https://w3id.org/np/RASizrS8-_GkTpmvhX9JPEgvxYw_rf0FkOgQFEY1TPlT0
.trig | .trig.txt | .jelly | .jelly.txt | .jsonld | .jsonld.txt | .nq | .nq.txt | .xml | .xml.txt
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sub:assertion {
<https://w3id.org/delphi/resource/assessment/wolfram2002-ch09-15--claim-test> a <https://w3id.org/delphi/ontology/Assessment>;
dc:description "analysis.py generates each graph implicitly (lattices, tree, Heisenberg Cayley graph) or explicitly (configuration-model random 4-regular graph, n = 300000, numpy seed 20261031) and counts the closed ball N(r) around the origin by breadth-first search. The growth exponent is the OLS slope of log N(r) on log r over r in [R/2, R] with R = 32 (Z^1..Z^3), 20 (Z^4), 13 (tree), 8 (random graph; largest r with N(r) <= n/10) and 24 (Heisenberg). For the tree and random graph the fit of log N(r) linear in r is compared with the log-log fit by R^2. Exact closed-form ball sizes of Z^d were used post hoc to check the BFS counts and to show how the slope estimator converges.";
<http://www.w3.org/2000/01/rdf-schema#comment> "Measured: [object Object] log-log slope of N(r) over r in [R/2, R] (growth-exponent estimate) (n = [object Object], exact counts; slopes are OLS fits with R^2 >= 0.988 (log-log) for every graph). Interpretation: Wolfram's 'number of nodes that appear out to distance r' is the size N(r) of the closed ball of graph-distance radius r around a node. 'Grows in some limiting sense like r^d' is operationalised as the local log-log slope of N(r) at the largest radii computed, which for an exact polynomial of degree d tends to d from below. 'Corresponds to ordinary d-dimensional space' is taken, for the primary test, as 'is the nearest-neighbour graph of Z^d'; for the secondary test we use the standard geometric-group-theory meaning (quasi-isometric to R^d) and rely on published theorems for the Heisenberg group's status. 'Tree-like or randomly connected' is operationalised as the infinite 3-regular tree (BFS-generated) and a configuration-model random 4-regular graph. All graphs are generated by analysis.py; there is no external download.";
<http://www.w3.org/2000/01/rdf-schema#label> "Claim test of wolfram2002-ch09-15: inconclusive, confidence 0.6";
<http://www.w3.org/2000/01/rdf-schema#seeAlso> <https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch09-15/result.json>;
prov:wasAttributedTo <https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1>;
<https://w3id.org/delphi/ontology/admissibleDatasets> "Planned datasets: nks-ch09-15-graph-family-generator";
<https://w3id.org/delphi/ontology/assessmentType> "claim-test/0.1.0";
<https://w3id.org/delphi/ontology/dataKind> "computational";
<https://w3id.org/delphi/ontology/disconfirmingObservation> "Pre-registered. 'refutes' if any lattice exponent differs from d by more than 0.5, or if either the tree or the random 4-regular graph has a slope below 3 over the stated range (i.e. appears to have a small finite dimension), or if log N(r) for the tree or random graph is better fit by a straight line in log r than in r. Any other outcome (e.g. a lattice exponent off by between 0.15 and 0.5) -> 'inconclusive' with reason 'other'.";
<https://w3id.org/delphi/ontology/evidenceAgainst> "'Corresponds to ordinary space' was not measured independently of the growth exponent; the status of each test graph was taken from its construction (lattice) or from the cited theorems (Heisenberg).",
"Every other pre-registered condition held: Z^1..Z^3 within 0.07 of d; the 3-regular tree and the random 4-regular graph have slopes above 6 that keep rising with r, and log N(r) is fit perfectly (R^2 = 1.000) by a line in r, i.e. exponential growth with no finite dimension, as the claim says of tree-like and random networks.",
"Post hoc (not pre-registered): the closed-form ball sizes of Z^d reproduce the BFS counts exactly, and the same slope estimator at R = 200 gives 0.997, 1.993, 2.989, 3.986 for d = 1..4, i.e. the estimator converges to d from below as the claim's 'limiting sense' requires. Had the tolerance been 0.2, the primary test would have passed and the pre-registered verdict would have been 'mixed'.",
"The Heisenberg Cayley graph has measured growth exponent 3.994 (degree 4, as proved by Bass 1972 and Guivarc'h 1973) although the group is not quasi-isometric to any Z^d (Pansu 1983). Under the strict reading of 'corresponds to ordinary d-dimensional space', r^d growth is therefore necessary but not sufficient; under Wolfram's own usage, in which dimension is defined by growth, the sentence is close to definitional. This bears on the 'precisely' (biconditional) wording and was the reason the pre-registered best outcome was 'mixed' rather than 'supports'.",
"The random 4-regular graph is a configuration-model multigraph; self-loops and multi-edges were not removed. They can only reduce N(r), so they bias against the exponential-growth conclusion that was nevertheless reached.",
"The verdict follows the pre-registered rule literally: the Z^4 slope at R = 20 (3.831) misses the pre-registered 0.15 tolerance by 0.019, which the rule maps to 'inconclusive'. The tolerance was set too tight for the finite radius chosen; this is a flaw of the pre-registration, not evidence against the claim.";
