{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp03_full320_20260801","cell_id":"invariant_mode_decomposition_design__education_pedagogy","trajectory_id":"R","attempt_index":0,"archetype_slug":"invariant_mode_decomposition_design","domain_slug":"education_pedagogy","decision":"CANDIDATE","problem_id":"hidden_coupled_skill_failure_under_aggregate_mastery","causal_lever_id":"coupled_skill_transition_modal_targeting","proposal":{"problem":"In sequential instruction, acceptable averages and separate skill subscores can conceal coupled patterns of prerequisite weakness that persist or grow across lessons. Teachers then remediate the most visibly weak skill, although later failure may arise from a combination of skills propagated by the instructional sequence.","actors_substrate":["students progressing through a bounded course unit","teachers and curriculum leads","weekly low-stakes item-level assessments","a locally estimated skill-state transition model","instructional task bundles available for remediation"],"observable_state":"For each weekly checkpoint, a student or small cohort has a vector of criterion-referenced skill estimates. Within one unit, the working model is x(t+1)=A x(t)+B u(t)+error, where A represents propagation among skills and u represents instructional support.","consequence":"A weakly damped or growing combination can remain inconspicuous in aggregate scores until it produces delayed non-mastery, while coordinate-wise remediation spends time on symptoms rather than the propagated pattern.","affected_objective":"Increase later independent mastery while limiting instructional time, missed prerequisite failures, and subgroup disparities.","structural_mapping":[{"archetype_element":"coupled transformation","domain_realization":"The lesson-to-lesson transition A maps a multiskill mastery vector to its next checkpoint state.","claim_kind":"HYPOTHESIS"},{"archetype_element":"invariant directions","domain_realization":"Approximate combinations of skills recur under A more consistently than individual skill coordinates.","claim_kind":"HYPOTHESIS"},{"archetype_element":"scalar response","domain_realization":"Each retained mode has an estimated persistence or amplification factor across checkpoints.","claim_kind":"INFERENCE"},{"archetype_element":"action-relevant modes","domain_realization":"Modes are prioritized by later-mastery sensitivity and harm, not merely current magnitude.","claim_kind":"HYPOTHESIS"},{"archetype_element":"residual and drift governance","domain_realization":"Out-of-sample prediction error, structured residuals, spectral separation, and basis rotation determine whether the model remains usable.","claim_kind":"INFERENCE"}],"component_map":[{"component":"Transformation Scope","status":"adapted","domain_realization":"One course unit, fixed assessment blueprint, and specified lesson sequence."},{"component":"State-Vector Definition","status":"adapted","domain_realization":"Weekly criterion-referenced estimates for named prerequisite and target skills, with uncertainty retained."},{"component":"Invariant Mode Basis","status":"adapted","domain_realization":"Eigenvectors of the estimated local skill-transition operator, traced back to signed skill loadings."},{"component":"Modal Gain Spectrum","status":"direct","domain_realization":"Estimated eigenvalues expressing decay, persistence, reversal, or growth per checkpoint."},{"component":"Dominant Mode Selection Rule","status":"adapted","domain_realization":"Retain modes exceeding preregistered later-mastery sensitivity or risk thresholds and passing residual checks."},{"component":"Stable/Unstable Mode Partition","status":"direct","domain_realization":"Classify by gain magnitude and uncertainty relative to the repeated-transition stability boundary."},{"component":"Modal Intervention Map","status":"adapted","domain_realization":"Estimate how available task bundles move modal coordinates through B."},{"component":"Reconstruction Residual Check","status":"direct","domain_realization":"Compare held-out observed skill vectors with reconstructions and inspect subgroup-specific residual structure."},{"component":"Mode Drift Monitor","status":"adapted","domain_realization":"Track mode angles, ordering, gains, and residuals across weeks and cohorts."},{"component":"Interpretation Scope Contract","status":"adapted","domain_realization":"Modes are local predictive combinations, not learner traits or proof of cognitive causes."},{"component":"Mode-Coupling Register","status":"adapted","domain_realization":"Record near-degeneracy, non-normality, correlated measurement error, and cross-effects of task bundles."},{"component":"Local Linearization Window","status":"direct","domain_realization":"The specified unit, curriculum version, checkpoint cadence, and score range used to fit A."},{"component":"Spectral Gap Threshold","status":"adapted","domain_realization":"Preregister a bootstrap-supported separation between retained and dropped modes; failure blocks reduction."}],"mechanism_dispositions":[{"slug":"eigendecomposition_workflow","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Decompose the explicit fitted A and report conditioning and incomplete-basis failures.","counterfactual_removal":"Without modes and gains, the proposal becomes ordinary multivariate skill tracking."},{"slug":"modal_sensitivity_sweep","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Simulate small feasible task-bundle changes and rank effects on later mastery, logging cross-mode movement.","counterfactual_removal":"Mode size would be mistaken for instructional leverage."},{"slug":"modal_stability_analysis","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Classify growth or damping only within the fitted instructional window.","counterfactual_removal":"The method could not distinguish transient weakness from propagating risk."},{"slug":"mode_shape_testing","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Its physical excitation protocol does not transfer cleanly; educational perturbations are consequential and B is estimated through the bounded pilot instead.","counterfactual_removal":"No change; empirical validation remains in the pilot and residual tests."