{
 "meta": {
  "seed": 20260729,
  "description": "Architecture-matched conservative (F=-grad E) vs direct force surrogates for a 6-particle Morse cluster: matched one-step force RMSE, wildly different Jacobian symmetry, loop work and MD energy drift.",
  "n_pass": 30,
  "n_total": 30
 },
 "system": {
  "n_particles": 6,
  "dim": 3,
  "morse_D_e": 1.0,
  "morse_a": 1.5,
  "morse_r_e": 1.0,
  "trap_k": 0.6,
  "equilibrium_energy": -13.649007101892641,
  "equilibrium_max_force": 1.6653345369377348e-15
 },
 "basis": {
  "n_radial": 6,
  "mu": [
   0.7,
   1.04,
   1.38,
   1.72,
   2.0599999999999996,
   2.4
  ],
  "sigma": 0.36,
  "r_cut": 3.0,
  "n_params_direct": 79,
  "n_params_conservative": 28,
  "conservative_is_subspace_of_direct": true
 },
 "data": {
  "n_configs": 1600,
  "n_train": 880,
  "n_val": 320,
  "n_test": 400,
  "force_label_noise_sigma": 0.03,
  "force_rms": 0.9412992507882301,
  "noise_fraction_of_force_rms": 0.03187084232233101,
  "pair_distance_min": 0.5652773215760463,
  "pair_distance_max": 1.9051530358086946,
  "pair_distance_mean": 1.0461950020801734,
  "fraction_pairs_beyond_cutoff": 0.0
 },
 "equivariance": {
  "max_relative_deviation_conservative": 1.494091049619226e-14,
  "max_relative_deviation_direct": 2.6159264334344125e-14,
  "note": "both models are exactly O(3)-equivariant; equivariance is not what distinguishes them"
 },
 "models": {
  "conservative": {
   "lam": 1e-11,
   "rmse_train_clean": 0.012211993170233951,
   "rmse_test_clean": 0.012939981682771502,
   "rmse_test_noisy": 0.032842053650509205
  },
  "direct": {
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   "rmse_train_clean": 0.010865052387261305,
   "rmse_test_clean": 0.011397006513125503,
   "rmse_test_noisy": 0.03236821663166589
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  "lambda_grid": [
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   1e-08,
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   4.641588833612773e-08,
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  "gradient_consistency_rel_error": 2.5838962903468864e-09,
  "fast_force_path_rel_error": 1.716456252541661e-14,
  "force_rmse_ratio_direct_over_conservative": 0.88075909166855
 },
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  "rms_relative_conservative": 1.4112569284376822e-08,
  "rms_relative_direct": 0.007315143372517744,
  "rms_absolute_conservative": 4.249664843098241e-08,
  "rms_absolute_direct": 0.021816861766092956,
  "rms_atomic_curl_conservative": 1.7353323291206739e-07,
  "rms_atomic_curl_direct": 0.06145439937905052,
  "ratio_direct_over_conservative": 518342.42405568913
 },
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  "n_loops_per_radius": 4,
  "quadrature_points": 2048,
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 },
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  "drift_frac_of_E0_total_conservative": 4.542927480529169e-07,
  "drift_frac_of_KE0_total_conservative": 9.834306959762316e-06,
  "drift_frac_of_E0_total_direct": 0.0013011841169074593,
  "drift_frac_of_KE0_total_direct": 0.028167396621846752,
  "drift_ratio_direct_over_conservative": 149.02733607325092,
  "n_trajectories_gaining_energy_direct": 15,
  "sign_bias_binomial_p_direct": 0.30745625495910645,
  "paired_frac_direct_worse": 1.0,
  "paired_secular_ratio_median": 209.71110127449668,
  "paired_secular_ratio_min": 37.52707506896779,
  "note": "median over 24 independent initial conditions; the mean is not used because a minority of direct-model trajectories run away and would dominate it (frac_runaway_direct records that minority)",
  "sign_note": "NEGATIVE RESULT, reported rather than tuned away: the direct model's drift is NOT a systematic heating. Signs are mixed (15/24 trajectories gain energy, binomial p = 0.307), so a non-conservative field is better described as pumping energy in or out depending on where in phase space the trajectory goes than as an effective friction. What is systematic is the magnitude: every single trajectory drifts faster under the direct model than the conservative one, by at least 38x."
