. . . . "analysis.py builds, for 3 seeds each, random geometric graphs with density 3183 and epsilon 0.045 on the flat 2-torus (40000 nodes), unit 2-sphere (40000), 2-sphere of radius 1.5 (90000) and hyperbolic plane (curvature -1, disk of hyperbolic radius 2.2, 71356 nodes), and with density 7600 and epsilon 0.086 on the flat 3-torus and unit 3-sphere (150018 nodes each); runs BFS from 500 sources per graph (hyperbolic: sources within hyperbolic radius 1.0 of the centre), averages the shell counts S(r), measures on the flat graphs the mean geodesic radius s(r) of hop-r nodes, and over hops with 0.25 <= s(r) <= 0.7 computes the relative correction Q(r) = S_curved/S_flat - 1, its sign, the log-log slope of S_curved - S_flat against r, and the through-origin coefficient k in Q = k s(r)^2 compared with the continuum value -R/(6d)." . . "Measured: 1.008 fitted curvature coefficient k divided by the continuum prediction -R/(6d) (hyperbolic plane); other geometries in extra (n = 4, shell-count SD across 3 graphs is 0.2-1% of S in the fit range (derived/measured.json S_sd)). Interpretation: Wolfram's count is of nodes reached by going r connections, i.e. the shell at graph distance r (claims.json paraphrases it as 'within distance r'; the test follows the book's wording and the r^(d+1) exponent, which is the shell exponent). 'Large network approximating d-dimensional space' is realised as random geometric graphs on constant-curvature model spaces, whose hop distance approximates geodesic distance up to a stretch factor that is measured on the flat graph of the same density and applied to the curved ones. Proportionality to the Ricci scalar is tested against the continuum coefficient -R/(6d) of the geodesic-sphere area expansion (Gray 2004, Tubes, ch. 3) across R = 8/9, 2, -2 (d = 2) and R = 6 (d = 3); only constant-curvature spaces are used, so the claim that no other curvature invariant enters at leading order is tested only through the coefficient, not through anisotropic examples." . "Claim test of wolfram2002-ch09-24: supports, confidence 0.85" . . . "Planned datasets: rgg-constant-curvature-generator" . "claim-test/0.1.0" . "computational" . "Pre-registered. 'Refutes' if the sign in (a) is wrong for any geometry over the fit range, or if any exponent in (b) lies outside [d, d+2], or if any ratio in (c) lies outside [0.4, 2.5] (the correction is not proportional to the Ricci scalar with the continuum coefficient). Other combinations (e.g. correct sign and exponent but coefficient off by 30-60%) are 'mixed'." . "Deviation from the pre-registration: the hyperbolic patch was pre-registered as a disk of hyperbolic radius 1.8 with sources within 0.2 of the centre. That region contains only about 400 nodes, so the 500 'sources' were essentially one overlapping sample and the hyperbolic shell counts fluctuated by 2-4% between seeds (the first run gave Q(8) = -0.001 and exponent 4.9, failing (a) and (b) on noise). The patch was enlarged to radius 2.2 with sources within radius 1.0 (about 10900 candidate nodes) before any threshold was changed; all other parameters and all criteria are as pre-registered. First-run hyperbolic numbers for the record: Q over hops 8-18 = [-0.0014, 0.0123, 0.0239, 0.0361, 0.0463, 0.0584, 0.0606, 0.0715, 0.0757, 0.0725, 0.0696], slope 4.93, k ratio 1.18, S(1) = 19.9 versus 20.4 on the torus." . "Hop distance differs from geodesic distance by a stretch factor (s(r)/r ~ 0.038 per hop versus epsilon = 0.045); using the flat graph's s(r) for the curved graphs assumes the stretch is curvature-independent, which holds to O(R epsilon^2) ~ 0.4%." . "Only constant-curvature spaces were used, so the statement that the coefficient involves the Ricci scalar and no other curvature invariant is tested only through agreement with the continuum coefficient -R/(6d) in d = 2 and d = 3; an anisotropic (e.g. product) manifold would be needed to separate the Ricci scalar from other invariants." . "The exponent fit is over a short range of hops (8-18 in d = 2, 5-10 in d = 3) and is sensitive to the discreteness of the hop shells; slopes of 2.8-3.5 (d = 2) and 4.25 (d = 3) are consistent with d+1 within the pre-registered tolerance but not a sharp determination." . "The fitted coefficients are 8% below the continuum value for the spheres and 1% above for the hyperbolic plane; the shortfall is within the pre-registered tolerance and is consistent with the finite width of hop shells and the O(s^4) terms, but was not modelled." . "Pre-registered. Random geometric graphs with identical node density (3183 per unit area in d = 2, 7600 per unit volume in d = 3) and connection radius (geodesic epsilon = 0.045 in d = 2, 0.086 in d = 3, mean degree about 20) on: flat torus (R = 0), unit 2-sphere (R = 2), 2-sphere