FA-65836 / Ecological population dynamics / Open access
Count-based population viability (diffusion approximation): time scaling · case 01
Uncertainty grows linearly instead of with the square root of time.
ROOT CAUSE
The standard deviation is scaled by T instead of sqrt(T).
THE FAILURE
The standard deviation is scaled by T instead of sqrt(T).
Unsuccessful approach: Using the variance directly confuses variance and standard deviation.
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[-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]),
('regression: 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: 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: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('regression: 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: 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])]]
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, 1.0] | [-0.062113, 0.004743, 0.688257] | Failed |
| regression: fluctuating stable | [0.009758, 0.046826, 0.650315] | [0.009758, 0.046826, 0.191687] | Failed |
| regression: growing | [0.157691, 0.003032, 3.1e-05] | [0.157691, 0.003032, 0.0] | Failed |
| 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, 1.0] | [-0.102165, 1.188964, 0.882919] | Failed |
SHA-256 / 79135e18c1c589d38bc840ef56378e680c909ab48e45a34c3ecd72aff48abd46
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(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 = 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]),
('regression: 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: 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: fluctuating stable', ([100, 120, 90, 110, 95, 105], 20, 50), [0.009758, 0.046826, 0.191687]),
('regression: 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: 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])]]
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, 0.878716] | [-0.062113, 0.004743, 0.688257] | Failed |
| regression: fluctuating stable | [0.009758, 0.046826, 0.334273] | [0.009758, 0.046826, 0.191687] | Failed |
| regression: 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, 1.0] | [-0.102165, 1.188964, 0.882919] | Failed |
SHA-256 / 818d0a63f5ed8f1548325433c8bdd76bcfed7bf923c4bb45ae4eccb6c3bb54a6
HELD IN THE MEMBER ARCHIVE
The verified repair and its recorded checks are member-only.
This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.
Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.
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Sign in to the archive ↗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 2026-09-29T14:47:37.778968+00:00.
Case digest / d5cc3a8e749729966bc1935bffc41789fcc2c91de1e541aef8ae95767e6169d0