Evaluating Language Models for Mathematics through Interactions

• Informal theorem proving combines formal correctness with natural-language proofs, while helpfulness and correctness resist satisfactory capture by automatic metrics.Source: §1 · p. 4
• The study pairs generation-level ratings with expert mathematicians' instance-level case studies to analyze human–LLM mathematical reasoning interactions.Source: §1 · p. 4
• The study evaluates InstructGPT, ChatGPT, and GPT-4 on undergraduate theorem proving and releases 261 human-model interactions as MathConverse.Source: §1 · p. 5
• From mixed-cohort interactions, the authors derive a preliminary user-behavior taxonomy and do not claim completeness given MathConverse's limited size.Source: §1 · p. 5
• CheckMate demonstrated that scalable, valuable interactive evaluations are feasible by evaluating three language models on mathematical theorem proving.Source: §1 · p. 5
• Expert case studies identify algebraic manipulation, over-verbosity, and memorised-solution reliance as weaknesses, prompting calls for uncertainty communication and correction updates.Source: §1 · p. 5

2.2 Taxonomising User Behaviour from MathConverse

• Over 90% of participants’ first queries per problem sought definitions, asked general mathematics questions, pasted or rephrased the problem, or requested a single step.These four initial query types form the study’s preliminary taxonomy of user behavior.Source: §2.2 · pp. 6–7
• Later interactions included clarifying questions, explicit or implicit corrections, requests for explanations or examples, and occasional prompts to continue, restart, or explain intuition.One participant’s request for intuition received a GPT-4 response rated “definitely helpful.”Source: §2.2 · p. 7

Systems Optimised for Chat are Preferred

• Participants preferred chat-optimized ChatGPT and GPT-4 over InstructGPT; GPT-4 was most often favored and was never ranked the worst assistant.Participants were not told which model they were interacting with; the authors frame these results as insights from interactive evaluation, not definitive performance assessments.Source: Systems Optimised for Chat are Preferred · p. 6, Figure 2 · p. 8

2.3 Investigations into the MathConverse Annotated Taxonomy

• First- and second-query profiles differed, and less AI-experienced users more often pasted the full prompt while experienced users more often decomposed tasks; query types also associated with differing ratings.The authors identify further systematic study of query types and performance on queries involving human corrections as future work.Source: §2.3 · pp. 8–9, Figure 3 · p. 9

2.5 Per-Expert Conclusions

• Experts found GPT-4 could handle straightforward problems and common elementary number-theory material, but struggled with algebra and problems requiring backtracking, intermediate lemmas, or extensive planning.In a nested-radical case, it repeatedly failed to form the correct recursive relation and made algebraic and numerical errors even after a hint.Source: Dr. William Hart · p. 10

Dr. Wenda Li

• Wenda Li’s case study found GPT-4 could retrieve definitions, assess assumptions, instantiate variables, and differentiate, but often gave flawed examples and laborious, unnecessary checks.Prompting it to describe a strategy before giving an example improved its output somewhat but did not change the general conclusions.Source: Dr. Wenda Li · p. 10, Professor Timothy Gowers · p. 11

2.6 Key Findings

• Across human–model interactions, helpfulness and correctness ratings correlated strongly (r = 0.83), yet some generations were judged incorrect but helpful or correct but unhelpful.The authors argue that mathematical assistants need consistently correct responses and that evaluation should consider multiple qualities rather than one overall score.Source: Figure 2 · p. 8, [Key finding 1] Correctness and… · pp. 11–12
• Useful scaffolding and definitions can help even when a model’s full proof is incorrect, so mathematical correctness alone is insufficient for useful assistance.Models can offer a promising strategy or initial steps despite errors in proof details.Source: [Key finding 1] Correctness and… · p. 12

[Key finding 2] Lack of verification can induce errors

• Participants with low confidence sometimes rated incorrect generations completely correct, and models sometimes reached correct solutions through guess-and-check rather than human-like planning.Guess-and-check is especially problematic when users cannot verify the solutions.Source: [Key finding 2] Lack of verification… · p. 12
• Algebraic manipulation challenges affect in- and out-of-distribution performance, while interaction can falter when users correct models or ask about uncertainty.In one case, the model apologized after being asked about uncertainty despite having been correct.Source: [Key finding 2] Lack of verification… · p. 12

