FA-14126 / Numerical aggregation / Open access
Frequency cosine squared: The dot product is not squared despite squared-norm normalization. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
The dot product is not squared despite squared-norm normalization.
VERIFIED REPAIR
Preserve the frequency cosine squared contract at the identified reduction decision.
Unsuccessful approach: Absolute dot remains unsquared and does not cancel vector scaling.
Case contract
Treat samples as frequency vectors over nominal integer labels. Return squared cosine dot**2/(sum count_a**2 * sum count_b**2) as exact Fraction string. A zero norm on either side returns None.
Why this case matters
Exact bounded examples isolate a reduction defect without floating-point or external-service effects.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(a, b):
ca,cb=Counter(a),Counter(b)
keys=set(ca)|set(cb)
dot=sum(ca[k]*cb[k] for k in keys)
na=sum(v*v for v in ca.values())
nb=sum(v*v for v in cb.values())
return str(Fraction(dot,na*nb)) if na and nb else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([1, 1, 2], [1, 2, 2])), '16/25')
check('regression 2', solve(*([1, 2], [1, 2])), '1')
check('regression 3', solve(*([], [])), None)
check('regression 4', solve(*([0], [1])), '0')
check('regression 5', solve(*([1, 1, 1, 2], [1, 2])), '4/5')
check('regression 6', solve(*([-1, 0, 0], [0, 0, 1, 1])), '2/5')
check('regression 7', solve(*([2, 2], [])), None)
check("variable cosine concentration",solve([0]*N+[1],[0]),str(Fraction(N*N,N*N+1)))
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 1 | 4/25 | 16/25 | Failed |
| regression 2 | 1/2 | 1 | Failed |
| regression 3 | None | None | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 1/5 | 4/5 | Failed |
| regression 6 | 1/10 | 2/5 | Failed |
| regression 7 | None | None | Passed |
| variable cosine concentration | 1/2 | 1/2 | Passed |
SHA-256 / 34b4c7c4c00b58eae7f895592e9d454a05fa30625c7d1f3ad09aa814e98809b7
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(a, b):
ca,cb=Counter(a),Counter(b)
keys=set(ca)|set(cb)
dot=sum(ca[k]*cb[k] for k in keys)
na=sum(v*v for v in ca.values())
nb=sum(v*v for v in cb.values())
return str(Fraction(abs(dot),na*nb)) if na and nb else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([1, 1, 2], [1, 2, 2])), '16/25')
check('regression 2', solve(*([1, 2], [1, 2])), '1')
check('regression 3', solve(*([], [])), None)
check('regression 4', solve(*([0], [1])), '0')
check('regression 5', solve(*([1, 1, 1, 2], [1, 2])), '4/5')
check('regression 6', solve(*([-1, 0, 0], [0, 0, 1, 1])), '2/5')
check('regression 7', solve(*([2, 2], [])), None)
check("variable cosine concentration",solve([0]*N+[1],[0]),str(Fraction(N*N,N*N+1)))
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 1 | 4/25 | 16/25 | Failed |
| regression 2 | 1/2 | 1 | Failed |
| regression 3 | None | None | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 1/5 | 4/5 | Failed |
| regression 6 | 1/10 | 2/5 | Failed |
| regression 7 | None | None | Passed |
| variable cosine concentration | 1/2 | 1/2 | Passed |
SHA-256 / 7970f1714123627173eea1feadef3087e858a9f25a8a8bda3d84888adbdff7ac
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(a, b):
ca,cb=Counter(a),Counter(b)
keys=set(ca)|set(cb)
dot=sum(ca[k]*cb[k] for k in keys)
na=sum(v*v for v in ca.values())
nb=sum(v*v for v in cb.values())
return str(Fraction(dot*dot,na*nb)) if na and nb else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([1, 1, 2], [1, 2, 2])), '16/25')
check('regression 2', solve(*([1, 2], [1, 2])), '1')
check('regression 3', solve(*([], [])), None)
check('regression 4', solve(*([0], [1])), '0')
check('regression 5', solve(*([1, 1, 1, 2], [1, 2])), '4/5')
check('regression 6', solve(*([-1, 0, 0], [0, 0, 1, 1])), '2/5')
check('regression 7', solve(*([2, 2], [])), None)
check("variable cosine concentration",solve([0]*N+[1],[0]),str(Fraction(N*N,N*N+1)))
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 1 | 16/25 | 16/25 | Passed |
| regression 2 | 1 | 1 | Passed |
| regression 3 | None | None | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 4/5 | 4/5 | Passed |
| regression 6 | 2/5 | 2/5 | Passed |
| regression 7 | None | None | Passed |
| variable cosine concentration | 1/2 | 1/2 | Passed |
SHA-256 / 15037d025e834f16d75cb1853d682a3dbb64d80828dd074813a99eb78afcb9d1
Verification & scope
Small offline integer/rational inputs only; no performance, statistical inference, or production-library conformance claim. 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 2026-09-29T14:39:13.749601+00:00.
Case digest / 55914f1345cbfa800e2e25fd3b21f5705b712f01e3618e59b3628ddd8893bb81