},{"slug":"network_spectral_centrality_analysis","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Skill importance is not node prestige, and a centrality ranking would not estimate temporal propagation.","counterfactual_removal":"No change; A, not a static adjacency ranking, is the transformation."},{"slug":"power_iteration_probe","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"A full spectrum and stability partition are needed, and the bounded skill matrix is not assumed too large to factor.","counterfactual_removal":"No change unless later scale makes full decomposition infeasible."},{"slug":"principal_component_analysis","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Variance directions need not be temporal propagation directions or intervention-relevant.","counterfactual_removal":"No change; covariance compression is not the causal lever."},{"slug":"reduced_order_model","disposition":"selected_supporting","contribution_type":"OPERATIONAL","adaptation_or_rejection":"Use retained modes as a local, auditable simulator for comparing feasible task bundles.","counterfactual_removal":"The causal hypothesis remains, but intervention comparison becomes slower and less explicit."},{"slug":"residual_reconstruction_test","disposition":"selected_load_bearing","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Test held-out reconstruction, prediction, residual structure, and subgroup error before acting.","counterfactual_removal":"Discarded educational behavior could be hidden by an apparently compact model."},{"slug":"singular_value_decomposition","disposition":"selected_supporting","contribution_type":"TEST_DESIGN","adaptation_or_rejection":"Use singular values and conditioning diagnostics to detect fragile or non-normal eigenvector interpretations, not as substitute invariant modes.","counterfactual_removal":"The core chain remains, but numerical fragility is harder to detect."},{"slug":"spectral_decomposition_report","disposition":"selected_supporting","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Document loadings, uncertainty, residuals, couplings, prohibited trait interpretations, and validity window.","counterfactual_removal":"Results become easier to overstate or detach from their scope."},{"slug":"spectral_gap_monitor","disposition":"selected_load_bearing","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Re-estimate gap and basis rotation at each checkpoint and suspend targeting on threshold failure.","counterfactual_removal":"A stale reduced basis could continue directing instruction after regime change."}],"causal_chain":["Repeated instruction and practice propagate a coupled skill state through an approximately local transition.","Decomposition identifies combinations estimated to persist or amplify despite unremarkable individual subscores.","Stability and sensitivity analyses distinguish consequential propagating modes from merely large or variable patterns.","Task bundles predicted to damp a risky mode are assigned in a bounded comparison.","Held-out mastery, residuals, subgroup errors, and modal drift determine whether targeting improves outcomes and remains valid."],"baseline":"Teachers use class averages and individual skill subscores to provide classwide reteaching or practice on the lowest visible skill.","nearest_rival":"Item-level prerequisite remediation targets the weakest causally ordered skill using formative assessment, without estimating coupled modes.","authority_safety":{"affected_parties":["participating students","teachers whose instructional time changes","students in subgroups for whom measurement error may differ"],"decision_authority":"The classroom teacher and curriculum lead jointly authorize the pilot under applicable school research and student-protection procedures.","authorized_first_step":"Within one unit, estimate the model from routine low-stakes checkpoints, then compare one short supplemental task block selected by modal targeting against the nearest rival using blocked random assignment where permitted; preserve normal required instruction.","excluded_actions":["high-stakes grading, placement, discipline, or exclusion based on modal scores","labeling a student with a fixed latent type","withholding required instruction or proven support","deployment beyond the fitted unit or curriculum version"],"halt_rollback":"Stop modal assignment and return to the ordinary baseline if the gap or conditioning gate fails, held-out residual exceeds the preregistered budget, subgroup error materially worsens, adverse workload appears, or required consent or oversight is withdrawn."}},"negative_tests":{"strongest_counterevidence":"Across held-out weeks or cohorts, the estimated modes rotate substantially, residuals remain structured, or ordinary prerequisite remediation matches or exceeds modal targeting at lower burden.","analogy_break":"Learners are adaptive agents, assessments alter behavior, instruction changes the operator, and mastery dynamics may be nonlinear and nonstationary; mathematical modes therefore need not be independent mechanisms or cognitive entities.","failure_condition":"No sufficiently stable, well-conditioned, spectrally separated local transition can be estimated within the unit, or available task bundles cannot selectively move the risky modes.","problem_falsifier":"The proposed problem is falsified if later non-mastery is adequately predicted and prevented by one directly observable skill or independent skill deficits, with no reproducible coupled propagation remaining in held-out data.","intervention_falsifier":"The intervention is falsified if, while model-validity gates pass, modal targeting does not improve later independent mastery or efficiency over the nearest rival, or worsens subgroup outcomes beyond the preregistered margin.","risks":["measurement noise may manufacture modes","mode labels may stigmatize students","targeting may narrow instruction toward assessed skills","teacher workload may exceed benefit","curriculum or cohort shifts may invalidate A","small samples may yield unstable spectra"]},"null_rationale":null,"classification":{"candidate_kind":"MECHANISM_ADAPTATION","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":0.72,"generator_notes":"Closed-book structural candidate. All educational effectiveness and stability claims remain hypotheses pending the bounded test; no novelty claim is made."}