{"accepted": true, "budget_feedback": "Already sufficient. If refining further, align indexing conventions exactly with the original J_n/n(x) definitions and note explicitly that the arbitrary-enumeration characterization is equivalent to one-example full-memory generation.", "budget_words": 200, "calibration": {"score": 4}, "candidate_id": "c200-f", "context": {"context_papers": [], "has_context": false}, "critique": "The reconstruction is faithful to the target result. It gets the model right: countable domain, infinite target languages, finite or countable collections, memoryless deterministic generators, set-valued outputs for the positive theorem, and success as eventual containment G(x_t)\u2286K. It also preserves the crucial finitely repeating assumption for the positive result and does not overclaim arbitrary-enumeration success.\n\nFor the main theorem, the construction is essentially exact. The definitions of J_n(x), n(x), and G(x) match the target mechanism, with only a slight indexing normalization via J_0(x)=N to guarantee existence of a maximizing index. The proof that B_K is finite is the same core proof as in the paper: bad points force n(x)=z then J_z(x) is finite; hence bad points lie in {=z infer J_z(x) is finite and therefore x lies in one of finitely many finite intersections among the first z languages; finitely repeating texts then eliminate those bad points after some stage. The arbitrary-enumeration theorem is recovered exactly as an iff condition on I_x being infinite, with the standard adversarial repetition argument for necessity and direct construction for sufficiency. The negative boundary for index-based outputs matches the paper-specific mod-4 example. The negative boundary for element-based outputs is also correct, though the proof method differs from the paper's case split; that is acceptable because it proves the same theorem under the same model. Overall, the reconstruction is faithful, specific, complete, and sufficient for the benchmark.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor presentation difference: defines J_0(x)=N to ensure existence of n(x), whereas the original audit observed nonemptiness already from J_1(x); this is harmless.", "The element-generator impossibility proof is reconstructed differently from the paper's described combinatorial case split, though it proves the stated claim under the same model."], "major_successes": ["Recovers the central positive theorem with the exact model: countable domain, finite/countable family of infinite languages, deterministic memoryless set-valued generator, finitely repeating texts, and eventual subset containment.", "Gives the paper's explicit construction via J_n(x), n(x)=max{n<=x: J_n(x) infinite}, and G(x)=J_{n(x)}(x).", "Main proof matches the original bad-set argument: for K=L_z, bad x implies n(x)[X]^infty, and the exact limit-success criterion of eventual subset containment along every finitely repeating enumeration/text. It also gives the paper-specific construction J_n(x)=intersection of all L_j with j<=n containing x, defines n(x) as the largest n<=x with J_n(x) infinite, and sets G(x)=J_{n(x)}(x). The proof then mirrors the original mechanism: define B_K={x in K: G(x) not subseteq K}; show any bad x must satisfy n(x)=z then J_z(x) must be finite; therefore x belongs to a finite union U_z of finite intersections from among L_1,...,L_z; hence B_K is finite; finally finitely repeating texts see each bad value only finitely often, implying eventual success. This is exactly the right proof architecture.\n\nFor the unrestricted-enumeration boundary theorem, the reconstruction also matches the original characterization: learnability from all texts iff every I_x, the intersection of all family members containing x, is infinite. The sufficiency proof G(x)=I_x and the necessity proof using infinite repetition of a single x are both correct and faithful.\n\nFor the necessity of set-based output, the reconstruction includes both required failures: element-based and index-based. The index-based result is especially faithful, using the same pair L_1={0,1 mod 4}, L_2={0,2 mod 4} and the same pigeonhole/use-the-other-target adversarial argument on the infinite common core 4N. The element-based result is stated in the correct novelty-output form and proved by a direct adversarial enumeration argument. This is slightly different from the paper's described combinatorial split proof, but it still establishes the stated impossibility.\n\nThe only real imperfections are cosmetic or presentational. The author switches to 0-based naturals and writes J_0(x)=N to justify existence of n(x), whereas the original benchmark description uses positive naturals and notes J_1(x) is infinite; this is harmless. The text also adds an unnecessary computability remark about n(x) being computable from a coded family, which is outside the target theorem. None of these affect theorem equivalence or proof validity. Accordingly, the reconstruction should pass.