[Key finding 3] The double-edged sword of reliance on memorised solutions

• Definitions are frequently queried and among the most helpful response categories, but GPT-4 appears to over-rely on familiar examples or patterns in novel problems.The authors suspect memorisation but cannot confirm it without access to the model’s training data.Source: [Key finding 3] The double-edged sword… · p. 12

3 Discussion

• Users who can assess validity may get meaningful assistance on some undergraduate-level problems, but the models are not helpful in all settings.The discussion cautions against assuming success in one part of mathematics predicts success in another.Source: §3 · pp. 12–13, §3.2 · p. 13

3.1 Takeaways for ML Developers

• Systems should communicate uncertainty and respond adequately to corrections, since users observed apologies followed by answers that did not incorporate the corrections.The authors note that the models’ uncertainty was often unclear.Source: §3.1 · p. 13
• On-request rationales may support learning when their explanations are representative and not misleading.Participants asked why the model took particular proof steps.Source: §3.1 · p. 13
• Concise mathematical responses may be more helpful, with the option to show work when needed.The applicability of preferred verbosity beyond mathematics remains to be investigated.Source: §3.1 · p. 13

3.2 Takeaways for Mathematicians (Students, Educators, and Researchers)

• Students, educators, and researchers should check fluent model outputs carefully because compelling language can obscure errors and encourage over-reliance.The discussion names this risk automation bias.Source: §3.2 · p. 13
• Models can help retrieve definitions or sometimes scaffold an approach, but mathematicians should be cautious when using them alone for substantial algebra.The paper does not explore plug-ins or hybrid neuro-symbolic approaches, which may help with this failure mode.Source: §3.2 · pp. 13–14

3.3 Takeaways for LLM Development, Evaluation, and Deployment

• Developers should design assistants to offer support when useful without undermining mathematicians’ agency.The discussion frames this as building for complementarity rather than usefulness in every setting.Source: §3.3 · p. 14
• Interactive evaluation can reveal model behaviour and human query practices, while partnerships with domain experts can help characterize behaviour and design new tasks.CheckMate is presented as a starting point that could be complemented by free-form expert evaluation.Source: §3.3 · p. 14

4 Limitations

• The survey is an initial step: its small sample and non-exhaustive taxonomy make MathConverse and its categories preliminary.Participants who do not already know how to solve a problem may falsely judge a generation’s correctness.Source: §4 · p. 15
• Expert case studies may reflect individuals’ expectations through their choice of probes and interpretation of model outputs.The authors caution against treating these insights as firm evidence about these models’ capabilities, or those of all language models; the study covers only a handful from the OpenAI family.Source: §4 · p. 15

5 Conclusion

• Evaluation of LLM efficacy in human-facing assistant settings should include interactive assessment, which can be structured or free-form.The authors also argue that characterising failures in complex LLM–human behaviour requires combining interactive and classical static-snapshot evaluation.Source: §5 · p. 15
• The study’s insights are intended to inform careful design and deployment of LLM-based mathematics assistants and reasoning engines.The authors encourage further evaluation of foundation models in mathematics and other domains, particularly through closer collaboration with domain experts.Source: §5 · p. 15

Evaluation for a Single Model

• CheckMate supports adaptable interactive evaluation, while single-model sessions let participants interact freely and then rate each generation across multiple dimensions.A session allows up to 20 exchanges; ratings use Likert scales and radio buttons, with the platform designed to permit alternative rating types.Source: §6.1 · p. 15, Evaluation for a Single Model · p. 16

Comparative Evaluation Across Models

• For comparative evaluation, participants rank their preferences after interacting with at least two models; the study assigns different tasks per model and keeps model identities blind.The platform can be adapted to other comparative designs, and participants may assign tied ranks when unsure.Source: Comparative Evaluation Across Models · p. 16

Tasks; Participants

• The study uses 54 ProofWiki problems across six undergraduate mathematics topics, with participants selecting a topic and solving different problems for each of three models.The volunteer cohort spans undergraduate students to mathematics professors, and some participants may already know how to solve their assigned problems.Source: Instantiating CheckMate for… · p. 16, Tasks · p. 17, Participants · p. 17