 },
 "dt_sweep": {
  "dt": [
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  "total_time": 50.0,
  "n_initial_conditions": 24,
  "runs": [
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    "n_steps": 40000,
    "total_time": 50.0,
    "E0": -12.316362042442924,
    "KE0": 0.5689505097989399,
    "median_slope_conservative": 7.545991333541362e-07,
    "mean_slope_conservative": 1.6896763714919083e-06,
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  "note": "A bounded energy error fitted over a window of length T yields an apparent rate ~ 1/T, so the conservative model's rate must fall by about the time ratio while its max|dE| stays put; a genuine non-conservative field has a T-independent rate and an error that accumulates linearly."
 },
 "browser": {
  "note": "Everything the tutorial page needs to re-run this experiment live: the two fitted coefficient vectors in the shared 79-feature raw basis, a reference configuration, and reference values the JavaScript reimplementation is checked against.",
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     0.11269433925437522
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     -0.4736737398068575,
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     0.5428202125096255,
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    [
     0.32914782426715145,
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    [
     0.13510686056427712,
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    ],
    [
     0.530029961746745,
     -0.37481490257693745,
     -0.13992130918303425
    ]
   ]
  ],
  "md_ic_V": [
   [
    [
     0.07372379426596665,
     -0.14572598935286107,
     0.4195657291223342
    ],
    [
     -0.14071806839288376,
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     0.5632914086856365
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     -0.17605711951168762,
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     -0.6126774821598141
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     -0.28039289081109986,
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     -0.014623483509470542
    ],
    [
     0.22050573187820333,
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     -0.01562172516116693
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     0.3029385525715012,
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     -0.28836084999727096,
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    [
     0.2280489609595295,
     -0.5794125920838713,
     0.12437753413840197
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    [
     0.05459395339006094,
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     0.1418311485216437
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     0.330253389209496,
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     -0.4535686240459935,
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   [
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     0.03434782465624449,
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    [
     -0.5267936233470502,
     0.040275273888279686,
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     0.22719424925125148
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     0.13081638447027705,
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     0.23470739824550124,
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     0.09719845026293975
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     0.006288976350976014,
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    ]
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   [
    [
     -0.16251692379328442,
     0.20932533149143473,
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    ],
    [
     -0.10916625517220102,
     -0.48144722365392867,
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    [
     -0.0020111881241471896,
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    [
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     -0.1267896046050341
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    [
     -0.033475100234386324,
     0.04812141320141307,
     0.14542071981339627
    ],
    [
     0.511435043605891,
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     0.1797663312477112
    ]
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   [
    [
     0.28299900120479343,
     0.5130676195997295,