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor indexing/presentation deviations: uses N starting at 0 and invokes J_0(x)=N rather than J_1(x) or the original positive-integer indexing; harmless but not verbatim.", "The element-generation impossibility proof proves a strong special formulation directly, rather than reproducing the paper's stated exhaustive combinatorial case split; theorem statement is effectively compatible, but proof fidelity there is slightly less exact.", "Says n(x) is 'computable from the coded family', which is extra and not part of the theorem; harmless but unnecessary.", "The reconstruction is a theorem package rather than a narrowly minimal theorem statement, but it still includes the required companion boundary statements."], "major_successes": ["Correctly states the central positive theorem for countable families of infinite languages over a countable domain in the memoryless set-based model under finitely repeating texts/enumerations.", "Recovers the explicit construction G(x)=J_{n(x)}(x) with J_n(x) as intersections over the initial segment and n(x)=max{n<=x: J_n(x) infinite}.", "Provides the original bad-set proof structure: define B_K, prove x in B_K implies n(x)=z infer J_z(x) finite and place x in a finite union U_z of finite intersections, yielding B_K finite.", "Correctly states and proves the characterization for arbitrary texts via infinitude of I_x=intersection of all languages containing x.", "Recovers the negative boundary results: impossibility for memoryless element-based generation with novelty requirement, and a two-language counterexample for memoryless index-based generation using the mod-4 languages."], "mean_score": 3.83, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "The reconstruction is highly faithful and benchmark-sufficient. The only small issues are minor notation/indexing shifts and a somewhat different presentation of the element-generator impossibility proof.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 94}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is very high because the reconstruction matches the original theorem package almost exactly: same model, same success notion, same finitely repeating restriction, same explicit generator, same arbitrary-enumeration characterization, and same necessity of set-based output with the same mod-4 index counterexample. Criterion 2 is also high because the proofs are logically sound and substantially complete. The main theorem proof follows the audited original line by line, including the bad set B_K, the deduction n(x)[X]^infty, and uses the exact success notion of eventual subset containment on every finitely repeating enumeration. The explicit construction J_n(x)=intersection of early languages containing x, n(x)=max{n<=x:J_n(x) infinite}, and G(x)=J_{n(x)}(x) is exactly the paper\u2019s mechanism. The proof that B_K is finite follows the same route as the target audit: if x is bad then n(x)=z then J_z(x) is finite; the finite intersections among the first z languages contribute only finitely many such x; finitely repeating texts eventually avoid this finite bad set. This part is benchmark-quality.\n\nThe arbitrary-enumeration characterization is also essentially correct. The sufficiency direction G(x)=I_x is exact. The necessity direction correctly uses the idea that if x can recur infinitely often in a text for each language containing x, then the fixed output G(x) must be contained in all such languages, hence in I_x; since G(x) is infinite, I_x must be infinite. This matches the original theorem, though the proof initially mentions the invalid constant text and then repairs it; that is harmless.\n\nWhere the reconstruction falls short is the boundary impossibility theorem package. For element-based generators, the theorem statement is the right one, but the proof is not complete. It oscillates between stage-by-stage and block constructions, and the key logical step is not established: from arranging that g(x_t) later appears in the text one does not directly get failure of the requirement g(x_t) in K\\{x_0,...,x_t} at time t. The paper\u2019s original proof uses a more careful exhaustive combinatorial split; the submitted proof does not adequately replace it.\n\nFor index-based generators, the theorem statement uses the correct two-language mod-4 example, but the proof is not the original and is not sound enough. The postponement argument shows only that a wrong response on a datum that can appear arbitrarily late defeats eventual correctness if that datum\u2019s response is wrong whenever it appears. But the proof\u2019s final case split on h(1), h(2), h(0) does not establish that one of the targets is forced to suffer infinitely many or arbitrarily late errors under a finitely repeating enumeration in every case. The original benchmark expects the sharper common-core argument on C=4N: among infinitely many core points, one index is chosen infinitely often; that repeated choice is incompatible with one of the two targets. The submitted argument does not provide that.