FAILURE MAP
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FA-91706 / Digital signal filters / Open access

Bilinear mapping flips the pole coefficient · case 01

The pole is mirrored to the other side of the z-plane, turning a lowpass into a resonance near Nyquist.

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

ROOT CAUSE

a1 is computed as (K - wc)/(K + wc).

VERIFIED REPAIR

Use a1 = (wc - K)/(K + wc).

Unsuccessful approach: The attempted repair divides by K instead of K + wc, leaving a0 unnormalized.

Case contract

Input [kind, wc, fs] with rational analog cutoff wc (rad/s) and sample rate fs. Map H(s) = wc/(s+wc) (lowpass) or s/(s+wc) (highpass) with s = K (1 - z^-1)/(1 + z^-1), K = 2 fs, no prewarping. Return {"b": [b0, b1], "a": [1, a1]} as fraction strings normalized so a0 = 1, or "bad-spec".

Why this case matters

The bilinear transform is the standard analog-to-digital mapping; K or sign slips move the cutoff or flip the filter type.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    kind, wc, fs = x[0], Fraction(x[1]), Fraction(x[2])
    if wc <= 0 or fs <= 0:
        return 'bad-spec'
    K = 2 * fs
    d = K + wc
    if kind == 'lowpass':
        b = [wc / d, wc / d]
    else:
        b = [K / d, -K / d]
    a = [Fraction(1), (K - wc) / d]
    return {'b': [str(v) for v in b], 'a': [str(v) for v in a]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: lowpass wc=1 fs=1', ['lowpass', '1', '1'], {'b': ['1/3', '1/3'], 'a': ['1', '-1/3']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}]], [['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}]], [['regression: highpass wc=1/2 fs=2', ['highpass', '1/2', '2'], {'b': ['8/9', '-8/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]]]
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 fixtureActualExpectedOutcome
regression: lowpass wc=100 fs=1000{'a': ['1', '19/21'], 'b': ['1/21', '1/21']}{'a': ['1', '-19/21'], 'b': ['1/21', '1/21']}Failed
regression: lowpass wc=1 fs=1{'a': ['1', '1/3'], 'b': ['1/3', '1/3']}{'a': ['1', '-1/3'], 'b': ['1/3', '1/3']}Failed
regression: lowpass wc=3 fs=1/2{'a': ['1', '-1/2'], 'b': ['3/4', '3/4']}{'a': ['1', '1/2'], 'b': ['3/4', '3/4']}Failed
control: bad specbad-specbad-specPassed
regression: lowpass wc=2000 fs=48000{'a': ['1', '47/49'], 'b': ['1/49', '1/49']}{'a': ['1', '-47/49'], 'b': ['1/49', '1/49']}Failed
regression: lowpass wc=1/2 fs=2{'a': ['1', '7/9'], 'b': ['1/9', '1/9']}{'a': ['1', '-7/9'], 'b': ['1/9', '1/9']}Failed
regression: highpass wc=100 fs=1000{'a': ['1', '19/21'], 'b': ['20/21', '-20/21']}{'a': ['1', '-19/21'], 'b': ['20/21', '-20/21']}Failed

SHA-256 / fbbd568c17f66bf557edb0806db60937ac6dee618c3898793d29fd7cacdbc9dd

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    kind, wc, fs = x[0], Fraction(x[1]), Fraction(x[2])
    if wc <= 0 or fs <= 0:
        return 'bad-spec'
    K = 2 * fs
    d = K + wc
    if kind == 'lowpass':
        b = [wc / d, wc / d]
    else:
        b = [K / d, -K / d]
    a = [Fraction(1), (wc - K) / K]
    return {'b': [str(v) for v in b], 'a': [str(v) for v in a]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: lowpass wc=1 fs=1', ['lowpass', '1', '1'], {'b': ['1/3', '1/3'], 'a': ['1', '-1/3']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}]], [['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}]], [['regression: highpass wc=1/2 fs=2', ['highpass', '1/2', '2'], {'b': ['8/9', '-8/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]]]
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 fixtureActualExpectedOutcome
regression: lowpass wc=100 fs=1000{'a': ['1', '-19/20'], 'b': ['1/21', '1/21']}{'a': ['1', '-19/21'], 'b': ['1/21', '1/21']}Failed
regression: lowpass wc=1 fs=1{'a': ['1', '-1/2'], 'b': ['1/3', '1/3']}{'a': ['1', '-1/3'], 'b': ['1/3', '1/3']}Failed
regression: lowpass wc=3 fs=1/2{'a': ['1', '2'], 'b': ['3/4', '3/4']}{'a': ['1', '1/2'], 'b': ['3/4', '3/4']}Failed
control: bad specbad-specbad-specPassed
regression: lowpass wc=2000 fs=48000{'a': ['1', '-47/48'], 'b': ['1/49', '1/49']}{'a': ['1', '-47/49'], 'b': ['1/49', '1/49']}Failed
regression: lowpass wc=1/2 fs=2{'a': ['1', '-7/8'], 'b': ['1/9', '1/9']}{'a': ['1', '-7/9'], 'b': ['1/9', '1/9']}Failed
regression: highpass wc=100 fs=1000{'a': ['1', '-19/20'], 'b': ['20/21', '-20/21']}{'a': ['1', '-19/21'], 'b': ['20/21', '-20/21']}Failed

