FAILURE MAP
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FA-87136 / Physics integrator stability / Open access

Verlet bootstrap step uses the full acceleration dt squared · case 01

The first displacement is too large and seeds a persistent velocity error.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

x1 adds a0*h0^2 instead of 0.5*a0*h0^2.

THE FAILURE

x1 adds a0*h0^2 instead of 0.5*a0*h0^2.

Unsuccessful approach: Dropping one h0 factor gives the wrong units.

Case contract

solve(x0, v0, w2, g, dts): position Verlet with variable steps, acceleration -w2*x+g. First step is Taylor x1=x0+v0*h0+0.5*a(x0)*h0^2; then x_{n+1}=x_n+(x_n-x_{n-1})*(h_n/h_{n-1})+a(x_n)*h_n*(h_n+h_{n-1})/2. Velocity estimate is (x_n-x_{n-1})/h_last. Empty dts returns the initial state. Return [x, v] rounded.

Why this case matters

Game and robotics physics loops depend on integrator update order, step control and stabilization terms; a wrong decision point turns a stable simulation into drifting or exploding motion.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x0, v0, w2, g, dts):
    def acc(p):
        return -w2 * p + g
    if not dts:
        return [round(x0, 6), round(v0, 6)]
    prev = x0
    x = x0 + v0 * dts[0] + acc(x0) * dts[0] * dts[0]
    for i in range(1, len(dts)):
        h, hp = dts[i], dts[i - 1]
        nxt = x + (x - prev) * (h / hp) + acc(x) * h * (h + hp) / 2
        prev, x = x, nxt
    v = (x - prev) / dts[-1]
    return [round(x, 6), round(v, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[-0.889, -0.08, 17.128, 2.0, [0.05, 0.01, 0.05, 0.01]], [-0.776487, 1.830242]], [[0.043, 0.281, 14.181, -9.81, [0.03, 0.05, 0.02, 0.02, 0.03, 0.02]], [-0.057871, -1.335146]], [[-0.399, -1.076, 18.618, 2.0, [0.03, 0.03, 0.03, 0.02, 0.01]], [-0.456288, 0.093019]], [[-1.409, 0.693, 7.348, -9.81, [0.02, 0.02, 0.05, 0.05, 0.05, 0.01]], [-1.266157, 0.69937]], [[0.617, -1.642, 8.295, -9.81, [0.05, 0.01, 0.02, 0.01, 0.03, 0.02]], [0.248614, -3.42338]], [[-1.559, 0.132, 13.212, 0.0, [0.05, 0.05, 0.02, 0.03, 0.01, 0.01]], [-1.249546, 3.301322]], [[1.122, 1.423, 17.941, -9.81, [0.03, 0.05, 0.02, 0.03, 0.03, 0.02]], [0.892542, -3.585623]], [[-1.222, -0.036, 10.12, -9.81, [0.02, 0.03]], [-1.220603, 0.053535]]], [[[-1.613, -0.622, 8.807, -9.81, [0.02, 0.03, 0.01, 0.05, 0.03]], [-1.655241, -0.043343]], [[1.227, 0.863, 11.687, 0.0, [0.05, 0.05, 0.01, 0.05, 0.03, 0.03]], [1.068817, -2.042643]], [[1.862, 1.422, 6.208, 2.0, [0.01, 0.03, 0.03, 0.01, 0.01]], [1.950431, 0.584063]], [[1.693, 1.184, 10.982, -9.81, [0.02, 0.03, 0.05, 0.05]], [1.550068, -2.368135]], [[0.833, -0.206, 3.523, 2.0, [0.05, 0.01]], [0.81897, -0.256194]], [[-1.624, 0.809, 19.846, 0.0, [0.03, 0.02, 0.05, 0.03, 0.02, 0.03]], [-0.997463, 5.451991]], [[0.644, -0.151, 13.871, -9.81, [0.02, 0.02, 0.01, 0.01, 0.05, 0.02, 0.03]], [0.387894, -2.717601]], [[0.487, 