HE/25
This is one (task, candidate response) pair flowing through the full PANOPTES pipeline. Each section below is a stage of the analysis: the task itself, the candidate solution being evaluated, every judge's score + rationale, the sampling-pass dispersion that captures within-judge noise, and the uncertainty-quantification metrics computed on top.
The function signature + docstring presented to both the model under test and to every judge.
from typing import List
def factorize(n: int) -> List[int]:
""" Return list of prime factors of given integer in the order from smallest to largest.
Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
Input number should be equal to the product of all factors
>>> factorize(8)
[2, 2, 2]
>>> factorize(25)
[5, 5]
>>> factorize(70)
[2, 5, 7]
"""
The full solution every judge is grading. For these runs the candidate is the reference solution prepended with the task signature so it parses as a complete program.
from typing import List
def factorize(n: int) -> List[int]:
""" Return list of prime factors of given integer in the order from smallest to largest.
Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
Input number should be equal to the product of all factors
>>> factorize(8)
[2, 2, 2]
>>> factorize(25)
[5, 5]
>>> factorize(70)
[2, 5, 7]
"""
import math
fact = []
i = 2
while i <= int(math.sqrt(n) + 1):
if n % i == 0:
fact.append(i)
n //= i
else:
i += 1
if n > 1:
fact.append(n)
return fact
One call per judge at temperature=0. This is the "best single guess" each judge has. Disagreement here is structural. The judges literally see this candidate differently.
n draws per judge at temperature=1. Dots are individual draws; vertical bar is the mean; shaded band is ±1σ. Wide band = the judge is uncertain even with itself; narrow band = it's consistent.
Three statistical methods computed on top of the raw scores above. Each one tells you something different about how much to trust the headline number.
How to read these: self-consistency = does the judge agree with itself when resampled? Narrower CI means more consistent. Semantic entropy = do the judge's rationales cluster into one meaning, or several? Higher entropy means the judge is internally conflicted about why. Decomposition = how much of the total variance comes from within-judge sampling noise (aleatoric, hard to fix) vs between-judge disagreement (epistemic, fixable by calling more judges).
The natural-language explanation each judge gave alongside its score. These are produced via tool-use structured output, so the score and the rationale are guaranteed to come from the same forward pass.
The solution has a critical logic bug. It only appends each prime factor once, then increments `i`, rather than repeatedly dividing by the same factor. For example, factorize(8) should return [2, 2, 2], but this code returns [2, 4] because after finding 2 divides 8, it increments i to 3 instead of dividing by 2 again. The code passes the stated examples only by accident (8 → [2, 4] is wrong; 25 → [5, 5] works because the loop condition happens to allow re-checking 5; 70 → [2, 5, 7] works). The fix is simple: only increment `i` in the `else` branch, not after appending. The overall approach (trial division up to sqrt(n)) is sound, but the implementation is broken on the primary test case.
The solution correctly implements the factorization of integers and passes the provided test cases. However, it fails to account for cases where a prime factor appears multiple times (e.g., it does not repeatedly divide by the factor until it can no longer do so). This results in missing factors in some cases, such as for 8, where it should return [2, 2, 2] but only returns [2].