FA-65826 / Ecological population dynamics / Open access
Count-based population viability (diffusion approximation): distance to threshold · case 01
Risk is computed from the first census rather than the current population.
ROOT CAUSE
The starting abundance is the first count.
VERIFIED REPAIR
Restore the distance to threshold rule: `d = math.log(counts[-1] / threshold)`.
Unsuccessful approach: Using the peak count overstates the distance to the threshold.
Case contract
Log growth increments ln(N[t+1]/N[t]); mu = mean, sigma^2 = sample variance (q-1); d = ln(N_last/threshold); d<=0 gives probability 1.0; zero variance gives 1.0 if mu*T <= -d else 0.0; otherwise P = Phi((-d-muT)/(s sqrtT)) + exp(-2 mu d/s^2) Phi((-d+muT)/(s sqrtT)) capped at 1; return [mu, var, P] rounded 6; None with fewer than three counts, non-positive counts or threshold.
Why this case matters
Population projections set harvest quotas, conservation status and pest-control timing; a wrong update order, boundary or rate conversion silently changes management advice.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(counts, threshold, horizon):
if len(counts) < 3 or min(counts) <= 0 or threshold <= 0:
return None
logs = [math.log(counts[t + 1] / counts[t]) for t in range(len(counts) - 1)]
q = len(logs)
mu = sum(logs) / q
var = sum((x - mu) ** 2 for x in logs) / (q - 1)
d = math.log(counts[0] / threshold)
if d <= 0:
return [round(mu, 6), round(var, 6), 1.0]
if var == 0:
return [round(mu, 6), 0.0, 1.0 if mu * horizon <= -d else 0.0]
sd = math.sqrt(var * horizon)
phi = lambda z: 0.5 * (1 + math.erf(z / math.sqrt(2)))
prob = phi((-d - mu * horizon) / sd) + math.exp(-2 * mu * d / var) * phi((-d + mu * horizon) / sd)
return [round(mu, 6), round(var, 6), round(min(prob, 1.0), 6)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])]]
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 |
|---|---|---|---|
| regression: grizzly census decline | [-0.062113, 0.004743, 0.19763] | [-0.062113, 0.004743, 0.688257] | Failed |
| regression: fluctuating stable | [0.009758, 0.046826, 0.203751] | [0.009758, 0.046826, 0.191687] | Failed |
| control: growing | [0.157691, 0.003032, 0.0] | [0.157691, 0.003032, 0.0] | Passed |
| control: already below threshold | [-0.660878, 0.130471, 1.0] | [-0.660878, 0.130471, 1.0] | Passed |
| control: constant decline | [-0.693147, 0.0, 1.0] | [-0.693147, 0.0, 1.0] | Passed |
| control: too short | None | None | Passed |
| regression: high variance | [-0.102165, 1.188964, 0.834919] | [-0.102165, 1.188964, 0.882919] | Failed |
SHA-256 / 418fcf748e3e75d094eb4861c456e39f04b4f108ebcca1387470acc15c3c547e
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(counts, threshold, horizon):
if len(counts) < 3 or min(counts) <= 0 or threshold <= 0:
return None
logs = [math.log(counts[t + 1] / counts[t]) for t in range(len(counts) - 1)]
q = len(logs)
mu = sum(logs) / q
var = sum((x - mu) ** 2 for x in logs) / (q - 1)
d = math.log(max(counts) / threshold)
if d <= 0:
return [round(mu, 6), round(var, 6), 1.0]
if var == 0:
return [round(mu, 6), 0.0, 1.0 if mu * horizon <= -d else 0.0]
sd = math.sqrt(var * horizon)
phi = lambda z: 0.5 * (1 + math.erf(z / math.sqrt(2)))
prob = phi((-d - mu * horizon) / sd) + math.exp(-2 * mu * d / var) * phi((-d + mu * horizon) / sd)
return [round(mu, 6), round(var, 6), round(min(prob, 1.0), 6)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])]]
