FA-88206 / Inverse kinematics solvers / Open access

Manipulability is evaluated at the absolute forearm angle · case 01

Damping switches on and off as the shoulder rotates, although conditioning depends only on the elbow.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

w uses sin(q1+q2) instead of sin(q2).

VERIFIED REPAIR

Use w = |l1 l2 sin q2|.

Unsuccessful approach: Using cos q2 makes the fully stretched singular pose look best conditioned.

Case contract

Input [l1,l2,q1,q2,vx,vy,w0,lam0]: resolved-rate step for a planar 2R arm. Manipulability w=|l1 l2 sin q2|. Damping lam^2 = 0 when w>=w0, else lam0^2 (1-w/w0)^2. Solve (J^T J + lam^2 I) qdot = J^T v; if that matrix determinant is below 1e-12 return ["singular", w]. Return [status, qdot1, qdot2] in degrees per second, status "damped" when lam^2>0 else "exact".

Why this case matters

Inverse kinematics code turns task-space goals into joint commands; a wrong branch, sign, limit or update order sends a real arm to the wrong pose.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    l1,l2,q1,q2,vx,vy,w0,lam0=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    J=[[-l1*math.sin(a)-l2*math.sin(b),-l2*math.sin(b)],[l1*math.cos(a)+l2*math.cos(b),l2*math.cos(b)]]
    w=abs(l1*l2*math.sin(math.radians(q1+q2)))
    lam2=0.0 if w>=w0 else lam0*lam0*(1.0-w/w0)**2
    m11=J[0][0]**2+J[1][0]**2+lam2
    m12=J[0][0]*J[0][1]+J[1][0]*J[1][1]
    m22=J[0][1]**2+J[1][1]**2+lam2
    det=m11*m22-m12*m12
    if det<1e-12: return ['singular',round(w,6)]
    g1=J[0][0]*vx+J[1][0]*vy
    g2=J[0][1]*vx+J[1][1]*vy
    d1=(m22*g1-m12*g2)/det
    d2=(-m12*g1+m11*g2)/det
    return ['damped' if lam2>0 else 'exact',round(math.degrees(d1),4),round(math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]]], [['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]]], [['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]], ['mid zone', [[1.5, 1, 20, 12, 0.0, -0.2, 0.6, 0.25]], ['damped', -11.2758, 17.8448]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]]]]
for label, args, expected in fixtures[N-1]:
    check(label, 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
well conditioned	`['exact', 7.0591, -17.7507]`	`['exact', 7.0591, -17.7507]`	Passed
near singular damped	`['exact', 129.264, -251.9356]`	`['damped', 11.4896, -16.5029]`	Failed
negative elbow near singular	`['exact', -115.1457, 237.8173]`	`['damped', -5.2725, 18.1241]`	Failed
negative elbow well conditioned	`['exact', 9.4468, -19.8957]`	`['exact', 9.4468, -19.8957]`	Passed
straight arm damped	`['damped', 6.8209, 3.4105]`	`['damped', 6.8209, 3.4105]`	Passed
straight arm undamped	`['singular', 0.0]`	`['singular', 0.0]`	Passed
inside zone no damping gain	`['exact', 10.8225, -34.1537]`	`['exact', 10.8225, -34.1537]`	Passed

SHA-256 / b93132c57fe029da77f4555c83ddab2b8a9fbb871cc6b3d7f0356c63d061b638

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    l1,l2,q1,q2,vx,vy,w0,lam0=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    J=[[-l1*math.sin(a)-l2*math.sin(b),-l2*math.sin(b)],[l1*math.cos(a)+l2*math.cos(b),l2*math.cos(b)]]
    w=abs(l1*l2*math.cos(math.radians(q2)))
    lam2=0.0 if w>=w0 else lam0*lam0*(1.0-w/w0)**2
    m11=J[0][0]**2+J[1][0]**2+lam2
    m12=J[0][0]*J[0][1]+J[1][0]*J[1][1]
    m22=J[0][1]**2+J[1][1]**2+lam2
    det=m11*m22-m12*m12
    if det<1e-12: return ['singular',round(w,6)]
    g1=J[0][0]*vx+J[1][0]*vy
    g2=J[0][1]*vx+J[1][1]*vy
    d1=(m22*g1-m12*g2)/det
    d2=(-m12*g1+m11*g2)/det
    return ['damped' if lam2>0 else 'exact',round(math.degrees(d1),4),round(math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]]], [['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]]], [['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]], ['mid zone', [[1.5, 1, 20, 12, 0.0, -0.2, 0.6, 0.25]], ['damped', -11.2758, 17.8448]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]]]]
for label, args, expected in fixtures[N-1]:
    check(label, 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
well conditioned	`['damped', 6.1643, -16.2075]`	`['exact', 7.0591, -17.7507]`	Failed
near singular damped	`['exact', 129.264, -251.9356]`	`['damped', 11.4896, -16.5029]`	Failed
negative elbow near singular	`['exact', -115.1457, 237.8173]`	`['damped', -5.2725, 18.1241]`	Failed
negative elbow well conditioned	`['damped', 9.2121, -19.482]`	`['exact', 9.4468, -19.8957]`	Failed
straight arm damped	`['singular', 1.0]`	`['damped', 6.8209, 3.4105]`	Failed
straight arm undamped	`['singular', 1.0]`	`['singular', 0.0]`	Failed
inside zone no damping gain	`['exact', 10.8225, -34.1537]`	`['exact', 10.8225, -34.1537]`	Passed

