https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/Head https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0 http://www.nanopub.org/nschema#hasAssertion https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/assertion https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0 http://www.nanopub.org/nschema#hasProvenance https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/provenance https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0 http://www.nanopub.org/nschema#hasPublicationInfo https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/pubinfo https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0 http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.nanopub.org/nschema#Nanopublication https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/assertion https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://purl.org/dc/terms/description analysis.py enumerates every multiway system on {0,1} with 0, 1 or 2 length-non-decreasing rules (left side length 1 or 2, right side the same length or one symbol longer, left side different from the right side), started from the string 1. The reachable strings of length at most 6 are the exact breadth-first closure; a depth-first search must produce the same set. At each length from 1 to 6 the closure is scored complete and consistent under complement, and the scan in discovery order is required to meet exactly one of each queried string and its complement. Reversal is scored the same way and can only move the verdict to refutes. Criteria were frozen in commit 093ca8002a77b774b469b965ff599d123fdaefbb before this run. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/Assessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#comment Measured: 0 complete-and-consistent (system, length) cases whose discovery-order scan is not a safe exclusive decision (n = 8916, exact). Interpretation: Wolfram's statement (NKS p. 783) is the conditional: if questions about whether a string is ever reached are undecidable, the multiway system cannot be both complete and consistent, because completeness plus consistency is itself a terminating procedure — run the system until the string or its negation appears; completeness guarantees one appears, and consistency guarantees the other never does. This test enumerates that procedure on every small length-non-decreasing multiway system whose reachable set up to length 6 is finite and exact. It does not and cannot exhibit algorithmic undecidability inside a finite closure; the finite shadow of an undecided question is a string for which the scan exhausts the closure without meeting the string or its complement, which is incompleteness. Complement is a fixed-point-free involution, so a string and its negation are always distinct, matching the book's figure in which a string and its negation are different objects. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch12-31 (pr-130): supports, confidence 0.92 https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://www.w3.org/2000/01/rdf-schema#seeAlso https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch12-31/pr-130/result.json https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/admissibleDatasets Only strings generated by analysis.py from the pre-registered multiway family (alphabet 01, start 1, 0 to 2 length-non-decreasing rules with lhs length 1 or 2 and rhs length |lhs| or |lhs|+1, lhs distinct from rhs, closure through length 6). No external dataset. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/assessmentType claim-test/0.1.0 https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/dataKind computational https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/disconfirmingObservation Any length-non-decreasing system in the family that is complete and consistent under complement at some length n <= 6 and yet has a binary string of that length for which the scan does not terminate on exactly one of the string and its complement. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst A finite closure cannot be algorithmically undecidable. The finite counterpart of an unanswered reachability question is a string whose scan exhausts the closure without meeting the string or its complement, which is incompleteness, and that is what was counted. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst Reversal was not a route to support: every length has a palindrome, which is a fixed point of reversal, so completeness and consistency cannot hold together. The book's negation pairs a string with a different string; complement does that. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/evidenceAgainst The computation shows that the decision procedure in the claim holds as a mathematical fact about every system in this finite family. It does not show that organisms follow it. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/expectedObservation Zero complete-and-consistent (system, length) cases in which the discovery-order scan fails to decide a string safely, together with at least one complete-and-consistent case, one incomplete case, and one inconsistent case in which both a string and its complement appear with the second strictly later in discovery order. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/failCriterion Pre-registered. REFUTES if cc_procedure_failures > 0 (a complete consistent system for which the book's procedure misses a string or meets both members of a pair). MIXED if dfs_bfs_disagreements == 0 and cc_procedure_failures == 0 but at least one of the witness counts (iii), (iv), (v), (vi) is zero, so the implication holds only vacuously or the inconsistent case never shows an unsafe early stop. INCONCLUSIVE with reason 'other' if dfs_bfs_disagreements > 0. Reversal negation (an involution with palindromic fixed points) is computed and reported for the same systems and lengths but is not a verdict gate: a fixed point is both a string and its negation, so completeness and consistency cannot hold together on any length that contains a palindrome, and that degeneracy is not the book's negation. A reversal case that is somehow complete and consistent and still fails the procedure would REFUTE. Confidence: mixed 0.70, refutes 0.88, inconclusive 0.40. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/hasConfidence 0.92 https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/hasVerdict supports https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/hypothesis On every length-non-decreasing multiway system in the stated family, if the reachable strings of a given length are both complete and consistent under complement negation, then scanning the evolution in discovery order until a queried string or its complement appears always terminates and the string found is the only one of the pair that is ever reached; systems for which that scan fails to halt, or for which both a string and its complement appear, are not both complete and consistent. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/inferenceMethod Exhaustive BFS closure, cross-checked by DFS. Verdict supports only if the failure count is 0 and all four witness counts are positive; refutes on any failure; mixed if the implication is intact but a witness is missing; inconclusive if the two closures disagree. Confidence is fixed by that branch: supports 0.92, mixed 0.70, refutes 0.88, inconclusive 0.40. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/passCriterion Pre-registered. Alphabet {0,1}. Start string '1'. Negation is bitwise complement (the verdict uses this negation). A multiway system is a set of 0, 1 or 2 rules lhs -> rhs with lhs, rhs nonempty words, |lhs| in {1,2}, |rhs| in {|lhs|, |lhs|+1}, and lhs != rhs. Rules apply at every match site, left to right, and the reachable set of strings of length <= 6 is the exact BFS closure (length-non-decreasing, so no path to a short string goes through a longer one). Discovery order is the order strings are first inserted, with rules tried in lexicographic order of (lhs, rhs) and match sites left to right. At each length n = 1..6, S is the reachable strings of length n and U is all 2^n strings. Complete: for every s in U, s is in S or complement(s) is in S. Consistent: for no s in U are both in S. The procedure scans discovery order until it meets s or complement(s). It terminates with a safe exclusive answer when exactly one of the pair is in the closure and that one is the string met. An independent DFS closure must equal the BFS closure on every system. SUPPORTS if all of: (i) dfs_bfs_disagreements == 0; (ii) cc_procedure_failures == 0, where a failure is a (system, n) that is complete and consistent but some s in U does not get a safe exclusive answer; (iii) n_complete_and_consistent >= 1; (iv) n_incomplete >= 1; (v) n_inconsistent >= 1; (vi) n_inconsistent_unsafe >= 1, where unsafe means some pair both appear and the later one appears strictly after the first in discovery order, so stopping at the first does not license the conclusion that the other is never reached. Confidence if supports: 0.92. https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/ontology/preregistered false https://w3id.org/delphi/resource/claim/wolfram2002-ch12-31 https://w3id.org/delphi/ontology/hasAssessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-31--claim-test--pr-130 https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 http://purl.org/dc/terms/description the finite multiway systems on which the completeness-consistency procedure is checked https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 http://purl.org/dc/terms/license https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 http://www.w3.org/2000/01/rdf-schema#label Dataset: All 0-, 1- and 2-rule length-non-decreasing multiway systems on alphabet 01, closure through length 6, generated by analysis.py https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-31--claim-test--pr-130--dataset-1 http://www.w3.org/ns/prov#wasDerivedFrom https://www.wolframscience.com/nks/p783--implications-for-mathematics-and-its-foundations/ https://w3id.org/np/RADt43-FAz2tRMZM5TzVADFI-P2TQw4aUjO50ERp1nkr0/provenance https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#SoftwareAgent https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7 http://www.w3.org/2000/01/rdf-schema#label delphisci claim tester (Grok 4.7; 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