\n\nThus: theorem recovery is strong and specific, proof fidelity for the positive/characterization parts is essentially the same as the original, but proof completeness for the whole target package is only partial. Strictly under the rubric, the reconstruction fails because Criterion 2 does not reach 85 and sufficient must be false.", "failure_tags": ["proof_gap"], "include_in_aggregate": true, "major_errors": ["The proof of the element-generator impossibility is not rigorous: it shifts between incorrect or insufficient criteria, gives only an adversarial sketch, and does not establish the quantified theorem as stated.", "The index-generator impossibility proof is materially flawed/incomplete: it relies on delaying a bad datum arbitrarily late rather than the original infinite-intersection/C-set pigeonhole mechanism, and the final case analysis does not prove failure for every memoryless index-generator.", "There are minor indexing inconsistencies (switching between x_0 and x_1 / N starting at 0 vs 1), though these are not fatal to theorem match.", "Because one companion boundary theorem lacks a sound proof, the theorem package as reconstructed is not fully supported at the benchmark standard."], "major_successes": ["Correctly recovers the central positive theorem for countable families of infinite languages over a countable domain under finitely repeating enumerations, with the explicit construction via J_n(x), n(x), and G(x)=J_{n(x)}(x).", "Correctly proves the finite bad-set argument for the main theorem using the target index z and finite unions of finite intersections among the first z languages.", "Correctly recovers the characterization under arbitrary enumerations via the pointwise intersections I_x being infinite.", "Recovers the two negative boundary statements: impossibility for memoryless element-based generation and existence of a 2-language counterexample for memoryless index-based generation.", "Uses the same overall proof strategy and key lemmas/mechanisms as the original for the positive theorem and arbitrary-enumeration characterization."], "mean_score": 3.7, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Main theorem recovery is strong; failure is due to proof quality on the boundary impossibility results, not theorem identification.", "phase": "measurement", "proof_completeness": "Partial", "proof_correctness": {"rubric_score": 3, "score": 78}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is high because the reconstructed statement matches the original target package closely: same setting, same success notion, same finitely repeating restriction, same explicit generator, same arbitrary-enumeration characterization, and the same intended negative boundary claims. I score 92 rather than 100 because there are minor presentational/indexing deviations and the negative theorems are stated a bit more loosely. Criterion 2 is below pass because although the main positive theorem and the arbitrary-enumeration theorem are proved well, the proof of the element-based impossibility is only a sketch with calibration notes, and the proof of the index-based impossibility does not actually establish the universal counterexample theorem in a rigorous way. Since the benchmark requires the central positive result together with the precise companion impossibility/characterization statements that delimit it, this proof package is not sufficient overall.", "scored_at": "2026-07-19T06:05:40+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 92}, "trial_id": "inspect-c200-f-200-20260719T060540186035Z", "_exp_budget": 200, "_exp_condition": "off", "_exp_run": 4} {"accepted": true, "budget_feedback": "Very strong reconstruction. If improving further, align indexing conventions exactly with the original presentation (positive integers / j<=n starting at 1) and explicitly note that every language appears at least once in the family enumeration, though this is effectively used.", "budget_words": 200, "calibration": {"score": 4}, "candidate_id": "c200-f", "context": {"context_papers": [], "has_context": false}, "critique": "The reconstruction tracks the original theorem package closely. On the main positive result, it sets the same countable-domain/countable-family framework, keeps the requirement that languages are infinite, uses deterministic memoryless set-valued outputs, and preserves the exact success criterion of eventual subset containment along every finitely repeating enumeration. It gives the same explicit construction G(x)=J_{n(x)}(x), where J_n(x) is the intersection of the first n listed languages containing x and n(x) is the largest m<=x for which J_m(x) remains infinite. The proof is essentially the same as the original: define the bad set B_K for K=L_z, prove badness implies n(x)=z deduce J_z(x) is finite, package all such points into a finite union U_z of finite intersections from the initial segment {L_1,...,L_z}, conclude B_K is finite, and finish using finite repetition. This is exactly the paper\u2019s mechanism.\n\nFor the unrestricted-enumeration boundary, the reconstruction correctly states and proves the iff characterization that memoryless set-generation under arbitrary texts is possible exactly when I_x, the intersection of all languages containing x, is infinite for every relevant x. This matches the original theorem and the associated intuition that arbitrary repetitions let an adversary amplify any mistake on a repeated point.