{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp06_four_proposal_generalization60_20260803","cell_id":"catalytic_pathway_enablement__mathematics","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_id":"verified_groebner_reduction_lane","proposal_index":1,"version":0,"title":"Verified Reusable Gröbner-Reduction Lane for Polynomial Identities","problem":"In a collaborative commutative-algebra project, researchers repeatedly need to determine whether candidate polynomial identities hold modulo the same fixed ideal. Although each determination is feasible, cases stall because the applicable ring, monomial order, generators, reductions, and proof trace are reconstructed or checked separately for every identity.","actors":["Researchers submitting candidate polynomial identities","A computational-algebra maintainer who stewards the reduction lane","An independent checker or reviewing mathematician","Downstream theorem authors who consume certified membership results"],"observable_state":"A queue contains candidate identities stated over one declared polynomial ring and ideal. Case records show repeated setup of the same ideal-membership context, variable waiting times, duplicated reductions, and outputs whose assumptions or proof traces require manual reconstruction before reuse in a theorem.","consequence":"The project delays decisions about which candidate identities can support downstream proofs, spends mathematical labor repeatedly rebuilding the same reduction context, and risks admitting results whose ring, ideal, coefficient field, or monomial-order assumptions do not match the intended theorem.","affected_objective":"Produce auditable ideal-membership or non-membership determinations for eligible candidate identities while preserving the project's exact algebraic assumptions and proof standard.","intervention":"Create a governed reduction lane centered on a versioned, independently verified Gröbner basis for one fixed ideal and monomial order. A published interface accepts only polynomials over the declared coefficient field and variable set, reduces each eligible polynomial to normal form, and emits a machine-checkable certificate tying the result back to the original ideal generators. A separate checker validates every certificate. Cases involving parameter specialization, radical membership, inequalities, changed generators, unsupported coefficient domains, or excessive resource demand leave the lane for ordinary mathematical analysis. The steward meters submissions against measured compute and checking capacity, monitors queue depth, certificate failures, coefficient growth, and version mismatches, and regenerates and revalidates the basis whenever its defining context changes.","structural_mapping":[{"archetype_element":"Target transformation specification","domain_realization":"Transform a candidate equality f=g into a certified determination of whether f-g belongs to a declared ideal I in a fixed polynomial ring, returning either a zero normal form with a generator-based membership certificate or a nonzero canonical remainder under the declared Gröbner basis."},{"archetype_element":"Activation barrier","domain_realization":"Each case otherwise reconstructs the same algebraic context, selects reductions, manages intermediate expressions, and rebuilds an auditable connection from the computation to the original ideal generators."},{"archetype_element":"Reusable facilitator","domain_realization":"A versioned verified Gröbner basis, its representations in the original generators, and a deterministic certificate-producing reducer are reused across multiple eligible polynomial inputs without being consumed by any result."},{"archetype_element":"Facilitator–substrate interface","domain_realization":"An intake contract fixes the coefficient field, variables, ideal version, monomial order, input syntax, resource bounds, and required output certificate before a polynomial can enter the lane."},{"archetype_element":"Selectivity rule","domain_realization":"The lane handles exact ideal membership only in the declared commutative polynomial context; superficially similar questions about radicals, special parameter values, inequalities, noncommutative rings, or changed ideals are rejected or rerouted."},{"archetype_element":"Turnover capacity","domain_realization":"Capacity is measured as independently checked determinations per basis version and per compute interval, together with queue time, reduction time, certificate-checking time, memory use, and coefficient growth."},{"archetype_element":"Regeneration cycle","domain_realization":"After a generator, field, variable, or monomial-order change—or evidence of a faulty basis—the steward withdraws the version, recomputes the basis and generator representations, reruns independent verification and reference cases, and releases a new immutable version."},{"archetype_element":"Saturation and inhibition monitoring","domain_realization":"Queue depth and compute occupancy indicate saturation, while rising certificate failures, pathological coefficient growth, malformed inputs, or version mismatches indicate interference rather than insufficient nominal capacity."},{"archetype_element":"Equilibrium neutrality","domain_realization":"The lane changes the route and time required to establish ideal membership; it does not change the ideal, the truth of the equality modulo that ideal, or the evidentiary standard for using the result in a theorem."},{"archetype_element":"Accountable stewardship and deactivation","domain_realization":"A named maintainer owns version integrity, fair intake, capacity limits, incident review, regeneration, and immediate withdrawal, but cannot declare a theorem proved merely because a reduction completed."