Checks · A dividing-plane barrier in the OpenAI forced Navier-Stokes blow-up construction

Recorded output of the verification program, October 1, 2026

The program wrote this file on October 1, 2026; its universal-time stamp, generated_utc, puts it on October 2. It holds 52 true-or-false checks, all true, and the summary flag all_checks_pass: 9 for part A, whose keys cite the OpenAI manuscript's equation numbers; 30 for the first half of part B (B1), two per integration, with the largest shear found (sup_a) and its bound (astar_at_end); 7 for the second half (B2); 6 for part C. Two limits: for h = 0 with the linearly growing strain the margin prints as 0.0, so the result rests on the sign of a derivative of 7.7 × 10⁻²¹, at the level of rounding (B1_note); and the series is compared with the integration only up to 0.144, half the radius its ratio test estimates (0.288).

Output of: The verification program for the barrier (checks/midplane_barrier.py), shown beside it on that page

Written by: midplane_barrier.py, a program by Claude Fable 5.1 (Anthropic)
Size: 7,228 bytes
SHA-256: 27753a1d4111c25dc8b17d934ceb6ea9eee2997c3798d5a0e9d0f9585a59eec2

The note's page Every file published with it This file on GitHub

Tables

generated_utc	2026-10-02T00:57:40Z
B1_note	a_below_2 is read from the sign of D_X H (min_DX_H > 0 and min_H > 0), since a = 2 - 2 D_X H / H; sup_a is a sampled double and prints as 2.0 where the gap 2 - a is below double precision (h = 0 cases)
cpu_seconds	1.015625
all_checks_pass	true

A

A1_material_derivative_is_4_14	true
A2_viscous_operator_is_4_over_r3_DX_DX_minus_1	true
A3_stress_free_equation_is_4_13	true
A4_midplane_ODE_from_full_equation	true
A5_a_equation_from_4_9	true
A5_riccati_equals_linear_ODE_over_H	true
A5_source_at_a_equals_2_is_minus_h	true
A6_lemma_4_3_axial_identity_4_16	true
A7_pressure_integration_by_parts	true

B1

	min_DX_H	min_H	sup_a	a_below_2	a_below_astar	astar_at_end	margin_2_minus_sup_a
h=0.0\|w=4	3.743049187636413e-12	1.999998500000001e-06	1.9999999999943854	true	true	2.0	5.614619880134342e-12
h=0.0\|w=0.5	2.000000500000001e-06	2.0000002500000006e-06	-2.499999691707444e-07	true	true	0.0	2.000000249999969
h=0.0\|w=1+3exp(-X)	1.999997000000001e-06	1.999998500000001e-06	0.8062183583685885	true	true	2.0	1.1937816416314115
h=0.0\|w=1+X/2	7.714999392078963e-21	2.0000000000000008e-06	2.0	true	true	2.0	0.0
h=0.0\|w=4+2sin(X)	2.0708986258795127e-12	1.999998500000001e-06	1.9999999999960834	true	true	2.0	3.916644786272627e-12
h=0.01\|w=4	1.9999970100000007e-06	1.9999985050000007e-06	1.9930954317926777	true	true	1.9933333333333334	0.006904568207322326
h=0.01\|w=0.5	2.0000005100000007e-06	2.000000255000001e-06	-2.549999673640002e-07	true	true	0.0	2.0000002549999674
h=0.01\|w=1+3exp(-X)	1.9999970100000007e-06	1.9999985050000007e-06	0.795004445902241	true	true	1.9933333266666633	1.204995554097759
h=0.01\|w=1+X/2	2.0000000100000006e-06	2.0000000050000007e-06	1.9979583174298992	true	true	1.998	0.0020416825701008445
h=0.01\|w=4+2sin(X)	1.9999970100000007e-06	1.9999985050000007e-06	1.9956626611623447	true	true	1.9959999850729215	0.004337338837655302
h=0.1\|w=4	1.999997100000001e-06	1.999998550000001e-06	1.9310311731884502	true	true	1.9333333333333333	0.06896882681154981
h=0.1\|w=0.5	2.000000600000001e-06	2.0000003000000006e-06	-2.9999995510010535e-07	true	true	0.0	2.000000299999955
h=0.1\|w=1+3exp(-X)	1.999997100000001e-06	1.999998550000001e-06	0.7076030486610181	true	true	1.9333332666666334	1.292396951338982
h=0.1\|w=1+X/2	2.0000001000000007e-06	2.0000000500000007e-06	1.9795851525523265	true	true	1.98	0.020414847447673523
h=0.1\|w=4+2sin(X)	1.999997100000001e-06	1.999998550000001e-06	1.9566659204232415	true	true	1.959999850729215	0.0433340795767585

