{"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_representation_transport_gateway","proposal_index":3,"version":0,"title":"Verified Representation-Transport Gateway for Linear-Algebra Theorems","problem":"A formal linear-algebra library maintains results expressed both as finite-dimensional linear maps and as matrices relative to ordered bases. When a proved result is needed in the other representation, contributors repeatedly reconstruct coordinate equivalences, align source and target bases, rewrite operations, discharge dimension conditions, and prove that the transported statement matches the original. The mathematical equivalence is available in principle, but the recurring transport burden delays usable counterpart theorems and encourages separately maintained proofs that can diverge.","actors":["Authors of formal linear-algebra theorems","Users needing matrix-form or linear-map-form counterparts","Maintainers of the formal mathematics library","Reviewers responsible for statement fidelity and proof-kernel acceptance"],"observable_state":"The library's issue and dependency records contain requests for counterparts of already-proved theorems in the alternate representation. Case histories show repeated basis setup, dimension alignment, coordinate rewrites, inverse-law proofs, manual repair, and rejection of attempted ports whose transported statement does not preserve the original theorem's exact assumptions or conclusion.","consequence":"Already-established mathematical results remain inconvenient or unavailable in one representation, contributors duplicate transport work, and separately proved counterparts can accumulate mismatched hypotheses or conclusions.","affected_objective":"Produce kernel-checked counterpart theorems across a declared matrix–linear-map equivalence while preserving mathematical meaning, explicit basis dependence, assumptions, and ordinary library-review standards.","intervention":"Build a versioned representation-transport gateway around independently verified equivalences between finite-dimensional linear maps and matrices relative to explicit ordered bases. Its contract accepts a proved source theorem, source and target type information, basis objects, dimension evidence, and a statement composed from supported operations. The gateway constructs the corresponding target statement, transports the proof through verified inverse and operation-preservation lemmas, and submits the result to the ordinary proof kernel. It releases the equivalence bundle unchanged after each case. Eligibility rules admit statements whose operations have validated transport laws and reroute basis-sensitive, unsupported, ambiguous, or theorem-strengthening cases to manual proof. Capacity monitoring distinguishes queued transports from type-inference failures, proof-term growth, or equivalence-version drift. When definitions or APIs change, the gateway is withdrawn until its inverse laws, preservation lemmas, and regression corpus are re-established.","structural_mapping":[{"archetype_element":"Target transformation specification","domain_realization":"Transform an already proved theorem about a finite-dimensional linear map into a kernel-checked matrix-form counterpart, or the reverse, under explicit bases and a fixed equivalence version, without changing the theorem's logical strength."},{"archetype_element":"Activation barrier","domain_realization":"Each case otherwise reconstructs coordinate maps, basis compatibility, dimension evidence, operation rewrites, inverse laws, and a proof that the target statement faithfully represents the source."},{"archetype_element":"Reusable facilitator","domain_realization":"A verified bundle of matrix–linear-map equivalences, inverse proofs, operation-preservation lemmas, and proof-transport combinators acts repeatedly on theorem cases without becoming part of or being consumed by any transported theorem."},{"archetype_element":"Facilitator–substrate interface","domain_realization":"The gateway requires a checked source theorem, declared coefficient field, finite-dimensional source and target spaces, explicit ordered bases, dimension evidence, transport direction, and a statement within a versioned supported grammar."},{"archetype_element":"Selectivity rule","domain_realization":"Automatic transport is limited to validated constructions such as equality, zero, addition, scalar multiplication, composition, identity, and explicitly supported structural predicates. Claims about particular entries, sparsity patterns, coordinate order, conditioning, or implicit basis choices are rejected unless separately validated."},{"archetype_element":"Permitted pathway boundary","domain_realization":"The gateway may transport an existing proof across an established equivalence; it may not invent a missing proof, strengthen a conclusion, suppress basis dependence, alter definitions, or substitute successful generation for kernel checking and human review."},{"archetype_element":"Turnover capacity","domain_realization":"Capacity is measured as accepted faithful transports per gateway version and compute interval, together with queue time, elaboration time, kernel-checking time, proof-term size, manual repair, rejection rate, and facilitator maintenance."},{"archetype_element":"Cofactor or complement map","domain_realization":"Required complements include explicit basis objects, compatible dimensions, stable source theorem declarations, operation-preservation lemmas, the proof kernel, and reviewer capacity. These enable the gateway but are not confused with the reusable equivalence facilitator."