{"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 recursively enumerable undecidable finite axioms theoremhood source","automated theorem prover proof search timeout not false first order logic"],"source_ids":["SRC1","SRC2","SRC3"],"no_result_note":"The theoretical impossibility is well supported, but no retained source establishes the hypothesized target service's universal binary requirement or timeout-as-NO behavior."},"closest_prior_art":{"queries":["SZS ontology theorem prover timeout unknown status official","automated theorem proving portfolio decidable fragments route first order proof search model finder unknown","Isabelle Sledgehammer Nitpick unknown timeout official manual counterexample theorem prover"],"source_ids":["SRC3","SRC4","SRC5","SRC6","SRC8"],"no_result_note":null},"historical_terminology":{"queries":["Church Entscheidungsproblem original paper first order logic undecidable history","Alonzo Church A Note on the Entscheidungsproblem 1936 DOI full text"],"source_ids":["SRC1","SRC2"],"no_result_note":null},"products_practices_standards":{"queries":["site:tptp.org SZS ontology status Timeout GaveUp Theorem CounterSatisfiable","site:smt-lib.org SMT-LIB 2.6 unknown response reason-unknown official","Vampire theorem prover output unknown time limit official documentation SZS","site:isabelle.in.tum.de sledgehammer timeout unknown proof reconstruction official"],"source_ids":["SRC3","SRC4","SRC5","SRC6","SRC8"],"no_result_note":null},"non_english_regional":{"queries":["Entscheidungsproblem Prädikatenlogik unentscheidbar semi-entscheidbar Satzbeweiser unbekannt Zeitüberschreitung","problème de décision logique du premier ordre indécidable semi-décidable démonstration automatique inconnu","entscheidbar Fragment Prädikatenlogik automatische Theorembeweiser Timeout unbekannt"],"source_ids":["SRC1","SRC7"],"no_result_note":null},"composition_subproblems":{"queries":["first-order logic validity recursively enumerable proof enumeration decidable fragments finite model finder","automated theorem prover portfolio proof certificate finite countermodel timeout status","Nitpick counterexample generator Isabelle finite model resource limit","SMT-LIB set-logic unknown reason-unknown proof response"],"source_ids":["SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"],"no_result_note":null}},"sources":[{"source_id":"SRC1","title":"A note on the Entscheidungsproblem","url":"https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/note-on-the-entscheidungsproblem/9461BEAD94BB16D56EC78933D7D67DEF","publisher":"Association for Symbolic Logic / Cambridge University Press","date_or_year":"1936","source_type":"PRIMARY_RESEARCH","language":"English, with historical German terminology","claims_supported":["The general Entscheidungsproblem for first-order logic is unsolvable.","The historical problem asks for an effective method determining whether an arbitrary expression is provable.","Effective enumeration and explicit formal-system assumptions are central to the boundary result."]},{"source_id":"SRC2","title":"The Undecidability of First-Order Logic","url":"https://www.cambridge.org/core/books/abs/computability-and-logic/undecidability-of-firstorder-logic/5AE8F37523C35D4E04EC22BDD38CF145","publisher":"Cambridge University Press","date_or_year":"2002","source_type":"SECONDARY_RESEARCH","language":"English","claims_supported":["A Turing-machine/input pair can be effectively mapped to a finite sentence set and conclusion preserving halting versus implication.","A total procedure deciding whether a finite set of first-order sentences implies another sentence would decide the halting problem.","The impossibility applies directly to the proposal's finite-axiom theoremhood class."]},{"source_id":"SRC3","title":"SZS Ontology","url":"https://tptp.org/UserDocs/SZSOntology/","publisher":"TPTP","date_or_year":"Undated; accessed 2026-08-04","source_type":"OFFICIAL_STANDARD","language":"English","claims_supported":["Automated-theorem-proving practice distinguishes semantic successes such as Theorem and CounterSatisfiable from no-success states such as Timeout and GaveUp.","Proofs, finite models, infinite models, and saturations are separately labeled justification forms.","Unknown means that a success value has not been established, so timeout is not a negative theoremhood judgment.","The standard also distinguishes verified, unverified, and failed-verification outputs."]},{"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","source_type":"OFFICIAL_STANDARD","language":"English","claims_supported":["The standard requires an explicit logic declaration and defines sat, unsat, and unknown responses.","A solver can report why a result is unknown, including resource exhaustion or incompleteness for the submitted formula class.","Resource limits may change an unknown result into sat or unsat without making unknown equivalent to either answer.","The language supplies proof-response and model-response contracts."]