{"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-conjecture-atlas","proposal_index":2,"version":0,"title":"Residual Conjecture Atlas for Large Finite Mathematical Searches","problem":"A mathematician investigating a conjecture over a large finite family may compute the same vector of invariants for every object or parameter point. Most results can conform to a stable family-level pattern, while the mathematically informative cases are localized departures: a candidate counterexample, a boundary between regimes, or evidence that the conjecture needs an additional condition. Full result tables consume storage, communication, and researcher attention, but a simple exception filter is unsafe because its definition of normal can suppress low-amplitude or structurally unfamiliar departures and can mistake a failed worker for confirmation.","actors":["mathematician responsible for the conjecture and search scope","operators of distributed enumeration or symbolic-computation workers","maintainer of the invariant predictor and routing policy","independent verifier of candidate witnesses and raw audit samples","research collaborators interpreting residual structure"],"observable_state":"For each object in one explicitly bounded finite family, workers produce a provenance-tagged vector containing the conjecture predicate, auxiliary invariants, computation status, uncertainty or rigor status, and any witness data. Observable symptoms are repeated invariant patterns across most objects, large full-result transfers and tables, researcher inspection dominated by expected cases, and informative departures distinguishable only after scanning or aggregating the complete output.","consequence":"Scarce computation-channel and mathematical attention capacity is spent restating expected invariant values, while a naive exceptions-only workflow could conceal counterexamples, regime boundaries, systematic model error, missing computations, or minority subclasses.","affected_objective":"Concentrate researcher attention and result-transfer capacity on mathematically informative departures while preserving reconstructibility, complete counterexample witnesses, independent raw coverage, and an explicit distinction between predicted results, computed results, and missing results.","intervention":"Build a versioned conjecture atlas for one finite search scope. Before each object is evaluated, a shared generative model predicts its invariant vector and confidence from declared structural features and its current region of the atlas. A worker independently computes the actual vector and forms a typed residual recording signed numerical differences, categorical changes, predicate disagreement, new structural features, and computation status. A consequence-aware gate transmits residuals preferentially, but any conjecture-predicate failure, rigor downgrade, unrecognized object class, missing result, invalid certificate, or new dependency bypasses suppression and carries the complete object, computation trace, and witness. The coordinator reconstructs each reported vector against the exact model version and maintains explicit heartbeats for suppressed conforming results. Validated residual clusters may propose new atlas regions or revised predictions through a separately reviewed update. Random and boundary-stratified complete results travel through an independent audit path; scheduled complete manifests resynchronize object coverage; and drift, checksum disagreement, unexplained residual structure, reconstruction failure, or budget exhaustion switches the affected region to full-result transmission.","structural_mapping":[{"archetype_element":"Prediction target and observation boundary","domain_realization":"For each enumerated object, the prediction target is the complete declared vector of conjecture truth value, auxiliary invariants, rigor status, computation status, and witness-presence indicators at the worker output boundary."},{"archetype_element":"Generative model state","domain_realization":"A versioned atlas partitions the finite family by mathematical features and assigns each region an expected invariant vector with uncertainty and an accountable owner."},{"archetype_element":"Scope and horizon","domain_realization":"Predictions apply only to the preregistered finite object family, feature vocabulary, invariant definitions, software environment, and one-object evaluation horizon."},{"archetype_element":"Expected and actual behavior","domain_realization":"The atlas prediction is fixed before evaluation; the actual vector is produced independently by the enumeration or symbolic-computation worker with source and completion provenance."},{"archetype_element":"Structured comparator and prediction-error signal","domain_realization":"The comparator preserves numerical direction, categorical substitutions, predicate disagreement, unknown features, rigor changes, witness changes, and missingness instead of reducing them to one anomaly score."},{"archetype_element":"Precision and consequence weighting","domain_realization":"Reliable small changes near a conjecture boundary can outrank large noisy auxiliary deviations, while predicate failures and compromised rigor bypass all ranking."