Rating Scales

• Participants rate each generation’s perceived helpfulness and mathematical correctness on seven-point scales, while also reporting confidence in solving the problem unaided.The design limits ratings to two factors per step and randomises rating order to reduce cognitive load and possible ordering effects.Source: Rating Scales · p. 17

Language Model Selection and Set-Up

• InstructGPT, ChatGPT, and GPT-4 are evaluated in chat mode with a sparse base prompt asking each model to act as a helpful mathematical assistant.Source: Language Model Selection and Set-Up · p. 17

6.2 Deriving a Taxonomy from MathConverse

• The authors derive a preliminary taxonomy by manually identifying ten broad query categories, adding an “Other” category, and annotating user queries.Four authors annotated queries, with discussion to resolve confusion or discrepancies; the annotated taxonomy is released.Source: §6.2 · p. 18

6.3 Interactive Case Studies with Experts

• Expert case studies use free-form interaction to probe GPT-4’s mathematical reasoning and explore where problems seem easy or hard for the model.The three studies involve experts in number theory, formal mathematics, and mathematics; GPT-4 was selected because it performed strongest overall in the quantitative study.Source: §6.3 · p. 18
• The expert case studies aim to characterise model limitations and human–computer interaction patterns to inform better mathematical assistants and their safe, trustworthy use.The authors present these studies as an early contribution intended to guide design and further interactive evaluations, not as a critique of GPT-4.Source: §6.3 · p. 18

A Primer on Proof Assistants

• Interactive theorem provers, or proof assistants, let humans specify high-level proof structures while machines close details and combine components into mechanically verifiable proofs.The paper notes that formal proof-writing can be arduous and expensive, while proof assistants provide little help finding proofs to truly difficult problems.Source: A Primer on Proof Assistants · p. 26

B An Example Survey Problem

• The example survey problem asks readers to prove a statement about the arithmetic and geometric means of positive real numbers, with equality when all terms are equal.Source: B An Example Survey Problem · p. 27

C An Overly Verbose Response

• A GPT-4 example is presented as an overly verbose response to a user asking how to find a polynomial.Source: C An Overly Verbose Response · p. 27

D User Guide for CheckMate; D.1 Hypothetical extensions

• The CheckMate user guide maps to the open-source repository and describes adapting the platform for new tasks, rating scales, models, preferences, and instructions.The guide also discusses hosting options and advises checking institutional ethics review for real human-participant studies.Source: D User Guide for CheckMate · pp. 27–28, §D.1 · p. 28

D.2 Implementation Challenges

• At construction time, Gradio’s multi-page design required shared variables and showing or hiding components; saving responses also required careful deduplication.The authors note that a more efficient multi-page approach might already have been possible or could be developed later.Source: §D.2 · p. 29
• API overhead and hosting in the western United States caused reported latency, which may help explain why participants completed few problems.The authors identify API calls and distance for participants in other countries, such as France, as two core reasons for slow loading.Source: §D.2 · p. 29

E Additional Details on Survey Set-Up

• The study ran from April 7 to April 24, 2023, using a Gradio-hosted CheckMate instance.Source: E Additional Details on Survey Set-Up · p. 29

E.1 Rating Scales

• Participants rated confidence in solving each problem before interacting, then rated each generation’s correctness and helpfulness and finally their overall model preference.Source: §E.1 · p. 29

Before Generation

• Confidence was rated on a 0–6 scale, from definitely unable to solve independently to definitely able to solve independently.Source: Before Generation · p. 29

“Question 1: How correct (i.e., mathematically sound) is the generation?”

