{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp03_full320_20260801","cell_id":"invariant_mode_decomposition_design__mathematics","trajectory_id":"R","attempt_index":0,"archetype_slug":"invariant_mode_decomposition_design","domain_slug":"mathematics","decision":"CANDIDATE","problem_id":"coupled_linear_recurrence_asymptotic_misclassification","causal_lever_id":"recurrence_invariant_mode_separation","proposal":{"problem":"For coupled linear recurrences, coordinate-wise formulas and finite iteration can conceal combinations of variables that grow, decay, persist, or oscillate, causing incorrect asymptotic classifications and incomplete stability arguments, especially when dominant roots are tied, nearly tied, or defective.","actors_substrate":["mathematicians proving asymptotic or stability results","students and reviewers checking recurrence arguments","finite-dimensional recurrence x_(n+1)=A x_n with explicit A","initial conditions and parameterized matrix families","proofs, symbolic calculations, and numerical checks"],"observable_state":"Several coordinates exhibit mixed or transient behavior; finite iterates suggest stability while some initial-condition directions grow, or a claimed single dominant term changes under small parameter or initial-condition changes.","consequence":"A theorem may omit exceptional initial conditions, misstate the dominant growth class, or treat coupled and generalized modes as independent.","affected_objective":"Correct, traceable, and economical classification of all long-run behaviors admitted by a coupled recurrence.","structural_mapping":[{"archetype_element":"many visible variables transformed together","domain_realization":"The recurrence matrix updates all sequence coordinates simultaneously.","claim_kind":"INFERENCE"},{"archetype_element":"invariant directions with scalar responses","domain_realization":"Eigenvectors or invariant generalized eigenspaces evolve according to eigenvalues, with Jordan-chain polynomial factors where necessary.","claim_kind":"INFERENCE"},{"archetype_element":"dominant and unstable directions","domain_realization":"Modes of maximal eigenvalue modulus determine generic asymptotic growth; modulus relative to one partitions discrete-time growth and decay.","claim_kind":"INFERENCE"},{"archetype_element":"intervention in modal coordinates","domain_realization":"Changing the initial condition or a matrix parameter changes modal coefficients and can suppress, excite, or couple asymptotic terms.","claim_kind":"HYPOTHESIS"},{"archetype_element":"residual and scope checks","domain_realization":"Reconstructed iterates are compared with exact direct iteration, while defective, repeated-root, and parameter-boundary cases remain explicit exceptions.","claim_kind":"INFERENCE"}],"component_map":[{"component":"Transformation Scope","status":"direct","domain_realization":"Fix A, its scalar field, dimension, parameter range, and whether the claim concerns exact or floating-point recurrence dynamics."},{"component":"State-Vector Definition","status":"direct","domain_realization":"Specify the ordered sequence components, units or normalization, and admissible initial-condition space."},{"component":"Invariant Mode Basis","status":"adapted","domain_realization":"Use eigenvectors when diagonalizable and invariant generalized eigenspaces or Jordan chains otherwise."},{"component":"Modal Gain Spectrum","status":"direct","domain_realization":"Record eigenvalues, algebraic and geometric multiplicities, and their moduli."},{"component":"Dominant Mode Selection Rule","status":"direct","domain_realization":"Retain every excited mode of maximal relevant modulus; never select a unique mode across a tie."},{"component":"Stable/Unstable Mode Partition","status":"direct","domain_realization":"Classify modulus below, above, or equal to one, adding oscillatory and Jordan-polynomial qualifications."},{"component":"Modal Intervention Map","status":"adapted","domain_realization":"Map initial-condition and permitted parameter changes to modal coefficients and asymptotic terms."},{"component":"Reconstruction Residual Check","status":"direct","domain_realization":"Compare modal reconstruction with exact direct iterates and inspect structured error."},{"component":"Mode Drift Monitor","status":"adapted","domain_realization":"Across a parameter family, track eigenspace rotation, eigenvalue crossings, multiplicity changes, and conditioning."},{"component":"Interpretation Scope Contract","status":"direct","domain_realization":"State field, parameter region, diagonalizability assumptions, exact-versus-numerical status, and excluded boundary cases."