Already uses predictor-corrector continuation, reuses continuation computations for rigorous validation, adapts step length, and quantifies validation overhead.","remaining_difference":"It does not present certified state as a versioned residual communication artifact with random cold-start audits, coverage manifests, checksum handshakes, and descendant invalidation.","source_ids":["S4"]},{"name":"General radii-polynomial framework for validated continuation","similarity":"Already provides a general, rigorous predictor-corrector continuation architecture with interval checks, adaptive step prediction, explicit error bounds, and implementation.","remaining_difference":"It focuses on proving branches rather than reducing certificate transmission, storage, and review through a separately reconstructible residual protocol.","source_ids":["S3"]},{"name":"Certified Krawczyk homotopy tracking","similarity":"Already certifies parameterized solution paths through interval methods and supplies correctness, termination, and implementation evidence.","remaining_difference":"It does not report random independent cold-start audits, full-state resynchronization, or residual certificate serialization as a governed message layer.","source_ids":["S5"]},{"name":"Robust adaptive path tracking","similarity":"Already predicts continuation steps, detects difficult regions and nearby paths, controls step size, and targets path-jumping prevention.","remaining_difference":"It is primarily numerical rather than an independently verified interval-certificate and audit protocol.","source_ids":["S6","S7"]}],"distinctive_claim_remaining":"On a fixed certified-continuation workload, adding a canonical, versioned residual-certificate representation with independent reconstruction, random and risk-stratified cold-start audits, full anchors, and automatic region decompression will reduce total solver, transfer, storage, verifier, and review cost relative to both full cold-start solving and established validated continuation with full certificates, while producing zero protected-event omissions and mathematically equivalent certified coverage.","confidence":"HIGH"},"implementation_evidence":{"support":"MODERATE","rationale":"The mathematical and software primitives are established: standardized interval arithmetic, guaranteed-enclosure libraries, predictor-corrector trackers, radii-polynomial validation, and Krawczyk certification. The untested work is the canonical residual/certificate schema, independent reconstruction across versions, audit design, branch-wide invalidation, and evidence that these additions yield net savings.","source_ids":["S1","S2","S3","S4","S5","S7"]},"scores":{"meaningful_impact":{"score":2,"rationale":"The intervention could improve trust and efficiency in a specialized but important computer-assisted-proof workflow; neither prevalence nor realized burden is quantified.","source_ids":["S4","S8"]},"stakeholder_pull":{"score":3,"rationale":"A credible international expert community is actively seeking shared software features and collaborative projects, but no adopter has requested or committed to this protocol.","source_ids":["S8"]},"incremental_advantage":{"score":2,"rationale":"Existing validated continuation already reuses predictor-corrector work with modest reported overhead, so the residual message layer must beat a strong incumbent rather than independent full solving alone.","source_ids":["S3","S4","S5"]},"distinctiveness_plausibility":{"score":2,"rationale":"The exact serialization, audit, resynchronization, and decompression bundle was not found, but most mathematical and algorithmic elements substantially collide with established continuation practice.","source_ids":["S3","S4","S5","S6","S7"]},"technical_implementability":{"score":4,"rationale":"Core arithmetic, certification, and predictor-corrector components exist in standards, research implementations, and maintained software; integration and canonical certificate semantics remain unproven.","source_ids":["S1","S2","S3","S5","S7"]},"adoption_authority_feasibility":{"score":4,"rationale":"A responsible mathematician can authorize a shadow workflow and an independent verifier can retain theorem authority without regulatory approval. Cross-project certificate semantics would require community governance.","source_ids":["S1","S8"]},"evidence_readiness":{"score":4,"rationale":"The candidate specifies a bounded mesh, comparators, protected events, artifacts, and falsifiers, and public implementations can support a reproducible benchmark.","source_ids":["S2","S3","S5","S7"]},"safety_net_benefit":{"score":1,"rationale":"No direct benefit to public safety-net populations or essential-service access is evidenced; benefits accrue primarily to a specialized mathematical research community.","source_ids":[]},"scalability":{"score":3,"rationale":"A canonical protocol could transfer across parameterized equations, but certificate semantics and branch logic vary substantially by equation class and validation method.","source_ids":["S3","S5","S8"]}},"score_confidence":"MODERATE","costs":{"first_evidence":{"band_2026_usd":"10K_TO_50K","scope":"Implement a shadow prototype on one public equation family and a fixed mesh, retaining full outputs and comparing against existing validated continuation and cold-start solves.","confidence":"MODERATE","assumptions":["Existing interval and continuation libraries are reused.","One researcher and one independent reviewer contribute approximately four to eight person-weeks.","Compute fits on existing workstation or institutional resources.","This is a resource-equivalent estimate, not a vendor quote."],"source_ids":["S2","S3","S4","S5","S7"]},"initial_deployment_startup":{"band_2026_usd":"50K_TO_250K","scope":"Build canonical residual and certificate schemas, deterministic reconstruction, provenance fingerprints, audit selection, fallback logic, replay tests, and continuous integration for one solver stack.","confidence":"LOW","assumptions":["Approximately 0.5 to 1.5 technical FTE-years are required.","Existing arithmetic and path-tracking code remains usable.","Formal verification of the implementation itself is excluded."],"source_ids":["S1","S2","S3","S5","S7"]},"operational_launch":{"band_2026_usd":"250K_TO_1M","scope":"Productionize across several equation classes or institutions with two verifier implementations, migration tooling, governance, documentation, security review, and replicated benchmark suites.","confidence":"LOW","assumptions":["Two to four specialist FTE-years plus independent