},{"archetype_element":"Saturation and interference monitoring","domain_realization":"Queue depth and concurrent elaborations indicate saturation, while increasing typeclass failures, proof-term growth, ambiguous basis inference, kernel rejection, or API mismatches indicate interference or degradation."},{"archetype_element":"Facilitator regeneration cycle","domain_realization":"After changes to representations, basis APIs, supported operations, or foundational definitions, maintainers withdraw the affected gateway version, re-prove inverse and preservation laws, replay the regression corpus, obtain independent review, and publish a new immutable version."},{"archetype_element":"Byproduct and side-pathway guardrail","domain_realization":"Statement-difference inspection, explicit basis display, kernel checking, and ordinary review detect altered quantifiers, lost assumptions, accidental basis dependence, unsupported simplification, unreadable output, and theorem duplication under misleading names."},{"archetype_element":"Equilibrium neutrality","domain_realization":"The gateway transports proof across an already established equivalence. It does not make an unproved proposition true, expand the class of valid models, or change the library's acceptance standard."},{"archetype_element":"Accountable catalyst steward","domain_realization":"A designated library maintainer owns the supported grammar, equivalence versions, capacity limits, regression corpus, incident response, withdrawal, and refresh, while ordinary maintainers retain merge and theorem-acceptance authority."}],"mechanism_mapping":[{"mechanism_slug":"interface_contract_design","role":"Declares the admissible spaces, bases, operations, proof inputs, target guarantees, protected invariants, exception route, and equivalence version for each transport.","counterfactual_removal":"Without the contract, basis and dimension assumptions can be inferred differently across cases, allowing the gateway to generate a formally related but unintended statement."},{"mechanism_slug":"prevalidated_transformation_template","role":"Encodes the verified equivalence, its inverse laws, and operation-preservation proofs so each theorem instantiates variable positions rather than rebuilding the transport argument.","counterfactual_removal":"Without a prevalidated template, every theorem again requires a bespoke equivalence proof, eliminating the catalytic reuse and permitting assumptions to drift."},{"mechanism_slug":"workflow_automation_or_macro","role":"Constructs the target statement and proof term, validates declared postconditions, invokes the kernel, and logs inputs, versions, outputs, and exceptions for every run.","counterfactual_removal":"Without deterministic execution and logging, maintainers must apply transport lemmas manually, reintroducing repetitive proof assembly and making failures difficult to reproduce."},{"mechanism_slug":"fast_track_with_eligibility_rules","role":"Routes statements built from validated operations through automatic transport while returning unsupported or basis-sensitive statements to ordinary theorem development.","counterfactual_removal":"Without eligibility and rerouting, throughput pressure could extend the gateway beyond its proved preservation envelope and produce misleading counterparts."},{"mechanism_slug":"catalyst_cofactor_system","role":"Maps and checks the explicit bases, dimension evidence, theorem declarations, preservation lemmas, kernel availability, and reviewer capacity needed for the gateway to function.","counterfactual_removal":"Without cofactor checks, missing or incompatible bases and supporting lemmas could be misdiagnosed as facilitator failure, or hidden manual work could be credited to the gateway."},{"mechanism_slug":"inhibitor_and_poison_screen","role":"Screens for ambiguous basis inference, incompatible dimensions, unsupported operations, stale theorem identifiers, reducibility assumptions, and equivalence-version mismatches before proof transport begins.","counterfactual_removal":"Without an upstream screen, incompatible cases occupy elaboration capacity and may fail late with opaque errors or accidentally use unintended instances."},{"mechanism_slug":"active_site_capacity_dashboard","role":"Displays queued and active transports, elaboration and kernel-checking time, proof-term growth, rejection classes, version health, and reviewer backlog.","counterfactual_removal":"Without joint queue and degradation visibility, maintainers may add more jobs when failures actually arise from drift, ambiguity, or proof-term explosion."},{"mechanism_slug":"turnover_and_selectivity_assay","role":"Compares accepted faithful counterparts per facilitator version against manual theorem transport while counting rejected cases, statement mismatches, repair work, resource use, and degradation across repeated cycles.","counterfactual_removal":"Without the assay, selection of simple theorems, added computation, or omitted review could be misattributed to reusable transport, and off-target statements would be hidden."},{"mechanism_slug":"catalyst_regeneration_protocol","role":"Defines withdrawal, inverse-law revalidation, preservation-lemma repair, regression replay, independent review, replacement, and retirement after representation or API drift.","counterfactual_removal":"Without regeneration and retirement, a stale equivalence bundle could remain nominally operational and repeatedly generate broken or semantically outdated counterpart theorems."