},{"component":"Mode-Coupling Register","status":"adapted","domain_realization":"Register Jordan chains, repeated eigenspaces, non-normality, and parameter points where modes cannot be treated independently."},{"component":"Local Linearization Window","status":"already_supplied","domain_realization":"For an exactly linear recurrence the representation is global; any extension to a nonlinear recurrence must name a neighborhood and remainder bound."},{"component":"Spectral Gap Threshold","status":"adapted","domain_realization":"Require an exact modulus separation for a unique dominant asymptotic term; otherwise retain the tied cluster and report numerical uncertainty."}],"mechanism_dispositions":[{"slug":"eigendecomposition_workflow","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Compute the full spectrum and eigenstructure of explicit A, with a diagonalizability check and generalized eigenspaces when a complete eigenbasis does not exist.","counterfactual_removal":"Without invariant spectral structure, coordinate mixtures remain unresolved and the proposed asymptotic classification loses its causal lever."},{"slug":"modal_sensitivity_sweep","disposition":"selected_supporting","contribution_type":"TEST_DESIGN","adaptation_or_rejection":"Perturb initial modal coefficients and bounded matrix parameters to expose exceptional initial conditions and mode coupling.","counterfactual_removal":"The core classification remains possible, but robustness and exception detection weaken."},{"slug":"modal_stability_analysis","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Interpret eigenvalue modulus, phase, multiplicity, and Jordan structure for discrete-time behavior.","counterfactual_removal":"The spectrum would not yield the required growing, decaying, marginal, and oscillatory verdicts."},{"slug":"mode_shape_testing","disposition":"incompatible","contribution_type":"NONE","adaptation_or_rejection":"The target is an explicitly specified formal operator, not an unknown physical system requiring empirical excitation.","counterfactual_removal":"No change; empirical identification is outside the mathematical substrate."},{"slug":"network_spectral_centrality_analysis","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"No node-importance question is posed; interpreting eigenvector entries as centrality would change the problem.","counterfactual_removal":"No change to recurrence classification."},{"slug":"power_iteration_probe","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Dominant-only approximation can miss ties, unexcited modes, instability outside the leading direction, and defective structure needed for an all-initial-conditions theorem.","counterfactual_removal":"No material loss because the bounded test uses a manageable explicit matrix and full analysis."},{"slug":"principal_component_analysis","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Variance directions of sampled trajectories are not invariant directions of the recurrence and depend on sampling and scaling.","counterfactual_removal":"No loss; covariance structure is not the causal object."},{"slug":"reduced_order_model","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"The objective is a correct theorem-level asymptotic classification, not a fast approximate simulator; truncation could erase exceptional modes.","counterfactual_removal":"No loss in the bounded low-dimensional setting."},{"slug":"residual_reconstruction_test","disposition":"selected_supporting","contribution_type":"TEST_DESIGN","adaptation_or_rejection":"Reconstruct exact iterates from the retained modal or generalized-modal expansion and compare against direct recurrence evaluation.","counterfactual_removal":"Algebraic omissions and unjustified truncation would be less detectable."},{"slug":"singular_value_decomposition","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Singular vectors measure one-step amplification but generally are not invariant under repeated application; use only as an auxiliary non-normality diagnostic, not the classification basis.","counterfactual_removal":"The eigenvalue-based asymptotic argument remains intact."},{"slug":"spectral_decomposition_report","disposition":"selected_supporting","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Turn assumptions, multiplicities, exceptional initial conditions, conditioning, and non-causal interpretations into an explicit proof-scope record.","counterfactual_removal":"The mathematics could still be computed, but numerical evidence and generic claims could more easily be overstated as universal proof."