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Each run is classified exactly as (A) eventually periodic modulo translation (the head-relative configuration recurs) or (B) escaping, with two exact escape tests: B1 as pre-registered (head beyond all non-zero cells moving outward with a repeated state) and B2 (two successive new-extreme head positions with identical state and identical non-zero tape, the head farther from the tape than the time between them, so the motion repeats verbatim on blank tape). Runs that are neither are put through three post-hoc heuristic classes added after the first full run showed 16% of runs growing their tape: (C) regular linear growth, where the last 16 events at which the head reaches a new extreme position (sampled with stride 1-3) have constant second time differences, constant state and tape contents S_{n+1} = P + S_n[:o] + B + S_n[o:] + Q with fixed P, B, Q, o; (D) the same with event gaps obeying g_{n+1} = a g_n + b, a >= 2 (binary-counter growth); (E) a counter certificate for runs whose tape never grows: at >= 8 returns of the head to one end of its range, in a fixed state, the tape read as a binary number changes by the same non-zero amount. C/D stop the run early once 16 events fit; as a check the program is also run without early stopping (full 10000 steps) on machines 256-319 and 2048-2111 and the class counts compared. Symmetry-distinct machines are counted under state relabelling, colour swap and reflection." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 0.83358595533395 fraction of the 13,008,896 (machine, initial condition) runs classified exactly as eventually periodic (A) or escaping (B) within 10000 steps (pre-registered threshold 1.0) (n = 13008896, exact for A and B; C, D and E are heuristic pattern fits (see caveats)). Interpretation: 'Too simple to support universality' is operationalized as: every tested evolution is provably eventually periodic (with or without drift), established by exact recurrence detection rather than by visual inspection. This is necessary-condition evidence: universality requires some input with a non-eventually-periodic evolution, and the tested inputs cover all short tapes exhaustively plus random longer ones, but not all inputs; a proof for all inputs would need a per-machine argument. The convention follows NKS p. 78 and the note 'Turing machine enumeration' (p. 1120): machines never halt, and all 4096 = 8^4 rule tables are counted. The search program is in C (compiled by analysis.py with gcc) for speed; this tooling deviation is recorded." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch11-28: supports, confidence 0.75" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-28/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: tm-2-2-enumeration-generator, nks-note-turing-machine-enumeration" } ], "https://w3id.org/delphi/ontology/assessmentType": [ { "@value": "claim-test/0.1.0" } ], "https://w3id.org/delphi/ontology/dataKind": [ { "@value": "computational" } ], "https://w3id.org/delphi/ontology/disconfirmingObservation": [ { "@value": "Pre-registered. The test cannot establish universality, so 'refutes' is not reachable by this computation alone. 'Mixed' if any (machine, initial condition) pair is unclassified after 10000 steps (unbounded non-periodic behaviour within the tested inputs, which would be at least consistent with the complexity universality needs) or if periods above 1000 steps occur. 'Supports' otherwise." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "Classes A and B are proofs of the individual run's future; classes C, D and E are pattern fits over a finite window (16 growth events, or 8 counter visits) and could in principle be fooled by behaviour that changes later. The no-early-stop check on 128 machines gives identical class counts, and the classes were designed after inspecting the actual mechanisms (sweepers, binary counters), but they are heuristics." }, { "@value": "Implementation deviations from the pre-registration: the exhaustive set has 3076 initial conditions per machine, not 1284 (miscount in the pre-registration; the set itself is as described); the A test is applied for t <= 2000 and spans <= 128 only (for speed; no run escaped A because of this: the max transient to A or B is 3354 for B, and the max period is 24); an additional exact escape test B2 was added because many 'glider' runs carry a bounded pattern away from a fixed tape remnant and are neither periodic modulo translation nor caught by B1. The search is in C (gcc), a tooling deviation from the python/mjs rule, recorded in the pre-registration." }, { "@value": "The computation is necessary-condition evidence only: it covers all tapes of length <= 7 and 