{"schema_version":1,"research_id":"eoa_inverse_innovation_exp03_external48_20260801","source_assessment_id":"eoa_inverse_innovation_exp03_opportunity320_20260801","cell_id":"invariant_mode_decomposition_design__mathematics","selection_stratum":"DEPLOYABLE_PRIORITY","search_queries":["site:netlib.org LAPACK Users Guide nonsymmetric eigenproblem condition eigenvectors nonnormal","matrix recurrence Jordan form asymptotic behavior generalized eigenvectors polynomial n","primary research nonnormal matrices transient growth eigenvalues stability Trefethen Embree paper","official documentation Jordan form matrix powers recurrence SageMath","site:epubs.siam.org matrix powers asymptotic Jordan form initial vector recurrence","site:math.mit.edu linear recurrence matrix powers eigenvalues Jordan form notes","site:siam.org peer review reviewer author correctness mathematical paper guidelines"],"sources":[{"source_id":"S1","title":"Matrix Powers in Finite Precision Arithmetic","publisher":"SIAM Journal on Matrix Analysis and Applications","url":"https://epubs.siam.org/doi/10.1137/S0895479893256347","source_class":"PRIMARY_RESEARCH","publication_date":"1995-04","accessed_at":"2026-08-02","claims_supported":["Even when the exact powers of a matrix converge, finite-precision powers may fail to converge.","Jordan canonical form and pseudospectral information are established tools for analyzing powers of nonnormal matrices and numerical convergence."]},{"source_id":"S2","title":"Generalizing Eigenvalue Theorems to Pseudospectra Theorems","publisher":"SIAM Journal on Scientific Computing","url":"https://epubs.siam.org/doi/10.1137/S1064827500373012","source_class":"PRIMARY_RESEARCH","publication_date":"2001-01","accessed_at":"2026-08-02","claims_supported":["Analysis of nonsymmetric matrix iterations based only on eigenvalues can be misleading.","Pseudospectral analysis is established prior art for exposing behavior omitted by eigenvalue-only reasoning."]},{"source_id":"S3","title":"Iterative Methods and Preconditioners, Section 11.3","publisher":"Massachusetts Institute of Technology","url":"https://math.mit.edu/~gs/linearalgebra/ila6/ila5preconditioners.pdf","source_class":"AUTHORITATIVE_SECONDARY","publication_date":"","accessed_at":"2026-08-02","claims_supported":["An initial vector decomposed into eigenvectors evolves through its modal coefficients.","When a complete eigenvector basis is unavailable, Jordan form is used to analyze matrix powers.","A size-two Jordan block produces an additional polynomial factor in the power, demonstrating why eigenvalue multiplicity and generalized modes matter."]},{"source_id":"S4","title":"Error Bounds for the Nonsymmetric Eigenproblem","publisher":"LAPACK / Netlib","url":"https://www.netlib.org/lapack/lug/node91.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"1999-10-01","accessed_at":"2026-08-02","claims_supported":["Computed nonsymmetric eigenvalues and eigenvectors require error and conditioning assessment.","LAPACK provides reciprocal condition estimates and approximate error bounds for computed eigenvalues and eigenvectors.","A numerical eigensolver can fail to converge and exposes a diagnostic status for that failure."]},{"source_id":"S5","title":"SageMath Matrix.jordan_form Documentation","publisher":"SageMath","url":"https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix2.html","source_class":"OFFICIAL_PRODUCT_DOCUMENTATION","publication_date":"","accessed_at":"2026-08-02","claims_supported":["SageMath can compute an exact Jordan form and an associated transformation matrix.","The implementation uses ranks of powers of A-xI and requires suitable factorization and coefficient-field support.","Jordan form is generally not implemented over inexact rings, illustrating a practical boundary between exact proof objects and floating-point calculations."]},{"source_id":"S6","title":"SIAM Publications Editorial Policy on Artificial Intelligence","publisher":"Society for Industrial and Applied Mathematics","url":"https://epubs.siam.org/artificial-intelligence","source_class":"OFFICIAL_GUIDANCE","publication_date":"2026-05","accessed_at":"2026-08-02","claims_supported":["Authors bear responsibility for correctness of mathematical content and testing of computational work.","Referees and editors must personally perform core review tasks and retain human judgment and accountability.","Computational tools do not acquire authority to approve a mathematical claim."]