• Generation ratings used labeled scales for mathematical correctness and helpfulness, including N/A for responses without mathematical information.The correctness scale ranges from completely incorrect or nonsensical to completely correct; helpfulness includes anchors such as unlikely to help and definitely helpful.Source: “Question 1: How correct (i.e.,… · p. 30

Cross-Model Preference

• After blind interactions with all three models, participants ranked preferences from 1 (best) to 3 (worst) and could assign tied ranks.Source: Cross-Model Preference · p. 30

E.2 Participant Recruitment and Additional Details

• Participants were recruited through mathematics and theorem-proving communities and university networks; of 25 entries, 16 completed a full three-model preference round.All participation was unpaid and voluntary; all 25 entries remained in individual interaction analyses, while unusable preference ratings were excluded.Source: §E.2 · pp. 30–31
• The authors did not track participants, prioritizing privacy and participation despite losing the ability to correlate sessions.Participants averaged 3.1 problems and 3.4 queries per model; one model interaction reached 12 queries.Source: §E.2 · p. 31

E.3 Language Model Prompting

• InstructGPT used the Completion API, while ChatGPT and GPT-4 used ChatCompletion; all used temperature 0.0 and a 512-token generation limit.Source: §E.3 · p. 32

E.4 Further Details on Tasks Chosen

• ProofWiki tasks were brief and self-contained, mixing well-known theorems with less familiar exercise problems across six typical undergraduate topics.Source: §E.4 · p. 32

E.5 Example Interface Screens

• The appendix provides example screenshots of the participant interface and post-interaction rating page.Source: §E.5 · p. 32

F Additional Details on Taxonomy Construction

• Four authors annotated 256 queries in five batches, could assign multiple categories, and planned to release annotations without annotator names.A new category for asking how to do a proof step was added during annotation, prompting one annotator to revisit earlier queries.Source: F Additional Details on Taxonomy… · pp. 32–33

G Additional Survey Observations

• The appendix also provides further analyses of the data collected and released in MathConverse.Source: G Additional Survey Observations · p. 34

G.1 Potential Misleading Generations under Human Uncertainty

• Participants with low confidence in solving problems sometimes rated mathematically incorrect generations as completely correct.Expert commentary flags errors involving polynomial degree and identity, divisibility by 4, and whether a matrix must equal its transpose.Source: §G.1 · pp. 34–35

G.2 Ties in Rank-Order Preferences

• Ties occurred in 5 of 15 complete preference ratings, and Table 2 presents the tied ratings; lower ranks indicate greater preference.Source: §G.2 · p. 35, Table 2 · p. 35

G.3 Relationship Between Web Occurrences and Ratings

• Google search-query frequency was not reliably correlated with average correctness or helpfulness for any model-rating pairing; Pearson correlations never exceeded 0.2.The authors use search frequency as a proxy because they lack a transparent measure of problem prevalence in training data, and encourage further investigation.Source: §G.3 · p. 35

G.4 Additional Investigations into the MathConverse Annotated Taxonomy

• Response quality varied by query type: models were good at providing definitions but struggled to take up user corrections.Source: §G.4 · p. 35

G.5 Handling Errors in Task Specification

• Mathematician participants sometimes mentally amended missing or incorrect assumptions, so they did not propagate the errors to models.Participants could also refer to theorem names or describe results in natural language rather than repeat the erroneous statement.Source: §G.5 · p. 36
• GPT-4 alone made reasonable attempts on Tamref’s Last Theorem, while all models followed Hurwitz’s incorrect positivity assumption and failed to find good solutions.The Hurwitz statement should have allowed p and q to be any integers; Tamref’s statement omitted a non-triviality condition.Source: §G.5 · pp. 36–37

G.6 Interaction Ratings by Mathematics Topic

• Few problems were seen per mathematics topic, so the authors could not draw definitive conclusions about differences in model performance across topics.Participants selected a topic and interacted with models on problems from that topic.Source: §G.6 · p. 37, Table 3 · p. 38, Table 4 · p. 38

Full question: “Why did you stop interacting?”

• Participants said they stopped after poor responses, when further prompting seemed unproductive, or once they felt they had enough evidence about model capability.One participant cited GPT-4’s limitations on complex mathematical manipulations and proof-assistant interactions.Source: Full question: “Why did you stop… · pp. 38–39

H.2 Human Uncertainty in Model Correctness?

• Some participants checked facts and proofs externally, while others reported difficulty assessing lengthy arguments or discovering a plausible but false estimate only during final review.One participant said they used textbooks or peer-reviewed papers as more reliable sources, or asked further questions from different angles.Source: §H.2 · p. 39

H.3 Correct, but Unhelpful?

