{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp06_four_proposal_generalization60_20260803","cell_id":"predictive_residual_processing__mathematics","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_id":"prp-math-residual-certificate-continuation","proposal_index":3,"version":0,"title":"Residual Certificate Continuation for Parameterized Mathematical Equations","problem":"A validated numerical study may need certified solution enclosures for many adjacent parameter cells of an equation F(x,λ)=0. Independently solving, transmitting, storing, and reviewing a complete enclosure and certificate at every cell repeats structure when the solution branch changes smoothly. Yet ordinary continuation can silently follow the wrong branch, cross a singularity, or miss the creation or loss of solutions. Reusing predictions without independent audits and full-solve fallback could therefore manufacture an apparently continuous branch that the equations do not support.","actors":["mathematician defining the parameterized equation and required theorem claim","validated-numerics worker computing certified enclosures","coordinator reconstructing the parameter-indexed solution atlas","independent certificate verifier","maintainer authorized to revise the continuation predictor and thresholds"],"observable_state":"For a preregistered compact parameter domain divided into finite cells, record the full equation specification, parameter cell, predicted enclosure, independently computed certified enclosure, root-count or existence status, conditioning diagnostics, certificate, solver version, and completion state. Relevant symptoms are small structured differences between neighboring certified enclosures, repeated certificate content, high transfer or review cost per cell, and occasional large or patterned discrepancies near singularities, branch changes, or predictor failure.","consequence":"Repeated full-state computation and review can consume the study's capacity, but an inadequately governed continuation shortcut can omit a branch transition or misrepresent incomplete computation as confirmation of the predicted solution.","affected_objective":"Represent and review a parameterized family of certified solution states with less redundant communication, storage, and correction work while preserving independently verifiable existence, uniqueness or root-count claims and reliable detection of branch changes.","intervention":"Create a versioned residual-certificate protocol over one bounded parameter mesh. For each next cell, a predictor uses the last full anchor, local branch model, and declared step to forecast the solution enclosure, root-count status, and conditioning range before the worker solves that cell. The worker performs a validated correction and emits a structured residual containing enclosure displacement, width correction, status changes, certificate amendments, and diagnostics. A coordinator with the identical predictor reconstructs the full state and accepts it into the study only if an independent verifier validates the reconstructed enclosure and attached certificate against the complete equation and parameter cell. Precision-weighted residuals determine review priority, while singular or near-singular Jacobian evidence, changed root count, failed inclusion tests, branch ambiguity, domain-boundary contact, unknown arithmetic conditions, missing output, or version mismatch bypass compression and trigger a cold-start full solve. Verified residuals may update local predictor uncertainty and step size, but structural predictor changes require separate approval. Random parameter cells and cells near suspected boundaries receive independent cold-start full solves; complete state anchors and coverage manifests periodically resynchronize the atlas.","structural_mapping":[{"archetype_element":"Prediction target and observation boundary","domain_realization":"The target is the complete certified solution state for one parameter cell: enclosure, existence or uniqueness status, root count where applicable, conditioning diagnostics, certificate identity, and explicit computation status."},{"archetype_element":"Generative model state","domain_realization":"A versioned local branch model predicts the next enclosure and diagnostic state from a full anchor, prior verified residuals, parameter displacement, and bounded derivative information."},{"archetype_element":"Model scope and horizon","domain_realization":"Each prediction is authorized only for one adjacent parameter step within a preregistered domain, equation version, arithmetic regime, branch identifier, and maximum step size."},{"archetype_element":"Expected and actual behavior","domain_realization":"The predicted solution state is frozen before the worker performs a validated correction; the actual state is the independently certified result of that correction or an explicit failure state."},{"archetype_element":"Prediction comparator","domain_realization":"A typed comparator preserves signed enclosure-center displacement, width changes, set-containment relations, root-count changes, diagnostic differences, certificate amendments, and missingness."},{"archetype_element":"Precision and consequence weighting","domain_realization":"Residual priority depends on validation margin, rounding guarantees, conditioning, source reliability, proximity to a boundary, and mathematical consequence rather than displacement magnitude alone."