{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp11_mechanism_context_external20_20260804","research_id":"eoa_inverse_innovation_exp11_external_scrutiny_20260804","cell_id":"computability_boundary_mapping__mathematics","opaque_id":"computability_boundary_mapping__mathematics__B","search_lanes":{"direct_problem":{"queries":["first-order logic validity undecidable recursively enumerable theoremhood proof enumeration Church theorem official source","theorem proving service timeout displayed no proof found unknown conjecture","automated theorem prover timeout unknown not false SZS ontology status official"],"source_ids":["SRC1","SRC2","SRC3","SRC8"],"no_result_note":"The theoretical boundary is strongly documented, but no retained source documents a specific mathematical-reasoning service that promises universal terminating binary theoremhood or converts timeout to an authoritative NO."},"closest_prior_art":{"queries":["site:tptp.org SZS ontology Timeout GaveUp Theorem CounterSatisfiable status","SMT-LIB standard unknown response reason-unknown timeout","site:microsoft.github.io/z3guide reason_unknown check returns unknown timeout official","site:cvc5.github.io docs unknown timeout incomplete check-sat result"],"source_ids":["SRC3","SRC4","SRC5","SRC6"],"no_result_note":null},"historical_terminology":{"queries":["Alonzo Church 1936 note on Entscheidungsproblem PDF Journal Symbolic Logic","Church An Unsolvable Problem of Elementary Number Theory 1936 PDF","Entscheidungsproblem semi-Entscheidungsverfahren Satzbeweis Nichtentscheidbarkeit Prädikatenlogik deutsch"],"source_ids":["SRC1","SRC2","SRC8"],"no_result_note":null},"products_practices_standards":{"queries":["site:tptp.org TPTP format proof output SZS official","site:smt-lib.org language standard unknown reason-unknown SMT-LIB 2.7","site:z3prover.github.io api reason_unknown unknown timeout Solver check","site:cvc5.github.io docs unknown timeout incomplete check-sat result"],"source_ids":["SRC3","SRC4","SRC5","SRC6"],"no_result_note":null},"non_english_regional":{"queries":["Entscheidungsproblem semi-Entscheidungsverfahren Satzbeweis Nichtentscheidbarkeit Prädikatenlogik deutsch","décidabilité logique du premier ordre semi-décidable démonstration automatique théorèmes inconnu délai"],"source_ids":["SRC8"],"no_result_note":"German university material was retained because it directly states the decision/semi-decision distinction, reduction direction, and nontermination behavior; French results were reviewed in search but were less authoritative or duplicative."},"composition_subproblems":{"queries":["Trakhtenbrot theorem Coq formalization decidability monadic first order primary paper","first-order theoremhood decidable fragments finite models proof enumeration","SMT-LIB logic declaration unknown proof production resource limit","automated theorem prover status ontology timeout proof model counterexample"],"source_ids":["SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"],"no_result_note":null}},"sources":[{"source_id":"SRC1","title":"A Note on the Entscheidungsproblem","url":"https://people.csail.mit.edu/brooks/idocs/church_ent.pdf","publisher":"Association for Symbolic Logic, Journal of Symbolic Logic","date_or_year":"1936","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Church's original paper states that the general Entscheidungsproblem is unsolvable for sufficiently expressive symbolic logic.","The historical result predates modern theorem-prover products and directly defeats a universal exact terminating procedure over unrestricted first-order inputs."]},{"source_id":"SRC2","title":"Alonzo Church","url":"https://plato.sydney.edu.au/entries/church/","publisher":"Stanford Encyclopedia of Philosophy","date_or_year":"2021; substantive revision 2022","source_type":"SECONDARY_RESEARCH","language":"English","claims_supported":["Identifies Church's theorem as the negative solution to the decision problem for first-order logic.","Confirms the historical terminology and distinguishes the computability result from later product practices."]},{"source_id":"SRC3","title":"SZS Ontology","url":"https://tptp.org/UserDocs/SZSOntology/","publisher":"TPTP World","date_or_year":"Current documentation, accessed 2026-08-04","source_type":"OFFICIAL_STANDARD","language":"English","claims_supported":["Defines separate theorem, counter-satisfiable, timeout, gave-up, and error statuses for automated theorem-proving results.","Requires a status to be reported separately from its justifying proof or model data.","Demonstrates an established guarantee/status-labeling practice that prevents timeout from being represented as a refutation."]},{"source_id":"SRC4","title":"The SMT-LIB Standard, Version 2.7","url":"https://smt-lib.org/papers/smt-lib-reference-v2.7-r2025-07-07.pdf","publisher":"SMT-LIB Initiative","date_or_year":"2025-07-07","source_type":"OFFICIAL_STANDARD","language":"English","claims_supported":["Standardizes sat, unsat, and unknown as distinct responses.","States that unknown means the search was inconclusive because of resource limits, incompleteness, or other reasons.","Provides logic declarations, resource limits, proof/model options, and an optional reason-unknown response including memory exhaustion or incompleteness."]