{"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__C","search_lanes":{"direct_problem":{"queries":["Hilbert's tenth problem undecidable integer polynomial equation algorithm no solution primary paper","Diophantine equation solver returns unknown integer polynomial software","site:maplesoft.com isolve NULL no integer solutions unable to find solutions"],"source_ids":["SRC1","SRC2","SRC3","SRC5","SRC6"],"no_result_note":null},"closest_prior_art":{"queries":["SMT-LIB unknown response nonlinear integer arithmetic incomplete official standard","Z3 nonlinear integer arithmetic incomplete unknown timeout official documentation","Incomplete SMT techniques nonlinear formulas integers linearization timeout"],"source_ids":["SRC3","SRC4","SRC5","SRC8"],"no_result_note":null},"historical_terminology":{"queries":["Hilbert 1902 Mathematical Problems tenth determination solvability Diophantine equation finite number operations original translation","Matiyasevich 1970 enumerable sets are Diophantine original paper English","Davis Putnam Robinson exponential Diophantine equation recursively enumerable 1961 original paper","Entscheidungsproblem Diophantine equation solvability historical terminology algorithm"],"source_ids":["SRC1","SRC2","SRC3"],"no_result_note":null},"products_practices_standards":{"queries":["site:smt-lib.org language standard unknown reason-unknown incomplete timeout SMT-LIB 2.7","site:microsoft.github.io/z3guide nonlinear integer arithmetic incomplete unknown","site:maplesoft.com isolve NULL no integer solutions unable to find solutions","site:docs.sympy.org diophantine solver linear diophantine equations limitations"],"source_ids":["SRC4","SRC5","SRC6","SRC7"],"no_result_note":null},"non_english_regional":{"queries":["десятая проблема Гильберта алгоритм диофантово уравнение неразрешима Матиясевич","Hilberts zehntes Problem Entscheidungsverfahren ganzzahlige Lösungen unentscheidbar","dixième problème de Hilbert algorithme équation diophantienne indécidable"],"source_ids":["SRC2"],"no_result_note":null},"composition_subproblems":{"queries":["formalization Hilbert's tenth problem undecidability Coq Lean reduction primary paper","linear Diophantine equations decidable Smith normal form integer solution algorithm primary source","Presburger arithmetic decidable linear integer arithmetic official solver certificate","semi-decidable Diophantine equations enumerate integer tuples witness verification"],"source_ids":["SRC3","SRC5","SRC7","SRC8"],"no_result_note":null}},"sources":[{"source_id":"SRC1","title":"Mathematical Problems","url":"https://www.astro.puc.cl/~rparra/tools/PAPERS/hilbert_1900.pdf","publisher":"Bulletin of the American Mathematical Society; mirror hosted by Pontificia Universidad Católica de Chile","date_or_year":"1902 English translation of 1900 address","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Hilbert's tenth problem requests a finite process deciding whether an arbitrary Diophantine equation with any number of unknowns and integer coefficients has an integer solution.","The historical specification is a universal, terminating decision requirement rather than merely a search for witnesses."]},{"source_id":"SRC2","title":"The Diophantineness of Enumerable Sets","url":"https://www.mathnet.ru/eng/dan35274","publisher":"Doklady Akademii Nauk SSSR / Math-Net.Ru","date_or_year":"1970","source_type":"PRIMARY_RESEARCH","language":"Russian","claims_supported":["Matiyasevich's result establishes the Diophantine representation of enumerable sets, completing the theoretical basis for the negative solution of Hilbert's tenth problem.","The source confirms the older Russian terminology and original regional publication context."]},{"source_id":"SRC3","title":"Hilbert’s Tenth Problem in Coq","url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.27","publisher":"Schloss Dagstuhl—Leibniz-Zentrum für Informatik","date_or_year":"2019","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["The undecidability of solvability of Diophantine equations over natural numbers has been mechanized in Coq.","The formalization proceeds through explicit computability reductions involving Minsky machines and FRACTRAN, demonstrating that a checked reduction is feasible boundary evidence."]