FA-86906 / Physics integrator stability / Open access

Stability discriminant uses raw stiffness instead of k over m · case 01

Heavy bodies are classified with the eigenvalues of a unit-mass system.

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

ROOT CAUSE

The discriminant is (c/m)^2 - 4k instead of (c/m)^2 - 4k/m.

VERIFIED REPAIR

Normalize stiffness by mass before forming the characteristic polynomial.

Unsuccessful approach: Using 2*k/m instead of 4*k/m misplaces the critical-damping boundary.

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 * k
    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 = [[[[48.787, 2.932, 7.285, 0.05], ['stable', 0.957792]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[39.354, 3.167, 6.495, 0.4], ['unstable', 1.472368]], [[39.672, 0.379, 2.933, 1.0], ['unstable', 9.896296]], [[46.563, 1.397, 1.187, 0.01], ['stable', 0.997415]], [[0.0, 3.279, 2.912, 0.2], ['marginal', 1.0]], [[42.742, 0.3, 1.248, 1.0], ['unstable', 11.803107]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[34.159, 0.297, 5.802, 0.2], ['unstable', 1.301333]], [[47.405, 3.962, 4.696, 0.05], ['stable', 0.985215]], [[0.0, 1.357, 2.319, 0.4], ['marginal', 1.0]], [[7.047, 1.493, 0.783, 0.8], ['unstable', 1.897698]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[25.383, 3.183, 5.311, 0.2], ['stable', 0.992609]], [[28.131, 2.644, 5.026, 0.2], ['unstable', 1.022448]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[33.332, 2.078, 0.851, 0.05], ['unstable', 1.009765]], [[0.0, 0.563, 2.488, 0.01], ['marginal', 1.0]], [[8.52, 3.836, 5.13, 0.4], ['stable', 0.90578]], [[15.636, 2.616, 6.979, 0.8], ['unstable', 1.640448]], [[34.687, 1.578, 6.074, 0.2], ['unstable', 1.053295]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[17.304, 2.161, 0.31, 0.4], ['unstable', 1.491242]]], [[[21.305, 1.348, 3.845, 0.2], ['unstable', 1.030398]], [[5.291, 2.418, 3.167, 0.01], ['stable', 0.99354]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[9.84, 2.992, 6.533, 0.4], ['stable', 0.807965]], [[0.0, 2.391, 6.417, 0.01], ['marginal', 1.0]], [[55.105, 1.329, 4.732, 0.8], ['unstable', 4.968721]], [[15.373, 0.488, 0.687, 0.8], ['unstable', 4.476057]]], [[[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]], [[41.507, 2.269, 5.919, 0.8], ['unstable', 3.258936]], [[0.0, 0.76, 4.988, 1.0], ['unstable', 5.563158]], [[48.542, 2.726, 7.357, 0.8], ['unstable', 3.199601]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[36.647, 2.751, 2.545, 0.01], ['stable', 0.996033]], [[14.174, 2.424, 0.649, 0.8], ['unstable', 2.127938]]]]
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	`['stable', 0.998867]`	`['stable', 0.957792]`	Failed
case 1	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 2	`['unstable', 2.544859]`	`['unstable', 1.472368]`	Failed
case 3	`['unstable', 5.738747]`	`['unstable', 9.896296]`	Failed
case 4	`['stable', 0.998078]`	`['stable', 0.997415]`	Failed
case 5	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 6	`['unstable', 6.291423]`	`['unstable', 11.803107]`	Failed
case 7	`['unstable', 1.0008]`	`['unstable', 1.0008]`	Passed

SHA-256 / 81dda2acaf59cabd5af09c7168d8bb8a671d1392ef82c9b982d5ee8907198821

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 / m
    b = k / m
    disc = a * a - 2 * 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 = [[[[48.787, 2.932, 7.285, 0.05], ['stable', 0.957792]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[39.354, 3.167, 6.495, 0.4], ['unstable', 1.472368]], [[39.672, 0.379, 2.933, 1.0], ['unstable', 9.896296]], [[46.563, 1.397, 1.187, 0.01], ['stable', 0.997415]], [[0.0, 3.279, 2.912, 0.2], ['marginal', 1.0]], [[42.742, 0.3, 1.248, 1.0], ['unstable', 11.803107]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[34.159, 0.297, 5.802, 0.2], ['unstable', 1.301333]], [[47.405, 3.962, 4.696, 0.05], ['stable', 0.985215]], [[0.0, 1.357, 2.319, 0.4], ['marginal', 1.0]], [[7.047, 1.493, 0.783, 0.8], ['unstable', 1.897698]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[25.383, 3.183, 5.311, 0.2], ['stable', 0.992609]], [[28.131, 2.644, 5.026, 0.2], ['unstable', 1.022448]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[33.332, 2.078, 0.851, 0.05], ['unstable', 1.009765]], [[0.0, 0.563, 2.488, 0.01], ['marginal', 1.0]], [[8.52, 3.836, 5.13, 0.4], ['stable', 0.90578]], [[15.636, 2.616, 6.979, 0.8], ['unstable', 1.640448]], [[34.687, 1.578, 6.074, 0.2], ['unstable', 1.053295]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[17.304, 2.161, 0.31, 0.4], ['unstable', 1.491242]]], [[[21.305, 1.348, 3.845, 0.2], ['unstable', 1.030398]], [[5.291, 2.418, 3.167, 0.01], ['stable', 0.99354]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[9.84, 2.992, 6.533, 0.4], ['stable', 0.807965]], [[0.0, 2.391, 6.417, 0.01], ['marginal', 1.0]], [[55.105, 1.329, 4.732, 0.8], ['unstable', 4.968721]], [[15.373, 0.488, 0.687, 0.8], ['unstable', 4.476057]]], [[[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]], [[41.507, 2.269, 5.919, 0.8], ['unstable', 3.258936]], [[0.0, 0.76, 4.988, 1.0], ['unstable', 5.563158]], [[48.542, 2.726, 7.357, 0.8], ['unstable', 3.199601]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[36.647, 2.751, 2.545, 0.01], ['stable', 0.996033]], [[14.174, 2.424, 0.649, 0.8], ['unstable', 2.127938]]]]
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	`['stable', 0.946872]`	`['stable', 0.957792]`	Failed
case 1	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 2	`['unstable', 1.083405]`	`['unstable', 1.472368]`	Failed
case 3	`['unstable', 6.752699]`	`['unstable', 9.896296]`	Failed
case 4	`['stable', 0.996579]`	`['stable', 0.997415]`	Failed
case 5	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 6	`['unstable', 8.250859]`	`['unstable', 11.803107]`	Failed
case 7	`['unstable', 1.0008]`	`['unstable', 1.0008]`	Passed