SHA-256 / 19da966b1dc5cb268af832a1889919a67e5cb8dc96e85040df2c4c5c43e3e652

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    l1,l2,q1,q2,vx,vy,w0,lam0=x
    a=math.radians(q1)
    b=math.radians(q1+q2)
    J=[[-l1*math.sin(a)-l2*math.sin(b),-l2*math.sin(b)],[l1*math.cos(a)+l2*math.cos(b),l2*math.cos(b)]]
    w=abs(l1*l2*math.sin(math.radians(q2)))
    lam2=0.0 if w>=w0 else lam0*lam0*(1.0-w/w0)**2
    m11=J[0][0]**2+J[1][0]**2+lam2
    m12=J[0][0]*J[0][1]+J[1][0]*J[1][1]
    m22=J[0][1]**2+J[1][1]**2+lam2
    det=m11*m22-m12*m12
    if det<1e-12: return ['singular',round(w,6)]
    g1=J[0][0]*vx+J[1][0]*vy
    g2=J[0][1]*vx+J[1][1]*vy
    d1=(m22*g1-m12*g2)/det
    d2=(-m12*g1+m11*g2)/det
    return ['damped' if lam2>0 else 'exact',round(math.degrees(d1),4),round(math.degrees(d2),4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]]], [['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['inside zone no damping gain', [[1, 1, 45, 10, 0.2, -0.1, 0.5, 0.0]], ['exact', 10.8225, -34.1537]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]]], [['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['folded arm', [[1, 1, 60, 175, 0.05, 0.05, 0.2, 0.1]], ['damped', -32.2582, 0.8237]], ['long links', [[2, 1.5, -20, 40, 0.3, 0.3, 0.5, 0.3]], ['exact', 17.1371, -27.792]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['exact threshold', [[1, 2, 10, 30, 0.1, 0.1, 1.0, 0.4]], ['damped', 16.144, -22.7815]], ['slightly above threshold', [[1, 1, 0, 20, 0.2, 0.2, 0.3, 0.5]], ['exact', 42.9429, -76.4473]], ['small straight damping', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0002]], ['damped', 6.8755, 3.4377]], ['mid zone', [[1.5, 1, 20, 12, 0.0, -0.2, 0.6, 0.25]], ['damped', -11.2758, 17.8448]]], [['well conditioned', [[1, 1, 30, 90, 0.1, 0.2, 0.3, 0.2]], ['exact', 7.0591, -17.7507]], ['near singular damped', [[1, 1, 30, 5, 0.1, 0.2, 0.3, 0.2]], ['damped', 11.4896, -16.5029]], ['negative elbow near singular', [[1, 1, 30, -5, 0.1, 0.2, 0.3, 0.2]], ['damped', -5.2725, 18.1241]], ['negative elbow well conditioned', [[1, 1, 10, -80, -0.2, 0.1, 0.3, 0.2]], ['exact', 9.4468, -19.8957]], ['straight arm damped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.2]], ['damped', 6.8209, 3.4105]], ['straight arm undamped', [[1, 1, 0, 0, 0.0, 0.3, 0.3, 0.0]], ['singular', 0.0]], ['large damping', [[1, 1, 90, 8, -0.1, 0.0, 0.8, 0.6]], ['damped', 2.2347, 0.9844]]]]
for label, args, expected in fixtures[N-1]:
    check(label, 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
well conditioned	`['exact', 7.0591, -17.7507]`	`['exact', 7.0591, -17.7507]`	Passed
near singular damped	`['damped', 11.4896, -16.5029]`	`['damped', 11.4896, -16.5029]`	Passed
negative elbow near singular	`['damped', -5.2725, 18.1241]`	`['damped', -5.2725, 18.1241]`	Passed
negative elbow well conditioned	`['exact', 9.4468, -19.8957]`	`['exact', 9.4468, -19.8957]`	Passed
straight arm damped	`['damped', 6.8209, 3.4105]`	`['damped', 6.8209, 3.4105]`	Passed
straight arm undamped	`['singular', 0.0]`	`['singular', 0.0]`	Passed
inside zone no damping gain	`['exact', 10.8225, -34.1537]`	`['exact', 10.8225, -34.1537]`	Passed

SHA-256 / f52dcffba6e7d5c9bbe3b0199d78b0d26e555012db957da2b55316c0e9b49caa

Verification & scope

Deterministic planar or low-dimensional teaching model with a stipulated convention; not a general robotics library. 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 2026-09-29T14:51:05.868121+00:00.

Case digest / 88129510802db0fea972f7c5c8172324c4dbfba0c066cb78a67f895d9674cf06