},{"mechanism_slug":"small_safe_to_fail_probe","role":"Tests the gateway on a bounded concealed set of archived theorem pairs, including ineligible cases and a simulated version change, without merging generated results into the live library.","counterfactual_removal":"Without a bounded counterfactual probe, maintainers could not attribute reduced transport work to the reusable equivalence or observe selectivity and regeneration failures before live use."}],"causal_chain":["A contributor submits an already-proved theorem, requested transport direction, explicit bases, dimension evidence, and target context through the versioned interface.","The preflight screen verifies that the source theorem and complements exist and that basis, dimension, and equivalence versions are compatible.","Eligibility analysis confirms that the statement uses only operations with validated transport laws; unsupported cases leave for manual proof.","The gateway instantiates the reusable equivalence and preservation template for the submitted theorem.","A deterministic transport procedure constructs the target statement and proof term while retaining explicit basis and assumption information.","The ordinary proof kernel checks the generated proof independently of the transport procedure.","A reviewer compares the source and target statements for intended meaning, readability, naming, basis dependence, and duplication before any merge decision.","The completed theorem is released while the equivalence bundle remains available for the next eligible case.","Queue, resource, rejection, repair, and statement-fidelity signals govern intake and distinguish saturation from facilitator degradation.","A representation or API change withdraws the affected version and triggers revalidation and regression replay before reuse resumes."],"baseline":"A contributor manually selects bases, expands the applicable matrix–linear-map equivalence, rewrites each operation, resolves type and dimension obligations, constructs the target theorem and proof, and submits it to normal kernel checking and review. Record human setup and repair time, elapsed time, elaboration and checking resources, proof-term size, statement revisions, rejected attempts, and reviewer findings.","nearest_rivals":["Maintaining separate manually proved matrix and linear-map theorem libraries, which can yield readable specialized results but duplicates proof and maintenance work.","Providing a collection of equivalence and rewrite lemmas for contributors to invoke manually, which shares ingredients but lacks a closed reusable transport cycle, eligibility gate, generated-statement contract, and regeneration regime.","Exposing only coordinate-free linear-map theorems and requiring users to instantiate matrix consequences locally, which avoids maintaining counterparts but leaves the recurring transport burden with each user.","Using a general-purpose simplifier, proof-search tactic, or elaborator automation, which may solve individual transports but does not define the permitted representation pathway or guarantee statement-level selectivity.","Redesigning the public theorem API around a single representation, which structurally removes some duplicate interfaces rather than repeatedly transporting results between representations that remain intentionally supported."],"remaining_contrastive_claim":"The untested contrastive hypothesis is that a verified representation equivalence can serve as a reusable facilitator across eligible theorem transports, removing repeated coordinate and proof-assembly work while keeping bases, logical strength, kernel checking, and review invariant. Evaluation must account for manual repair, proof-term growth, rejected cases, cofactor preparation, and maintenance so the result is distinguishable from ordinary tactic automation, extra computation, or a decision to support only one representation.","authority_safety":{"decision_authority":"The formal library's maintainers may authorize an experimental non-exported module, approve its equivalence contract, and appoint its steward. Existing code owners and reviewers retain exclusive authority to merge, name, deprecate, or accept transported theorems.","authorized_first_step":"Implement one isolated gateway version for a fixed field and declared class of finite-dimensional spaces, then replay a bounded concealed set of archived theorem pairs without changing the live library.","excluded_actions":["Merging generated theorems automatically","Treating kernel acceptance alone as confirmation that the generated statement is the intended counterpart","Inferring or hiding bases when more than one compatible choice exists","Transporting entry-specific, sparsity, ordering, conditioning, or other unsupported basis-sensitive claims","Adding assumptions or weakening conclusions to make transport succeed","Changing canonical definitions, public APIs, or source theorems during the bounded probe","Bypassing the ordinary proof kernel or human review","Expanding the supported grammar without new preservation proofs and regression cases"],"halt_rollback":"Stop all jobs and withdraw the gateway version upon a kernel rejection attributed to the facilitator, an unintended target statement, an ambiguous or incorrect basis choice, a lost or added assumption, a broken inverse law, an unbounded proof-term expansion, or a stale-version execution. Discard unmerged outputs, retain diagnostic logs, restore cases to manual status, and require revalidation plus independent review before restart."