\n\nFor the negative output-model boundary, the reconstruction includes both required statements. The index-based counterexample is recovered exactly using L_1={0,1 mod 4} and L_2={0,2 mod 4}, with the same pigeonhole-style argument on their infinite intersection. The element-based impossibility is also recovered under the correct success notion requiring eventual novelty. The proof given is not verbatim the same as the audit\u2019s exhaustive fiber/image case split, but it is sound: either infinitely many x satisfy g(x)<=x, yielding immediate failure on an increasing enumeration of that infinite set, or infinitely many x satisfy g(x)>x, allowing construction of an infinite target K={a_t} with a_{t+1}>g(a_t), which forces g(a_t) outside K. This proves a stronger statement than needed, so it is acceptable.\n\nThe only discrepancies are negligible presentational choices: the reconstruction works over N starting at 0 and uses J_0(x)=N to guarantee existence of n(x), whereas the original writeup starts indexing at 1. It also says the language listing has repetitions allowed rather than explicitly stating every language appears at least once, though that is naturally implicit in 'enumeration of the family' and is enough for the proof. None of these alter the theorem. Overall this is a benchmark-quality recovery.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor indexing/notation differences only: uses N={0,1,2,...} and allows J_0(x)=N to witness existence of n(x), whereas the original writes positive integers and starts at 1.", "The element-generator impossibility proof is not the same combinatorial case split as in the audit, but it proves a stronger impossibility statement under the same success notion.", "Main theorem package is presented as a bundled reconstruction rather than singling out Theorem 1.1 first, but the target result and delimiters are still present."], "major_successes": ["Recovers the central positive theorem exactly: every finite/countable family of infinite languages over a countable domain is memorylessly set-generable under finitely repeating texts.", "Includes the explicit construction via J_n(x), n(x)=max{m<=x: J_m(x) infinite}, and G(x)=J_{n(x)}(x).", "Proves the finite bad-set argument with the same core lemmas: x bad implies n(x)=z this forces J_z(x) finite; therefore bad points lie in a finite union of finite intersections.", "Recovers the exact characterization for arbitrary repetitions via the one-point intersections I_x being infinite.", "Recovers both negative boundary results: impossibility for memoryless element-generators and a two-language counterexample for memoryless index-generators using the mod-4 pair."], "mean_score": 4.0, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Reconstruction is faithful and specific to the paper. Differences are harmless presentational choices or stronger proofs of negative parts.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 92}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is near-perfect because the reconstruction matches the target theorem package in model, assumptions, quantifiers, success notion, and sharp boundary statements. It states the positive bounded-memory/memoryless set-based result for all countable families of infinite languages on countable domains under finitely repeating texts, gives the explicit generator, includes the arbitrary-text iff characterization via I_x, and includes the element-based and index-based impossibility results. Criterion 2 is high because the proof of the positive theorem is complete and sound, reproducing the original bad-set/U_z mechanism. The arbitrary-text characterization proof is correct. The index-based impossibility proof is correct. The element-based impossibility proof differs from the original audited case split but is logically valid and even establishes a stronger impossibility for the class of all infinite subsets. Since theorem match exceeds 90, proof quality exceeds 85, and the benchmark target is fully met, this passes.", "scored_at": "2026-07-19T06:41:22+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 98}, "trial_id": "inspect-c200-f-200-20260719T064122230325Z", "_exp_budget": 200, "_exp_condition": "off", "_exp_run": 5} {"accepted": false, "budget_feedback": "Repair the arbitrary-enumeration theorem first: align the definition of text/enumeration with the proof, or prove necessity under the paper\u2019s actual requirement that every target element appears at least once. Remove or clearly separate the extra 'fresh generation with auxiliary memory' consequence so the reconstruction stays within the paper\u2019s model.", "budget_words": 200, "calibration": {"score": 2}, "candidate_id": "c200-f", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "The reconstruction is impressively specific and largely faithful on the main positive result. It identifies the countable-domain/countable-family setting, memoryless set-based output, finitely repeating enumerations, the exact construction J_n(x)=\u2229{L_j:j\u2264n and x\u2208L_j}, the cutoff n(x)=max{n\u2264x:J_n(x) infinite}, and the proof via the finite bad set B_K. The proof that x in B_K implies n(x)=1 in the definition of J_n/n(x), and explicitly note that every language appears at least once in the chosen indexing of the family.", "budget_words": 200, "calibration": {"score": 4}, "candidate_id": "c200-f", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "This is a highly faithful reconstruction. For Theorem 1.1, it gets the