     -0.1037922386647647
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     0.10236296994267322,
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     -0.1420977690626222,
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     0.3483043261256311
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     -0.21384285958735302,
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     -0.06327354008882853,
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     -0.1368151173575211,
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     -0.10283151521985703,
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    [
     -0.291428896596344,
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     0.4205266278836301
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    [
     0.18863083310372586,
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     0.29601503889051617
    ],
    [
     -0.03955470016422928,
     0.048596258491091934,
     -0.48062330455539104
    ],
    [
     0.3819993962342256,
     0.24887687755397364,
     0.18312250969484872
    ]
   ],
   [
    [
     0.3450682700398572,
     0.029880973089699968,
     0.19213788318558095
    ],
    [
     0.01824177603983055,
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     0.11352043033873178
    ],
    [
     0.08346850302767647,
     -0.11885739670697568,
     -0.12919547149799887
    ],
    [
     0.3885094857998149,
     0.36678786154350207,
     -0.31382603867152165
    ],
    [
     -0.5249390691453731,
     -0.36383962416101784,
     0.002995043812107606
    ],
    [
     -0.31034896576180604,
     -0.017915548004203946,
     0.1343681528331001
    ]
   ],
   [
    [
     0.01033867148547199,
     -0.0004450154398643186,
     -0.03315290836518523
    ],
    [
     0.25200999173160965,
     0.05792115312248889,
     -0.247744408190904
    ],
    [
     0.044979406640233366,
     -0.36604380229968775,
     -0.04324680992395146
    ],
    [
     -0.2586447029477967,
     -0.2932923420824824,
     0.04828432938615507
    ],
    [
     -0.01579943946332865,
     0.146785982709232,
     0.46536148009096
    ],
    [
     -0.032883927446189606,
     0.4550740239903135,
     -0.18950168299707437
    ]
   ]
  ],
  "md_ic_note": "the first 8 of the 24 initial conditions behind the MD aggregates above, in the same order",
  "reference_jacobian_relative_asymmetry_conservative": 9.382901833708656e-09,
  "reference_atomic_curl_conservative": 1.6267678801758474e-07,
  "reference_jacobian_relative_asymmetry_direct": 0.01700048357877038,
  "reference_atomic_curl_direct": 0.22332323756544967,
  "reference_loop": {
   "U": [
    [
     -0.15160381351533297,
     0.41572876004506415,
     -0.2422900540892567
    ],
    [
     0.20878450720620798,
     -0.10708869856205365,
     0.1425988041661171
    ],
    [
     -0.21984863305283223,
     -0.16924037761910182,
     0.10310968497150488
    ],
    [
     0.13532180477000164,
     0.3839188059210564,
     0.38807866020811627
    ],
    [
     0.38221550228118983,
     -0.005774754345470982,
     0.24574876574221538
    ],
    [
     0.0699439971406269,
     -0.23395448186574727,
     0.005353099222722795
    ]
   ],
   "V": [
    [
     0.1187953843319186,
     -0.06962450240665435,
     -0.34140203687128734
    ],
    [
     0.37215890909937216,
     0.4655431473502355,
     -0.20752040613678097
    ],
    [
     0.1327792157762417,
     -0.028884526005541325,
     -0.2547592426053544
    ],
    [
     0.2772613149781314,
     -0.24034191059621288,
     0.266756766616146
    ],
    [
     -0.11828618134436214,
     0.264558059286522,
     -0.03883487172402047
    ],
    [
     0.2821544090314049,
     -0.01603620172834902,
     0.10800085155530195
    ]
   ],
   "radius": 0.35,
   "n_quadrature": 2048,
   "net_work_conservative": 1.743934249004316e-16,
   "gross_work_conservative": 3.7396410375388736,
   "net_work_direct": -0.024644271773434503,
   "gross_work_direct": 3.7569942823262155
  },
  "reference_trajectory": {
   "dt": 0.005,
   "n_steps": 1000,
   "record_stride": 50,
   "t": [
    0.0,
    0.25,
    0.5,
    0.75,
    1.0,
    1.25,
    1.5,
    1.75,
    2.0,
    2.25,
    2.5,
    2.75,
    3.0,
    3.25,
    3.5,
    3.75,
    4.0,
    4.25,
    4.5,
    4.75,
    5.0
   ],
   "e_total_conservative": [
    -11.647070771218704,
    -11.649196931074922,
    -11.649544683369932,