},{"source_id":"S7","title":"SIAM Journal on Mathematical Analysis: Instructions for Referees","publisher":"Society for Industrial and Applied Mathematics","url":"https://epubs.siam.org/journal/sima/instructions-for-referees","source_class":"OFFICIAL_GUIDANCE","publication_date":"","accessed_at":"2026-08-02","claims_supported":["Referees examine manuscripts and submit reports and recommendations.","Editors, rather than referees alone, make the formal publication decision.","Supplementary examples and code can document research, but such materials generally require separate scrutiny."]}],"problem_evidence":{"support":"MODERATE","rationale":"The existence and mathematical mechanism of the problem are externally supported: defective blocks introduce polynomial factors, eigenvalue-only analysis of nonsymmetric iterations can mislead, and finite-precision matrix powers can behave differently from exact powers. The search did not establish how often published recurrence arguments are actually misclassified, so prevalence remains unverified.","source_ids":["S1","S2","S3","S4"]},"stakeholder_evidence":{"support":"WEAK","rationale":"Official SIAM guidance confirms that authors are responsible for mathematical correctness and that referees and editors provide independent review, making these actors credible adopters or authorizers. No survey, adoption inquiry, usage data, or documented demand for this additional workflow was found.","source_ids":["S6","S7"]},"prior_art":{"proximity":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"Exact Jordan-form analysis of matrix powers","similarity":"It represents x_n as A^n x_0, decomposes the state into generalized modes, handles missing eigenvectors, and retains the polynomial factors created by Jordan blocks. This matches the candidate's central asymptotic mechanism.","remaining_difference":"The candidate packages the mathematics as an auditable workflow with explicit dominance rules, all-initial-condition subspace reporting, reconstruction checks, parameter-boundary tests, and halt criteria. Evidence that this packaging improves results is absent.","source_ids":["S1","S3","S5"]},{"name":"Pseudospectral and finite-precision analysis of nonnormal matrix iterations","similarity":"It addresses misleading eigenvalue-only conclusions, nonnormality, perturbation sensitivity, and discrepancies between exact and computed powers—the candidate's principal numerical safeguards.","remaining_difference":"The cited work analyzes matrix behavior and numerical reliability; it does not establish the proposed recurrence-specific author/reviewer checklist or a proof-complexity advantage over complete existing methods.","source_ids":["S1","S2","S4"]},{"name":"Exact symbolic Jordan computation plus conditioned numerical eigensolving","similarity":"Existing software already supplies exact Jordan transformations where supported, numerical eigenpairs, condition estimates, error bounds, residual-style checks, and failure diagnostics.","remaining_difference":"The products do not automatically classify every initial-condition subspace, adjudicate asymptotic claims across parameter crossings, or compare proof workflows under a declared rubric.","source_ids":["S4","S5"]}],"distinctive_claim_remaining":"The only credible remaining claim is not a new decomposition method: it is that a standardized, recurrence-specific reporting and halt-rule workflow measurably reduces missed exceptional subspaces or proof-review effort relative to complete Jordan/characteristic-root, generating-function, Schur, and pseudospectral practice.","confidence":"HIGH"},"implementation_evidence":{"support":"STRONG","rationale":"For explicit 5–20 dimensional matrices over supported exact fields, existing software can compute Jordan structure and transformations; established numerical software provides eigenvalue/eigenvector condition estimates and error bounds. Implementation must branch to exact, Schur/invariant-subspace, or pseudospectral analysis when an inexact Jordan form is unavailable or conditioning is inadequate.","source_ids":["S1","S2","S4","S5"]},"scores":{"meaningful_impact":{"score":3,"rationale":"The documented failure modes can change convergence and stability conclusions, so correcting one is meaningful. The frequency and downstream incidence of such errors were not established.","source_ids":["S1","S2","S3"]},"stakeholder_pull":{"score":2,"rationale":"Authors and reviewers have explicit correctness responsibilities, but no evidence shows demand for, or willingness to adopt, this additional structured workflow.","source_ids":["S6","S7"]},"incremental_advantage":{"score":2,"rationale":"The mathematical and numerical components are established practice. Any advantage is confined to workflow integration and remains untested against strong