0.867, 0.777, -9.81, [0.05, 0.05, 0.03, 0.01, 0.02, 0.01, 0.05]], [0.430783, -1.12279]]], [[[-0.686, 1.111, 17.91, -9.81, [0.05, 0.03, 0.03, 0.03]], [-0.5152, 1.255652]], [[0.106, -1.937, 2.608, 0.0, [0.03, 0.01, 0.03, 0.05, 0.01, 0.05, 0.05, 0.02]], [-0.374085, -1.858207]], [[-1.478, -0.428, 14.024, 2.0, [0.01, 0.02, 0.01]], [-1.476913, 0.36867]], [[-0.233, -0.829, 19.928, -9.81, [0.01, 0.03, 0.02, 0.01]], [-0.302753, -1.125418]], [[0.153, -1.246, 15.582, 0.0, [0.01, 0.05, 0.02]], [0.046962, -1.364051]], [[-0.197, 1.673, 9.677, 2.0, [0.03, 0.03, 0.05, 0.05, 0.03, 0.05, 0.02, 0.05]], [0.420067, 2.034313]], [[1.671, -1.397, 3.602, 0.0, [0.01, 0.05, 0.03, 0.02, 0.05, 0.01, 0.02, 0.03]], [1.228659, -2.495971]], [[1.606, 1.967, 9.66, 0.0, [0.05, 0.03, 0.01, 0.02, 0.03, 0.05, 0.01]], [1.674706, -1.223548]]], [[[-1.994, -0.111, 14.711, -9.81, [0.03, 0.05, 0.02, 0.01, 0.01]], [-1.868671, 2.069192]], [[1.703, 1.989, 16.348, -9.81, [0.01, 0.05, 0.02, 0.05, 0.01, 0.03, 0.01, 0.03]], [1.291487, -5.197734]], [[0.774, -1.188, 7.384, 2.0, [0.03, 0.05, 0.05, 0.02]], [0.55902, -1.610601]], [[-0.434, -1.688, 15.899, 0.0, [0.03, 0.05, 0.01, 0.02]], [-0.573341, -0.885589]], [[-0.101, -1.02, 19.226, 2.0, [0.05, 0.01]], [-0.154839, -0.776626]], [[1.819, 0.853, 6.729, 0.0, [0.05, 0.02, 0.01, 0.01, 0.05, 0.01, 0.05, 0.03, 0.02]], [1.648448, -2.056524]], [[0.396, 0.753, 11.545, 2.0, [0.01, 0.03, 0.03, 0.05, 0.02]], [0.472972, 0.357792]], [[-0.516, 0.344, 13.779, -9.81, [0.01, 0.03, 0.01, 0.01, 0.03, 0.05]], [-0.495793, 0.012391]]], [[[-0.855, -0.987, 10.095, 2.0, [0.05, 0.02, 0.01]], [-0.899493, -0.170438]], [[-1.434, -0.348, 10.741, -9.81, [0.03, 0.03, 0.03, 0.01, 0.05, 0.03, 0.03, 0.03, 0.01, 0.01]], [-1.336072, 1.031162]], [[1.471, -0.746, 12.484, 2.0, [0.01, 0.03, 0.02, 0.02, 0.02, 0.01, 0.03, 0.05, 0.05, 0.05]], [0.658164, -4.181996]], [[-0.907, 1.374, 17.435, 0.0, [0.02, 0.01, 0.02, 0.03, 0.03, 0.01]], [-0.637033, 2.967008]], [[1.097, 0.642, 16.162, 2.0, [0.01, 0.03]], [1.110042, 0.246936]], [[-0.083, 1.642, 5.418, -9.81, [0.01, 0.05, 0.01, 0.05]], [0.04482, 0.722016]], [[-0.233, 1.656, 10.109, -9.81, [0.02, 0.03, 0.01]], [-0.147526, 1.223195]], [[1.94, 0.689, 18.476, 0.0, [0.05, 0.01, 0.03, 0.01, 0.05, 0.02, 0.03, 0.02, 0.02]], [1.133627, -6.57541]]]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("case %d" % i, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
case 0[-0.726611, 2.212964][-0.776487, 1.830242]Failed
case 1[-0.082732, -1.46395][-0.057871, -1.335146]Failed
case 2[-0.440028, 0.217392][-0.456288, 0.093019]Failed
case 3[-1.26512, 0.704061][-1.266157, 0.69937]Failed
case 4[0.197687, -3.770824][0.248614, -3.42338]Failed
case 5[-1.167173, 3.726155][-1.249546, 3.301322]Failed
case 6[0.81905, -3.923233][0.892542, -3.585623]Failed
case 7[-1.219328, 0.078972][-1.220603, 0.053535]Failed