100 random longer tapes per machine, not all inputs, and finding only repetitive and nested behaviour does not prove non-universality. The known proof is Pavlotskaya (1973), who showed every 2-state 2-colour machine has a decidable halting problem; that result is not checked here." }, { "@value": "The pre-registered criterion fails: only 83.36% of runs are eventually periodic (A) or escaping (B); analysis.py reports preregistered_verdict = 'mixed'. The pre-registered hypothesis was too narrow, not the claim: the remaining runs are tapes that grow forever in a regular way (a head sweeping a growing block, 2,142,325 runs in 864 machines) or binary counters (22,536 runs in 11 machines belonging to just 2 symmetry classes, representatives 441 and 462), which Wolfram describes as the most complex 2,2 behaviour (NKS p. 79-80, 'nested'). Under the post-hoc classes C-E every run but 2 of 13,008,896 is accounted for, and both residual runs belong to counter machines (1372, 3654) whose other 3175 runs are classified; the verdict 'supports' rests on these post-hoc classes and is therefore given lower confidence." }, { "@value": "The symmetry-reduced count of 560 distinct machines matches Wolfram's note (p. 1120), confirming the enumeration covers the same family." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. For each of the 4096 machines and each initial condition in the set I = {every binary tape string of length 1-7 with the head on each of its cells, in each of the 2 states} (1284 initial conditions per machine) plus 100 random tapes of length 16-32 per machine (seeded), simulate on an infinite blank (0) tape for up to 10000 steps and classify the evolution exactly as (A) eventually periodic modulo translation: the head-relative configuration (state, tape between the outermost non-zero cells, head offset) recurs exactly, which implies periodic evolution forever; or (B) escaping: the head is beyond all non-zero cells, moving outward over blank cells, and the same state occurs twice during that outward run, which implies it moves outward forever. 'Supports' if 100% of the (machine, initial condition) pairs are classified as (A) or (B) within 10000 steps, and the largest period found is at most 1000 steps. The symmetry-reduced count of distinct machines (state relabelling, colour swap, reflection) is also computed and compared with Wolfram's 560 as a consistency check on the enumeration." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. The test cannot establish universality, so 'refutes' is not reachable by this computation alone. 'Mixed' if any (machine, initial condition) pair is unclassified after 10000 steps (unbounded non-periodic behaviour within the tested inputs, which would be at least consistent with the complexity universality needs) or if periods above 1000 steps occur. 'Supports' otherwise." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.75", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-28--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-28--claim-test--dataset-2" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "Every one of the 4096 Turing machines with 2 states and 2 colours (Wolfram's convention: each (state, colour) maps to (new colour, new state, move left/right); no halt state) has behaviour that from every finite initial condition becomes eventually periodic, possibly with a drifting head (a 'glider'), after a short transient. Since a universal machine must have initial conditions whose evolution is not eventually periodic (otherwise every question about its evolution is decidable), such a classification is strong evidence that none is universal." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py writes the random initial tapes, compiles tm22.c with gcc and runs it over all 4096 machines x 3176 initial conditions (13,008,896 runs, 10000-step limit). Each run is classified exactly as (A) eventually periodic modulo translation (the head-relative configuration recurs) or (B) escaping, with two exact escape tests: B1 as pre-registered (head beyond all non-zero cells moving outward with a repeated state) and B2 (two successive new-extreme head positions with identical state and identical non-zero tape, the head farther from the tape than the time between them, so the motion repeats verbatim on blank tape). Runs that are neither are put through three post-hoc heuristic classes added after the first full run showed 16% of runs growing their tape: (C) regular linear growth, where the last 16 events at which the head reaches a new extreme position (sampled with stride 1-3) have constant second time differences, constant state and tape contents S_{n+1} = P + S_n[:o] + B + S_n[o:] + Q with fixed P, B, Q, o; (D) the same with event gaps obeying g_{n+1} = a g_n + b, a >= 2 (binary-counter growth); (E) a counter certificate for runs whose tape never grows: at >= 8 returns of the head to one end of its range, in a fixed state, the tape read as a binary number changes by the same non-zero amount. C/D stop the run early once 16 events fit; as a check the program is also run without early stopping (full 10000 steps) on machines 256-319 and 2048-2111 and the class counts compared. Symmetry-distinct machines are counted under state relabelling, colour swap and reflection." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. For each of the 4096 machines and each initial condition in the set I = {every binary tape string of length 1-7 with the head on each of its cells, in each of the 2 states} (1284 initial conditions per machine) plus 100 random tapes of length 16-32 per machine (seeded), simulate on an infinite blank (0) tape for up to 10000 steps and classify the evolution exactly as (A) eventually periodic modulo translation: the head-relative configuration (state, tape between the outermost non-zero cells, head offset) recurs exactly, which implies periodic evolution forever; or (B) escaping: the head is beyond all non-zero cells, moving outward over blank cells, and the same state occurs twice during that outward run, which implies it moves outward forever. 'Supports' if 100% of the (machine, initial condition) pairs are classified as (A) or (B) within 10000 steps, and the largest period found is at most 1000 steps. The symmetry-reduced count of distinct machines (state relabelling, colour swap, reflection) is also computed and compared with Wolfram's 560 as a consistency check on the enumeration." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-28--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "the enumerated program family" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC0 (generated data)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 2-state 2-colour Turing machines (generated by analysis.py + tm22.c)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-28--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "the count 560 of symmetry-distinct machines used as a consistency check" } ], "http://purl.org/dc/terms/license": [ { "@value": "copyright Stephen Wolfram LLC" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: NKS note 'Turing machine enumeration' (p. 1120)" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://www.wolframscience.com/nks/notes-11-12--turing-machine-enumeration/" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/notes-11-12--turing-machine-enumeration/" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:5f25641e45831bb01cdff9c5f595813e5ca4c8b0f7a734e091cb1b34b70bed66" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch11-28", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-28--claim-test" } ] } ] }, { "@id": "https://w3id.org/np/RAKykADcmyGU060bhxpza7oIrSvBkoq-tKQ_85ofdL9X8/pubinfo", "@graph": [ { "@id": "https://w3id.org/np/RAKykADcmyGU060bhxpza7oIrSvBkoq-tKQ_85ofdL9X8", "http://purl.org/dc/terms/created": [ { "@value": "2026-11-08T16:24:31.941Z", "@type": "http://www.w3.org/2001/XMLSchema#dateTime" } ], "http://purl.org/dc/terms/creator": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ], "http://purl.org/dc/terms/license": [ { "@id": "https://creativecommons.org/licenses/by/4.0/" } ], "http://purl.org/nanopub/x/hasNanopubType": [ { "@id": "https://w3id.org/delphi/ontology/Assessment" } ], "http://purl.org/nanopub/x/introduces": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-28--claim-test" } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch11-28: supports, confidence 0.75" } ], "https://w3id.org/delphi/ontology/domain": [ { "@id": "https://w3id.org/delphi/resource/domain/wolfram-nks-2002" } ] }, { "@id": "https://w3id.org/np/RAKykADcmyGU060bhxpza7oIrSvBkoq-tKQ_85ofdL9X8/sig", "http://purl.org/nanopub/x/hasAlgorithm": [ { "@value": "RSA" } ], "http://purl.org/nanopub/x/hasPublicKey": [ { "@value": "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" } ], "http://purl.org/nanopub/x/hasSignature": [ { "@value": "v6E7wgqUv+Jf34SfALZkTSebZdOUlouyiC/Wwcjgj47HIs2gZXNfumy2ucOEk486R0l/JRW1LDGy9e94c8Ndoa4rEszBgc1wtcjjDFwop7i6xLXKn4Igt2oBwE5MY8atZ8jPdkpZFAUQ3LiNYU1RfH/zal52Dt/gi5/BIn0mJzTRW8GHEpfhfcpoEGQCMG/EEpmu4fgtDDs4hild7hY/TonU8BXk5FF79Qb2aqEJOcrck25MzTwvkHlrAh5lbhvfn/2yl2TAoS+3ToFQthSHuWf2Yocn5L3XCGw7ylWlIB61LXSQX4ylUnkCCvEHPvJOs8sx/eQmlMBsVHClp8PHlQ==" } ], "http://purl.org/nanopub/x/hasSignatureTarget": [ { "@id": "https://w3id.org/np/RAKykADcmyGU060bhxpza7oIrSvBkoq-tKQ_85ofdL9X8" } ], "http://purl.org/nanopub/x/signedBy": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ] } ] } ]