{"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__A","search_lanes":{"direct_problem":{"queries":["arithmetic truth undecidable standard model natural numbers decision procedure","formal mathematics repository conjecture triage unknown timeout certificate proof assistant","true arithmetic not recursively enumerable standard model natural numbers"],"source_ids":["SRC1","SRC2","SRC6","SRC8"],"no_result_note":null},"closest_prior_art":{"queries":["official SMT-LIB standard unknown reason-unknown incomplete solver response","TPTP SZS ontology status unknown theorem prover timeout official","theorem prover service statuses proved disproved unknown certificate timeout"],"source_ids":["SRC3","SRC4","SRC5","SRC8"],"no_result_note":null},"historical_terminology":{"queries":["Turing 1936 computable numbers Entscheidungsproblem original paper","Hilbert tenth problem undecidable Diophantine equations Matiyasevich primary paper","Church theorem Entscheidungsproblem arithmetic undecidable"],"source_ids":["SRC1","SRC2","SRC6"],"no_result_note":null},"products_practices_standards":{"queries":["official SMT-LIB check-sat unknown reason incomplete","TPTP SZS ontologies problem solution logical status online service","Lean proof terms independently checkable kernel tactic failure"],"source_ids":["SRC3","SRC4","SRC5"],"no_result_note":null},"non_english_regional":{"queries":["indecidibilidad verdad aritmética números naturales procedimiento decisión español","aritmetica verità standard indecidibile procedura decisione congetture italiano","indecidibilità aritmetica Peano algoritmo teoria formalizzata"],"source_ids":["SRC6","SRC7"],"no_result_note":null},"composition_subproblems":{"queries":["halting reduction arithmetic sentence truth classifier","decidable arithmetic fragment quantifier elimination versus Peano arithmetic","bounded theorem proof search timeout unknown certificate checking","parser enforced logic fragment solver incomplete unknown"],"source_ids":["SRC1","SRC2","SRC3","SRC5","SRC6"],"no_result_note":null}},"sources":[{"source_id":"SRC1","title":"On Computable Numbers, with an Application to the Entscheidungsproblem","url":"https://trhvidsten.com/docs/classics/Turing-1936.pdf","publisher":"Proceedings of the London Mathematical Society","date_or_year":"1936–1937","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["There is no general finite procedure for deciding whether an encoded machine is circle-free.","A hypothetical general decision procedure can be refuted by constructing and applying a diagonal machine.","The Entscheidungsproblem has no uniform effective solution."]},{"source_id":"SRC2","title":"Gödel’s Incompleteness Theorems","url":"https://plato.stanford.edu/entries/goedel-incompleteness/","publisher":"Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University","date_or_year":"2013; current online revision accessed 2026","source_type":"SECONDARY_RESEARCH","language":"English","claims_supported":["Sufficiently strong formal arithmetic theories are undecidable, and Church's theorem establishes undecidability of first-order predicate logic.","Standard and nonstandard models must be distinguished when specifying arithmetic semantics.","Undecidability is scope-sensitive: nontrivial theories such as real-closed fields are decidable."]},{"source_id":"SRC3","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":["A standard solver response may be unknown rather than sat or unsat.","The standard provides a reason-unknown query after an unknown response.","Standard reasons distinguish resource exhaustion from known incompleteness for the submitted formula class."]},{"source_id":"SRC4","title":"The TPTP World","url":"https://www.tptp.org/","publisher":"TPTP Project, University of Miami","date_or_year":"2026","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["TPTP is established automated-theorem-proving infrastructure containing problem and solution libraries.","The SZS ontologies provide standardized values for logical statuses of problems and solutions.","TPTP operates online services for submitting ATP problems and examining solutions."]},{"source_id":"SRC5","title":"Lean Language Reference: Tactic Proofs","url":"https://lean-lang.org/doc/reference/latest/Tactic-Proofs/","publisher":"Lean Project","date_or_year":"2026","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["Tactics may fail, leave subgoals, or complete a proof rather than forcing every attempt into a truth-valued answer.","Successful tactics construct proof terms that are checked by Lean's kernel.","Proof terms are independently checkable evidence and can be checked by independent implementations."]},{"source_id":"SRC6","title":"Diophantine Sets","url":"https://www.mathnet.ru/eng/rm5112","publisher":"Russian Mathematical Surveys / Math-Net.Ru","date_or_year":"1972","source_type":"PRIMARY_RESEARCH","language":"English and Russian","claims_supported":["Hilbert's tenth problem asks for a uniform method deciding solvability of arbitrary Diophantine equations.","No such uniform method exists.","An undecidable Diophantine subclass supplies a direct obstruction to any classifier covering all arithmetic sentences."]