},{"slug":"spectral_gap_monitor","disposition":"selected_supporting","contribution_type":"OPERATIONAL","adaptation_or_rejection":"For parameterized families, track dominance gaps, eigenspace rotation, and crossings rather than temporal data drift.","counterfactual_removal":"A single fixed matrix remains classifiable, but parameter-range claims may silently cross regimes where the selected dominant mode changes."}],"causal_chain":["Represent the coupled recurrence as an explicit linear transformation on a declared state space.","Extract eigenvalues and invariant eigenspaces, replacing incomplete eigenbases with generalized eigenspaces.","Translate initial conditions into modal coefficients and register ties, Jordan coupling, and non-normal conditioning.","Classify each excited term by modulus, phase, and polynomial factor, retaining all terms not separated by the dominance rule.","Reconstruct direct iterates and test parameter perturbations, residuals, and boundary cases.","Issue only the asymptotic statement supported within the declared scope."],"baseline":"Expand or simulate each coordinate separately, infer the apparent dominant term from a finite horizon, and handle cancellations or exceptional initial conditions ad hoc.","nearest_rival":"Derive coordinate generating functions or solve the characteristic polynomial directly, retaining every root and multiplicity without organizing the result as an intervention-oriented modal workflow.","authority_safety":{"affected_parties":["authors relying on the classification","reviewers and readers","downstream users of the theorem or algorithm"],"decision_authority":"The mathematician responsible for the proof decides the formal claim; a reviewer independently accepts or rejects it.","authorized_first_step":"On one exact 5–20 dimensional recurrence, compute the full generalized spectral structure, classify all initial-condition subspaces, and compare reconstructed with direct iterates for a fixed finite horizon plus symbolic asymptotics.","excluded_actions":["Treating floating-point eigenpairs as a proof","discarding tied, defective, or low-amplitude modes without a bound","generalizing from generic initial conditions to all initial conditions","extending conclusions across an eigenvalue crossing or outside a stated nonlinear neighborhood"],"halt_rollback":"Halt if reconstruction exceeds the declared tolerance, eigenstructure is ill-conditioned enough to change the classification, or a defective/tied case is unhandled; revert to unreduced exact recurrence, invariant-subspace, Jordan, or generating-function analysis."}},"negative_tests":{"strongest_counterevidence":"A complete eigenvector basis may not exist, and for nearly defective or strongly non-normal matrices computed eigenvectors can be so ill-conditioned that small perturbations radically change coefficients despite unchanged short-run observations.","analogy_break":"Unlike independent physical modes, generalized eigenvectors and non-normal eigendirections can interact through Jordan chains and transient amplification; scalar eigenvalue gain alone then does not describe finite-time behavior.","failure_condition":"The approach fails if the operator is unknown, dimension or coefficient field prevents usable invariant-subspace computation, or no stable parameter region supports the claimed modal identity.","problem_falsifier":"The target problem is absent if the recurrence is already decoupled, one directly observable scalar determines every admissible asymptotic behavior, or no competing growth directions or exceptional initial-condition subspaces exist.","intervention_falsifier":"The proposal is falsified if, on the bounded corpus, it neither corrects any classification nor reduces proof complexity, or if its declared residual, gap, and conditioning checks fail to flag deliberately included tied and defective cases.","risks":["Numerical eigenvalue error may be mistaken for exact separation.","Generic-case language may conceal measure-zero but theorem-relevant initial conditions.","Eigenvalue modulus may obscure non-normal transient growth.","Parameter sweeps may miss narrow crossing or defect sets.","A report may give conjectural modal interpretations undeserved authority."]},"null_rationale":null,"classification":{"candidate_kind":"MECHANISM_COMPOSITION","prior_art_status":"UNSEARCHED","evidence_maturity":"HYPOTHESIS"},"revision_change_log":{"revision_kind":"ORIGINAL","prior_problem_id":null,"prior_causal_lever_id":null,"problem_changed":false,"causal_lever_changed":false,"conceptual_changes":[],"operational_changes":[],"repairs_addressed":[]},"confidence":0.84,"generator_notes":"Closed-book structural inference from the supplied packet. The candidate is restricted to explicit finite-dimensional coupled recurrences; empirical effectiveness and novelty are unsearched."}