},{"archetype_element":"Residual propagation and reconstruction","domain_realization":"Workers send selected typed residuals with model identity and object coordinates; the coordinator reconstructs the invariant vector as atlas prediction plus structured correction and verifies a digest."},{"archetype_element":"Learning from mismatch","domain_realization":"Reviewed residual clusters can split an atlas region, revise its expected vector, or narrow the conjecture, but updates are versioned and do not rewrite observations already collected."},{"archetype_element":"Synchronization and freshness","domain_realization":"Worker and coordinator compare atlas, invariant-definition, enumerator, and arithmetic-environment fingerprints, and complete coverage manifests periodically re-anchor the search."},{"archetype_element":"Independent raw-state audit","domain_realization":"Uniformly random objects and objects sampled near known region boundaries are transmitted with their complete computed vectors through a path whose selection does not depend solely on the production predictor."},{"archetype_element":"Safety-critical bypass and fallback","domain_realization":"Counterexample candidates, missing results, unrecognized classes, rigor downgrades, invalid certificates, version conflicts, and reconstruction failures force full transmission and independent verification."},{"archetype_element":"Attention and residual-error budgets","domain_realization":"The pilot budgets transfer volume, researcher triage time, suppressed residual mass, audit coverage, reconstruction disagreement, and predictor-maintenance cost separately; no budget permits suppression of protected events."}],"mechanism_mapping":[{"mechanism_slug":"predictive_codec","role":"Keep compatible atlas predictors at workers and coordinator so complete invariant vectors can be reconstructed from model-relative residuals.","counterfactual_removal":"Without matched prediction and reconstruction, the intervention becomes an exception list rather than predictive residual processing."},{"mechanism_slug":"anomaly_detection_model","role":"Score departures from the expected invariant pattern within each declared atlas region and surface candidate boundary cases or misspecification.","counterfactual_removal":"The system would lack a systematic way to distinguish routine residual variation from cases warranting mathematical investigation."},{"mechanism_slug":"event_triggered_residual_reporting","role":"Transmit a result event when its precision-weighted typed residual crosses a declared threshold, while using heartbeats and manifests to make silence interpretable.","counterfactual_removal":"Workers would continue sending complete vectors for every object, eliminating the residual channel's communication focus."},{"mechanism_slug":"precision_weighted_error_gate","role":"Rank nonprotected residuals using reliability, boundary relevance, consequence, novelty, and transfer cost under an explicit capacity budget.","counterfactual_removal":"Large but unreliable numerical deviations could crowd out small, exact changes that alter the conjecture's plausible boundary."},{"mechanism_slug":"confidence_threshold_table","role":"Version the mapping from residual class and confidence to suppress, transmit residual, transmit full result, or halt region processing.","counterfactual_removal":"Suppression authority would be implicit and could drift toward a desired queue size without an attributable error budget."},{"mechanism_slug":"model_version_checksum_handshake","role":"Confirm that worker and coordinator share the same atlas, invariant definitions, and prediction-affecting environment before interpreting a residual.","counterfactual_removal":"A residual could be reconstructed against incompatible definitions and still appear syntactically valid."},{"mechanism_slug":"periodic_full_state_resynchronization","role":"Exchange complete object-coverage manifests and selected region snapshots to bound accumulated omissions or ordering errors.","counterfactual_removal":"Lost events or divergent coverage maps could persist indefinitely and make incomplete enumeration look complete."},{"mechanism_slug":"shadow_raw_channel_sampling","role":"Send uniform random and boundary-stratified full object results through an independent audit route for comparison with reconstructed or suppressed results.","counterfactual_removal":"The atlas would be unable to measure cases that its own feature partition systematically fails to recognize."},{"mechanism_slug":"residual_comparison_test","role":"Test residuals for feature-dependent structure and compare the atlas with a simpler constant-within-region predictor and raw samples.","counterfactual_removal":"Systematic mathematical structure left unexplained could be mislabeled as harmless computational noise."},{"mechanism_slug":"model_drift_monitoring","role":"Track residual distributions, unknown-feature frequency, audit disagreement, environment changes, and elapsed validity of each atlas region.","counterfactual_removal":"A once-adequate partition could retain suppression authority after the search enters a different mathematical regime."},{"mechanism_slug":"raw_signal_fallback_switch","role":"Restore full-result transmission for any region when validity, synchronization, rigor, coverage, or reconstruction conditions fail.","counterfactual_removal":"The search could remain compressed precisely where the atlas is least trustworthy."