mathematical review are required.","Certificate interoperability is harder than arithmetic interoperability.","No specialized hardware acquisition is required."],"source_ids":["S1","S3","S5","S8"]},"annual_recurring":{"band_2026_usd":"50K_TO_250K","scope":"Maintain solver and arithmetic compatibility, rerun regression and cold-start audits, review threshold changes, preserve artifacts, and support users.","confidence":"LOW","assumptions":["Approximately 0.25 to 1 specialist FTE plus compute and independent review is required annually.","Usage remains research-scale rather than a high-volume commercial service.","Audit frequency is consequence-weighted."],"source_ids":["S1","S2","S7","S8"]}},"verified_pipeline_gates":{"externally_supported_problem":{"status":"YES","reason":"Primary research directly documents the cost-validity tradeoff, repeated parameter studies, and continuation failures including path jumping.","source_ids":["S3","S4","S5","S6"]},"externally_credible_adopter_or_authorizer":{"status":"YES","reason":"The ICMS validated-numerics community, participating mathematicians, and maintainers of validated-numerics software constitute identifiable, technically credible adopters and authorizers, although adoption of this exact protocol is not committed.","source_ids":["S2","S7","S8"]},"distinct_testable_incremental_claim":{"status":"YES","reason":"The remaining claim is explicitly contrastive against both cold-start solving and established validated continuation and can be falsified by cost, reconstruction, coverage, or protected-event failures.","source_ids":["S3","S4","S5"]},"bounded_next_evidence_step":{"status":"YES","reason":"A fixed-mesh shadow benchmark with frozen software, public artifacts, two incumbent comparators, planted failures, and preregistered acceptance criteria is bounded and reversible.","source_ids":["S2","S3","S5","S7"]},"no_unresolved_safety_or_authority_stop":{"status":"YES","reason":"For a shadow evaluation, theorem authority remains with the mathematician and independent verifier; full artifacts remain intact and any discrepancy forces rollback to cold-start computation. Production replacement is not authorized.","source_ids":["S1","S3","S5"]},"credible_cost_scope_and_range":{"status":"UNCERTAIN","reason":"The scope is decomposed and existing components bound some effort, but no direct 2026 labor quote, implementation estimate, or operating-cost dataset was found among the eight sources.","source_ids":["S2","S3","S4","S7"]}},"next_evidence_step":"Pre-register a shadow benchmark on one public parameterized equation and a 256-cell mesh. Freeze equation, arithmetic, solver, predictor, certificate schema, step policy, random-audit seed, protected-event rules, and resource accounting. Compare: A) independent cold-start certified solve with full certificate at every cell; B) established validated predictor-corrector continuation with full certificates; and C) the proposed residual-certificate protocol. Include blinded singular or near-singular cells, branch proximity, changed root-count cases, missing outputs, version mismatches, and planted bad anchors. Independently verify every reconstructed state and cold-start audit. Measure certified coverage, branch/root-count agreement, enclosure containment or equality under declared semantics, certificate bytes, solver work, wall time, verifier time, human review time, fallback rate, audit detection, and descendant invalidation radius. Falsify the claim upon any protected-event omission, false certificate acceptance, reconstruction mismatch, unbounded invalidation, missing-result-as-confirmation behavior, or failure to reduce total resource-equivalent cost by at least 20% versus the better incumbent comparator at equal certified coverage. A pass authorizes only a second shadow benchmark.","blocking_evidence":["No field measurement shows that certificate transmission, storage, or review duplication is a material cost beyond solver and validation computation.","No benchmark demonstrates net advantage over modern validated continuation or certified Krawczyk tracking, which already reuse predictions and corrections.","No live test establishes canonical reconstruction across equation, solver, arithmetic, and certificate versions.","No independent audit has tested wrong-branch propagation, protected-event recall, or descendant invalidation.","No named institution or software project has committed to implement or adopt the exact protocol.","The 2026 USD estimates lack direct labor-rate, compute-cost, and maintenance evidence."],"research_disposition":"PARTNERED_RESEARCH_PROGRAM","world_novelty_boundary":"World novelty, patentability, freedom to operate, market size, and realized impact were not measured. The eight-source search found substantial prior art for validated predictor-corrector continuation, adaptive step control, certified homotopy tracking, and standardized interval arithmetic. It did not establish whether the exact combination of canonical residual certificates, independent reconstruction, random cold-start audits, full-anchor resynchronization, and automatic region decompression exists elsewhere; absence from this bounded search is not evidence of novelty.","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_version":0,"controller_recommendation":{"action":"STOP_EMPIRICAL_RESEARCH_NEEDED","repairable":false,"material_progress_observed":true,"progress_targets":["Obtain an adopter-authorized shadow benchmark on one fixed public equation family and 256-cell mesh.","Demonstrate zero protected-event omissions, zero false certificate acceptances, and exact declared reconstruction semantics against independently verified full states.","Demonstrate at least 20% lower total resource-equivalent cost than the better of full cold-start solving and established validated continuation with full certificates at equal certified coverage.","Show that planted bad anchors, version mismatches, missing results, singularities, and branch hazards trigger bounded invalidation and full-solve fallback.","Produce direct labor, compute, verification, and maintenance measurements sufficient to replace the current low-confidence cost estimates."],"reason":"Web research establishes the problem, adopter community, implementable primitives, and substantial collision with established validated continuation. The decisive incremental claim is comparative performance and failure containment; it requires live computation, independent verification, and workflow measurement rather than additional bounded web search."},"proposal_index":3}