<https://w3id.org/delphi/ontology/expectedObservation> "Pre-registered. Primary (necessity and the tree/random clause): for the hypercubic lattices Z^1, Z^2, Z^3, Z^4 the growth exponent, estimated as the ordinary-least-squares slope of log N(r) against log r over r in [R/2, R] (R = 32 for d <= 3, R = 20 for d = 4), lies within 0.15 of d; for the infinite 3-regular tree (R = 13) and for a random 4-regular graph on n = 300000 nodes (R = largest radius with N(R) <= n/10) the same slope estimated over r in [R/2, R] exceeds 5, and log N(r) is better fit (higher R^2) by a straight line in r than in log r. If every one of these six conditions holds the primary test passes. Secondary (sufficiency / 'precisely'): the Cayley graph of the discrete Heisenberg group with generators x, y and inverses is known not to be quasi-isometric to any Z^d (Pansu 1983), yet has polynomial growth of degree 4 (Bass 1972 / Guivarc'h 1973). If its measured slope over r in [12, 24] lies in [3.5, 4.5], then a network with r^d growth exists that does not correspond to ordinary d-dimensional space in the strict sense. Verdict rule: primary passes and secondary slope in [3.5, 4.5] -> 'mixed' (the necessity direction and the tree/random clause hold; the sufficiency direction holds only if 'corresponds to ordinary d-dimensional space' is read as 'has growth dimension d'); primary passes and secondary slope outside [3.5, 4.5] -> 'supports'.";
<https://w3id.org/delphi/ontology/failCriterion> "Pre-registered. 'refutes' if any lattice exponent differs from d by more than 0.5, or if either the tree or the random 4-regular graph has a slope below 3 over the stated range (i.e. appears to have a small finite dimension), or if log N(r) for the tree or random graph is better fit by a straight line in log r than in r. Any other outcome (e.g. a lattice exponent off by between 0.15 and 0.5) -> 'inconclusive' with reason 'other'.";
<https://w3id.org/delphi/ontology/hasConfidence> "0.6"^^xsd:double;
<https://w3id.org/delphi/ontology/hasEvidence> <https://w3id.org/delphi/resource/evidence/wolfram2002-ch09-15--claim-test--dataset-1>;
<https://w3id.org/delphi/ontology/hasVerdict> "inconclusive";
<https://w3id.org/delphi/ontology/hypothesis> "For graphs that are discretisations of ordinary d-dimensional space (hypercubic lattices Z^d) the number of nodes within graph distance r grows like r^d, while for tree-like and randomly connected graphs the growth is exponential, so no finite exponent d exists; the biconditional ('precisely') additionally requires that polynomial growth of degree d occur only for networks corresponding to ordinary d-dimensional space.";
<https://w3id.org/delphi/ontology/inferenceMethod> "analysis.py generates each graph implicitly (lattices, tree, Heisenberg Cayley graph) or explicitly (configuration-model random 4-regular graph, n = 300000, numpy seed 20261031) and counts the closed ball N(r) around the origin by breadth-first search. The growth exponent is the OLS slope of log N(r) on log r over r in [R/2, R] with R = 32 (Z^1..Z^3), 20 (Z^4), 13 (tree), 8 (random graph; largest r with N(r) <= n/10) and 24 (Heisenberg). For the tree and random graph the fit of log N(r) linear in r is compared with the log-log fit by R^2. Exact closed-form ball sizes of Z^d were used post hoc to check the BFS counts and to show how the slope estimator converges.";
<https://w3id.org/delphi/ontology/passCriterion> "Pre-registered. Primary (necessity and the tree/random clause): for the hypercubic lattices Z^1, Z^2, Z^3, Z^4 the growth exponent, estimated as the ordinary-least-squares slope of log N(r) against log r over r in [R/2, R] (R = 32 for d <= 3, R = 20 for d = 4), lies within 0.15 of d; for the infinite 3-regular tree (R = 13) and for a random 4-regular graph on n = 300000 nodes (R = largest radius with N(R) <= n/10) the same slope estimated over r in [R/2, R] exceeds 5, and log N(r) is better fit (higher R^2) by a straight line in r than in log r. If every one of these six conditions holds the primary test passes. Secondary (sufficiency / 'precisely'): the Cayley graph of the discrete Heisenberg group with generators x, y and inverses is known not to be quasi-isometric to any Z^d (Pansu 1983), yet has polynomial growth of degree 4 (Bass 1972 / Guivarc'h 1973). If its measured slope over r in [12, 24] lies in [3.5, 4.5], then a network with r^d growth exists that does not correspond to ordinary d-dimensional space in the strict sense. Verdict rule: primary passes and secondary slope in [3.5, 4.5] -> 'mixed' (the necessity direction and the tree/random clause hold; the sufficiency direction holds only if 'corresponds to ordinary d-dimensional space' is read as 'has growth dimension d'); primary passes and secondary slope outside [3.5, 4.5] -> 'supports'.";
<https://w3id.org/delphi/ontology/preregistered> "false"^^xsd:boolean .
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<https://w3id.org/delphi/resource/assessment/wolfram2002-ch09-15--claim-test> .
<https://w3id.org/delphi/resource/evidence/wolfram2002-ch09-15--claim-test--dataset-1>
a <https://w3id.org/delphi/ontology/EvidenceItem>;
dc:description "ball sizes N(r) measured by breadth-first search from the origin node; radii up to R = 32 (Z^1..Z^3), 20 (Z^4), 24 (Heisenberg), and until N(r) > n/10 for the tree and random graph";
dc:license "generated";
<http://www.w3.org/2000/01/rdf-schema#label> "Dataset: Graph families generated by analysis.py: hypercubic lattices Z^1..Z^4 (nearest-neighbour, L1 adjacency), infinite 3-regular tree, random 4-regular configuration-model graph (n = 300000, numpy seed 20261031), discrete Heisenberg group Cayley graph (generators x, y and inverses)" .
}
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<http://www.w3.org/2000/01/rdf-schema#label> "Claim test of wolfram2002-ch09-15: inconclusive, confidence 0.6";
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