},{"mechanism_slug":"prediction_error_replay_buffer","role":"Retain residuals with complete object features, model version, and selected raw context for later region refinement and regression tests.","counterfactual_removal":"Surprising cases would not form a durable, attributable basis for revising the atlas or conjecture boundary."},{"mechanism_slug":"bayesian_model_update","role":"Represent uncertainty over competing atlas partitions and revise it from verified residual evidence without collapsing immediately to one explanation.","counterfactual_removal":"Atlas revision would rely on ad hoc reactions to individual surprises rather than bounded uncertainty-aware accumulation of evidence."}],"causal_chain":["A versioned atlas predicts the complete invariant vector for each object from its declared structural region before computation.","An enumeration or symbolic-computation worker independently evaluates the object and records an explicit completion or missingness state.","A typed comparator subtracts predictable numerical and categorical content while preserving predicate, rigor, witness, and unknown-feature changes.","Protected changes travel in full; other residuals compete for limited transfer and researcher-attention capacity using precision and mathematical consequence.","The coordinator reconstructs results against the matching atlas version, verifies digests, and uses heartbeats plus complete manifests to distinguish conformity from worker or channel failure.","Researchers examine residual structure to identify candidate counterexamples, regime boundaries, or missing conjecture conditions rather than repeatedly scanning conforming objects.","Only independently verified residual patterns can revise the atlas partition or its uncertainty, and prior observations remain tied to the version that predicted them.","Raw audit samples, rival residual comparisons, drift tests, and full manifests expose systematic suppression; any validity failure decompresses the affected region to full results."],"baseline":"Transmit, store, and inspect the complete computed invariant vector for every object in the finite family, accompanied by explicit worker completion records and ordinary sorting or aggregation. Compare this baseline with the intervention on identical computations and review tasks.","nearest_rivals":["A full enumeration table with filters and sortable columns, which preserves every result but does not predict and encode expected vectors as reconstructible residuals.","A fixed counterexample search that reports only predicate failures, which can find declared failures but does not surface model-boundary evidence, maintain uncertainty, audit suppressed results, or reconstruct the full invariant stream.","Uniform sampling of the object family, which reduces computation or inspection volume but does not use a shared predictive model or learn from structured mismatch.","Clustering or dimensionality reduction over completed results, which summarizes global structure after collection but does not create a synchronized event-triggered residual channel with raw fallback.","A static anomaly threshold on one invariant, which lacks typed mathematical comparison, protected predicate bypasses, model-version provenance, and atlas revision."],"remaining_contrastive_claim":"The proposal's testable contrast is that object-level mathematical results are represented and routed as typed errors against a synchronized, reconstructive conjecture atlas; verified errors can revise that atlas, while independent full-result samples, coverage manifests, and automatic decompression constrain its authority. No claim is made that this package is novel or beneficial before the bounded comparison.","authority_safety":{"decision_authority":"The responsible mathematician defines the conjecture, invariant semantics, finite search scope, and protected event classes. An independent verifier determines whether a candidate witness reproduces. The residual system may prioritize investigation but may neither declare a theorem nor certify a counterexample.","authorized_first_step":"Run a read-only replay over one bounded finite family for which complete results will also be retained. Fit the initial atlas only on a preregistered subset, freeze thresholds, and compare residual-mode reconstruction and triage with the complete-result baseline on an untouched subset.","excluded_actions":["declaring a theorem from absence of transmitted residuals","declaring a counterexample without independent full-witness reproduction","suppressing predicate failures, missing computations, rigor downgrades, invalid certificates, unknown object classes, or changed invariant definitions","discarding complete baseline results during the first evaluation","automatically changing the conjecture, atlas partition, protected classes, or thresholds","treating a matching model checksum as evidence that the mathematical prediction is correct","extending the atlas outside the preregistered finite family without a new authorization"],"halt_rollback":"Immediately disable residual-only routing for an affected region after any protected-event omission, reconstruction digest failure, checksum conflict, missing heartbeat, coverage-manifest discrepancy, unexplained audit disagreement, invalid witness path, or residual-budget breach. Roll back to full-result transmission and review using the retained computations and immutable versioned provenance."