},{"archetype_element":"Residual propagation and reconstruction","domain_realization":"The worker sends a model-tagged correction and certificate amendment; the coordinator reconstructs the full enclosure and status against the matching prediction and verifies a canonical digest."},{"archetype_element":"Update rule","domain_realization":"Verified innovations update local branch slope, uncertainty, and permitted step size, while proposed changes to branch structure or predictor form are reviewed and versioned separately."},{"archetype_element":"Synchronization and provenance","domain_realization":"Every residual identifies the equation, parameter cell, branch, predictor, arithmetic library, solver, certificate format, and last full anchor; incompatible fingerprints prohibit residual interpretation."},{"archetype_element":"Freshness, drift, and residual-error budget","domain_realization":"Limits apply to parameter distance from the anchor, cumulative enclosure correction, validation-margin decay, residual correlation, and time or cell count since the last cold-start solve."},{"archetype_element":"Independent raw audit","domain_realization":"Uniformly random cells and risk-stratified cells near high curvature, weak validation margins, or suspected branch boundaries are solved from cold start without using the production prediction as the initial representation."},{"archetype_element":"Fallback and decompression","domain_realization":"Validity failure restores complete enclosure, diagnostic, and certificate transmission and invokes a cold-start solve for the affected region."},{"archetype_element":"Safety-critical bypass","domain_realization":"Evidence of singularity, changed root count, branch ambiguity, failed validation, boundary contact, unrecognized arithmetic behavior, or missing computation always travels in full."},{"archetype_element":"Attention and bandwidth budget","domain_realization":"The study accounts for certificate bytes, solver corrections, verification time, human review time, audits, resynchronization, and predictor maintenance alongside reconstruction fidelity."}],"mechanism_mapping":[{"mechanism_slug":"innovation_residual_filter","role":"Predict the next certified state, compute its innovation against the validated result, carry uncertainty, and update the local continuation state from verified corrections.","counterfactual_removal":"The workflow would lose its recursive prediction-error teaching loop and reduce to independent solving or unstructured differencing."},{"mechanism_slug":"predictive_codec","role":"Keep worker and coordinator predictors aligned and encode each solution state as prediction plus a structured certified correction.","counterfactual_removal":"Residuals would not provide a reconstructible representation of the parameter-indexed solution atlas."},{"mechanism_slug":"precision_weighted_error_gate","role":"Prioritize residuals using validation margin, conditioning, reliability, boundary proximity, mathematical consequence, and review cost.","counterfactual_removal":"Large benign displacements could displace small but reliable signs of lost uniqueness or branch change."},{"mechanism_slug":"confidence_threshold_table","role":"Version the rules mapping uncertainty and residual classes to routine verification, enhanced review, reduced step size, full solve, or halt.","counterfactual_removal":"Continuation and fallback policy would be implicit and vulnerable to adjustment for convenience rather than certificate risk."},{"mechanism_slug":"model_version_checksum_handshake","role":"Verify equation, predictor, solver, arithmetic, certificate, and anchor compatibility before a correction is reconstructed.","counterfactual_removal":"A valid-looking residual could be applied against a different equation or numerical baseline."},{"mechanism_slug":"periodic_full_state_resynchronization","role":"Insert complete certified anchors and reconcile coverage manifests after a fixed number of cells or an early drift trigger.","counterfactual_removal":"Small reconstruction or ordering errors could propagate without a bounded recovery point."},{"mechanism_slug":"model_drift_monitoring","role":"Track cumulative correction, residual correlation, validation-margin decay, fallback frequency, unknown conditions, and anchor age.","counterfactual_removal":"A deteriorating local branch model could retain residual-processing authority until after a mathematical transition was missed."},{"mechanism_slug":"anomaly_detection_model","role":"Identify residual patterns inconsistent with the current smooth-branch model and route them for boundary or misspecification investigation.","counterfactual_removal":"The protocol would react only to individual threshold crossings and could miss a sustained structured departure composed of modest corrections."},{"mechanism_slug":"residual_comparison_test","role":"Compare residual structure with a simpler local predictor, a challenger predictor, and cold-start audit solutions.","counterfactual_removal":"Correlated or biased errors could be dismissed as numerical noise without evidence that the model had extracted the predictable structure."},{"mechanism_slug":"shadow_raw_channel_sampling","role":"Run independent cold-start full solves at random and risk-stratified parameter cells and compare them with reconstructed states.","counterfactual_removal":"The continuation path would be auditing itself and could preserve a shared wrong branch across all compressed cells."},{"mechanism_slug":"raw_signal_fallback_switch","role":"Suspend residual mode and restore full computation and transmission when certification, scope, synchronization, or branch-validity conditions fail.","counterfactual_removal":"The system could remain compressed exactly where the predictor is mathematically least defensible."