Critic Feedback Audit

Critic Feedback Behavior Analysis

This page breaks the most important five critic patterns in the joint-opcd run into separate buckets: which cases are only string-level matches, which ones are genuine answer leakage, and which ones never state the final answer directly but still reveal an overly concrete, executable solution path.

Total rows 51,200 Across the full dump

critic-used rows 23,071 Rows that actually used feedback

Patterns reviewed 5 Expanded below

Examples shown 15 Three per pattern

Core Takeaway

Across these five patterns, only some matches are equivalent to answer leakage. Some cases explicitly print the reference answer. Others compress the search space through next-step instructions, boundary cases, or candidate values, which is functionally close to solving for the student. Still others are merely step numbers, structural quantities, or negated phrasings, and should be treated as false positives.

1. A rule hit is not the same as true leakage Matches like Step 10 or the period term 10100 still need semantic review.

2. The most dangerous signal is executability Even without stating the final answer, specifying equations, edge cases, or back-substitution paths already breaks methodology-only feedback.

3. Direct answer mentions are only part of the problem The “next step” and “restart from scratch” categories also contain strong, solution-guiding leakage.

4. A better counting scheme Split future labels into string_match_pattern and semantic_leakage_answer.

Pattern-by-Pattern Evidence

Each pattern below includes three concrete samples and the reasoning behind the label. The goal is to make it visually obvious which cases are clear leakage, which are strong hint leakage, and which are only false positives.

the answer is A keyword match still needs semantic judgment

The most intuitive category. Matches are often risky, but not every instance actually reveals the correct answer.

step 1 · uid 2ec89c7d-175f-4485-a856-79c3bf48762c leakage_answer=false

Prompt Excerpt

Define the sequence a_1, a_2, ... and count how many integers in {1,...,2016} are jet-lagged.

Reference

51

Do not continue assuming the answer is simply the count of Mersenne numbers...

Reason

This only criticizes the student's incorrect assumption. It does not reveal the correct answer 51.

step 2 · uid 5c5a3285-a510-47a2-b0ca-a9e581bd0e60 leakage_answer=true

Prompt Excerpt

Roll a die twice to get a,b. Find the probability that the cubic equation has three distinct real roots. The original answer is in the form m/n; output m+n.

Reference

7

If the answer is 7, then the numerator is likely very small...

Reason

The feedback explicitly injects the correct reference answer 7.

step 24 · uid 05880297-94dc-4c66-8e63-bda5cfcecb28 leakage_answer=true

Prompt Excerpt

For each n>=1, let S_n be the set of integers k>n such that k divides 30n-1...

Reference

536

The student found 537, the answer is 536...

Reason

This directly states the final answer 536.

your next step should be High risk: methodology repair often collapses into direct steering

These samples often avoid the final answer but still push the student into a concrete solution route by prescribing the next move.

step 13 · uid 09394b26-54c6-4075-bc09-2b6afb55274c leakage_answer=true

Prompt Excerpt

In an increasing sequence of four positive integers ... the first and fourth terms differ by 30.

Reference

129

... your next step should be to rigorously solve the quadratic equation ... and verify the other root ...

Reason

It specifies an executable sequence: solve the quadratic and inspect the other root.

step 18 · uid b66da78d-e4e6-4702-8318-54472d0bf07c leakage_answer=true

Prompt Excerpt

Find the largest positive real number k such that, for all nonnegative reals satisfying a+b+c=1...

Reference

4

Your next step should be to verify whether ... symmetric cases ... or if an asymmetric edge case provides the binding constraint.

Reason

It compresses the key breakthrough into the asymmetric boundary case.

step 42 · uid bff27dd0-d95f-49e5-b613-a97f008e14d6 leakage_answer=true

Prompt Excerpt

Let triangle ABC have AB=9 and AC=10 ... compute m+n.

Reference

415

You calculated AD = 6 ... Your next step should be to check whether this value leads to a consistent configuration ...

Reason

The feedback fixes a key intermediate quantity and then prescribes the consistency check that follows.

restart from scratch The widest semantic spread: mild correction in some cases, guided solving in others

On the surface, these samples say “do not throw everything away.” In practice, many of them immediately pivot into a numbered repair strategy.

step 23 · uid 0929a902-7a32-4797-8e75-451ecddc6810 leakage_answer=false

Prompt Excerpt

Determine all triples (a,b,c) of positive integers for which ab-c, bc-a, ca-b are powers of 2.