}],"mechanism_mapping":[{"mechanism_slug":"interface_contract_design","role":"Publishes the exact algebraic preconditions, certificate postconditions, protected invariants, exception route, and immutable version identifiers for the reduction lane.","counterfactual_removal":"Without the contract, inputs from incompatible rings, ideals, or orders could be processed under unstated assumptions, making apparent acceleration non-selective and results difficult to audit."},{"mechanism_slug":"prevalidated_transformation_template","role":"Carries forward the once-verified Gröbner basis, its link to the original generators, and the permitted reduction structure while leaving only the submitted polynomial variable.","counterfactual_removal":"Without the template, each case must reconstruct or revalidate the pathway, eliminating the intended multi-case reuse and allowing proof assumptions to drift."},{"mechanism_slug":"workflow_automation_or_macro","role":"Performs deterministic normal-form reduction, emits a certificate, validates basic output conditions, and logs the basis version and execution record for every case.","counterfactual_removal":"Without repeatable execution and logging, the basis remains a mathematical reference rather than an operational facilitator, and manual transcription or undocumented choices reintroduce the barrier."},{"mechanism_slug":"turnover_and_selectivity_assay","role":"Compares checked determinations per facilitator version, elapsed work, correct target classifications, exceptions, and incorrect or unverifiable outputs against an ordinary per-case pathway.","counterfactual_removal":"Without the assay, faster easy cases, added computation, or omitted checking could be misattributed to reuse, and off-target or unverifiable outputs would be hidden."},{"mechanism_slug":"active_site_capacity_dashboard","role":"Displays queue depth, active reductions, checking backlog, cycle time, memory pressure, coefficient-growth alerts, and basis-version health so intake can be matched to usable capacity.","counterfactual_removal":"Without joint queue and degradation visibility, saturation could be confused with a defective or mismatched basis, leading to more submissions when the lane should instead pause."},{"mechanism_slug":"inhibitor_and_poison_screen","role":"Rejects malformed polynomials, undeclared coefficients, version mismatches, unsupported domains, and predicted resource-limit breaches before they occupy or corrupt the shared lane.","counterfactual_removal":"Without an upstream screen, incompatible cases can consume capacity, trigger misleading failures, or produce results under the wrong algebraic context."},{"mechanism_slug":"catalyst_regeneration_protocol","role":"Defines withdrawal, recomputation, independent basis verification, reference-case replay, replacement, and retirement when the ideal context changes or integrity cannot be restored.","counterfactual_removal":"Without regeneration and retirement, a stale or faulty basis could remain nominally available and repeatedly accelerate invalid determinations."},{"mechanism_slug":"small_safe_to_fail_probe","role":"Tests the lane on a bounded concealed set with a facilitator-off comparison, precommitted correctness and stop criteria, and no authority to alter theorem status.","counterfactual_removal":"Without a bounded counterfactual probe, the project would lack attributable first evidence and could expose live proof dependencies to an unvalidated shared pathway."}],"causal_chain":["Researchers submit polynomial differences that conform to one versioned ring-and-ideal contract.","The intake screen admits only compatible cases and reroutes mismatches or exceptional mathematical questions.","The verified Gröbner basis supplies a previously established reduction pathway instead of requiring each case to reconstruct that pathway.","The reducer computes a normal form and releases the shared basis unchanged for the next eligible polynomial.","A separate checker validates the certificate against the original ideal generators and declared basis version.","Validated results proceed to theorem authors; failures and exceptions return to ordinary mathematical analysis rather than being forced through the lane.","Turnover, queueing, resource use, exceptions, and incorrect or unverifiable outputs are compared with the per-case baseline.","Capacity signals meter intake, while integrity or compatibility signals pause the lane and initiate diagnosis.","A changed or degraded algebraic context triggers recomputation and independent revalidation before the facilitator returns to service."],"baseline":"For each candidate identity, a researcher independently sets up the ring and ideal, chooses a computer-algebra or hand-reduction route, and reconstructs enough evidence for another mathematician to check the result. Record setup time, reduction time, checking time, total elapsed time, compute resources, classification, certificate completeness, and exceptions without weakening the proof standard.","nearest_rivals":["Per-case Gröbner-basis recomputation with an independently checked certificate, which preserves rigor but does not amortize basis construction across cases.","Direct theorem-specific proof using substitutions, structural lemmas, elimination, or syzygies, which may exploit mathematical structure that generic normal-form reduction misses.","An ungoverned shared computer-algebra worksheet or cached basis, which offers reuse but lacks a fixed interface, independent certificate checking, capacity controls, and explicit refresh or withdrawal rules.","Additional compute workers running the existing per-case process, which expands bulk capacity but does not establish whether a reusable pathway removes repeated