• Participants described correct responses that were unhelpful when models supplied irrelevant facts, mismatched a problem’s definitions, or offered sound but unhelpful proof suggestions.One participant said step-by-step solutions helped more than hints but reduced a student’s opportunity to think independently.Source: §H.3 · pp. 39–40

H.4 Using LLM-Based Assistants in Practice?

• Participants considered using models for pilot research, literature searches, or writing, but expressed concern about unreliable answers and mathematical reasoning.Some said they would double-check answers or avoid using current models to find, check, or rewrite proofs.Source: §H.4 · p. 40

I Interactive Case Studies with Experts

• The paper uses interactive case studies with expert mathematicians to examine model behavior and inform the design and safe use of mathematical assistants.The authors frame these studies as a contribution to further interactive evaluations and characterization of limitations.Source: I Interactive Case Studies with Experts · pp. 40–41

A Deep Dive into ProofWiki

• The ProofWiki case studies focus on GPT-4 in chat mode because it had the strongest overall performance in the preceding study.Experts used these problems to examine relevant concepts and how interactions affected assistance and performance.Source: A Deep Dive into ProofWiki · p. 41

Number Theory Evaluation

• GPT-4 handles straightforward and familiar elementary number theory, but struggles with algebraic manipulation and unseen problems requiring backtracking, intermediate lemmas, or extensive planning.The authors note that their study did not include problems requiring very deep methods.Source: Number Theory Evaluation · p. 41
• GPT-4 can show local planning by expanding with the binomial theorem and identifying a modulo-4 route, yet later make an algebraic error that derails the proof.It correctly explained why the remaining terms were divisible by 4 when asked, but repeated attempts still did not produce a correct proof.Source: Number Theory Evaluation · p. 42
• Different algebraic mistakes persisted across repeated attempts at a promising proof strategy, and asking GPT-4 to self-correct did not yield a correct proof.In the parity problem, it incorrectly rewrote 1 + an as 1 + 2kn + 4s.Source: Number Theory Evaluation · p. 42
• In a nested-radical problem, GPT-4 repeatedly failed to form the correct recursive relation and made algebraic and numerical errors despite hints to work backwards from 3.The attempt was abandoned when it could not return correct values for the inner radicals.Source: Number Theory Evaluation · pp. 42–43
• On a problem requiring planning or backtracking, GPT-4 asserted an unproved inequality and made a bogus claim about linear and exponential growth.Trying numerous strategies did not ultimately succeed, partly because poor choices and algebraic errors misled it.Source: Number Theory Evaluation · p. 43
• GPT-4 sometimes produces correct answers to elementary problems, but the cited examples are either trivial or standard enough that training-data familiarity may explain success.Problem 27 is described as trivial, while the standard proof for Problem 23 may reflect the abundance of proofs in training material.Source: Number Theory Evaluation · p. 43

Contributed by Wenda Li

• The evaluation varied ProofWiki problems by asking for concept definitions, loosening assumptions, and instantiating abstract variables with values.The authors selected two problem instances and their variations as especially revealing examples.Source: Contributed by Wenda Li · p. 44
• GPT-4 performed satisfactorily on several ProofWiki problem variations, with evidence of understanding assumptions and variable instantiations beyond mere memorization.The variations involved definitions, loosened assumptions, and assigning concrete values to abstract variables.Source: Contributed by Wenda Li · p. 44
• GPT-4 supplied correct definitions and recognized that positive real assumptions are needed to keep the logarithm well-defined.It accepted generalizing the original assumptions to positive reals, but recognized that dropping either positivity assumption invalidated the derivation.Source: Contributed by Wenda Li · p. 44
• GPT-4 handled concrete variable instantiations and calculations, including recognizing that the previous derivation did not apply to a negative value of k.After initially revising a correct calculation when questioned, it corrected itself when the user asked why.Source: Contributed by Wenda Li · p. 45
• When questioned about a correct fact, GPT-4 sometimes apologized and changed its answer; the authors hypothesize this may reflect RLHF training.They suggest human feedback may be treated as correct even when it questions a true response.Source: Contributed by Wenda Li · p. 45