},{"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 check-sat response vocabulary distinguishes sat, unsat, and unknown.","The standard supports a reason-unknown response and predefines memout and incomplete, explicitly separating resource failure or solver incompleteness from unsatisfiability."]},{"source_id":"SRC5","title":"Arithmetic | Online Z3 Guide","url":"https://microsoft.github.io/z3guide/docs/theories/Arithmetic/","publisher":"Microsoft Research / Z3 Project","date_or_year":"Accessed 2026-08-04","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["Z3's first-party documentation states that nonlinear integer arithmetic is undecidable and that no procedure can be both correct and terminating with sat or unsat for every input.","Z3 may return unknown or fail to terminate on nonlinear problems while still solving many instances.","Z3 routes linear integer arithmetic and other syntactic fragments to specialized solving engines."]},{"source_id":"SRC6","title":"isolve — Maple Help","url":"https://www.maplesoft.com/support/help/Maple/view.aspx?path=isolve","publisher":"Maplesoft","date_or_year":"Accessed 2026-08-04","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["Maple's isolve attempts to solve equations over the integers.","Its documented NULL result conflates two materially different conditions: no integer solutions and inability to find solutions.","This is direct product evidence for the proposed problem of operationally collapsing nonexistence with unresolved computation."]},{"source_id":"SRC7","title":"Diophantine — SymPy 1.14.0 Documentation","url":"https://docs.sympy.org/latest/modules/solvers/diophantine.html","publisher":"SymPy Project","date_or_year":"2025 documentation, accessed 2026-08-04","source_type":"FIRST_PARTY_PRODUCT","language":"English","claims_supported":["SymPy classifies a Diophantine equation and dispatches to solvers for supported equation types.","The documented supported classes include linear Diophantine equations and selected quadratic or special-form subclasses rather than an unrestricted universal solver.","The documentation provides a concrete first-party precedent for syntactic fragment admission and specialized procedures."]},{"source_id":"SRC8","title":"Incomplete SMT Techniques for Solving Non-Linear Formulas over the Integers","url":"https://arxiv.org/abs/2008.13601","publisher":"arXiv; Borralleras, Larraz, Oliveras, Rodriguez-Carbonell, and Rubio","date_or_year":"2020","source_type":"PRIMARY_RESEARCH","language":"English","claims_supported":["Quantifier-free nonlinear integer arithmetic consists of integer polynomial constraints and is approached with explicitly incomplete SMT methods.","The method reduces nonlinear terms to linear arithmetic, imposes artificial finite bounds, enlarges them iteratively, and may end in timeout.","This provides close prior art for combining a decidable linear core with bounded or incomplete search outside that core."]}],"problem_evidence":{"status":"PARTLY_SUPPORTED","finding":"The mathematical impossibility is decisively supported: the requested full-class Boolean decider is Hilbert's tenth problem, and both classical and mechanized results rule it out. The operational failure mode also exists in a current product: Maple documents one NULL value for both proven nonexistence and inability to find solutions. However, the retained sources do not show a deployed service explicitly advertising a correct, terminating yes/no guarantee for every encoded integer polynomial; that exact promise remains hypothetical.","source_ids":["SRC1","SRC2","SRC3","SRC5","SRC6"],"uncertainty":"The sources establish the impossibility and a real status-collapse analogue, but not the prevalence of unrestricted guarantees or downstream proof corruption."},"adopter_evidence":{"status":"SUPPORTED","finding":"Identifiable adopters and authorizers include Maplesoft's Maple maintainers, whose documented isolve interface collapses two states, and maintainers of Z3 or SymPy-style solver services that already control grammar admission, fragment dispatch, and result vocabularies. A service owner can authorize an advisory, read-only evaluation and an interface narrowing subject to technical review.","source_ids":["SRC5","SRC6","SRC7"],"uncertainty":"No source identifies a specific individual decision-maker or confirms willingness to run the proposed pilot."