    -11.648934964424368,
    -11.648916103350246,
    -11.647989286873319,
    -11.648187803009384,
    -11.647961069127827,
    -11.666005333978475,
    -11.64374398573526,
    -11.649246037316253,
    -11.645848263287613,
    -11.650231726776632,
    -11.642930705220936,
    -11.643756339280078,
    -11.64493641951512,
    -11.644156072020722,
    -11.64749349143675,
    -11.6466062233161,
    -11.649766414358131,
    -11.646940229996321
   ],
   "e_total_direct": [
    -11.647070771218704,
    -11.644291480677571,
    -11.645273126469359,
    -11.647691598683918,
    -11.647434679323881,
    -11.642177726282233,
    -11.64075690284895,
    -11.640970052767083,
    -11.640362883321698,
    -11.607180895706545,
    -11.612120334147638,
    -11.609396330670311,
    -11.611833717206641,
    -11.60904271788954,
    -11.613944466333724,
    -11.613717135497533,
    -11.610313524084395,
    -11.614538132277241,
    -11.616181910735415,
    -11.613537745804843,
    -11.611729287663259
   ],
   "note": "deliberately short: the dynamics are chaotic, so a second implementation tracks this trajectory to roundoff over 5 time units but cannot over the 200 used for the statistics"
  }
 },
 "leaderboard_context": {
  "note": "TRANSCRIBED FROM PUBLISHED SOURCES, NOT COMPUTED BY THIS SCRIPT. These are the only numbers in this experiment that were typed in rather than measured; they are recorded here with their provenance and their counterexamples so that the caveats travel with them.",
  "metric": "kappa_SRME: symmetric relative mean error of the predicted phonon thermal conductivity, 0 perfect, 2 worst possible",
  "worst_possible_kappa_srme": 2.0,
  "association": "On the Matbench Discovery leaderboard every submitted direct-force model sits above kappa_SRME 1.6, and every submission below 1.0 predicts forces as an energy gradient.",
  "entries": [
   {
    "model": "eqV2 M",
    "forces": "direct",
    "f1": 0.917,
    "kappa_srme": 1.7707,
    "rmsd": 0.0691,
    "source": "Matbench Discovery leaderboard"
   },
   {
    "model": "CHGNet",
    "forces": "gradient",
    "f1": null,
    "kappa_srme": 1.7167,
    "rmsd": null,
    "source": "Matbench Discovery leaderboard (targets EFS_GM)"
   },
   {
    "model": "M3GNet",
    "forces": "gradient",
    "f1": null,
    "kappa_srme": 1.4094,
    "rmsd": null,
    "source": "Matbench Discovery leaderboard (targets EFS_G)"
   },
   {
    "model": "AlphaNet-v1-MPtrj",
    "forces": "gradient",
    "f1": null,
    "kappa_srme": 1.3046,
    "rmsd": null,
    "source": "Matbench Discovery leaderboard"
   },
   {
    "model": "orb-v3-direct-inf-mpa",
    "forces": "direct",
    "f1": 0.883,
    "kappa_srme": 0.348,
    "rmsd": null,
    "source": "Rhodes et al., arXiv:2504.06231, Table 1"
   }
  ],
  "refutation": "Non-conservativeness is neither necessary nor sufficient for a bad kappa_SRME: CHGNet is conservative and scores 1.7167, essentially the same as eqV2 M, while a direct-force variant of orb-v3 reaches 0.348.",
  "mechanism": "kappa_SRME depends on force constants -- second derivatives of the energy surface -- which direct-force training does not constrain, and which a conservative model can also get badly wrong."
 },
 "punchline": {
  "force_rmse_conservative": 0.012939981682771502,
  "force_rmse_direct": 0.011397006513125503,
  "force_rmse_ratio": 0.88075909166855,
  "loop_work_relative_ratio": 11822849067219.045,
  "jacobian_asymmetry_ratio": 518342.42405568913,
  "md_drift_ratio": 149.02733607325092,
  "frac_direct_trajectories_runaway": 0.125,
  "statement": "one-step force RMSE differs by a factor 0.881 (the direct model is the *better* one); median MD energy drift rate differs by a factor 149"
 },
 "checks": [
  {
   "name": "analytic gradient of E_theta matches finite differences",
   "pass": true,
   "detail": "rel error 2.58e-09"
  },
  {
   "name": "contracted force evaluator reproduces the explicit 79-feature basis",
   "pass": true,
   "detail": "max rel dev 1.72e-14"
  },
  {
   "name": "all training pairs lie inside the radial cutoff",
   "pass": true,
   "detail": "max pair distance 1.905 vs r_cut 3.0"
  },
  {
   "name": "conservative model is O(3)-equivariant: F(RX) = R F(X)",
   "pass": true,
   "detail": "max rel dev 1.49e-14"
  },
  {
   "name": "direct model is O(3)-equivariant: F(RX) = R F(X)",
   "pass": true,
   "detail": "max rel dev 2.62e-14"
  },
  {