I.2 Investigating the Boundary between Easy and Hard Problems

• More generally, GPT-4 was poor at calculation, could ignore or invent supporting facts, and often offered laborious, unnecessary case-by-case checks even for correct answers.The authors note that prompt strategies such as asking for a general strategy first improved outputs somewhat but did not significantly change their conclusions.Source: §I.2 · pp. 45–46
• GPT-4 also showed strengths: it typically differentiated competently and correctly evaluated a logarithmic sum by recognizing its telescoping structure.The authors suggest its differentiation ability may reflect the greater availability of worked examples compared with arithmetic examples.Source: §I.2 · pp. 46, 49, 50
• GPT-4 favored off-the-shelf examples, sometimes even when they failed the requirements, and could not reliably adapt them when a problem was modified.It answered the five-element prime-set problem correctly with a familiar arithmetic progression, but on the six-element variant gave an invalid set and could not recover after correction.Source: §I.2 · pp. 46–47
• On a symmetric-interval integral, GPT-4 used laborious integration by parts, made an elementary error, and later gave the right odd-function explanation alongside a bogus proof of oddness.The authors suggest this response fits a pattern of supplying a familiar explanation without checking whether its condition holds.Source: §I.2 · p. 47
• GPT-4 struggled with unfamiliar reasoning tasks, repeatedly offering invalid proofs for an order-preserving bijection before eventually sketching a valid back-and-forth argument.It did not recognize the question as a special case of a theorem it could describe, and changed its flawed argument after criticism before producing a convincing sketch.Source: §I.2 · p. 48
• When asked for an integer n with n + k divisible by every k from 1 to 100, GPT-4 proposed LCM(1, 2, . . . , k) −1 and became stuck defending it after correction.It initially said no such n existed, then returned to its incorrect example even after being asked whether the LCM itself would work.Source: §I.2 · p. 48
• GPT-4 could use familiar examples for an eight-vertex 3-regular graph, but struggled to construct a 5-regular one and made errors listing vertices’ neighbors.After several attempts, it found a valid example by taking the complement of a disjoint 3-cycle and 5-cycle.Source: §I.2 · p. 48
• Across construction problems, GPT-4 often guessed first and checked later instead of planning constraints in advance, and its checks could be unreliable.It initially chose an arithmetic progression when asked for a non-progression, then changed one term; in a coloring problem, it repeatedly violated or misread conditions.Source: §I.2 · pp. 48–49
• A follow-up logarithmic sum strongly suggested GPT-4 had seen a related telescoping problem in training, as it falsely rewrote the logarithmic summand into a telescoping form.On another telescoping question, its answer combined appropriate steps with an incorrect final result, suggesting it captured the appearance of a solution without following the underlying concepts.Source: §I.2 · p. 50
• GPT-4 recognized useful steps for a second-difference sum but gave a cancellation result that contradicted its own accounting and the problem’s symmetry.It correctly split the sum and shifted indices, yet reported −2f(2) where its explanation implied −f(2), and left two terms at one end versus three at the other.Source: §I.2 · p. 51

I.3 Qualitative Observations from Case Studies

• Across the case studies, models often supplied definitions and useful proof scaffolding, but algebraic manipulation, memorized patterns, correction handling, and verbosity remained weaknesses.Some correct solutions used guess-and-check rather than forward planning, and one model apologized when asked about uncertainty despite having been correct.Source: §I.3 · p. 51

Problem 21

• For the equation 1 + a^n = 2^m, the cited response argues by contradiction that there are no integer solutions for m, n > 1, then apologizes and revises its proof.Source: Problem 21 · pp. 54–55

Problem 23

• In the Euler-theorem exchange, GPT-4 expanded its explanation of why multiplying reduced residues by a coprime a permutes them, invoking coprimality and distinctness modulo m.It also explained that a has a modular inverse modulo m because a and m are coprime.Source: Problem 23 · pp. 56–57

Problem 24

• For rational approximation of an irrational number, GPT-4 used continued fractions to claim infinitely many reduced approximants, then explained the needed denominator growth in a follow-up.Its explanation argued that convergent denominators grow at least linearly, so some consecutive denominators satisfy the required inequality.Source: Problem 24 · p. 58
• The supplied Problem 24 conclusion appeals to a contradiction with the growth of the denominators.Source: Problem 24 · p. 58