},{"source_id":"SRC5","title":"The Vampire Diary","url":"https://arxiv.org/abs/2506.03030","publisher":"arXiv","date_or_year":"2025","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Vampire accepts TPTP and SMT-LIB inputs, attempts refutation, and emits a step-by-step proof when successful.","It can establish satisfiability through finite-model finding or saturation in applicable cases.","Its practical operation combines multiple reasoning engines, adaptive schedules, proof search, counterexample capabilities, and explicit time limits.","The system has identifiable maintainers and deployments in mathematics, proof assistants, and software verification."]},{"source_id":"SRC6","title":"Sledgehammer: Let Automatic Theorem Provers Write Your Isabelle Scripts","url":"https://isabelle.in.tum.de/website-Isabelle2009/sledgehammer.html","publisher":"Isabelle Project","date_or_year":"2009","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["Isabelle Sledgehammer routes a proof goal to multiple automated provers in parallel and keeps the first successful result.","Successful external results are converted into Isabelle proof commands for replay by Metis.","The product exposes a time limit and does not describe failure to find a proof as proof of falsity.","The Isabelle project and prover maintainers are identifiable adopters and authorizers for routing policy."]},{"source_id":"SRC7","title":"Chapter 4: Propositional Inference – Computational Semantics","url":"https://compsem.phil.hhu.de/kapitel/chapter-4-propositional-inference/","publisher":"Heinrich Heine University Düsseldorf","date_or_year":"Undated; accessed 2026-08-04","source_type":"OFFICIAL_GUIDANCE","language":"German and English","claims_supported":["The German-language teaching material states that general first-order satisfiability and validity-related checking are undecidable.","It contrasts general predicate logic with a decidable quantifier-free fragment.","It explains proof-theoretic and model-building approaches and the distinction between soundness and completeness."]},{"source_id":"SRC8","title":"Nitpick: A Counterexample Generator for Isabelle/HOL Based on the Relational Model Finder Kodkod","url":"https://easychair.org/publications/paper/zXQs/open","publisher":"EasyChair EPiC Series","date_or_year":"2010","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Nitpick searches finite countermodels by translating proof obligations to relational logic and SAT.","If a finite counterexample exists, enumeration eventually finds it unless resources run out.","Infinite types and some recursive constructs are handled through explicitly bounded approximations rather than universal negative guarantees.","Nitpick was integrated into Isabelle and routinely used to catch false conjectures, demonstrating an operational bounded-counterexample fallback."]}],"problem_evidence":{"status":"PARTLY_SUPPORTED","finding":"The computability diagnosis is correct: exact terminating theoremhood for every finite first-order axiom set and formula is impossible, while proof discovery is one-sidedly effective. Standards also show that timeout and incompleteness should be represented separately from semantic NO. However, the preserved proposal names no actual service, and this search found no direct evidence that a deployed service is contractually required to provide the impossible universal binary answer or currently converts timeout into NO.","source_ids":["SRC1","SRC2","SRC3","SRC4"],"uncertainty":"The central operational failure remains a hypothesis until a named service's requirements, accepted grammar, implementation, logs, and user interface are audited."},"adopter_evidence":{"status":"SUPPORTED","finding":"Concrete adopter and authorizer classes are identifiable: TPTP/SZS and SMT-LIB maintainers define result contracts; Vampire and Isabelle maintainers operate proof-search, portfolio, replay, timeout, and countermodel mechanisms; a target service owner could authorize only a local routing-policy change subject to logic and checker review.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC8"],"uncertainty":"No specific noncompliant mathematical reasoning service or accountable owner was identified in the proposal."},"implementation_evidence":{"status":"PARTLY_SUPPORTED","finding":"Most technical components already exist in deployed or standardized form: declared logics, three-valued or richer status vocabularies, timeout and incompleteness reasons, proof-bearing success, proof replay, finite countermodel search, resource bounds, and portfolio scheduling. The retained sources do not demonstrate the proposal's entire governance bundle—an independently checked reduction certificate, versioned guarantee record, machine-enforced fragment router, and recheck triggers—as one integrated service.","source_ids":["SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"],"uncertainty":"Integration feasibility is strong, but correctness for a particular service depends on its exact language, proof calculus, trusted checker, fragment recognizers, and treatment of finite countermodels."},"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":"Directly separates theorem/countermodel claims from Timeout, GaveUp, Unknown, and verification status, while attaching proof or model data forms.","remaining_difference":"It is an output ontology, not a complete organizational process for proving and versioning a service-specific computability boundary."