FA-12056 / Sensor fusion consistency / Open access
Correlated estimates lose shared uncertainty during fusion · case 01
Reported fused uncertainty is smaller than the shared sensor noise.
ROOT CAUSE
Cross covariance is omitted from the best linear unbiased variance.
VERIFIED REPAIR
Use (p*r-c*c)/(p+r-2*c) for the declared two-estimate covariance.
Unsuccessful approach: Adding covariance to the numerator without correcting the denominator remains inconsistent.
Case contract
Given positive p,r and cross covariance c with c*c<p*r and p+r-2*c>0, return exact variance of the minimum-variance unbiased linear combination as a fraction string.
Why this case matters
Deterministic sensor-fusion model isolating one consistency contract; no hardware or production estimator is simulated.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(p, r, c):
return str(Fraction(p*r,p+r))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive shared noise', solve(2*N,2*N,N), str(Fraction(3*N,2)))
check('negative correlation', solve(2*N,2*N,-N), str(Fraction(N,2)))
check('independent equal sensors', solve(N,N,0), str(Fraction(N,2)))
check('unequal independent sensors', solve(N,3*N,0), str(Fraction(3*N,4)))
check('unequal shared noise', solve(2*N,5*N,N), str(Fraction(9*N,5)))
check('unequal opposing noise', solve(2*N,5*N,-N), str(N))
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 |
|---|---|---|---|
| positive shared noise | 1 | 3/2 | Failed |
| negative correlation | 1 | 1/2 | Failed |
| independent equal sensors | 1/2 | 1/2 | Passed |
| unequal independent sensors | 3/4 | 3/4 | Passed |
| unequal shared noise | 10/7 | 9/5 | Failed |
| unequal opposing noise | 10/7 | 1 | Failed |
SHA-256 / 8834dc9a809f4e101cea58fb75700e103d9be2e3fcd21252af5547a6f3f1d861
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(p, r, c):
return str(Fraction(p*r-c*c,p+r))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive shared noise', solve(2*N,2*N,N), str(Fraction(3*N,2)))
check('negative correlation', solve(2*N,2*N,-N), str(Fraction(N,2)))
check('independent equal sensors', solve(N,N,0), str(Fraction(N,2)))
check('unequal independent sensors', solve(N,3*N,0), str(Fraction(3*N,4)))
check('unequal shared noise', solve(2*N,5*N,N), str(Fraction(9*N,5)))
check('unequal opposing noise', solve(2*N,5*N,-N), str(N))
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 |
|---|---|---|---|
| positive shared noise | 3/4 | 3/2 | Failed |
| negative correlation | 3/4 | 1/2 | Failed |
| independent equal sensors | 1/2 | 1/2 | Passed |
| unequal independent sensors | 3/4 | 3/4 | Passed |
| unequal shared noise | 9/7 | 9/5 | Failed |
| unequal opposing noise | 9/7 | 1 | Failed |
SHA-256 / 97ed123f15a802e7430ba362b2bd6c31bd3530c9fb7033ce1116efcf137233ee
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(p, r, c):
return str(Fraction(p*r-c*c,p+r-2*c))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive shared noise', solve(2*N,2*N,N), str(Fraction(3*N,2)))
check('negative correlation', solve(2*N,2*N,-N), str(Fraction(N,2)))
check('independent equal sensors', solve(N,N,0), str(Fraction(N,2)))
check('unequal independent sensors', solve(N,3*N,0), str(Fraction(3*N,4)))
check('unequal shared noise', solve(2*N,5*N,N), str(Fraction(9*N,5)))
check('unequal opposing noise', solve(2*N,5*N,-N), str(N))
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 |
|---|---|---|---|
| positive shared noise | 3/2 | 3/2 | Passed |
| negative correlation | 1/2 | 1/2 | Passed |
| independent equal sensors | 1/2 | 1/2 | Passed |
| unequal independent sensors | 3/4 | 3/4 | Passed |
| unequal shared noise | 9/5 | 9/5 | Passed |
| unequal opposing noise | 1 | 1 | Passed |
SHA-256 / 62943b5a6e4848148906cd095f245bf1ae88d3851ce05a3a196bb2e830c32a11
Verification & scope
Exact small scalar or two-axis models; no nonlinear dynamics, numerical conditioning, or real sensor noise simulation. 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 2026-09-29T14:38:53.466020+00:00.
Case digest / ae546bad65da3f7c9d20aca3e0e70b788f079ed97eb90612241a896b8b8df54d