},{"mechanism_slug":"prediction_error_replay_buffer","role":"Retain verified corrections, failed validations, complete context, and model versions for predictor review and regression testing.","counterfactual_removal":"Boundary evidence and failure cases would not become durable tests for later predictor versions."},{"mechanism_slug":"surprise_to_action_bridge","role":"Translate a validated singularity, root-count change, or branch ambiguity into an owned task to isolate the region, run full solves, and review the mathematical claim.","counterfactual_removal":"Consequential residuals could be displayed without causing the additional computation and scope revision they require."}],"causal_chain":["A full certified anchor and bounded local branch model predict the mathematical state of one adjacent parameter cell.","A worker independently performs a validated correction or records an explicit failure rather than treating prediction as observation.","A typed comparator represents the result as enclosure, status, diagnostic, and certificate residuals against the frozen prediction.","The coordinator reconstructs the full state using the matching model version, and an independent verifier checks it against the complete equation and parameter cell.","Precision-weighted residuals focus routine review, while singularity, branch, root-count, validation, missingness, and compatibility signals bypass compression.","Verified innovations update local uncertainty and step size, allowing prediction to adapt without granting it authority over certification.","Residual-distribution monitoring and comparisons with challenger predictions detect systematic local-model failure rather than isolated numerical noise.","Random and boundary-stratified cold-start solves test what continuation may have explained away, while complete anchors bound accumulated divergence.","Any failed validity condition decompresses the affected region to complete states and cold-start solving before continuation can resume."],"baseline":"Solve every parameter cell from a common cold start, transmit and store its complete enclosure, diagnostics, and certificate, and have the verifier process each full result without reusing a predictive representation. Measure full costs and mathematical outcomes on the identical finite mesh.","nearest_rivals":["Ordinary numerical continuation, which predicts and corrects a branch but need not make residuals the reconstructible communication unit or provide version handshakes, independent raw audits, and automatic full-state decompression.","Independent validated solving at every parameter cell, which minimizes shared-model dependence but repeats complete work and representation.","Interpolation or surrogate modeling over parameter space, which estimates solutions between computed points but does not require certified residual reconstruction or full-solve safety bypasses.","Adaptive mesh refinement driven by a local error estimate, which allocates cells by estimated approximation error but does not synchronize predictor copies or encode certified solution states as model-relative residuals.","Delta storage between neighboring enclosures, which records changes but lacks a forward generative model, uncertainty-aware update loop, drift detection, and independent cold-start audit."],"remaining_contrastive_claim":"The proposal's testable contrast is a governed residual representation for certified parameter continuation: the predicted solution state is shared, the validated correction is the primary message and teaching signal, reconstructed states remain independently certifiable, and cold-start audits plus automatic decompression constrain branch-model failure. This is an architectural contrast, not a claim of novelty or superior performance.","authority_safety":{"decision_authority":"The responsible mathematician defines the equation, domain, theorem-level claim, protected conditions, and acceptable certificate semantics. The independent verifier determines whether a reconstructed or full result satisfies those semantics. The predictor may choose review priority and propose a smaller step, but it cannot establish existence, uniqueness, root count, branch identity, or a theorem.","authorized_first_step":"Run a read-only shadow evaluation on one bounded parameter mesh while retaining the complete cold-start baseline. Freeze the equation, predictor, thresholds, audit selection rule, and fallback triggers before scoring the untouched evaluation cells.","excluded_actions":["asserting a theorem or complete parameter coverage from residual silence","accepting an enclosure without independent certificate verification against the full equation and parameter cell","suppressing singularity evidence, root-count changes, branch ambiguity, failed inclusion tests, boundary contact, unknown arithmetic conditions, or missing results","discarding full baseline states during the first evaluation","automatically changing the equation, mathematical claim, protected classes, branch structure, or certificate semantics","continuing residual mode after a model, solver, arithmetic, equation, or anchor fingerprint mismatch","extending conclusions outside the preregistered parameter domain or mesh"],"halt_rollback":"Halt residual continuation for the affected region after any protected-event omission, certificate-verification failure, reconstruction-digest mismatch, audit disagreement, coverage gap, incompatible version, missing heartbeat, unexplained structured residual, or error-budget breach. Roll back to the most recent verified full anchor, invalidate dependent residual reconstructions, and recompute the region using cold-start full solves; the shadow pilot leaves the baseline artifacts unchanged."