{"schema_version":1,"assessment_id":"eoa_inverse_innovation_exp03_opportunity320_20260801","source_experiment_id":"eoa_inverse_innovation_exp03_full320_20260801","cell_id":"invariant_mode_decomposition_design__mathematics","archetype_slug":"invariant_mode_decomposition_design","domain_slug":"mathematics","title":"Invariant-mode workflow for classifying coupled linear recurrences","opportunity_summary":"Evaluate a structured workflow that decomposes explicit finite-dimensional coupled linear recurrences into invariant or generalized modes, tracks exceptional initial-condition subspaces, and checks tied, defective, and non-normal cases before issuing asymptotic claims. The candidate could improve proof correctness and traceability relative to coordinate-wise finite iteration, but its prevalence, realized advantage, and distinctiveness from established spectral and generating-function practice are unsupported.","adopter_authorizer":"The mathematician responsible for an asymptotic or stability proof can adopt the workflow; a reviewer independently authorizes acceptance of the resulting claim.","scores":{"meaningful_impact":{"score":4,"rationale":"Preventing omitted exceptional initial conditions, incorrect dominant-growth claims, and incomplete stability arguments would materially improve theorem correctness, although the packet does not establish how frequently these errors occur."},"stakeholder_pull":{"score":3,"rationale":"Authors, students, and reviewers have a recognizable interest in correct and traceable recurrence arguments, but the sealed candidate supplies no observed requests, adoption behavior, or problem-prevalence evidence."},"incremental_advantage":{"score":3,"rationale":"The proposed workflow explicitly organizes generalized eigenspaces, modal coefficients, dominance ties, conditioning, reconstruction, and boundary tests beyond the coordinate-wise baseline. However, the nearest rival retains every root and multiplicity and may produce the same classifications, so comparative proof-effort or error-reduction advantage remains untested."},"distinctiveness_plausibility":{"score":2,"rationale":"The composition is coherent, but eigenvalue analysis, generalized eigenspaces, Jordan chains, invariant subspaces, perturbation checks, and generating functions are all presented as nearby methods, while prior art is explicitly unsearched. No affirmative basis establishes a distinctive contribution."},"technical_implementability":{"score":4,"rationale":"The authorized test on one explicit 5–20 dimensional recurrence is bounded and uses available symbolic, exact, and numerical operations. Implementability is reduced by nearly defective, strongly non-normal, unknown-operator, or coefficient-field cases that can prevent stable modal identification."},"adoption_authority_feasibility":{"score":5,"rationale":"The responsible mathematician directly controls use in the proof, and the reviewer has a clearly separated acceptance role. The proposal requires no broader institutional authority for the bounded first step."},"evidence_readiness":{"score":3,"rationale":"The packet supplies a concrete comparison, reconstruction checks, edge cases, falsifiers, halt rules, and a bounded test object, but reports no completed comparison, benchmark corpus, or observed correction of a classification."},"safety_net_benefit":{"score":4,"rationale":"Residual, spectral-gap, conditioning, perturbation, and boundary checks can expose unsupported classifications, while explicit halt rules revert to exact recurrence, invariant-subspace, Jordan, or generating-function analysis. The benefit is limited when the checks themselves cannot reliably resolve ill-conditioned structure."},"scalability":{"score":4,"rationale":"A reusable workflow could apply across many explicit finite-dimensional coupled recurrences and parameter families. Scaling is constrained when dimensions, coefficient fields, conditioning, or eigenvalue crossings make generalized spectral computation or verification impractical."}},"score_confidence":"MODERATE","costs":{"first_evidence":{"band_2026_usd":"UNDER_10K","scope":"Analyze one preselected exact 5–20 dimensional recurrence, classify its initial-condition subspaces, reconstruct finite iterates, compare symbolic asymptotics, and document whether planted tied or defective cases are detected.","confidence":"MODERATE","assumptions":["The recurrence and initial-condition family are already specified.","Existing symbolic and numerical software is sufficient.","One mathematician can complete the analysis and independent check within a limited number of working days.","No paid proprietary dataset, specialized equipment, or formal compliance review is required."]},"initial_deployment_startup":{"band_2026_usd":"10K_TO_50K","scope":"Create a reusable analysis template, exact-versus-numerical checking procedures, a small edge-case benchmark corpus, documentation, and review criteria for a research group or course.","confidence":"LOW","assumptions":["Deployment remains limited to explicit finite-dimensional linear recurrences.","Most resource use is expert mathematical and software labor.","Existing algebra systems and computing equipment are reused.","The benchmark corpus is curated rather than newly commissioned at large scale."]},"operational_launch":{"band_2026_usd":"10K_TO_50K","scope":"Introduce the workflow within one research or instructional group, train users, run several supervised cases, and independently review the resulting asymptotic claims and failure reports.","confidence":"LOW","assumptions":["Launch involves a small group rather than a field-wide standardization effort.","No production-critical software certification is required.","Reviewers already possess linear algebra and recurrence-analysis expertise.","Ill-conditioned cases are escalated to exact analysis rather than supported through new numerical research."]