},{"source_id":"SRC5","title":"Z3 Solver Class Reference","url":"https://z3prover.github.io/api/html/classz3py_1_1_solver.html","publisher":"Z3 Project / Microsoft Research","date_or_year":"Current API documentation, accessed 2026-08-04","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["Z3 exposes an explicit unknown result and a reason_unknown method.","Z3 can return an incompleteness reason for an unresolved arithmetic query.","Z3 exposes proof retrieval when proof construction is enabled, showing practical separation of result labels and certificates."]},{"source_id":"SRC6","title":"cvc5 Output Tags","url":"https://cvc5.github.io/docs-ci/docs-main/output-tags.html","publisher":"cvc5 Project","date_or_year":"Current documentation, accessed 2026-08-04","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["cvc5 reports unknown for incomplete quantified reasoning and can print the incompleteness reason.","A timeout-core example returns unknown rather than unsat.","The product can expose trusted proof steps, supporting certificate-aware result modes."]},{"source_id":"SRC7","title":"Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory","url":"https://arxiv.org/abs/2004.07390","publisher":"arXiv; Dominik Kirst and Dominique Larchey-Wendling","date_or_year":"2020","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Provides a mechanized many-one reduction chain proving finite first-order satisfiability undecidable when the signature has a binary relation.","Proves decidability for specified monadic signatures and enumerability more generally.","Shows that 'finite-model fragment' alone is not a sufficient decidability boundary; a fixed bound or genuinely decidable syntactic fragment is required."]},{"source_id":"SRC8","title":"3.4 Unentscheidbarkeit / 3.5 Herbrand-Theorie","url":"https://www.informatik.uni-leipzig.de/~brewka/papers/PL2.pdf","publisher":"Leipzig University, Institute of Computer Science","date_or_year":"Undated course material, accessed 2026-08-04","source_type":"OTHER","language":"German","claims_supported":["Defines a decision procedure as terminating with a correct YES/NO answer on every input and a semi-decision procedure as guaranteeing termination only on YES instances.","States Church's theorem and the undecidability of first-order validity, satisfiability, and consequence.","Gives a Herbrand/Gilmore enumeration procedure that terminates correctly for the recognized side and may run forever otherwise, directly supporting one-sided recognition."]}],"problem_evidence":{"status":"PARTLY_SUPPORTED","finding":"The mathematical core is supported: unrestricted first-order validity and consequence do not admit a total correct binary decision procedure, while the positive theoremhood side is semi-decidable. However, the operational diagnosis remains hypothetical. No retained source identifies a real service whose enforced contract promises universal terminating YES/NO or whose interface converts timeout into authoritative NO. Modern ATP and SMT standards instead explicitly distinguish timeout, gave-up, and unknown from negative results.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC5","SRC6","SRC8"],"uncertainty":"The search cannot determine whether an unnamed target service has the alleged requirement or interface behavior. A requirements and interface audit of a named service is necessary."},"adopter_evidence":{"status":"PARTLY_SUPPORTED","finding":"Relevant authorizers and implementers are identifiable by role and precedent: theorem-service owners, TPTP/SZS and SMT-LIB maintainers, and solver teams such as Z3 and cvc5 already control closely analogous status, scope, and certificate policies. No specific organization is identified as the intended adopter of this proposal.","source_ids":["SRC3","SRC4","SRC5","SRC6"],"uncertainty":"The public evidence establishes capable adopter classes and existing implementations, not commitment by a named deployment owner."},"implementation_evidence":{"status":"SUPPORTED","finding":"The principal fallback mechanisms are technically demonstrated: explicit UNKNOWN/Timeout/GaveUp states, reason codes, declared logics, resource bounds, proofs or models, and decidable-fragment routing all have standards, product, or formal-research precedents. The proposal must replace the vague phrase 'finite-model fragments' with fixed finite bounds or a proven decidable fragment, because unrestricted finite first-order satisfiability remains undecidable for common signatures.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"],"uncertainty":"The evidence supports mechanism feasibility, not the proposed 60-case sample size, an 80% utility threshold, independent-checker compatibility for every calculus, or usability in an unnamed service."},"prior_art":{"disposition":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"TPTP SZS success and no-success ontologies","source_ids":["SRC3"],"same_problem":true,"same_causal_lever":true,"overlap":"For automated first-order theorem proving, SZS distinguishes Theorem and CounterSatisfiable from Timeout, GaveUp, and Error, and associates successful statuses with proof or model data.","remaining_difference":"It is an output/status ontology rather than the proposal's full organizational package of versioned decision records, independent logic review, recheck triggers, and a local shadow-mode evaluation."},{"name":"SMT-LIB 2.7 response and logic contracts","source_ids":["SRC4"],"same_problem":false,"same_causal_lever":true,"overlap":"SMT-LIB standardizes declared logics, sat/unsat/unknown outputs, reasons for unknown, resource limits, and optional proofs or models. This is substantially the proposed scope-and-guarantee labeling pattern.","remaining_difference":"Its primary problem is satisfiability modulo theories rather than arbitrary theoremhood from any finite first-order axiom presentation, and reason-unknown support is optional."