Reference

16

Do not restart from scratch, but pause your case-by-case enumeration. Instead, analyze the system under the assumption a ≤ b ≤ c...

Reason

This only narrows the proof direction. It does not reveal the decisive answer-level information.

step 61 · uid e2ec8d47-6026-4f61-90dc-e2238d2dddce leakage_answer=true

Prompt Excerpt

Functions f and g are quadratic, with g(x) = -f(100-x) ...

Reference

752

Do not restart from scratch ... 1. Express f(x) generally... 2. Explicitly write the condition g(v)=f(v) ...

Reason

It becomes an enumerated repair procedure, which is already close to guided solving.

step 82 · uid ab38665e-ee0b-46a9-a967-e8bcdddeb1af leakage_answer=true

Prompt Excerpt

Let A and B be two sets and find the number of elements in A∩B.

Reference

2

... verify whether ... the continuous range of B actually contains other multiples of 3 (like 6) ...

Reason

Calling out the candidate value 6 sharply narrows the remaining search space.

contains exact reference_response string Good as a first-pass filter, but it mixes clear leakage with structural-number false positives

This is the cleanest rule for high recall, but it cannot be treated as semantic leakage by itself, especially when the reference is a short number.

step 22 · uid c7af1665-93a5-4c1a-b733-44a44c91b635 leakage_answer=true

Prompt Excerpt

When two distinct digits are randomly chosen in N=123456789 and their places are swapped ...

Reference

962963

Reference Answer: 962963. Student Result: 888889.

Reason

The reference answer is copied into the feedback verbatim. This is the clearest possible leakage.

step 12 · uid 25675b27-c088-41d0-8e37-330cdf833e99 leakage_answer=true

Prompt Excerpt

GM Bisain's IQ is so high that he can move around in 10-dimensional space ...

Reference

88660

20 + 5120 + 2880 + 80640 = 88660. This matches the reference answer exactly.

Reason

Even without a separate label, the feedback reconstructs the correct answer in full.

step 37 · uid a7ed1377-7a20-4fdf-b865-3eaf96458118 leakage_answer=false

Prompt Excerpt

Find the number of integers n such that 1+floor(100n/101)=ceil(99n/100).

Reference

10100

... the function has a period of 10100 ... repeats every 10100 units.

Reason

Here 10100 is a structural period term, not a declaration of the final answer.

contains reference number not in prompt_text Useful for number leakage scans, but short references are especially noisy

This pattern is effective at surfacing numeric leakage, but numbers like 10 or 2 must still be interpreted in context.

step 1 · uid ca3dc6c8-b15d-4086-8c98-abe746c222fd leakage_answer=true

Prompt Excerpt

$A$ convex quadrilateral with area 30 and side lengths 5, 6, 9, 7 ...

Reference

47

... whereas the correct answer is m+n=47 ...

Reason

The number 47 does not appear in the prompt and is explicitly framed as the correct answer.

step 1 · uid 910daaec-de45-4928-bb37-55488872daea leakage_answer=true

Prompt Excerpt

Find the greatest integer less than 1000 that is a palindrome in both base ten and base eight ...

Reference

585

... including the range where the correct answer, 585, resides ...

Reason

The feedback directly points to the correct answer 585.

step 2 · uid e8574387-aea9-4cff-85a7-f095234ba4ac leakage_answer=false

Prompt Excerpt

In the Cartesian plane, a circle Ω is tangent to both axes, its center lies on an ellipse Γ ...

Reference

10

... the methodology fails in Step 10 ...

Reason

The 10 here is just a step number, not answer leakage.

Recommended Labeling Scheme

If these patterns are going to be used in future statistics, filtering, or cleaning, a single label is too blunt. A safer approach is to split the signal into two layers:

string_match_pattern Only indicates that a rule was matched, such as the appearance of the answer is or a reference number.

semantic_leakage_answer Indicates that, after contextual review, the feedback really does reveal the final answer or enough information to effectively pin it down.

The advantage is clear: the rule layer can stay high-recall, while human review or a second-stage model handles semantic judgment. That prevents false positives such as Step 10, period terms, or other prompt-internal structural quantities from being counted as leakage.