   "name": "conservative model force RMSE is small vs force scale",
   "pass": true,
   "detail": "RMSE 0.0129 = 1.37% of force RMS 0.941"
  },
  {
   "name": "direct model force RMSE is small vs force scale",
   "pass": true,
   "detail": "RMSE 0.0114 = 1.21% of force RMS 0.941"
  },
  {
   "name": "the two force RMSEs agree within 20%",
   "pass": true,
   "detail": "ratio direct/conservative 0.8808"
  },
  {
   "name": "conservative model Jacobian is symmetric to the finite-difference floor",
   "pass": true,
   "detail": "rms ||asym||/||J|| 1.41e-08"
  },
  {
   "name": "direct model's Jacobian asymmetry is of the same order as its own relative force error -- nothing suppresses it",
   "pass": true,
   "detail": "rms ||asym||/||J|| 7.315e-03 vs relative force error 1.211e-02"
  },
  {
   "name": "asymmetry separates the models by >1e4",
   "pass": true,
   "detail": "ratio 5.183e+05"
  },
  {
   "name": "direct model has nonzero per-atom curl, conservative does not",
   "pass": true,
   "detail": "curl rms direct 6.145e-02 vs conservative 1.74e-07"
  },
  {
   "name": "conservative model does zero net work on every loop",
   "pass": true,
   "detail": "max |net|/gross over 20 loops 2.24e-15"
  },
  {
   "name": "direct model does nonzero net work around closed loops",
   "pass": true,
   "detail": "median |net|/gross 3.627e-03"
  },
  {
   "name": "loop-work path dependence separates the models by >1e5",
   "pass": true,
   "detail": "ratio 1.182e+13"
  },
  {
   "name": "conservative model conserves its own E_theta + KE (Verlet floor)",
   "pass": true,
   "detail": "median rel excursion 1.46e-07"
  },
  {
   "name": "conservative true energy oscillates but does not drift (secular change << fluctuation)",
   "pass": true,
   "detail": "|secular|/fluct 0.124, drift +0.0000% of |E0| over T=200"
  },
  {
   "name": "direct model true energy drifts secularly, not just noisily",
   "pass": true,
   "detail": "|secular|/fluct 4.587, drift +0.130% of |E0| (+2.82% of KE0)"
  },
  {
   "name": "EVERY paired trajectory drifts faster under the direct model",
   "pass": true,
   "detail": "24/24; secular |dE| ratio median x210, worst case still x38"
  },
  {
   "name": "median drift rate separates the models by >20x at T=200",
   "pass": true,
   "detail": "ratio 149.0 (2.041e-04 vs 1.369e-06 per unit time)"
  },
  {
   "name": "NEGATIVE RESULT recorded: the direct model's drift is bidirectional, not a systematic heating",
   "pass": true,
   "detail": "15/24 trajectories gain energy, binomial p = 0.307 -- consistent with a coin flip, so this is energy pumping in both directions, not an effective friction"
  },
  {
   "name": "a minority of direct trajectories are destroyed outright; no conservative trajectory is",
   "pass": true,
   "detail": "|dE| > 1.0 for 12% of direct and 0% of conservative trajectories at T=200"
  },
  {
   "name": "conservative true-energy error stays inside a fixed envelope as t grows",
   "pass": true,
   "detail": "median max|dE| grows x1.41 when the run is x4 longer; linear accumulation would give x4 (0.0073 -> 0.0103)"
  },
  {
   "name": "conservative apparent drift rate falls like 1/T, as a bounded error must",
   "pass": true,
   "detail": "rate x0.155 for a x4 longer run; 1/T predicts x0.25"
  },
  {
   "name": "direct drift rate is T-independent, so its energy error accumulates without bound",
   "pass": true,
   "detail": "rate x0.951 for a x4 longer run (2.146e-04 -> 2.041e-04)"
  },
  {
   "name": "conservative model's own invariant fails only as dt^2 (pure Verlet error)",
   "pass": true,
   "detail": "E_theta+KE excursion grows x65.8 as dt grows x8; dt^2 predicts x64"
  },
  {
   "name": "conservative true-energy error is dt-independent, i.e. bounded model error not accumulating integrator error",
   "pass": true,
   "detail": "median max|dE| varies only x1.12 over an x8 range in dt"
  },
  {
   "name": "direct model drift survives dt -> 0 (a property of the field, not of the integrator)",
   "pass": true,
   "detail": "median |slope| varies x1.55 across dt = [0.00125, 0.0025, 0.005, 0.01]: 3.29e-04, 2.73e-04, 2.15e-04, 3.33e-04"
  },
  {
   "name": "the model separation holds at every timestep tested",
   "pass": true,
   "detail": "drift ratios 62, 52, 24, 32"
  },
  {
   "name": "force error ratio ~1 while drift ratio is orders of magnitude",
   "pass": true,
   "detail": "force RMSE x0.881, drift x149"
  }
 ]
}