},{"name":"SMT-LIB logic and result contracts","source_ids":["SRC4"],"same_problem":true,"same_causal_lever":true,"overlap":"Declares the accepted logic, returns sat/unsat/unknown, reports incompleteness or resource reasons, and supports proof and model responses.","remaining_difference":"It standardizes solver interaction rather than requiring an independent impossibility certificate or a theoremhood-specific governance decision record."},{"name":"Isabelle Sledgehammer with Nitpick","source_ids":["SRC6","SRC8"],"same_problem":true,"same_causal_lever":true,"overlap":"Routes conjectures among multiple proof tools, replays successful proofs, applies bounded finite-countermodel search, and respects resource limits without treating unsuccessful proof search as falsity.","remaining_difference":"The sources do not show a single machine-enforced classification lattice covering every submitted first-order axiom presentation or a versioned reduction certificate."},{"name":"Vampire theorem prover","source_ids":["SRC3","SRC4","SRC5"],"same_problem":true,"same_causal_lever":true,"overlap":"Uses standardized inputs and statuses, proof-producing refutation, finite-model or saturation results where available, portfolios of strategies, and explicit resource limits.","remaining_difference":"Its emphasis is high-performance reasoning; the retained documentation does not establish the proposal's independent review, recheck-trigger, and service-level guarantee-record process."}],"contrastive_claim_remaining":"Only a context-specific incremental claim remains: for a demonstrably noncompliant theorem-status service, adding a machine-enforced fragment classifier plus an independently reviewed, versioned computability certificate and automatic recheck triggers will eliminate timeout-as-NO labels while preserving useful decisions. This is an integration and governance claim, not a new computability or result-status concept.","contrastive_claim_falsifier":"The claim is falsified if the interface audit finds the service already distinguishes no-proof-found from false and enforces only certified decidable inputs, or if the shadow test produces any unsound YES/NO, any scope escape, or more than 80% UNKNOWN without improved decision utility.","confidence":"HIGH","search_limitations":"This was a bounded eight-source public-web review. It did not inspect proprietary requirements, source code, production logs, patents, every theorem prover, or every language community. Search results establish close public practice but cannot show universal adoption."},"researchability_gates":{"externally_supported_problem":{"status":"INDETERMINATE","rationale":"The mathematical impossibility and correct UNKNOWN semantics are externally supported, but the alleged universal service requirement and timeout-as-NO behavior are not tied to an evidenced deployment.","source_ids":["SRC1","SRC2","SRC3","SRC4"]},"identifiable_adopter_or_authorizer":{"status":"PASS","rationale":"Service owners and formal-methods reviewers are identifiable authorizer roles, and TPTP, SMT-LIB, Vampire, and Isabelle provide concrete maintainer communities that already govern analogous mechanisms.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC8"]},"distinct_testable_incremental_claim":{"status":"PASS","rationale":"Although the core package is established practice, the service-specific combination of machine-enforced routing, a versioned checked boundary certificate, and automatic recheck triggers has a distinct measurable claim against a noncompliant baseline.","source_ids":["SRC3","SRC4","SRC5","SRC6","SRC8"]},"bounded_next_evidence_step":{"status":"PASS","rationale":"First audit one named service's contract, grammar, result labels, and timeout handling; if the diagnosis is confirmed, the authorized 60-input shadow test is bounded and has explicit failure criteria.","source_ids":["SRC3","SRC4","SRC6","SRC8"]},"no_unresolved_safety_or_authority_stop":{"status":"PASS","rationale":"Shadow mode, independent review, prohibition of production NO outside certified fragments, and rollback on any unsound result keep the next step within service-owner authority and avoid an unresolved safety stop.","source_ids":["SRC3","SRC4","SRC6","SRC8"]},"adequate_search_evidence":{"status":"PASS","rationale":"All six required lanes were searched adversarially using eight opened sources from independent publishers, including primary research, official standards, and first-party product documentation.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"]}},"strict_success":false,"screen_survival":false,"remaining_research_value":"MODERATE","recommended_next_step":"Identify one target reasoning service and perform a read-only requirements/interface audit for its accepted logic, quantifiers, output semantics, timeout handling, proof checking, and enforced fragments. Proceed to the proposed 60-case shadow test only if that audit confirms a universal binary overclaim or UNKNOWN-to-NO coercion.","world_novelty_boundary":"No world-novelty conclusion is made. This bounded eight-source search cannot establish novelty, patentability, freedom to operate, market size, realized impact, or the absence of additional prior art; it only supports the reported comparison with the retained public sources."}