B2

order	12
series_cpu_s	0.03125
eta_truncation_degree_K	14
U_n_at_0_all_zero	true
parity_U_odd_phi_even	true
phi_n_at_0	["1", "-299/400", "-171899/480000", "581711201/1152000000", "800893912001/4608000000000", "-5846553603409499/27648000000000000", "-22746132400203036299/232243200000000000000", "41589561486870482641867/867041280000000000000000", "1627869481088614805265127201/37456183296000000000000000000", "2109364251703765409071012036501/674211299328000000000000000000000", "-22990137257032348659941243268197357/2118949797888000000000000000000000000", "-862454272997921909605136252556968654933/130527307549900800000000000000000000000000", "-280233205060808740627787053845268051262057/1745336569524387840000000000000000000000000000"]
w_n_at_0	["4", "351/40", "5063/1200", "17463/6400", "5591060201/5760000000", "11501866471/43200000000", "11621292009114199/145152000000000000", "39602288202034721/1161216000000000000", "7112103618232405846289/1040449536000000000000000", "-77403953563075669229563/35115171840000000000000000", "-6781589928400379939772942160007/9270405365760000000000000000000000", "50694816580907623135845418067671/81579567218688000000000000000000000", "8909706035690970286187669727159736277/27970137332121600000000000000000000000000"]
midplane_ODE_residual_coefficients_0_to_N	["0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"]
midplane_ODE_holds_to_order_N	true
radius_estimate_from_ratio_test	0.2883597618942812
X_max_tested	0.1441798809471406
sup_a_series_on_[0,Xmax]	0.26601861065173016
a_series_below_2	true
max_abs_diff_series_vs_ODE	3.897743239278384e-12
series_matches_ODE_1e-6	true
asymmetric_U0_at_0	1/20
asymmetric_U1_at_0	451/4000
asymmetric_breaks_hypothesis	true
S_partial_midplane_negative	true
S_partial_midplane_value	-0.008901958759995906

C

M_candidate_solves_exterior_condition	true
M_solution_even	true
M_solution_matches_(1-eta2)^D	true
S_candidate_solves_tail_condition	true
S_solution_matches_(1-eta2)^(-2h)	true
S_candidate_exponent_negative_for_h_positive	true
note	first-order linear homogeneous ODEs: one-dimensional solution spaces