SHA-256 / f54c37cc1ff30f9902909952383a18305585d5fee221a9ab3d2997e45efe9591

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    kind, wc, fs = x[0], Fraction(x[1]), Fraction(x[2])
    if wc <= 0 or fs <= 0:
        return 'bad-spec'
    K = 2 * fs
    d = K + wc
    if kind == 'lowpass':
        b = [wc / d, wc / d]
    else:
        b = [K / d, -K / d]
    a = [Fraction(1), (wc - K) / d]
    return {'b': [str(v) for v in b], 'a': [str(v) for v in a]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: lowpass wc=1 fs=1', ['lowpass', '1', '1'], {'b': ['1/3', '1/3'], 'a': ['1', '-1/3']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}]], [['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}]], [['regression: highpass wc=1/2 fs=2', ['highpass', '1/2', '2'], {'b': ['8/9', '-8/9'], 'a': ['1', '-7/9']}], ['regression: lowpass wc=100 fs=1000', ['lowpass', '100', '1000'], {'b': ['1/21', '1/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: lowpass wc=1/2 fs=2', ['lowpass', '1/2', '2'], {'b': ['1/9', '1/9'], 'a': ['1', '-7/9']}], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]], [['regression: lowpass wc=3 fs=1/2', ['lowpass', '3', '1/2'], {'b': ['3/4', '3/4'], 'a': ['1', '1/2']}], ['regression: lowpass wc=2000 fs=48000', ['lowpass', '2000', '48000'], {'b': ['1/49', '1/49'], 'a': ['1', '-47/49']}], ['regression: highpass wc=2000 fs=48000', ['highpass', '2000', '48000'], {'b': ['48/49', '-48/49'], 'a': ['1', '-47/49']}], ['control: bad spec', ['lowpass', '0', '1'], 'bad-spec'], ['regression: highpass wc=100 fs=1000', ['highpass', '100', '1000'], {'b': ['20/21', '-20/21'], 'a': ['1', '-19/21']}], ['regression: highpass wc=1 fs=1', ['highpass', '1', '1'], {'b': ['2/3', '-2/3'], 'a': ['1', '-1/3']}], ['regression: highpass wc=3 fs=1/2', ['highpass', '3', '1/2'], {'b': ['1/4', '-1/4'], 'a': ['1', '1/2']}]]]
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 fixtureActualExpectedOutcome
regression: lowpass wc=100 fs=1000{'a': ['1', '-19/21'], 'b': ['1/21', '1/21']}{'a': ['1', '-19/21'], 'b': ['1/21', '1/21']}Passed
regression: lowpass wc=1 fs=1{'a': ['1', '-1/3'], 'b': ['1/3', '1/3']}{'a': ['1', '-1/3'], 'b': ['1/3', '1/3']}Passed
regression: lowpass wc=3 fs=1/2{'a': ['1', '1/2'], 'b': ['3/4', '3/4']}{'a': ['1', '1/2'], 'b': ['3/4', '3/4']}Passed
control: bad specbad-specbad-specPassed
regression: lowpass wc=2000 fs=48000{'a': ['1', '-47/49'], 'b': ['1/49', '1/49']}{'a': ['1', '-47/49'], 'b': ['1/49', '1/49']}Passed
regression: lowpass wc=1/2 fs=2{'a': ['1', '-7/9'], 'b': ['1/9', '1/9']}{'a': ['1', '-7/9'], 'b': ['1/9', '1/9']}Passed
regression: highpass wc=100 fs=1000{'a': ['1', '-19/21'], 'b': ['20/21', '-20/21']}{'a': ['1', '-19/21'], 'b': ['20/21', '-20/21']}Passed

SHA-256 / a1ab277fe8b5a952e42206b1813f85875fae5bd276a1633b5d2f45e7d5ac470c

Verification & scope

A deterministic bounded teaching model with a stipulated toy contract; exact rational arithmetic or fixed-decimal rounding keeps outputs strict JSON. It is not a production DSP library and claims no standards conformance. 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 2026-09-29T14:51:38.426725+00:00.

Case digest / a860f54dbb439c99691d94063ba34fbc37b35f379d309aeef2104497166538b3