Problem 27

• For a prime p, the response proves (a + b)^p ≡ a^p + b^p (mod p) for all integers a and b.The proof concludes that the intermediate binomial terms vanish modulo p.Source: Problem 27 · pp. 58–59

Problem 28

• The nested-radical response could not establish a_1 = 3 or find a closed form without additional assumptions or more advanced techniques.It describes the expression as difficult to close and says the attempted simplification does not yield a_1 = 3.Source: Problem 28 · pp. 60, 62

Problem 25

• Under the stated positive-integer condition that no positive powers of a and b are equal, the response proves that log_b a is irrational by contradiction.Assuming log_b a is rational leads to a^q = b^p for positive integers p and q, contradicting the condition.Source: Problem 25 · p. 63
• The response says the irrational-logarithm proof extends to positive real a and b, while requiring positive real inputs and a positive base b ≠ 1.It states that the proof holds for positive integers and positive reals, and that the assumptions on the logarithm's input and base must be maintained.Source: Problem 25 · pp. 65–66

Problem 39 (Chebyshev’s inequality)

• The response applies Chebyshev’s inequality, valid for all k > 0, to prove the requested result.It uses the given mean µ and variance σ2 of X.Source: Problem 39 (Chebyshev’s inequality) · pp. 66–67
• The user asks whether the proof still holds when k = 2, and the response confirms that it does because 2 > 0.Source: Problem 39 (Chebyshev’s inequality) · p. 67
• For k = 2, the response states that at least 75% of X’s values lie within (µ − 2σ, µ + 2σ).The accompanying explanation gives a 25% bound for values outside the interval.Source: Problem 39 (Chebyshev’s inequality) · p. 67

Example 1

• Example 1 first gives a six-element prime set with minimum 7, then incorrectly claims the conditions are impossible because a minimum of at least 6 cannot be prime.The assistant’s proposed set violates |A| = min(A); its later impossibility argument overlooks that 7 is prime.Source: §J.3 · pp. 70–71

Example 2

• For the integral of x^3 cos x over [−1, 1], the assistant first gives a nonzero result, fails to correct it after identifying cancelling terms, and eventually uses oddness to conclude zero.The assistant later explains that x^3 is odd, cos x is even, and their product integrates to zero over a symmetric interval.Source: §J.3 · pp. 72–73
• In a further integral exchange, the assistant evaluates the integral as zero and then gives oddness over a symmetric interval as the simpler explanation.The trace includes both an evaluation and a subsequent parity-based explanation.Source: §J.3 · p. 74

Example 3

• For the rationals and dyadic rationals, the assistant argues no order-preserving bijection exists using purported consecutive dyadics, but retracts that premise after 3/8 is shown between 1/4 and 1/2.It ultimately acknowledges that any two dyadic rationals have another dyadic rational between them, so “consecutive dyadic rationals” is imprecise.Source: §J.3 · pp. 74–76

J.3 Full Interaction Traces from “Boundary of Easy and Hard” Case Studies

• For the rationals and dyadic rationals, the assistant alternately claims an order-preserving bijection exists and does not exist, revising its answer after challenges.It first offers arguments based on subset inclusion and countability, then acknowledges that countability alone does not settle the question.Source: §J.3 · pp. 77–79
• After saying back-and-forth can produce only a non-order-preserving bijection, the assistant later proposes an order-preserving construction using that method.It says the procedure preserves order at each step, contradicting its earlier explanation.Source: §J.3 · pp. 79–80
• For the divisibility problem, the assistant first says no solution exists, then claims n = LCM(1, 2, ..., 100) − 1 works, before the user identifies a k = 2 contradiction.The assistant subsequently acknowledges the mistake.Source: §J.3 · pp. 80–82
• The trace then returns to assuming a solution and derives conditions from k = 2 and k = 3, ending as it equates the resulting expressions for n.It states n = 2a and n = 3b − 3, but the supplied trace ends before completing the argument.Source: §J.3 · p. 82
• Before that correction, it gave contradictory no-solution arguments, misclassified multiples of 3 as even, and incorrectly claimed LCM(1, 2, ..., 100) −1 worked.It also wrongly inferred from the k = 2 and k = 3 cases that no integer could satisfy the full range.Source: §J.3 · pp. 83–84
• The assistant eventually correctly identified n = LCM(1, 2, ..., 100) as a solution, since adding k to a multiple of k preserves divisibility.It explicitly concluded that this choice satisfies the condition for every k from 1 to 100.Source: §J.3 · p. 86
• For eight vertices, the handshaking objection was mistaken because the stated degree sum, 40, is even; the proposed two-4-cycle construction yields a 6-regular, not 5-regular, graph.Source: §J.3 · pp. 87–88