},"annual_recurring":{"band_2026_usd":"UNDER_10K","scope":"Maintain templates and examples, update software compatibility, review occasional difficult cases, and record false alarms or missed edge cases for a small adopting group.","confidence":"LOW","assumptions":["Usage volume is modest.","The workflow does not become a hosted service.","Existing personnel and software licenses absorb routine use.","Major algorithm development or extensive independent auditing is excluded."]}},"research_burden":"MODERATE","earliest_credible_horizon":"0_TO_3_MONTHS","pipeline_gates":{"recognizable_externally_supportable_problem":{"status":"YES","reason":"The candidate specifies observable asymptotic misclassification mechanisms: mixed coordinate behavior, finite-horizon stability masking growing directions, and omitted tied, defective, or exceptional modes."},"identifiable_adopter_or_authorizer":{"status":"YES","reason":"The proof author is identified as the adoption decision-maker, while the reviewer independently accepts or rejects the resulting formal claim."},"distinct_testable_incremental_claim":{"status":"YES","reason":"The workflow can be compared with coordinate-wise finite iteration or direct characteristic-root analysis on the same recurrence, with falsification if it neither corrects a classification nor reduces proof complexity or fails to flag planted tied and defective cases. Whether this claim is historically distinctive remains uncertain."},"bounded_next_evidence_step":{"status":"YES","reason":"The candidate authorizes analysis of one exact 5–20 dimensional recurrence using generalized spectral structure, initial-condition classification, finite-horizon reconstruction, and symbolic asymptotics."},"no_unresolved_safety_or_authority_stop":{"status":"YES","reason":"Proof and review authority are separated, numerical eigenpairs are denied proof status, prohibited extrapolations are explicit, and reconstruction or conditioning failures trigger a halt and return to exact methods."},"implementation_cost_scope_and_range":{"status":"UNCERTAIN","reason":"The sealed candidate bounds the mathematical object and activities but supplies no labor duration, software requirements, case volume, or deployment scale from which an externally supported implementation range can be established."}},"blocking_evidence":["No evidence establishes how often coupled-recurrence proofs are misclassified by the stated baseline or how strongly authors and reviewers demand a new workflow.","No completed comparison shows that the workflow corrects classifications or reduces proof complexity relative to coordinate generating functions, full characteristic-root analysis, or existing invariant-subspace practice.","Prior art is unsearched, so the composition's distinctiveness and any defensible incremental contribution are unknown.","No evidence shows that the residual, gap, and conditioning checks reliably flag the deliberately tied, defective, and strongly non-normal cases without excessive false reassurance.","Resource estimates lack observed labor time and software or review requirements."],"next_evidence_step":"On one predeclared exact 5–20 dimensional recurrence containing at least one tied or defective case and an exceptional initial-condition subspace, compare the proposed modal workflow with coordinate-wise finite iteration and full characteristic-root or generating-function analysis. Reconstruct direct iterates over a fixed horizon and verify symbolic asymptotics. Treat the intervention as falsified for this case if it does not correct a classification or reduce documented proof complexity, or if its residual, gap, and conditioning checks fail to flag the planted edge case; do not proceed to live or broader adoption from this single result.","research_questions":["How often do the specified asymptotic errors occur in the intended class of proofs, and which actors report material difficulty detecting them?","Does the composed workflow improve classification accuracy, proof traceability, or proof effort against the nearest rival on a predeclared bounded corpus?","Which parts of the workflow, if any, are distinct from established recurrence, Jordan-form, invariant-subspace, perturbation, and generating-function practice?","Under what conditioning, coefficient-field, dimension, and parameter-crossing conditions do the checks cease to support a stable classification?","What expert labor, software, and independent-review resources are actually required per case and for reusable deployment?"],"recommendation":"PRIOR_ART_RESEARCH","uncertainty_constraints":["Closed-book assessment cannot establish prior art, novelty, prevalence, stakeholder demand, market size, realized impact, or exact cost.","The candidate is limited to explicit finite-dimensional linear recurrences and does not support extension to unknown operators or nonlinear systems beyond a separately justified neighborhood.","Finite numerical reconstruction and floating-point eigenpairs cannot establish the formal asymptotic claim.","Nearly defective and strongly non-normal matrices may make modal coefficients unstable even when short-run observations remain similar.","The favorable authority assessment applies to individual proof development and review, not to adoption as a disciplinary standard."],"closed_book_prior_art_boundary":"No conclusion is made about novelty, historical precedence, prevalence, or existing adoption. The packet itself identifies recurrence, Jordan-form, generalized-eigenspace, invariant-subspace, perturbation, and generating-function methods as nearby practices; external prior-art research is required to determine whether the proposed workflow contributes anything distinctive beyond their organization."}