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 |
|---|---|---|---|
| regression: grizzly census decline | [-0.062113, 0.004743, 0.19763] | [-0.062113, 0.004743, 0.688257] | Failed |
| regression: fluctuating stable | [0.009758, 0.046826, 0.161525] | [0.009758, 0.046826, 0.191687] | Failed |
| control: growing | [0.157691, 0.003032, 0.0] | [0.157691, 0.003032, 0.0] | Passed |
| control: already below threshold | [-0.660878, 0.130471, 1.0] | [-0.660878, 0.130471, 1.0] | Passed |
| control: constant decline | [-0.693147, 0.0, 1.0] | [-0.693147, 0.0, 1.0] | Passed |
| control: too short | None | None | Passed |
| regression: high variance | [-0.102165, 1.188964, 0.795455] | [-0.102165, 1.188964, 0.882919] | Failed |
SHA-256 / 627c0d761025acd2d7a6d6246a0c91d07dc493373fef2d3b0671528b806637dd
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(counts, threshold, horizon):
if len(counts) < 3 or min(counts) <= 0 or threshold <= 0:
return None
logs = [math.log(counts[t + 1] / counts[t]) for t in range(len(counts) - 1)]
q = len(logs)
mu = sum(logs) / q
var = sum((x - mu) ** 2 for x in logs) / (q - 1)
d = math.log(counts[-1] / threshold)
if d <= 0:
return [round(mu, 6), round(var, 6), 1.0]
if var == 0:
return [round(mu, 6), 0.0, 1.0 if mu * horizon <= -d else 0.0]
sd = math.sqrt(var * horizon)
phi = lambda z: 0.5 * (1 + math.erf(z / math.sqrt(2)))
prob = phi((-d - mu * horizon) / sd) + math.exp(-2 * mu * d / var) * phi((-d + mu * horizon) / sd)
return [round(mu, 6), round(var, 6), round(min(prob, 1.0), 6)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: already below threshold', ([30, 20, 8], 10, 10), [-0.660878, 0.130471, 1.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])],
[('regression: grizzly census decline',
([45, 42, 44, 38, 36, 35, 31], 10, 20),
[-0.062113, 0.004743, 0.688257]),
('regression: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('control: growing', ([30, 36, 40, 50, 55, 66], 10, 30), [0.157691, 0.003032, 0.0]),
('control: constant decline', ([80, 40, 20], 5, 5), [-0.693147, 0.0, 1.0]),
('control: too short', ([10, 9], 5, 5), None),
('regression: high variance', ([200, 100, 260, 90, 300, 120], 30, 25), [-0.102165, 1.188964, 0.882919]),
('regression: peak then decline', ([60, 120, 80, 70, 65], 15, 40), [0.020011, 0.222192, 0.539603])]]
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 |
|---|---|---|---|
| regression: grizzly census decline | [-0.062113, 0.004743, 0.688257] | [-0.062113, 0.004743, 0.688257] | Passed |
| regression: fluctuating stable | [0.009758, 0.046826, 0.191687] | [0.009758, 0.046826, 0.191687] | Passed |
| control: growing | [0.157691, 0.003032, 0.0] | [0.157691, 0.003032, 0.0] | Passed |
| control: already below threshold | [-0.660878, 0.130471, 1.0] | [-0.660878, 0.130471, 1.0] | Passed |
| control: constant decline | [-0.693147, 0.0, 1.0] | [-0.693147, 0.0, 1.0] | Passed |
| control: too short | None | None | Passed |
| regression: high variance | [-0.102165, 1.188964, 0.882919] | [-0.102165, 1.188964, 0.882919] | Passed |
SHA-256 / e1d333b24f53737090baae6f35c2f9920ffc82b1734110f282ea5a14456371b0
Verification & scope
Deterministic bounded teaching model with a stipulated contract; not a validated scientific or public-health modelling 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 2026-09-29T14:47:37.646034+00:00.
Case digest / 6ab03821cec0d38366250db3c6a67e3c7a8cf7d87da138444903583e1792be58