},"implementation_evidence":{"status":"PARTLY_SUPPORTED","finding":"Nearly every load-bearing component has public precedent: a mechanized undecidability reduction, standardized sat/unsat/unknown and incompleteness reasons, fragment-specific dispatch, decidable linear handling, and incomplete bounded search for nonlinear integer constraints. The exact composite—degree-one admission, general witness verification, five-way public status separation, certificate replay, guarantee versioning, recheck triggers, and independent review—was not found as one deployed mathematics-service package.","source_ids":["SRC3","SRC4","SRC5","SRC7","SRC8"],"uncertainty":"The sources do not test user comprehension, certificate replay across checker versions, or the proposed 60-case operating procedure."},"prior_art":{"disposition":"ESTABLISHED_PRACTICE","closest_analogues":[{"name":"SMT-LIB result contract plus Z3 arithmetic fragment routing","source_ids":["SRC4","SRC5"],"same_problem":true,"same_causal_lever":true,"overlap":"Integer polynomial root existence is a satisfiability problem in nonlinear integer arithmetic. SMT-LIB standardizes sat, unsat, and unknown, including incompleteness reasons, while Z3 explicitly states the undecidability boundary and uses specialized engines for linear fragments.","remaining_difference":"The proposal adds a dedicated mathematics-service governance wrapper: a specifically enforced degree-one grammar, certificate lifecycle records, recheck triggers, claim quarantine, and independent-logician approval."},{"name":"Incomplete SMT methods for nonlinear integer arithmetic","source_ids":["SRC5","SRC8"],"same_problem":true,"same_causal_lever":true,"overlap":"Existing research deliberately combines reduction to linear arithmetic with finite-domain search that expands until a solution is found or the procedure times out, without treating timeout as unsatisfiability.","remaining_difference":"The proposal emphasizes public semantic states and organizational controls rather than improved nonlinear-solving performance."},{"name":"SymPy equation classification and specialized Diophantine solvers","source_ids":["SRC7"],"same_problem":true,"same_causal_lever":true,"overlap":"SymPy classifies equations and invokes solvers only for named supported forms, including linear equations, closely matching grammar-based admission to decidable or otherwise supported subclasses.","remaining_difference":"The documentation does not expose the proposal's complete UNKNOWN/TIMEOUT/OOS/ERROR/NO contract or a checked full-class impossibility certificate."},{"name":"Maple isolve result behavior","source_ids":["SRC6"],"same_problem":true,"same_causal_lever":false,"overlap":"Maple is an identifiable integer-equation service and directly exhibits the target baseline by returning the same NULL value for no solutions and inability to find solutions.","remaining_difference":"It is counterexample prior art rather than the proposed remedy: its documented interface does not separate nonexistence from unresolved computation."},{"name":"Mechanized DPRM/Hilbert's tenth undecidability proof","source_ids":["SRC3"],"same_problem":true,"same_causal_lever":true,"overlap":"The Coq development supplies a machine-checkable reduction-based impossibility result for Diophantine solvability, matching the proposal's demand for checked boundary evidence.","remaining_difference":"It is a research artifact, not an operational service contract, fragment gate, result UI, or guarantee-lifecycle process."}],"contrastive_claim_remaining":"For a named CAS or mathematics service whose current result contract conflates inability with nonexistence, enforcing a machine-checked linear-fragment gate and distinct UNKNOWN, TIMEOUT, OOS, ERROR, and NO states will produce zero unsupported Boolean negatives on a preregistered labeled workload while retaining every correctly verified witness. This is a local implementation and usability claim, not a new computability result.","contrastive_claim_falsifier":"The claim is falsified if the target's current public contract already provides equivalent enforced scope and state separation, or if a preregistered pilot yields any false fragment answer, converts an unrestricted timeout to NO, rejects a valid witness, leaks a Boolean outside the proved fragment, or fails a blinded user test distinguishing UNKNOWN from mathematical nonexistence.","confidence":"HIGH","search_limitations":"The bounded eight-source search covered the original formulation, Russian primary terminology, modern mechanized proof, a current standard, two major product ecosystems, and incomplete-NIA research. It did not inspect proprietary implementations, exhaust all CAS products, run solver experiments, search patents, or establish how routinely governance features such as guarantee quarantine and independent review are deployed."