setting right: countable domain X, finite or countable family of infinite languages, deterministic memoryless set-valued generator, success defined as eventual containment G(x_t)\u2286K, and the finitely repeating restriction. It reproduces the explicit construction G(x)=J_{n(x)}(x) with the same core definitions. The proof uses exactly the original mechanism: fix K=L_z, define the bad set B_K, show bad points have n(x)=1; this is a harmless presentation inconsistency rather than a substantive mathematical error.", "The element-generator impossibility is stated a bit more strongly/generally ('for all infinite target languages') than the paper's phrasing, but the provided construction indeed establishes the needed impossibility and does not weaken the benchmark theorem."], "major_successes": ["Recovers the central positive theorem with the correct model: countable domain, finite/countable family of infinite languages, deterministic memoryless set-based generator, finitely repeating enumerations, and eventual subset containment.", "Gives the paper's explicit construction via J_n(x), n(x)=max{n<=x : J_n(x) infinite}, and G(x)=J_{n(x)}(x).", "Uses the correct bad-set proof strategy: define B_K, show x in B_K implies n(x)=z, and conclude B_K is finite via a finite union of finite intersections.", "Recovers the exact characterization for arbitrary enumerations using the one-point intersections I_x and proves both necessity and sufficiency.", "Recovers the sharp negative boundary results for memoryless element-based and index-based generators, including the mod-4 two-language counterexample."], "mean_score": 4.0, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Faithful reconstruction with near-verbatim theorem package and proof mechanism.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 94}, "proof_fidelity": "Same", "reasoning": "The reconstruction matches the original theorem package essentially exactly. It states the main positive bounded-memory/memoryless set-generation theorem under finitely repeating enumerations with the correct quantifiers and explicit construction. It also includes the precise arbitrary-enumeration iff characterization via I_x and the necessity of set-based output through the element-based and index-based impossibility results. The proofs are sound and closely track the audited original arguments. The only issues are minor presentation choices: an indexing inconsistency about allowing n=0 implicitly, and a slightly stronger phrasing of one negative result that is nonetheless supported by the given proof. These do not alter the model or benchmark target. Hence both theorem match and proof quality clear the strict pass thresholds.", "scored_at": "2026-07-19T06:51:20+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 98}, "trial_id": "inspect-c200-f-200-20260719T065120323166Z-ctx", "_exp_budget": 200, "_exp_condition": "on", "_exp_run": 2} {"accepted": true, "budget_feedback": "Very strong reconstruction. If improving further, explicitly mention the equivalent 'one-example full-memory generation' characterization in the arbitrary-enumerations theorem, and note that the original element-generator impossibility used a different exhaustive case split.", "budget_words": 200, "calibration": {"score": 4}, "candidate_id": "c200-f", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "Main theorem: The reconstruction exactly captures the positive bounded-memory/memoryless set-generation result. It uses the same hypotheses (countable X; finite or countable family of infinite languages; finitely repeating texts), same success notion, and same explicit generator G(x)=J_{n(x)}(x) with n(x) the largest n<=x such that J_n(x) remains infinite. The proof is faithful: for K=L_z, define bad points B_K, prove badness implies n(x)=z infer J_z(x) finite, represent J_z(x) as one of finitely many intersections among the first z languages, and collect all finite such intersections into a finite set U_z. This yields B_K finite and hence eventual success under finite repetition. This is exactly the paper's mechanism.\n\nBoundary characterization under arbitrary repetitions: The reconstruction gives the correct iff theorem with I_x = intersection of all languages containing x, and the proof is standard and correct. It uses constant texts for necessity and chooses G(x)=I_x for sufficiency. This matches the original theorem. The only omission is the paper's stated equivalence to one-example full-memory set-generation, but that is explanatory context rather than a missing theorem condition.\n\nNecessity of set-based outputs: The reconstruction correctly states that no memoryless element-generator can succeed even under finitely repeating texts, and proves a stronger statement: for any g one can build an infinite target K and repetition-free text a_1,a_2,... on which g fails at every stage. This is a valid strengthening under the same model, even if it is not the original proof. For index-generators, it reproduces the exact two-language mod-4 example and the adversarial argument showing failure on one of the two targets.\n\nSpecificity and calibration: The answer is highly specific to the paper and appropriately flags in the notes that the exact original proof for the element-generator theorem may have differed. It does not hallucinate extra assumptions or alter the model. Overall this meets the benchmark cleanly.