SHA-256 / b2ab26a57b254a62fdc0b8026fccb4d9cda73b5e5301fb1334756af1339bfb0b

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 = [[[[48.787, 2.932, 7.285, 0.05], ['stable', 0.957792]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[39.354, 3.167, 6.495, 0.4], ['unstable', 1.472368]], [[39.672, 0.379, 2.933, 1.0], ['unstable', 9.896296]], [[46.563, 1.397, 1.187, 0.01], ['stable', 0.997415]], [[0.0, 3.279, 2.912, 0.2], ['marginal', 1.0]], [[42.742, 0.3, 1.248, 1.0], ['unstable', 11.803107]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[34.159, 0.297, 5.802, 0.2], ['unstable', 1.301333]], [[47.405, 3.962, 4.696, 0.05], ['stable', 0.985215]], [[0.0, 1.357, 2.319, 0.4], ['marginal', 1.0]], [[7.047, 1.493, 0.783, 0.8], ['unstable', 1.897698]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[25.383, 3.183, 5.311, 0.2], ['stable', 0.992609]], [[28.131, 2.644, 5.026, 0.2], ['unstable', 1.022448]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]]], [[[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[33.332, 2.078, 0.851, 0.05], ['unstable', 1.009765]], [[0.0, 0.563, 2.488, 0.01], ['marginal', 1.0]], [[8.52, 3.836, 5.13, 0.4], ['stable', 0.90578]], [[15.636, 2.616, 6.979, 0.8], ['unstable', 1.640448]], [[34.687, 1.578, 6.074, 0.2], ['unstable', 1.053295]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[17.304, 2.161, 0.31, 0.4], ['unstable', 1.491242]]], [[[21.305, 1.348, 3.845, 0.2], ['unstable', 1.030398]], [[5.291, 2.418, 3.167, 0.01], ['stable', 0.99354]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[9.84, 2.992, 6.533, 0.4], ['stable', 0.807965]], [[0.0, 2.391, 6.417, 0.01], ['marginal', 1.0]], [[55.105, 1.329, 4.732, 0.8], ['unstable', 4.968721]], [[15.373, 0.488, 0.687, 0.8], ['unstable', 4.476057]]], [[[1.0, 1.0, 3.0, 0.1], ['stable', 0.961803]], [[41.507, 2.269, 5.919, 0.8], ['unstable', 3.258936]], [[0.0, 0.76, 4.988, 1.0], ['unstable', 5.563158]], [[48.542, 2.726, 7.357, 0.8], ['unstable', 3.199601]], [[0.0, 1.0, 2.0008, 1.0], ['unstable', 1.0008]], [[0.0, 1.0, 1.0, 0.5], ['marginal', 1.0]], [[36.647, 2.751, 2.545, 0.01], ['stable', 0.996033]], [[14.174, 2.424, 0.649, 0.8], ['unstable', 2.127938]]]]
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	`['stable', 0.957792]`	`['stable', 0.957792]`	Passed
case 1	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 2	`['unstable', 1.472368]`	`['unstable', 1.472368]`	Passed
case 3	`['unstable', 9.896296]`	`['unstable', 9.896296]`	Passed
case 4	`['stable', 0.997415]`	`['stable', 0.997415]`	Passed
case 5	`['marginal', 1.0]`	`['marginal', 1.0]`	Passed
case 6	`['unstable', 11.803107]`	`['unstable', 11.803107]`	Passed
case 7	`['unstable', 1.0008]`	`['unstable', 1.0008]`	Passed

SHA-256 / 891153dd6f7b7c2b2080b6795e7b090072aa4e6d7eaa89773efb5c90f02fbe37

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 2026-09-29T14:50:53.875683+00:00.

Case digest / d7732dcee4048383a1fd4d4e07b27892b2b50ce43e5b546ad5f0cc5351dacb18