},"researchability_gates":{"externally_supported_problem":{"status":"PASS","rationale":"Primary and mechanized research establish the full-class impossibility, Z3 documents the corresponding nonlinear-integer limitation, and Maple directly documents an interface that collapses no-solution with solver inability. The exact universal commercial promise is unverified, so the support is for the underlying problem and mechanism rather than its prevalence.","source_ids":["SRC1","SRC2","SRC3","SRC5","SRC6"]},"identifiable_adopter_or_authorizer":{"status":"PASS","rationale":"Maplesoft is a concrete potential adopter for status separation, while Z3 and SymPy identify comparable maintainer-controlled solver interfaces. The relevant authorizer is the service or package owner, with independent mathematical review before changing guarantees.","source_ids":["SRC5","SRC6","SRC7"]},"distinct_testable_incremental_claim":{"status":"PASS","rationale":"Although the core approach is established in SMT practice, a falsifiable incremental claim remains for migrating a CAS-style ambiguous result contract to an enforced fragment gate and explicit states, measured by unsupported Boolean outputs, witness retention, and user state discrimination.","source_ids":["SRC4","SRC5","SRC6","SRC7","SRC8"]},"bounded_next_evidence_step":{"status":"PASS","rationale":"A read-only 60-case replay is finite and technically feasible because existing sources supply linear-fragment solvers, incomplete nonlinear behavior, explicit unknown semantics, and mechanized boundary evidence. It can be run without changing production answers.","source_ids":["SRC3","SRC4","SRC5","SRC7","SRC8"]},"no_unresolved_safety_or_authority_stop":{"status":"PASS","rationale":"The authorized first step is advisory and read-only, entails no destructive action, and leaves production claims unchanged. Product maintainers are identifiable authorizers, and independent proof review addresses the principal epistemic risk. No source reveals a legal, physical-safety, or external-authority barrier.","source_ids":["SRC3","SRC5","SRC6","SRC7"]},"adequate_search_evidence":{"status":"PASS","rationale":"All six mandated lanes were searched adversarially. Exactly eight retained sources were opened; they span eight publishing or project contexts and include multiple primary-research, official-standard, and first-party product sources. Direct phrase misses were not treated as novelty evidence.","source_ids":["SRC1","SRC2","SRC3","SRC4","SRC5","SRC6","SRC7","SRC8"]}},"strict_success":false,"screen_survival":false,"remaining_research_value":"MODERATE","recommended_next_step":"Select an identifiable CAS-style target with an ambiguous unresolved result, freeze its accepted polynomial grammar, and run the proposed read-only 60-case pilot against a wrapper that admits only a reviewed linear fragment for Boolean decisions, verifies unrestricted witnesses, and emits distinct UNKNOWN, TIMEOUT, OOS, ERROR, and NO states. Preregister zero unsupported Boolean outputs and a blinded state-comprehension check; do not claim a new computability result.","world_novelty_boundary":"This bounded public search finds the central problem–intervention mechanism already established across computability theory, SMT-LIB, Z3, specialized Diophantine solvers, and incomplete nonlinear-integer methods. It cannot establish world novelty, patentability, freedom to operate, market size, deployment prevalence, or realized impact. The remaining boundary is a context-specific implementation and evaluation claim for a named mathematics service, not novelty of the underlying theorem, fragment strategy, or unknown-result contract."}