SHA-256 / bc2f9b8608baf962861fbbbe30f2e76e288acda160e56b11a0d7ad894ffffed7

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x0, v0, w2, g, dts):
    def acc(p):
        return -w2 * p + g
    if not dts:
        return [round(x0, 6), round(v0, 6)]
    prev = x0
    x = x0 + v0 * dts[0] + 0.5 * acc(x0) * dts[0]
    for i in range(1, len(dts)):
        h, hp = dts[i], dts[i - 1]
        nxt = x + (x - prev) * (h / hp) + acc(x) * h * (h + hp) / 2
        prev, x = x, nxt
    v = (x - prev) / dts[-1]
    return [round(x, 6), round(v, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[-0.889, -0.08, 17.128, 2.0, [0.05, 0.01, 0.05, 0.01]], [-0.776487, 1.830242]], [[0.043, 0.281, 14.181, -9.81, [0.03, 0.05, 0.02, 0.02, 0.03, 0.02]], [-0.057871, -1.335146]], [[-0.399, -1.076, 18.618, 2.0, [0.03, 0.03, 0.03, 0.02, 0.01]], [-0.456288, 0.093019]], [[-1.409, 0.693, 7.348, -9.81, [0.02, 0.02, 0.05, 0.05, 0.05, 0.01]], [-1.266157, 0.69937]], [[0.617, -1.642, 8.295, -9.81, [0.05, 0.01, 0.02, 0.01, 0.03, 0.02]], [0.248614, -3.42338]], [[-1.559, 0.132, 13.212, 0.0, [0.05, 0.05, 0.02, 0.03, 0.01, 0.01]], [-1.249546, 3.301322]], [[1.122, 1.423, 17.941, -9.81, [0.03, 0.05, 0.02, 0.03, 0.03, 0.02]], [0.892542, -3.585623]], [[-1.222, -0.036, 10.12, -9.81, [0.02, 0.03]], [-1.220603, 0.053535]]], [[[-1.613, -0.622, 8.807, -9.81, [0.02, 0.03, 0.01, 0.05, 0.03]], [-1.655241, -0.043343]], [[1.227, 0.863, 11.687, 0.0, [0.05, 0.05, 0.01, 0.05, 0.03, 0.03]], [1.068817, -2.042643]], [[1.862, 1.422, 6.208, 2.0, [0.01, 0.03, 0.03, 0.01, 0.01]], [1.950431, 0.584063]], [[1.693, 1.184, 10.982, -9.81, [0.02, 0.03, 0.05, 0.05]], [1.550068, -2.368135]], [[0.833, -0.206, 3.523, 2.0, [0.05, 0.01]], [0.81897, -0.256194]], [[-1.624, 0.809, 19.846, 0.0, [0.03, 0.02, 0.05, 0.03, 0.02, 0.03]], [-0.997463, 5.451991]], [[0.644, -0.151, 13.871, -9.81, [0.02, 0.02, 0.01, 0.01, 0.05, 0.02, 0.03]], [0.387894, -2.717601]], [[0.487, 0.867, 0.777, -9.81, [0.05, 0.05, 0.03, 0.01, 0.02, 0.01, 0.05]], [0.430783, -1.12279]]], [[[-0.686, 1.111, 17.91, -9.81, [0.05, 0.03, 0.03, 0.03]], [-0.5152, 1.255652]], [[0.106, -1.937, 2.608, 0.0, [0.03, 0.01, 0.03, 0.05, 0.01, 0.05, 0.05, 0.02]], [-0.374085, -1.858207]], [[-1.478, -0.428, 14.024, 2.0, [0.01, 0.02, 0.01]], [-1.476913, 0.36867]], [[-0.233, -0.829, 19.928, -9.81, [0.01, 0.03, 0.02, 0.01]], [-0.302753, -1.125418]], [[0.153, -1.246, 15.582, 0.0, [0.01, 0.05, 0.02]], [0.046962, -1.364051]], [[-0.197, 1.673, 9.677, 2.0, [0.03, 0.03, 0.05, 0.05, 0.03, 0.05, 0.02, 0.05]], [0.420067, 2.034313]], [[1.671, -1.397, 3.602, 0.0, [0.01, 0.05, 0.03, 0.02, 0.05, 0.01, 0.02, 0.03]], [1.228659, -2.495971]], [[1.606, 