},{"source_id":"SRC7","title":"Indecidibilità","url":"https://www.treccani.it/enciclopedia/indecidibilita_%28Enciclopedia-della-Matematica%29/","publisher":"Istituto della Enciclopedia Italiana Treccani","date_or_year":"2013","source_type":"SECONDARY_RESEARCH","language":"Italian","claims_supported":["Italian mathematical terminology defines an undecidable formal theory by the absence of an algorithm deciding theoremhood for every well-formed formula.","Formal Peano arithmetic is given as an example of an undecidable theory.","The source distinguishes the undecidability of a theory from an individual formula's independence."]},{"source_id":"SRC8","title":"Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics","url":"https://arxiv.org/abs/2605.13171","publisher":"arXiv / paper authors","date_or_year":"2026","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["A real collaborative repository exposes a structured interface connecting conjecture formalizers with human and automated solvers.","The repository explicitly retains large numbers of open conjectures alongside solved problems.","Proofs and disproofs are used as auditing evidence to improve formalization fidelity, demonstrating identifiable maintainers, contributors, and reviewers."]}],"problem_evidence":{"status":"PARTLY_SUPPORTED","finding":"The core problem is real conditional on the declared unrestricted scope: a total effective classifier for truth of every arithmetic sentence would decide known undecidable subclasses, including Diophantine solvability, and conflicts with classical computability results. Standards and theorem-proving systems also demonstrate why unknown, incompleteness, resource failure, and certified success must remain distinct. No retained source verifies that the particular hypothetical repository currently collapses timeouts into false answers, so that operational allegation remains unverified.","source_ids":["SRC1","SRC2","SRC3","SRC5","SRC6"],"uncertainty":"The impossibility conclusion depends on the accepted syntax genuinely containing a reduction-preserving undecidable fragment, standard-natural-number semantics, and an ordinary uniform effective computation model. A finite corpus, decidable fragment, or oracle-backed service would change the conclusion."},"adopter_evidence":{"status":"PARTLY_SUPPORTED","finding":"Repository maintainers and formalization reviewers are identifiable adopter classes, and existing TPTP, Lean, and Formal Conjectures infrastructure shows real organizations and communities performing closely related stewardship. The proposal assigns authorization to a repository governance board and independent logicians, but no publicly named board for the hypothetical repository was found.","source_ids":["SRC4","SRC5","SRC8"],"uncertainty":"Role-level adopters and authorizers are clear, but the specific organization, accountable individuals, and approval process are not externally identified."},"implementation_evidence":{"status":"PARTLY_SUPPORTED","finding":"The principal implementation mechanisms already exist separately and in combination: standardized unknown and incompleteness responses, theorem-proving status ontologies, proof-term certificates, independent checking, open-conjecture retention, and restricted decidable theories. This supports feasibility of a bounded certificate-bearing fallback, but no retained study tests the proposed 100-formula pilot or measures whether its status lattice reduces false closure in the target repository.","source_ids":["SRC2","SRC3","SRC4","SRC5","SRC8"],"uncertainty":"Parser-enforced scope, the exact machine-to-sentence reduction, standard-semantics fidelity, certificate throughput, and user interpretation of unknown would require local validation."},"prior_art":{"disposition":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"Classical undecidability analysis from the Entscheidungsproblem through arithmetic and Diophantine reductions","source_ids":["SRC1","SRC2","SRC6","SRC7"],"same_problem":true,"same_causal_lever":true,"overlap":"It tests a universal effective decision promise by reduction to a known undecidable problem and rejects unrestricted totality when the reduction is valid.","remaining_difference":"The classical results do not themselves specify a modern repository interface, parser gate, multi-status fallback, pilot, or governance record."},{"name":"SMT-LIB unknown and reason-unknown protocol","source_ids":["SRC3"],"same_problem":false,"same_causal_lever":true,"overlap":"It operationalizes abstention rather than forcing a Boolean result and distinguishes resource failure from solver incompleteness relative to a formula class.","remaining_difference":"It concerns satisfiability modulo declared theories rather than truth of every arithmetic sentence in the standard natural numbers, and it does not require a repository-specific impossibility certificate."},{"name":"TPTP/SZS result ontologies and online ATP services","source_ids":["SRC4"],"same_problem":true,"same_causal_lever":true,"overlap":"It supplies standardized problem and solution statuses, online automated-theorem-proving services, and separate solution artifacts instead of treating every unsuccessful run as false.","remaining_difference":"Its statuses are generally relative to submitted axioms and ATP outcomes, not a total standard-arithmetic truth service, and the retained overview does not establish the proposed governance workflow."},{"name":"Lean tactic failure, residual goals, and independently checked proof terms","source_ids":["SRC5"],"same_problem":true,"same_causal_lever":true,"overlap":"Automation may fail or leave unresolved goals, while successful closure requires a kernel-checked certificate; this closely matches explicit unresolved states and certificate-bearing positive outputs.","remaining_difference":"Lean is an interactive proof assistant rather than the proposed triage service and checks derivability in a declared environment rather than directly deciding standard-model truth."