},{"name":"Z3 and cvc5 production solver interfaces","source_ids":["SRC5","SRC6"],"same_problem":false,"same_causal_lever":true,"overlap":"Both products operationalize UNKNOWN and diagnostic reasons; cvc5 explicitly reports timeout-related cases as unknown, and both expose proof-related facilities.","remaining_difference":"They do not by themselves establish the proposed governance record, independent mathematical-logic authorization, or utility of routing in a particular mathematical service."},{"name":"Mechanized computability-boundary and fragment classification","source_ids":["SRC7","SRC8"],"same_problem":true,"same_causal_lever":true,"overlap":"These sources use explicit reductions, enumeration/semi-decision procedures, and proven decidable subclasses to map the boundary before selecting algorithms.","remaining_difference":"They are research and teaching artifacts, not an end-user theorem-status router or operational decision-record system."}],"contrastive_claim_remaining":"For a specifically audited theorem service that currently coerces resource exhaustion into NO, a machine-enforced router combining certified fragment deciders, witness-bearing positive proof search, explicit UNKNOWN reasons, and versioned scope records will eliminate unsound binary labels while providing greater decision utility than merely increasing prover timeouts. This is a local integration-and-governance claim, not a new computability result or a new UNKNOWN convention.","contrastive_claim_falsifier":"The claim is falsified if the target service audit finds no universal binary promise or timeout-to-NO coercion; or if a pre-registered shadow test produces any unsound YES/NO, scope escape, certificate-check failure, or no material utility improvement over the service's already status-aware baseline.","confidence":"HIGH","search_limitations":"This was a bounded public-web search across English, German, and French terminology. It retained exactly eight direct sources and did not inspect patents, proprietary requirements, internal service behavior, unpublished implementations, or every theorem prover. It cannot establish world novelty, patentability, freedom to operate, market size, or realized impact."},"researchability_gates":{"externally_supported_problem":{"status":"INDETERMINATE","rationale":"Undecidability and the danger of treating inconclusive search as a negative result are externally established, but no evidence confirms that an actual target service makes the alleged universal promise or timeout-as-NO error.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC8"]},"identifiable_adopter_or_authorizer":{"status":"PASS","rationale":"Service owners and logic/solver maintainers are identifiable authorizer classes, with TPTP, SMT-LIB, Z3, and cvc5 providing concrete precedents.","source_ids":["SRC3","SRC4","SRC5","SRC6"]},"distinct_testable_incremental_claim":{"status":"PASS","rationale":"After narrowing away from the already-standard UNKNOWN convention, the remaining context-specific claim—machine-enforced routing plus versioned guarantees and independent certificate checks improves label soundness and utility in a named overclaiming service—is falsifiable by audit and shadow comparison.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC7"]},"bounded_next_evidence_step":{"status":"PASS","rationale":"A requirements/interface audit followed, only if the diagnosis is confirmed, by a shadow-mode corpus spanning certified fragments, proof-bearing positives, bounded countermodels, and forced UNKNOWN cases is bounded and measurable.","source_ids":["SRC3","SRC4","SRC5","SRC6"]},"no_unresolved_safety_or_authority_stop":{"status":"PASS","rationale":"Shadow mode is reversible and does not require production relabeling. Independent review, certificate checking, prohibition of unqualified NO outside certified fragments, and rollback on unsound labels adequately bound authority risk for the next step.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC7"]},"adequate_search_evidence":{"status":"PASS","rationale":"All six required lanes were searched adversarially, including historical Entscheidungsproblem terminology, standards and products, German and French terminology, and component combinations. Eight retained sources include multiple independent publishers and six primary, official, standards, or first-party sources.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"]}},"strict_success":false,"screen_survival":false,"remaining_research_value":"MODERATE","recommended_next_step":"Name the intended service and first conduct a read-only requirements, accepted-language, computation-model, and interface audit. Stop if it already enforces a decidable fragment or explicitly means 'no proof found.' If the overclaim is confirmed, pre-register and run the proposed shadow test, but define finite fallbacks as fixed-cardinality bounds or formally proven decidable fragments rather than unrestricted finite-model reasoning.","world_novelty_boundary":"The search supports a classical computability boundary and finds the central status-labeling intervention already established in ATP/SMT standards and products. It leaves only a potentially useful context-specific governance and integration claim. No conclusion is made about world novelty, patentability, freedom to operate, market size, or realized impact."}