1.967, 9.66, 0.0, [0.05, 0.03, 0.01, 0.02, 0.03, 0.05, 0.01]], [1.674706, -1.223548]]], [[[-1.994, -0.111, 14.711, -9.81, [0.03, 0.05, 0.02, 0.01, 0.01]], [-1.868671, 2.069192]], [[1.703, 1.989, 16.348, -9.81, [0.01, 0.05, 0.02, 0.05, 0.01, 0.03, 0.01, 0.03]], [1.291487, -5.197734]], [[0.774, -1.188, 7.384, 2.0, [0.03, 0.05, 0.05, 0.02]], [0.55902, -1.610601]], [[-0.434, -1.688, 15.899, 0.0, [0.03, 0.05, 0.01, 0.02]], [-0.573341, -0.885589]], [[-0.101, -1.02, 19.226, 2.0, [0.05, 0.01]], [-0.154839, -0.776626]], [[1.819, 0.853, 6.729, 0.0, [0.05, 0.02, 0.01, 0.01, 0.05, 0.01, 0.05, 0.03, 0.02]], [1.648448, -2.056524]], [[0.396, 0.753, 11.545, 2.0, [0.01, 0.03, 0.03, 0.05, 0.02]], [0.472972, 0.357792]], [[-0.516, 0.344, 13.779, -9.81, [0.01, 0.03, 0.01, 0.01, 0.03, 0.05]], [-0.495793, 0.012391]]], [[[-0.855, -0.987, 10.095, 2.0, [0.05, 0.02, 0.01]], [-0.899493, -0.170438]], [[-1.434, -0.348, 10.741, -9.81, [0.03, 0.03, 0.03, 0.01, 0.05, 0.03, 0.03, 0.03, 0.01, 0.01]], [-1.336072, 1.031162]], [[1.471, -0.746, 12.484, 2.0, [0.01, 0.03, 0.02, 0.02, 0.02, 0.01, 0.03, 0.05, 0.05, 0.05]], [0.658164, -4.181996]], [[-0.907, 1.374, 17.435, 0.0, [0.02, 0.01, 0.02, 0.03, 0.03, 0.01]], [-0.637033, 2.967008]], [[1.097, 0.642, 16.162, 2.0, [0.01, 0.03]], [1.110042, 0.246936]], [[-0.083, 1.642, 5.418, -9.81, [0.01, 0.05, 0.01, 0.05]], [0.04482, 0.722016]], [[-0.233, 1.656, 10.109, -9.81, [0.02, 0.03, 0.01]], [-0.147526, 1.223195]], [[1.94, 0.689, 18.476, 0.0, [0.05, 0.01, 0.03, 0.01, 0.05, 0.02, 0.03, 0.02, 0.02]], [1.133627, -6.57541]]]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("case %d" % i, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
case 0[0.171156, 9.101969][-0.776487, 1.830242]Failed
case 1[-0.861709, -5.499827][-0.057871, -1.335146]Failed
case 2[0.069466, 4.114399][-0.456288, 0.093019]Failed
case 3[-1.21536, 0.929263][-1.266157, 0.69937]Failed
case 4[-0.719014, -10.024821][0.248614, -3.42338]Failed
case 5[0.31554, 11.373151][-1.249546, 3.301322]Failed
case 6[-1.483708, -14.50169][0.892542, -3.585623]Failed
case 7[-1.158155, 1.29995][-1.220603, 0.053535]Failed

SHA-256 / accc176b4f29f2677eeee50ba2da287a801ca7716bd8c58d4b36caa058284e68

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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Verification & scope

A deterministic bounded teaching model with stipulated toy conventions and rounded float output; not a production physics engine or a proof of numerical stability. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:50:55.999458+00:00.

Case digest / 2e999ca8118b61a9d675e44acb059e28103b23e31ed898f324266e888baf741f