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor presentational deviation: theorem statement for arbitrary texts omits the paper's equivalent phrasing about one-example/full-memory generation, though the core iff characterization is correctly stated.", "Element-generator impossibility proof is not the same combinatorial case split as the audited original proof, though it does prove the required theorem and even a stronger universal failure statement."], "major_successes": ["Recovers the central positive theorem exactly: every countable family of infinite languages over a countable domain admits a deterministic memoryless set-based generator under finitely repeating enumerations/texts.", "Includes the explicit construction via J_n(x), n(x)=max{n<=x: J_n(x) infinite}, and G(x)=J_{n(x)}(x).", "Proof of the main theorem matches the original mechanism: define bad set B_K, show x in B_K implies n(x)=z deduce J_z(x) finite and hence x lies in a finite union of finite intersections, yielding B_K finite.", "Recovers the exact characterization for arbitrary enumerations: generation is possible iff every pointwise intersection I_x of languages containing x is infinite.", "Recovers the two boundary impossibility results: no memoryless element-based generator succeeds even under finitely repeating texts, and the mod-4 two-language example defeats every memoryless index-based generator."], "mean_score": 4.0, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "The reconstruction is faithful and slightly stronger in one negative theorem. It preserves the model, assumptions, and success notions.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 94}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is very high because the reconstruction states the same model and the same central theorem package: countable domain, countable family of infinite languages, deterministic memoryless set-valued generator, finitely repeating enumerations, eventual subset containment, explicit J_n/n(x) construction, exact arbitrary-enumeration characterization, and the element/index negative boundaries. The only missing paper-specific nuance is the equivalent formulation relating Theorem 3.1 to one-example generation, which is not essential to theorem equivalence. Criterion 2 is also high because the proofs are sound and essentially complete. The main theorem proof mirrors the original audited proof step-for-step. The arbitrary-enumeration characterization proof is complete. The index-generator negative proof is complete. The element-generator negative proof differs from the paper's proof but correctly establishes the required impossibility, and a different valid proof is acceptable. Since theorem match >=90, proof quality >=85, and the benchmark target is met, this is a pass.", "scored_at": "2026-07-19T06:55:48+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 98}, "trial_id": "inspect-c200-f-200-20260719T065548093233Z-ctx", "_exp_budget": 200, "_exp_condition": "on", "_exp_run": 3} {"accepted": true, "budget_feedback": "If more budget were available, the main repair would be to align the element-generator impossibility proof more closely with the original paper's combinatorial case split and avoid extraneous citation/context notes. But as a benchmark reconstruction, the theorem package is already sufficient.", "budget_words": 200, "calibration": {"score": 3}, "candidate_id": "c200-f", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "The central positive bounded-memory result is recovered faithfully. The reconstruction uses the right setting: countable domain X, finite or countable family of infinite languages, deterministic memoryless set-generator G:X->[X]^infty, and success meaning eventual subset containment on every finitely repeating enumeration of the target. It also recovers the exact concrete construction by intersecting all early languages containing x and taking the deepest still-infinite such intersection up to index x. The proof structure matches the original closely: define B_K, show badness implies n(x)=z derive finiteness of J_z(x), place such x into a finite union U_z of finite intersections, conclude B_K finite, and finally use finite repetition to obtain eventual success.\n\nThe companion theorem for arbitrary repetitions is also essentially exact: the iff condition that every one-point consistency intersection I_x be infinite is the same characterization as in the target theorem, and the proof correctly uses repeated presentation of x to derive necessity and direct choice G(x)=I_x for sufficiency.\n\nThe necessity of set-based output is also recovered correctly. For index-based generation, the exact mod-4 two-language witness from the target appears, and the proof is standard and sound. For element-based generation, the target answer key mentions a case-split combinatorial proof for arbitrary K; the reconstruction instead proves a stronger one-language impossibility for K=N by constructing an injective enumeration adapted to the functional digraph of g so that every output has already appeared. This is acceptable because it establishes the stated impossibility under the same model and assumptions, though it is not the same proof.