},{"name":"Formal Conjectures open/solved repository with evidence-driven auditing","source_ids":["SRC8"],"same_problem":true,"same_causal_lever":true,"overlap":"A formal-mathematics repository keeps open problems unresolved, exposes a structured solver interface, and uses proofs and disproofs as auditing evidence.","remaining_difference":"It is a benchmark and collaborative repository, not a documented attempt at an unrestricted total truth classifier, and it does not foreground a halting reduction or parser-enforced decidable-fragment map."}],"contrastive_claim_remaining":"The credible remaining claim is local rather than a new general method: for a repository whose present contract really accepts an undecidable arithmetic class and demands Boolean closure, adding an enforceable scope gate plus certificate-checked proved/refuted outputs and distinct unknown, timeout, failure, and out-of-scope states will reduce unsupported Boolean labels without materially reducing certified closures under a fixed budget.","contrastive_claim_falsifier":"The claim is falsified if a requirements audit shows the accepted domain is already finite or decidable, or if a preregistered replay/pilot finds no unsupported Boolean labels at baseline, the new states are still collapsed downstream, certificate checking fails, or certified closure falls beyond a prespecified tolerance.","confidence":"HIGH","search_limitations":"The bounded search retained exactly eight sources and covered six lanes, including Italian/Russian terminology, historical computability language, standards, products, repositories, and component combinations. It did not exhaust every theorem prover, repository policy, patent, product, language, or unpublished implementation. The evidence establishes close and routinely used component practices, not universal adoption of the exact governance bundle."},"researchability_gates":{"externally_supported_problem":{"status":"PASS","rationale":"Classical primary results establish undecidable decision problems reducible to arithmetic, while current standards explicitly accommodate unknown and incompleteness. Thus the universal exact terminating promise is a supported problem whenever the stated unrestricted scope and computation assumptions hold.","source_ids":["SRC1","SRC2","SRC3","SRC6"]},"identifiable_adopter_or_authorizer":{"status":"PASS","rationale":"Repository maintainers, formalization reviewers, and the body approving public guarantees are operationally identifiable roles; comparable repositories and ATP services demonstrate that these roles and substrates exist, although the target organization is unnamed.","source_ids":["SRC4","SRC5","SRC8"]},"distinct_testable_incremental_claim":{"status":"PASS","rationale":"Although the general intervention is established practice, its local incremental effect is distinct and falsifiable: compare unsupported Boolean closure, certified closure, status preservation, and user interpretation before and after an enforceable scope/status/certificate contract.","source_ids":["SRC3","SRC4","SRC5","SRC8"]},"bounded_next_evidence_step":{"status":"PASS","rationale":"A read-only, preregistered 100-formula replay can test parsing, fragment membership, certificate verification, status separation, reduction assumptions, and baseline-versus-fallback label behavior within a fixed budget.","source_ids":["SRC3","SRC4","SRC5"]},"no_unresolved_safety_or_authority_stop":{"status":"PASS","rationale":"The proposed first step is read-only and withholds public truth labels. Existing proof-checking and collaborative-audit practices support quarantine and review. Execution should remain contingent on named repository approval and independent review of the reduction, but that is a manageable prerequisite rather than an unresolved stop.","source_ids":["SRC5","SRC8"]},"adequate_search_evidence":{"status":"PASS","rationale":"The search covered all six required lanes with exactly eight opened direct sources from independent publishers, including multiple primary research papers, an official standard, and first-party theorem-proving documentation and infrastructure.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"]}},"strict_success":false,"screen_survival":false,"remaining_research_value":"MODERATE","recommended_next_step":"First conduct a requirements audit that freezes the accepted grammar, standard-model semantics, computation model, and theory version, then have two independent logicians verify a reduction into the admitted language. If the unrestricted promise survives that audit, run a preregistered read-only 100-formula replay comparing the current wrapper with a parser-enforced, certificate-checked multi-status wrapper; halt and quarantine outputs on any unknown/false conflation or certificate failure.","world_novelty_boundary":"This bounded search supports the impossibility diagnosis and finds the proposed intervention's main mechanisms already established across computability theory, SMT/ATP standards, proof assistants, and formal-conjecture repositories. It does not establish world novelty, patentability, freedom to operate, market size, realized impact, or that no single system implements every governance detail."}