of radius 1.5 (R = 8/9), hyperbolic plane of curvature -1 (R = -2; sampled within hyperbolic radius 1.8, sources within 0.2 of the centre), flat 3-torus (R = 0) and unit 3-sphere (R = 6); 3 independent graphs per geometry, 500 BFS sources per graph. Let S_g(r) be the mean number of nodes at hop distance r, s(r) the mean geodesic distance of hop-r nodes measured in the flat graph of the same dimension, and Q_g(r) = S_g(r)/S_flat(r) - 1. Fit range: hops r with 0.25 <= s(r) <= 0.7. (a) Sign: Q_g(r) < 0 for every r in the fit range for the three positively curved geometries and > 0 for the hyperbolic plane. (b) Exponent: the least-squares slope of log|S_g(r) - S_flat(r)| against log r over the fit range lies within d+1 +/- 0.6 for every curved geometry. (c) Proportionality to R: the coefficient k_g of the through-origin fit Q_g(r) = k_g s(r)^2 over the fit range satisfies 0.7 <= k_g / (-R_g/(6d)) <= 1.3 for every curved geometry, and k(unit sphere)/k(sphere 1.5) is within 30% of 2.25. 'Supports' requires (a), (b) and (c)." . "Pre-registered. 'Refutes' if the sign in (a) is wrong for any geometry over the fit range, or if any exponent in (b) lies outside [d, d+2], or if any ratio in (c) lies outside [0.4, 2.5] (the correction is not proportional to the Ricci scalar with the continuum coefficient). Other combinations (e.g. correct sign and exponent but coefficient off by 30-60%) are 'mixed'." . "0.85"^^ . . "supports" . "In a network that approximates a d-dimensional Riemannian space, the number of nodes at graph distance r grows like r^(d-1), and the leading curvature correction to it is proportional to the Ricci scalar R times r^(d+1): in the continuum the geodesic sphere of radius s has area d*omega_d*s^(d-1)*(1 - R s^2/(6d) + O(s^4)), so for a random geometric graph of fixed local density and connection radius the shell count on a curved space divided by the flat shell count equals 1 - R s(r)^2/(6d) to leading order, where s(r) is the geodesic radius reached in r hops; the correction is negative for positive curvature and positive for negative curvature." . "analysis.py builds, for 3 seeds each, random geometric graphs with density 3183 and epsilon 0.045 on the flat 2-torus (40000 nodes), unit 2-sphere (40000), 2-sphere of radius 1.5 (90000) and hyperbolic plane (curvature -1, disk of hyperbolic radius 2.2, 71356 nodes), and with density 7600 and epsilon 0.086 on the flat 3-torus and unit 3-sphere (150018 nodes each); runs BFS from 500 sources per graph (hyperbolic: sources within hyperbolic radius 1.0 of the centre), averages the shell counts S(r), measures on the flat graphs the mean geodesic radius s(r) of hop-r nodes, and over hops with 0.25 <= s(r) <= 0.7 computes the relative correction Q(r) = S_curved/S_flat - 1, its sign, the log-log slope of S_curved - S_flat against r, and the through-origin coefficient k in Q = k s(r)^2 compared with the continuum value -R/(6d)." . "Pre-registered. Random geometric graphs with identical node density (3183 per unit area in d = 2, 7600 per unit volume in d = 3) and connection radius (geodesic epsilon = 0.045 in d = 2, 0.086 in d = 3, mean degree about 20) on: flat torus (R = 0), unit 2-sphere (R = 2), 2-sphere of radius 1.5 (R = 8/9), hyperbolic plane of curvature -1 (R = -2; sampled within hyperbolic radius 1.8, sources within 0.2 of the centre), flat 3-torus (R = 0) and unit 3-sphere (R = 6); 3 independent graphs per geometry, 500 BFS sources per graph. Let S_g(r) be the mean number of nodes at hop distance r, s(r) the mean geodesic distance of hop-r nodes measured in the flat graph of the same dimension, and Q_g(r) = S_g(r)/S_flat(r) - 1. Fit range: hops r with 0.25 <= s(r) <= 0.7. (a) Sign: Q_g(r) < 0 for every r in the fit range for the three positively curved geometries and > 0 for the hyperbolic plane. (b) Exponent: the least-squares slope of log|S_g(r) - S_flat(r)| against log r over the fit range lies within d+1 +/- 0.6 for every curved geometry. (c) Proportionality to R: the coefficient k_g of the through-origin fit Q_g(r) = k_g s(r)^2 over the fit range satisfies 0.7 <= k_g / (-R_g/(6d)) <= 1.3 for every curved geometry, and k(unit sphere)/k(sphere 1.5) is within 30% of 2.25. 'Supports' requires (a), (b) and (c)." . "false"^^ . . "networks whose shell counts are measured" . "CC0 (generated data)" . . "Dataset: Random geometric graphs on flat tori, spheres and the hyperbolic plane (generated by analysis.py)" . . "delphisci claim tester (Claude Fable 5.1 (Cursor cloud agent); cursor/claim-tests-wolfram-nks-f0e1)" . . . "Claude Fable 5.1 (Cursor cloud agent)" . . . . . . . . . "2026-11-08T16:24:31.934Z"^^ . . . . . "Claim test of wolfram2002-ch09-24: supports, confidence 0.85" . . "RSA" . 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