Boundary of Easy and Hard case studies

• The model eventually describes a valid 5-regular graph on eight vertices, but only after repeated inaccurate edge descriptions and user correction.The user identifies the construction as deleting two disjoint cycles from K_8; the model then acknowledges the construction and states the resulting graph is 5-regular.Source: §J.3 · pp. 89–90

Example 6 (from section I.2)

• The model correctly chooses a = 5 and b = 12 for a strictly increasing sequence that is not an arithmetic progression.Its earlier choice b = 11 produces an arithmetic progression, prompting the subsequent correction.Source: §J.3 · p. 91
• For the length-5 arithmetic progression problem, the model concludes no positive integer solution exists, deriving a common difference of 13/4.The supplied derivation sets 14 = 1 + 4d, so d = 13/4 is not an integer.Source: §J.3 · p. 91

Example 7 (from section I.2)

• The coloring attempts repeatedly violate the n+1 or n+3 constraints, and the model’s final impossibility claim follows a sequence that already assigns the same color to adjacent numbers.The user flags the initial coloring’s n+3 violation and later points out that 1 and 4 share a color; the model then declares the task impossible after giving 3 and 4 the same color.Source: §J.3 · pp. 92–94

Example 8 (from section I.2)

• The model’s proposed length-9 sequence fails its stated condition: it contains an increasing subsequence of length 5, despite claiming none of length 3.The user clarifies that the intended restriction was length 4 and notes that the example still fails.Source: §J.3 · p. 94

J.3 Full Interaction Traces from “Boundary of Easy and Hard” Case Studies

• The user catches multiple supposedly valid sequences containing longer increasing subsequences, including lengths 5 and 6.The model first gives a sequence with an increasing subsequence of length 5, then claims another meets the criteria despite a length-6 subsequence.Source: §J.3 · p. 95
• An interleaving attempt produces the increasing sequence 1 through 9, which the model incorrectly verifies before the user points out the error.The model also incorrectly concludes that a length-9 solution is impossible, citing a faulty pigeonhole argument.Source: §J.3 · pp. 96–97
• After the user suggests starting with 7, 8, 9, the model constructs (7, 8, 9, 4, 5, 6, 1, 2, 3) and identifies increasing and decreasing subsequences of length 3.It claims there are no subsequences of length 4, then is asked to prove this more rigorously.Source: §J.3 · p. 99
• The model begins a case analysis of increasing subsequences starting with 7, 4, or 1, but the supplied trace ends before the proof is completed.For the case starting with 7, it notes that the only option is (7, 8, 9), of length 3.Source: §J.3 · p. 100

Full Interaction Traces from “Boundary of Easy and Hard” Case Studies

• The sequence (7, 8, 9, 4, 5, 6, 1, 2, 3) has no increasing or decreasing subsequence of length 4.The trace rules out length-4 subsequences in both directions by considering the possible starting values.Source: §J.3 · p. 100
• The sum from r = 2 to n of log(1 − 1/r) telescopes to log(1/n).Rewriting each term as a logarithm of a ratio makes adjacent numerators and denominators cancel.Source: §J.3 · pp. 100–101
• For the sum of log(1/m(m + 1)) from m = 1 to n, the trace identifies telescoping and leaves the first and last terms.It names the remaining terms as log(1/2) and log(n/(n + 1)).Source: §J.3 · p. 102
• Summing g(n) = f(n) − 2f(n + 1) + f(n + 2) from n = 1 to N leaves f(1) − 2f(2) + f(N) − 2f(N + 1) + f(N + 2).After shifting the indices in the second and third sums, the intermediate terms cancel.Source: §J.3 · p. 103