baselines.","source_ids":["S1","S2","S3","S4","S5"]},"distinctiveness_plausibility":{"score":2,"rationale":"The checklist, reporting convention, and halt rules may be a distinguishable pedagogical or review artifact, but the substantive decomposition, conditioning, and perturbation mechanisms substantially collide with prior art.","source_ids":["S1","S2","S3","S4","S5"]},"technical_implementability":{"score":4,"rationale":"A bounded exact-matrix benchmark is readily implementable with current symbolic and numerical tools. Unsupported coefficient fields, nonsplitting characteristic polynomials, near defects, and nonnormality require alternate branches rather than routine eigendecomposition.","source_ids":["S4","S5"]},"adoption_authority_feasibility":{"score":4,"rationale":"Authors can use the workflow without transferring claim authority, while referees recommend and editors decide publication. Adoption is procedurally feasible, although the candidate's statement that a reviewer alone accepts or rejects a claim overstates the formal publication role.","source_ids":["S6","S7"]},"evidence_readiness":{"score":4,"rationale":"Exact matrices with known Jordan structures can supply ground truth, and existing software supports reconstruction, conditioning, and error checks. No preregistered corpus or comparison result currently exists.","source_ids":["S3","S4","S5"]},"safety_net_benefit":{"score":4,"rationale":"Exact reconstruction, condition estimates, eigensolver diagnostics, and pseudospectral checks can prevent unsupported reliance on floating-point eigenpairs. Their integrated detection performance still requires testing.","source_ids":["S1","S2","S4","S5"]},"scalability":{"score":3,"rationale":"The approach is reusable for modest explicit recurrences, but exact Jordan computation depends on coefficient-field support and can become impractical; numerically difficult cases need Schur or invariant-subspace methods and additional expert judgment.","source_ids":["S4","S5"]}},"score_confidence":"MODERATE","costs":{"first_evidence":{"band_2026_usd":"10K_TO_50K","scope":"Design and execute a non-live benchmark of twelve exact 5–20 dimensional recurrences, including specialist mathematical labor, synthetic ground-truth construction, existing symbolic and numerical software, workstation compute, blinded comparison, independent evaluation, documentation, and basic confidentiality handling.","confidence":"LOW","assumptions":["No direct cost study was found; this is a calendar-2026 resource-equivalent estimate.","Matrices and initial conditions are synthetic or openly shareable, so no data acquisition fee is required.","Existing workstations and general mathematics software are sufficient.","One analyst and one independent checker can finish the bounded study without custom theorem-proving software.","No regulated data or live-system compliance review is involved."],"source_ids":["S4","S5","S6"]},"initial_deployment_startup":{"band_2026_usd":"10K_TO_50K","scope":"Create reusable templates, exact and numerical analysis scripts, edge-case fixtures, reviewer instructions, software-environment controls, training materials, validation records, and confidentiality guidance for a small mathematics group.","confidence":"LOW","assumptions":["Existing SageMath and LAPACK-class tools are reused rather than replaced.","Startup covers specialist labor, coordination, software configuration, documentation, validation, training, and limited compliance review.","The scope remains explicit finite-dimensional linear recurrences.","No production-grade automated proof-approval system is built."],"source_ids":["S4","S5","S6","S7"]},"operational_launch":{"band_2026_usd":"50K_TO_250K","scope":"Run a controlled multi-user pilot on a bounded corpus of completed or synthetic recurrence arguments, with participant coordination, secure manuscript handling, software and compute support, training, blinded baseline comparison, independent mathematical review, and evaluation of error detection and proof effort.","confidence":"LOW","assumptions":["No confidential manuscript is supplied to an unauthorized automated service.","Human authors, reviewers, and editors retain all decision authority.","The band includes coordination and evaluation labor, not only computation.","Difficult defective and nonnormal cases require specialist review.","No live theorem approval or journal workflow integration is implied."],"source_ids":["S4","S5","S6","S7"]},"annual_recurring":{"band_2026_usd":"10K_TO_50K","scope":"Maintain software environments and test fixtures, update documentation, train new users, audit a modest case sample, coordinate human review, preserve confidentiality controls, and