},"negative_tests":{"strongest_counterevidence":"The certified solution state may change too irregularly for one-step prediction to remain trustworthy, or detecting branch structure may require global information unavailable to local residuals. Full certificates may also be irreducible, making prediction, synchronization, audits, and fallback costlier than independent full solves.","problem_falsifier":"The problem is unsupported if adjacent certified states show little reusable structure, complete cold-start solving and review fit within the declared capacity, or repeated full certificates contain no material redundancy under the required semantics.","intervention_falsifier":"Reject residual continuation if any protected transition is suppressed; any reconstructed enclosure, status, or certificate differs from its full result; a missing computation is treated as confirmation; cold-start audits reveal an omitted solution branch; structured residuals persist without decompression; invalidated states are not bounded by the last full anchor; or total solver, verification, transfer, storage, audit, and maintenance cost is not lower than the baseline at equal certified coverage.","risks":["Local prediction can remain smooth while tracking the wrong solution branch.","A shared predictor can create correlated worker and coordinator errors.","Canonical enclosure deltas may omit certificate semantics needed for independent verification.","Near-singular behavior may emerge between sampled parameter cells.","Adaptive step reduction can create unstable fallback cycles or exhaust the computation budget.","Risk-stratified audits can overfocus known boundary types and miss unfamiliar transitions.","A green checksum proves compatibility, not mathematical correctness.","Repeated predictor updates may absorb a regime change rather than exposing it.","Invalidating descendants after a bad anchor may require extensive recomputation.","Residual compression may save representation cost while adding more verification complexity than it removes."]},"next_evidence_step":"Pre-register one finite parameter mesh, a full cold-start solver baseline, a frozen initial predictor, exact reconstruction rules, and protected-event tests. On untouched cells, compare residual continuation with the baseline using independently verified enclosure equality or declared containment semantics, branch and root-count agreement, blinded singularity and missing-result cases, random and boundary-stratified cold-start audits, invalidation radius after planted anchor errors, solver work, certificate bytes, verification time, human review time, fallback frequency, and total maintenance cost. The result can authorize only another bounded shadow test, not theorem claims or replacement of complete certificates.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Proposal 1 predicts sequential proof-assistant states to compress the human audit transcript of a formal derivation; its decision is where reviewers inspect proof-state changes, and kernel acceptance remains external. Proposal 2 predicts invariant vectors across a finite family to concentrate exploratory attention on counterexample candidates and conjecture boundaries; its learning action repartitions a conjecture atlas. This proposal instead governs certified numerical continuation for a parameterized equation: it predicts a neighboring solution enclosure, uses a validated numerical correction as both reconstructive message and local model update, and falls back to cold-start solving when branch or certification conditions fail. Its primary scarce resources are validated solver work, certificate transfer, and certificate verification; its central hazard is following or certifying the wrong solution branch; and its downstream decision is whether a parameter region requires full recomputation. It neither reviews proof commands nor screens a population of independently characterized objects, and it can be adopted without either a proof-state transcript system or a conjecture-atlas search, making it independently adoptable from proposals 1 and 2.","revision_record":{"parent_version":null,"progress_targets_addressed":["Initial complete formulation for proposal_index 3","Material independence from sealed proposals 1 and 2"],"conceptual_changes":["Defined certified parameter continuation as a prediction-correction-reconstruction problem with branch validity as the protected mathematical property.","Separated predictive continuation from independent certificate authority and theorem authority."],"operational_changes":["Specified one-cell prediction horizons, typed enclosure and certificate residuals, version handshakes, full anchors, coverage manifests, cold-start audits, descendant invalidation, and region-level fallback.","Bound the first step to a shadow comparison retaining a complete cold-start baseline."],"evidence_changes":["Specified untouched-cell evaluation, blinded singularity and missingness cases, branch agreement, reconstruction checks, audit solves, invalidation-radius measurement, and complete cost accounting.","No external or prior-art evidence was consulted."],"claim_changes":["Limited the proposal to a falsifiable architectural contrast and made no novelty, prevalence, demand, or effect-size claim."]}}