setup."],"remaining_contrastive_claim":"The untested contrastive hypothesis is that, for cases genuinely sharing one ring, ideal, and monomial order, reuse of an independently verified basis and certificate contract removes repeated setup and validation work while leaving the membership criterion unchanged. The comparison must separately account for computation, human checking, case eligibility, and exceptions; otherwise the intervention is indistinguishable from ordinary automation, case selection, or added capacity.","authority_safety":{"decision_authority":"The project's designated algebra lead may authorize a bounded shadow-mode evaluation and appoint the basis steward. Only the project's ordinary theorem-review authority may accept a result into a proof or publication.","authorized_first_step":"Build one immutable facilitator version for a non-live fixed ideal and run it in shadow mode on a bounded set of already adjudicated identities, with outcomes concealed from operators until classifications and certificates are recorded.","excluded_actions":["Changing the ideal, coefficient field, monomial order, or theorem statement to make a candidate pass","Treating a successful reduction as authority to approve or publish a theorem","Skipping independent certificate checking","Sending unsupported or exceptional cases through the lane solely to increase utilization","Expanding beyond the bounded evaluation before review of all failures and exceptions","Using the lane for radical membership, parameter-specialized claims, inequalities, or noncommutative problems without separate validation and authorization"],"halt_rollback":"Immediately stop intake and withdraw the facilitator version upon a wrong classification, an unverifiable certificate, a basis-integrity failure, an undeclared context mismatch, or resource behavior exceeding the probe's preset bound. Preserve logs, return every affected case to its prior adjudicated status, and require recomputation plus independent revalidation before any restart."},"negative_tests":{"strongest_counterevidence":"Representative cases may share the same written ideal yet obtain their tractability from theorem-specific structure; generic Gröbner reduction could create coefficient or intermediate-expression growth that consumes more compute and checking effort than direct proofs, while only a small eligible subset benefits from reuse.","problem_falsifier":"A case audit falsifies the proposed problem if repeated reconstruction of a shared ideal-membership context is not a material part of the recorded work, if few cases actually share an invariant ring and ideal, or if downstream delay is instead caused by unresolved theorem statements, missing mathematical ideas, or reviewer capacity.","intervention_falsifier":"The intervention is falsified for the tested scope if it produces any undetected wrong classification, cannot emit independently checkable certificates, fails to reduce total setup-plus-reduction-plus-checking burden against the fixed baseline, requires regeneration nearly as often as cases arrive, or shifts the dominant delay to certificate checking or downstream theorem review.","risks":["A stale basis version may silently answer a different mathematical question.","The eligibility contract may exclude difficult cases and create a misleading comparison dominated by easy inputs.","Coefficient growth or memory exhaustion may saturate the lane despite low case counts.","A bug shared by the reducer and its checker may create correlated false assurance.","Researchers may defer to canonical normal forms and overlook shorter or more explanatory proofs.","The steward may become a gatekeeper for access or version changes.","A faster stream of determinations may overload downstream proof integration and review.","Logs or dashboards may report utilization while omitting certificate quality or exception burden."]},"next_evidence_step":"Select twelve previously adjudicated polynomial identities spanning zero and nonzero normal forms, ordinary and high-expression-growth cases, all under one unchanged ring, ideal, and order. Conceal adjudications; pre-register eligibility, resource ceilings, certificate requirements, and immediate-stop conditions. Have one operator process all twelve through the versioned lane and a separate mathematician check certificates, while another operator applies the documented per-case baseline without the shared basis. Compare complete work time, compute use, correct classifications, certificate completeness, exception handling, queue behavior, and facilitator reuse. Include one deliberately mismatched-context input to test rejection and one simulated version withdrawal to test regeneration and rollback. This probe supplies bounded local evidence only and authorizes no production use.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Not assessed; exactly one proposal is in scope.","revision_record":{"parent_version":null,"progress_targets_addressed":["Initial schema-complete reverse-innovation candidate","Concrete mathematics-domain transformation with an explicit reusable facilitator cycle","Operational authority, safeguards, falsifiers, rivals, and bounded first evidence"],"conceptual_changes":["Initial version; mapped catalytic reuse to repeated certified ideal-membership determinations under one fixed algebraic context."],"operational_changes":["Initial version; specified intake, certificate checking, capacity monitoring, exception routing, regeneration, and withdrawal."],"evidence_changes":["Initial version; defined a twelve-case concealed shadow probe with a facilitator-off baseline and integrity tests."],"claim_changes":["Initial version; limited the claim to an untested contrastive hypothesis and marked prior art unsearched."]}}