periodically re-evaluate false-halt and missed-halt performance.","confidence":"LOW","assumptions":["Use is limited to a small team and modest annual case volume.","Existing equipment and core software remain adequate.","The estimate includes maintenance, coordination, compliance, and evaluation labor.","Case-specific theorem-proving labor beyond the workflow is charged to the underlying research project."],"source_ids":["S4","S5","S6","S7"]}},"verified_pipeline_gates":{"externally_supported_problem":{"status":"YES","reason":"Primary research and authoritative instruction independently show that nonsymmetric eigenvalue-only reasoning can mislead, finite-precision powers can diverge from exact behavior, and defective blocks require polynomially weighted generalized modes.","source_ids":["S1","S2","S3"]},"externally_credible_adopter_or_authorizer":{"status":"YES","reason":"Official SIAM guidance places correctness responsibility on authors, review responsibility on referees, and the publication decision with editors. These are credible adoption and authorization roles, provided the reviewer is not described as the sole formal decision-maker.","source_ids":["S6","S7"]},"distinct_testable_incremental_claim":{"status":"YES","reason":"After removing any implication of mathematical novelty, the remaining claim is testable: the integrated workflow should reduce missed classifications or review effort relative to both a weak finite-iteration baseline and a complete established analytical baseline.","source_ids":["S1","S2","S3","S4","S5"]},"bounded_next_evidence_step":{"status":"YES","reason":"A small, synthetic, exact-matrix benchmark can compare methods without live deployment, confidential data, or delegated proof authority.","source_ids":["S4","S5","S6"]},"no_unresolved_safety_or_authority_stop":{"status":"YES","reason":"The test can retain human mathematical authority, prohibit floating-point output as proof, use explicit conditioning and failure diagnostics, and revert to exact or invariant-subspace analysis when checks fail.","source_ids":["S2","S4","S5","S6"]},"credible_cost_scope_and_range":{"status":"YES","reason":"The analytical, software, review, coordination, confidentiality, and evaluation tasks are bounded sufficiently for broad resource-equivalent bands, although no direct cost observations were found and confidence is low.","source_ids":["S4","S5","S6","S7"]}},"next_evidence_step":"Pre-register a non-live benchmark containing twelve exact 5–20 dimensional recurrences: three spectrally separated and diagonalizable, three with tied dominant moduli, three defective, and three rational parameter families that become nearly defective or strongly nonnormal. Blind two qualified analysts to the planted cases. Compare (A) coordinate-wise finite iteration, (B) a complete exact Jordan/characteristic-root or generating-function solution augmented by Schur/pseudospectral checks where numerical conditioning is tested, and (C) the proposed integrated workflow. Measure asymptotic classification errors, missed exceptional initial-condition subspaces, analyst time and proof steps, false halts, and missed halts; verify symbolic reconstruction and direct iterates for n=0 through 50. Falsify the incremental claim if C corrects no error relative to A and produces no declared effort reduction relative to B, or if C fails to halt on any planted tied, defective, or ill-conditioned case.","blocking_evidence":["No external evidence establishes the prevalence of asymptotic misclassification in published coupled-recurrence arguments.","No stakeholder inquiry establishes demand for the additional reporting and halt-rule workflow.","No controlled comparison shows an accuracy or proof-effort advantage over complete Jordan, characteristic-root, generating-function, Schur, or pseudospectral analysis.","The proposed residual, gap, conditioning, and perturbation checks have not been evaluated for false-halt or missed-halt rates on a declared corpus.","The cost bands are reasoned resource estimates rather than observed project costs."],"research_disposition":"PRIOR_ART_DIFFERENTIATION_STUDY","world_novelty_boundary":"This bounded search found the core mathematical mechanism—generalized-mode/Jordan analysis of matrix powers—and its principal numerical safeguards—conditioning, Schur/invariant-subspace methods, and pseudospectra—to be established practice. It did not exhaust books, patents, unpublished teaching materials, or every recurrence-specific checklist. The only unresolved distinctiveness concerns the integrated reporting and halt-rule workflow and its measurable effect; no world-novelty claim is made."}