},"negative_tests":{"strongest_counterevidence":"The invariant vectors may not be predictable enough within defensible regions, or the supposedly routine full results may themselves be necessary for recognizing new mathematical structure. Atlas construction, synchronization, audits, and residual triage may also consume at least as much capacity as complete collection and inspection.","problem_falsifier":"The proposed problem is unsupported if the bounded family has little repeated invariant structure, complete results fit comfortably within the declared transfer and attention budgets, researchers identify informative cases from the baseline without measurable scanning burden, or missing and protected events are already isolated completely by a simpler procedure.","intervention_falsifier":"Reject residual routing if any planted or naturally occurring protected event is suppressed; any audited invariant vector fails exact reconstruction; incomplete worker coverage is interpreted as conformity; residuals show persistent feature-dependent structure without triggering fallback; atlas updates become unstable or self-confirming; or total computation, transfer, storage, audit, maintenance, and review cost is not lower than the full-result baseline at equal protected-event detection and reconstruction fidelity.","risks":["A wrong atlas partition could suppress an entire mathematically meaningful subclass.","Adaptive predictions could normalize a gradual approach toward counterexamples.","Researcher attention could become anchored to the atlas's existing feature vocabulary.","Rare unknown structures may evade both risk-stratified sampling and residual thresholds.","A residual may expose a distinctive unpublished object or witness more readily than a bulk table.","Incomplete computations could appear conforming if heartbeat and manifest logic fail.","Bayesian or other uncertainty updates could become confidently wrong under a misspecified likelihood.","Compression incentives could encourage narrowing audit coverage or weakening protected-event definitions.","Exact numerical reconstruction may not preserve semantic equivalence when canonicalization or rigor metadata is incomplete."]},"next_evidence_step":"On one preregistered finite family, retain complete results while simulating residual routing. Freeze an atlas from a training partition and evaluate an untouched partition containing blinded predicate flips, missing-worker events, new feature classes, boundary cases, and environment-version mismatches. Measure exact vector reconstruction, protected-event recall, false triage, residual structure, audit disagreement, transfer volume, researcher task time, fallback frequency, and total model-maintenance cost. This evidence can authorize only another bounded evaluation, not residual-only retention or mathematical claims.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Proposal 1 addresses sequential auditing of machine-checked derivations: it predicts each post-command proof state so reviewers can focus on changes within a proof while the kernel remains the acceptance authority. This proposal addresses exploration of a large finite family of mathematical objects: it predicts parallel object-level invariant vectors so residual structure can reveal counterexample candidates and conjecture boundaries. Its primary actors include enumeration workers and conjecture investigators rather than proof authors and proof-state reviewers; its scarce channel is object-result transfer and exploratory attention rather than derivation-transcript review; its maintained model is a feature-partitioned conjecture atlas rather than a one-step proof-state transition model; and its learning action revises family regions or conjecture conditions rather than improving reconstruction of tactic transitions. It can be adopted for computational conjecture exploration without a proof assistant or residual proof transcript, so it is independently adoptable rather than a feature or implementation variant of proposal 1.","revision_record":{"parent_version":null,"progress_targets_addressed":["Initial complete formulation for proposal_index 2","Material independence from sealed proposal 1"],"conceptual_changes":["Defined a finite-family conjecture exploration problem centered on object-level invariant residuals and atlas refinement.","Separated exploratory prioritization from theorem and counterexample authority."],"operational_changes":["Specified typed residuals, explicit missingness, protected full-result classes, worker-coordinator version handshakes, coverage manifests, independent raw samples, region-level fallback, and immutable observation provenance.","Bound the first step to a full-retention replay with a frozen training partition and untouched evaluation partition."],"evidence_changes":["Added blinded protected-event, missing-worker, new-class, boundary, and version-mismatch tests against a complete-result baseline.","No external or prior-art evidence was consulted."],"claim_changes":["Made only a falsifiable architectural contrast and no claim about novelty, prevalence, demand, or effect size."]}}