},"negative_tests":{"strongest_counterevidence":"The coordinate-transport steps may be a small and readable part of each proof, while generic transport produces large, opaque proof terms and unnatural theorem statements. Eligible cases may be too heterogeneous for one supported grammar, making manual specialized proofs or a single-representation API easier to maintain.","problem_falsifier":"The proposed problem is falsified if an audit finds few requested counterpart theorems, little repeated coordinate setup, or that failed ports are caused primarily by missing mathematical arguments rather than representation transport. It is also falsified if one representation can meet the project's legitimate uses without recurring local transport.","intervention_falsifier":"The intervention is falsified for the tested scope if generated statements fail blinded fidelity review, proof terms fail kernel checking, total setup-plus-repair-plus-review burden does not improve against manual transport, most representative cases are ineligible, regeneration is required nearly as often as transports occur, or maintenance and downstream review become the dominant burden.","risks":["Implicit basis selection may produce a valid theorem different from the requested counterpart.","A supported operation may preserve syntax while obscuring a mathematically important side condition.","Generated proof terms or statements may become too large or opaque to maintain.","Shared bugs in transport-side normalization and statement comparison may create correlated false assurance.","Representation or library API drift may invalidate preservation lemmas without obvious surface changes.","Eligibility may be broadened under pressure, admitting basis-sensitive claims without proof of preservation.","The gateway may create redundant counterparts that worsen theorem discovery and naming consistency.","Hidden labor in constructing bases or missing preservation lemmas may be miscredited to the facilitator.","Faster theorem generation may overload reviewers and library maintenance.","Dependence on one equivalence bundle may create a common-mode failure across generated theorems."]},"next_evidence_step":"Select ten archived, accepted matrix–linear-map theorem pairs covering the initially supported operations, plus two archived cases whose conclusions depend on particular entries or basis order and should be rejected. Conceal the target proofs and names. Have one contributor use the gateway and another perform documented manual transport under equal access to the source theorem, bases, and library. Precommit kernel acceptance, exact assumption preservation, explicit-basis requirements, statement-fidelity criteria, resource ceilings, and immediate-stop conditions. Blinded reviewers compare generated and archived target statements, record readability and repair needs, and inspect all rejections. Measure complete human work, elapsed processing, elaboration and checking resources, proof-term size, exceptions, and reviewer findings. After half the cases, introduce a controlled change to a non-live representation API and test withdrawal, regeneration, regression replay, and rollback. This produces bounded local evidence and authorizes no live merge or expansion.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Proposal 1 transforms candidate polynomial identities into exact ideal-membership or non-membership determinations through a verified Gröbner basis and normal-form certificates. This proposal begins with an already-proved theorem and transports that proof across a matrix–linear-map equivalence; it performs no ideal reduction and makes no new truth classification. Proposal 2 transforms informally stated cross-subfield obstructions into lemma handoff packets through a rotating human clinic, trust relationships, interpretation, and brokerage. This proposal operates only after definitions and equivalence conditions are formally fixed; its facilitator is a deterministic verified equivalence bundle, not a human coordination service, and its output is a kernel-checked counterpart theorem rather than a work packet. Its causal path is equivalence instantiation, operation-preserving proof transport, kernel validation, release, and artifact regeneration. It can be adopted by a single formal library without either the polynomial-reduction lane or the cross-subfield clinic.","revision_record":{"parent_version":null,"progress_targets_addressed":["Initial schema-complete proposal at index 3","A problem, intervention, and causal path independent of sealed proposals 1 and 2","Explicit facilitator reuse, selectivity, turnover, cofactor checks, regeneration, authority, falsifiers, rivals, and bounded evidence"],"conceptual_changes":["Initial version; defined repeated proof transport across a fixed mathematical representation equivalence as the target transformation."],"operational_changes":["Initial version; specified an explicit-basis interface, supported operation grammar, preflight screen, proof-kernel validation, statement review, capacity monitoring, withdrawal, and regeneration."],"evidence_changes":["Initial version; defined a twelve-case archival probe with manual comparison, deliberately ineligible cases, blinded fidelity review, and a controlled regeneration test."],"claim_changes":["Initial version; limited the claim to an untested catalytic-reuse hypothesis, distinguished it from prior sealed proposals, made no effect-size claim, and marked prior art unsearched."]}}