JSON

{
  "generated_utc": "2026-10-02T00:57:40Z",
  "A": {
    "A1_material_derivative_is_4_14": true,
    "A2_viscous_operator_is_4_over_r3_DX_DX_minus_1": true,
    "A3_stress_free_equation_is_4_13": true,
    "A4_midplane_ODE_from_full_equation": true,
    "A5_a_equation_from_4_9": true,
    "A5_riccati_equals_linear_ODE_over_H": true,
    "A5_source_at_a_equals_2_is_minus_h": true,
    "A6_lemma_4_3_axial_identity_4_16": true,
    "A7_pressure_integration_by_parts": true
  },
  "B1": {
    "h=0.0|w=4": {
      "min_DX_H": 3.743049187636413e-12,
      "min_H": 1.999998500000001e-06,
      "sup_a": 1.9999999999943854,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 2.0,
      "margin_2_minus_sup_a": 5.614619880134342e-12
    },
    "h=0.0|w=0.5": {
      "min_DX_H": 2.000000500000001e-06,
      "min_H": 2.0000002500000006e-06,
      "sup_a": -2.499999691707444e-07,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 0.0,
      "margin_2_minus_sup_a": 2.000000249999969
    },
    "h=0.0|w=1+3exp(-X)": {
      "min_DX_H": 1.999997000000001e-06,
      "min_H": 1.999998500000001e-06,
      "sup_a": 0.8062183583685885,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 2.0,
      "margin_2_minus_sup_a": 1.1937816416314115
    },
    "h=0.0|w=1+X/2": {
      "min_DX_H": 7.714999392078963e-21,
      "min_H": 2.0000000000000008e-06,
      "sup_a": 2.0,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 2.0,
      "margin_2_minus_sup_a": 0.0
    },
    "h=0.0|w=4+2sin(X)": {
      "min_DX_H": 2.0708986258795127e-12,
      "min_H": 1.999998500000001e-06,
      "sup_a": 1.9999999999960834,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 2.0,
      "margin_2_minus_sup_a": 3.916644786272627e-12
    },
    "h=0.01|w=4": {
      "min_DX_H": 1.9999970100000007e-06,
      "min_H": 1.9999985050000007e-06,
      "sup_a": 1.9930954317926777,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.9933333333333334,
      "margin_2_minus_sup_a": 0.006904568207322326
    },
    "h=0.01|w=0.5": {
      "min_DX_H": 2.0000005100000007e-06,
      "min_H": 2.000000255000001e-06,
      "sup_a": -2.549999673640002e-07,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 0.0,
      "margin_2_minus_sup_a": 2.0000002549999674
    },
    "h=0.01|w=1+3exp(-X)": {
      "min_DX_H": 1.9999970100000007e-06,
      "min_H": 1.9999985050000007e-06,
      "sup_a": 0.795004445902241,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.9933333266666633,
      "margin_2_minus_sup_a": 1.204995554097759
    },
    "h=0.01|w=1+X/2": {
      "min_DX_H": 2.0000000100000006e-06,
      "min_H": 2.0000000050000007e-06,
      "sup_a": 1.9979583174298992,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.998,
      "margin_2_minus_sup_a": 0.0020416825701008445
    },
    "h=0.01|w=4+2sin(X)": {
      "min_DX_H": 1.9999970100000007e-06,
      "min_H": 1.9999985050000007e-06,
      "sup_a": 1.9956626611623447,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.9959999850729215,
      "margin_2_minus_sup_a": 0.004337338837655302
    },
    "h=0.1|w=4": {
      "min_DX_H": 1.999997100000001e-06,
      "min_H": 1.999998550000001e-06,
      "sup_a": 1.9310311731884502,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.9333333333333333,
      "margin_2_minus_sup_a": 0.06896882681154981
    },
    "h=0.1|w=0.5": {
      "min_DX_H": 2.000000600000001e-06,
      "min_H": 2.0000003000000006e-06,
      "sup_a": -2.9999995510010535e-07,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 0.0,
      "margin_2_minus_sup_a": 2.000000299999955
    },
    "h=0.1|w=1+3exp(-X)": {
      "min_DX_H": 1.999997100000001e-06,
      "min_H": 1.999998550000001e-06,
      "sup_a": 0.7076030486610181,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.9333332666666334,
      "margin_2_minus_sup_a": 1.292396951338982
    },
    "h=0.1|w=1+X/2": {
      "min_DX_H": 2.0000001000000007e-06,
      "min_H": 2.0000000500000007e-06,
      "sup_a": 1.9795851525523265,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.98,
      "margin_2_minus_sup_a": 0.020414847447673523
    },
    "h=0.1|w=4+2sin(X)": {
      "min_DX_H": 1.999997100000001e-06,
      "min_H": 1.999998550000001e-06,
      "sup_a": 1.9566659204232415,
      "a_below_2": true,
      "a_below_astar": true,
      "astar_at_end": 1.959999850729215,
      "margin_2_minus_sup_a": 0.0433340795767585
    }
  },
  "B1_note": "a_below_2 is read from the sign of D_X H (min_DX_H > 0 and min_H > 0), since a = 2 - 2 D_X H / H; sup_a is a sampled double and prints as 2.0 where the gap 2 - a is below double precision (h = 0 cases)",
  "B2": {
    "order": 12,
    "series_cpu_s": 0.03125,
    "eta_truncation_degree_K": 14,
    "U_n_at_0_all_zero": true,
    "parity_U_odd_phi_even": true,
    "phi_n_at_0": [
      "1",
      "-299/400",
      "-171899/480000",
      "581711201/1152000000",
      "800893912001/4608000000000",
      "-5846553603409499/27648000000000000",
      "-22746132400203036299/232243200000000000000",
      "41589561486870482641867/867041280000000000000000",
      "1627869481088614805265127201/37456183296000000000000000000",
      "2109364251703765409071012036501/674211299328000000000000000000000",
      "-22990137257032348659941243268197357/2118949797888000000000000000000000000",
      "-862454272997921909605136252556968654933/130527307549900800000000000000000000000000",
      "-280233205060808740627787053845268051262057/1745336569524387840000000000000000000000000000"
    ],
    "w_n_at_0": [
      "4",
      "351/40",
      "5063/1200",
      "17463/6400",
      "5591060201/5760000000",
      "11501866471/43200000000",
      "11621292009114199/145152000000000000",
      "39602288202034721/1161216000000000000",
      "7112103618232405846289/1040449536000000000000000",
      "-77403953563075669229563/35115171840000000000000000",
      "-6781589928400379939772942160007/9270405365760000000000000000000000",
      "50694816580907623135845418067671/81579567218688000000000000000000000",
      "8909706035690970286187669727159736277/27970137332121600000000000000000000000000"
    ],
    "midplane_ODE_residual_coefficients_0_to_N": [
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0",
      "0"
    ],
    "midplane_ODE_holds_to_order_N": true,
    "radius_estimate_from_ratio_test": 0.2883597618942812,
    "X_max_tested": 0.1441798809471406,
    "sup_a_series_on_[0,Xmax]": 0.26601861065173016,
    "a_series_below_2": true,
    "max_abs_diff_series_vs_ODE": 3.897743239278384e-12,
    "series_matches_ODE_1e-6": true,
    "asymmetric_U0_at_0": "1/20",
    "asymmetric_U1_at_0": "451/4000",
    "asymmetric_breaks_hypothesis": true,
    "S_partial_midplane_negative": true,
    "S_partial_midplane_value": -0.008901958759995906
  },
  "C": {
    "M_candidate_solves_exterior_condition": true,
    "M_solution_even": true,
    "M_solution_matches_(1-eta2)^D": true,
    "S_candidate_solves_tail_condition": true,
    "S_solution_matches_(1-eta2)^(-2h)": true,
    "S_candidate_exponent_negative_for_h_positive": true,
    "note": "first-order linear homogeneous ODEs: one-dimensional solution spaces"
  },
  "cpu_seconds": 1.015625,
  "all_checks_pass": true
}