FA-86931 / Physics integrator stability / Open access

Damping rate multiplies by mass instead of dividing · case 01

Heavy bodies appear strongly damped and light ones barely damped.

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

ROOT CAUSE

The normalized damping is computed as c*m.

VERIFIED REPAIR

Use c/m as the linear coefficient.

Unsuccessful approach: Using c/(2m) confuses the rate with the damping ratio numerator.

Case contract

solve(k, m, c, dt): classify explicit Euler on x'=v, v'=(-k*x-c*v)/m. Eigenvalues are the roots of s^2+(c/m)s+k/m; amplification is max |1+dt*lambda|. Return ["marginal" if |amp-1|<=1e-9, else "stable" if amp<1, else "unstable", amp rounded to 6].

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(k, m, c, dt):
    a = c * m
    b = k / m
    disc = a * a - 4 * b
    if disc >= 0:
        r = math.sqrt(disc)
        lams = [complex((-a + r) / 2, 0), complex((-a - r) / 2, 0)]
    else:
        r = math.sqrt(-disc)
        lams = [complex(-a / 2, r / 2), complex(-a / 2, -r / 2)]
    amp = max(abs(1 + dt * lam) for lam in lams)
    if abs(amp - 1) <= 1e-9:
        label = 'marginal'
    elif amp < 1:
        label = 'stable'
    else:
        label = 'unstable'
    return [label, round(amp, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.221, 2.418, 7.819, 0.01], ['stable', 0.984586]], [[16.494, 1.905, 4.155, 0.05], ['stable', 0.955296]], [[28.666, 0.459, 5.82, 0.1], ['stable', 0.597125]], [[42.548, 3.763, 4.806, 0.1], ['stable', 0.992649]], [[0.0, 3.907, 1.097, 1.0], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 0.421, 0.691, 0.05], ['marginal', 1.0]]], [[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[38.018, 1.276, 0.385, 0.2], ['unstable', 1.459946]], [[59.677, 2.374, 3.266, 1.0], ['unstable', 4.976144]], [[6.203, 0.678, 1.745, 0.05], ['stable', 0.945614]], [[0.0, 0.907, 6.001, 0.01], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.043, 3.236, 0.05], ['marginal', 1.0]], [[18.985, 2.12, 2.861, 1.0], ['unstable', 2.933541]]], [[[48.986, 0.701, 3.497, 0.1], ['unstable', 1.095419]], [[49.989, 2.073, 1.922, 0.8], ['unstable', 3.961243]], [[0.0, 1.763, 0.013, 0.8], ['marginal', 1.0]], [[17.741, 0.849, 5.676, 0.01], ['stable', 0.967075]], [[20.646, 3.925, 4.888, 0.2], ['stable', 0.980477]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.196, 3.443, 4.277, 0.8], ['unstable', 2.801748]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]]], [[[41.131, 0.314, 6.187, 0.4], ['unstable', 3.751925]], [[23.134, 3.325, 2.902, 0.05], ['stable', 0.98679]], [[3.87, 2.12, 0.308, 0.05], ['stable', 0.998649]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 3.102, 2.755, 0.05], ['marginal', 1.0]], [[9.079, 3.699, 5.728, 0.05], ['stable', 0.963696]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[57.413, 2.505, 6.221, 0.2], ['unstable', 1.191674]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.889, 7.242, 0.8], ['unstable', 2.06702]], [[31.247, 2.981, 5.247, 0.01], ['stable', 0.991689]], [[29.999, 0.715, 0.909, 0.01], ['stable', 0.995732]], [[18.339, 0.83, 4.998, 0.4], ['unstable', 1.458271]], [[20.299, 1.465, 6.715, 0.05], ['stable', 0.897474]], [[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]]]]
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 fixture	Actual	Expected	Outcome
case 0	`['unstable', 1.0008]`	`['unstable', 1.0008]`	Passed
case 1	`['stable', 0.990263]`	`['stable', 0.984586]`	Failed
case 2	`['stable', 0.934449]`	`['stable', 0.955296]`	Failed
case 3	`['unstable', 1.165072]`	`['stable', 0.597125]`	Failed
case 4	`['stable', 0.935154]`	`['stable', 0.992649]`	Failed
case 5	`['unstable', 3.285979]`	`['marginal', 1.0]`	Failed
case 6	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 7	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed

SHA-256 / a2238a2cd5a3d0fdfcc05da9d2721c20e080f2a4cfa116f60e9e0dff4083d0e6

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(k, m, c, dt):
    a = c / (2 * m)
    b = k / m
    disc = a * a - 4 * b
    if disc >= 0:
        r = math.sqrt(disc)
        lams = [complex((-a + r) / 2, 0), complex((-a - r) / 2, 0)]
    else:
        r = math.sqrt(-disc)
        lams = [complex(-a / 2, r / 2), complex(-a / 2, -r / 2)]
    amp = max(abs(1 + dt * lam) for lam in lams)
    if abs(amp - 1) <= 1e-9:
        label = 'marginal'
    elif amp < 1:
        label = 'stable'
    else:
        label = 'unstable'
    return [label, round(amp, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.221, 2.418, 7.819, 0.01], ['stable', 0.984586]], [[16.494, 1.905, 4.155, 0.05], ['stable', 0.955296]], [[28.666, 0.459, 5.82, 0.1], ['stable', 0.597125]], [[42.548, 3.763, 4.806, 0.1], ['stable', 0.992649]], [[0.0, 3.907, 1.097, 1.0], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 0.421, 0.691, 0.05], ['marginal', 1.0]]], [[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[38.018, 1.276, 0.385, 0.2], ['unstable', 1.459946]], [[59.677, 2.374, 3.266, 1.0], ['unstable', 4.976144]], [[6.203, 0.678, 1.745, 0.05], ['stable', 0.945614]], [[0.0, 0.907, 6.001, 0.01], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.043, 3.236, 0.05], ['marginal', 1.0]], [[18.985, 2.12, 2.861, 1.0], ['unstable', 2.933541]]], [[[48.986, 0.701, 3.497, 0.1], ['unstable', 1.095419]], [[49.989, 2.073, 1.922, 0.8], ['unstable', 3.961243]], [[0.0, 1.763, 0.013, 0.8], ['marginal', 1.0]], [[17.741, 0.849, 5.676, 0.01], ['stable', 0.967075]], [[20.646, 3.925, 4.888, 0.2], ['stable', 0.980477]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.196, 3.443, 4.277, 0.8], ['unstable', 2.801748]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]]], [[[41.131, 0.314, 6.187, 0.4], ['unstable', 3.751925]], [[23.134, 3.325, 2.902, 0.05], ['stable', 0.98679]], [[3.87, 2.12, 0.308, 0.05], ['stable', 0.998649]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 3.102, 2.755, 0.05], ['marginal', 1.0]], [[9.079, 3.699, 5.728, 0.05], ['stable', 0.963696]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[57.413, 2.505, 6.221, 0.2], ['unstable', 1.191674]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.889, 7.242, 0.8], ['unstable', 2.06702]], [[31.247, 2.981, 5.247, 0.01], ['stable', 0.991689]], [[29.999, 0.715, 0.909, 0.01], ['stable', 0.995732]], [[18.339, 0.83, 4.998, 0.4], ['unstable', 1.458271]], [[20.299, 1.465, 6.715, 0.05], ['stable', 0.897474]], [[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]]]]
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 fixture	Actual	Expected	Outcome
case 0	`['marginal', 1.0]`	`['unstable', 1.0008]`	Failed
case 1	`['stable', 0.992763]`	`['stable', 0.984586]`	Failed
case 2	`['stable', 0.983422]`	`['stable', 0.955296]`	Failed
case 3	`['stable', 0.995261]`	`['stable', 0.597125]`	Failed
case 4	`['unstable', 1.02431]`	`['stable', 0.992649]`	Failed
case 5	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 6	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 7	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed

SHA-256 / ff1773c0422c791eed15e9c0f8a29330c04259305e88d1a4fc5917f6146ef17f

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(k, m, c, dt):
    a = c / m
    b = k / m
    disc = a * a - 4 * b
    if disc >= 0:
        r = math.sqrt(disc)
        lams = [complex((-a + r) / 2, 0), complex((-a - r) / 2, 0)]
    else:
        r = math.sqrt(-disc)
        lams = [complex(-a / 2, r / 2), complex(-a / 2, -r / 2)]
    amp = max(abs(1 + dt * lam) for lam in lams)
    if abs(amp - 1) <= 1e-9:
        label = 'marginal'
    elif amp < 1:
        label = 'stable'
    else:
        label = 'unstable'
    return [label, round(amp, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.221, 2.418, 7.819, 0.01], ['stable', 0.984586]], [[16.494, 1.905, 4.155, 0.05], ['stable', 0.955296]], [[28.666, 0.459, 5.82, 0.1], ['stable', 0.597125]], [[42.548, 3.763, 4.806, 0.1], ['stable', 0.992649]], [[0.0, 3.907, 1.097, 1.0], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 0.421, 0.691, 0.05], ['marginal', 1.0]]], [[[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[38.018, 1.276, 0.385, 0.2], ['unstable', 1.459946]], [[59.677, 2.374, 3.266, 1.0], ['unstable', 4.976144]], [[6.203, 0.678, 1.745, 0.05], ['stable', 0.945614]], [[0.0, 0.907, 6.001, 0.01], ['marginal', 1.0]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.043, 3.236, 0.05], ['marginal', 1.0]], [[18.985, 2.12, 2.861, 1.0], ['unstable', 2.933541]]], [[[48.986, 0.701, 3.497, 0.1], ['unstable', 1.095419]], [[49.989, 2.073, 1.922, 0.8], ['unstable', 3.961243]], [[0.0, 1.763, 0.013, 0.8], ['marginal', 1.0]], [[17.741, 0.849, 5.676, 0.01], ['stable', 0.967075]], [[20.646, 3.925, 4.888, 0.2], ['stable', 0.980477]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[42.196, 3.443, 4.277, 0.8], ['unstable', 2.801748]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]]], [[[41.131, 0.314, 6.187, 0.4], ['unstable', 3.751925]], [[23.134, 3.325, 2.902, 0.05], ['stable', 0.98679]], [[3.87, 2.12, 0.308, 0.05], ['stable', 0.998649]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 3.102, 2.755, 0.05], ['marginal', 1.0]], [[9.079, 3.699, 5.728, 0.05], ['stable', 0.963696]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[57.413, 2.505, 6.221, 0.2], ['unstable', 1.191674]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.889, 7.242, 0.8], ['unstable', 2.06702]], [[31.247, 2.981, 5.247, 0.01], ['stable', 0.991689]], [[29.999, 0.715, 0.909, 0.01], ['stable', 0.995732]], [[18.339, 0.83, 4.998, 0.4], ['unstable', 1.458271]], [[20.299, 1.465, 6.715, 0.05], ['stable', 0.897474]], [[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]]]]
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 fixture	Actual	Expected	Outcome
case 0	`['unstable', 1.0008]`	`['unstable', 1.0008]`	Passed
case 1	`['stable', 0.984586]`	`['stable', 0.984586]`	Passed
case 2	`['stable', 0.955296]`	`['stable', 0.955296]`	Passed
case 3	`['stable', 0.597125]`	`['stable', 0.597125]`	Passed
case 4	`['stable', 0.992649]`	`['stable', 0.992649]`	Passed
case 5	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 6	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 7	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed

SHA-256 / 43a5f6724b1b78616a06def5eb4d5ed34f5939cf27a3ba251b768aef54c06888

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 2026-09-29T14:50:54.077404+00:00.

Case digest / e929f502ad2ea29de7709d4dfa66bf058645271b11c058822a2835e969bab18f