\n\nI do not see a theorem-level mismatch. The only small deviations are notation/indexing choices (starting J_n at n=0) and some extra contextual prose referring to another paper, neither of which harms the recovered theorem package. Overall this is a successful reconstruction.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor indexing/presentation deviation: uses J_0(x)=N to justify existence of n(x), whereas the original target formulation defined J_n for positive indices and notes J_1(x) is infinite; this is harmless.", "The element-generator impossibility proof is reconstructed via a different, stronger functional-digraph argument rather than the paper's described case split, so proof fidelity for that subresult is not literally identical.", "Slightly overstates reliance on external paper context in notes, but the mathematical content itself is self-contained enough."], "major_successes": ["Accurately states the central positive theorem for countable families of infinite languages on a countable domain with deterministic memoryless set-valued output under finitely repeating texts.", "Recovers the paper's explicit construction via J_n(x), n(x)=max{n<=x: J_n(x) infinite}, and G(x)=J_{n(x)}(x).", "Proves the bad-set finiteness argument using x in B_K => n(x)=z obtain finiteness of J_z(x), place x in a finite union U_z of finite intersections, conclude B_K finite, and use finite repetition to finish.\n\nFor the arbitrary-enumeration characterization, the statement is exact. The sufficiency proof via G(x)=I_x is immediate and correct. The necessity proof is also correct: if I_x is finite, then by considering texts with a long initial block of x's, success on arbitrary texts would force G(x) to be contained in every language containing x, hence in I_x, contradicting infinitude of G(x). This is equivalent in content to the target theorem's one-example characterization.\n\nFor the boundary results, the reconstruction correctly states both negative claims. The index-based obstruction matches the target example exactly and the proof is sound. The element-based proof is not the same combinatorial case split highlighted in the audit, but it is a valid direct diagonal construction establishing the same impossibility claim under finitely repeating texts; that is acceptable because different proofs are allowed if they prove the same theorem under the same assumptions.\n\nThe only noticeable imperfections are minor. The preliminaries define a language as a subset of X rather than explicitly as an infinite subset, though every theorem later quantifies over infinite languages, so the setting is effectively recovered. The notes section mentions external paper context and confidence levels; this is extraneous but harmless. Overall, this is a benchmark-level recovery of both theorem and proof package.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Defines languages in preliminaries as arbitrary subsets rather than explicitly infinite subsets, though all theorem statements later impose infinitude where needed.", "In Theorem 3.1 converse, the proof packages the adversarial repetition argument slightly differently from the original one-sample characterization; sound but somewhat more indirect.", "Includes citation/context-paper remarks not needed for theorem recovery; harmless but not part of the target theorem."], "major_successes": ["Correctly states the central positive theorem for countable families of infinite languages over a countable domain in the deterministic memoryless set-based model under finitely repeating enumerations/texts.", "Recovers the explicit construction G(x)=J_{n(x)}(x) with J_n(x) as intersections of low-index languages containing x and n(x)=max{n<=x: J_n(x) infinite}.", "Gives the correct bad-set proof that for target K=L_z, bad points satisfy n(x)=z, lie in a finite union of finite intersections among L_1,...,L_z.", "Correctly includes the sharp characterization for arbitrary repetitions/texts via infinitude of I_x=\u22c2{L:x\u2208L}.", "Correctly includes the two boundary impossibility results: no memoryless element-generator can universally succeed on finitely repeating texts, and a two-language family defeats every memoryless index-generator.", "Uses essentially the same proof strategy and even the same explicit mod-4 two-language obstruction as the target paper."], "mean_score": 3.83, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Very strong reconstruction. Minor definitional looseness in preliminaries does not alter the theorem statements or proofs.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 92}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is well above threshold because the reconstruction matches the target theorem package almost exactly: same model, same finitely repeating restriction, same explicit construction, same arbitrary-enumeration iff characterization, and same element/index negative boundary statements. Criterion 2 is above threshold because the positive theorem proof is complete and sound, the arbitrary-text characterization proof is correct, and the negative results are adequately proved. The element-based impossibility proof differs from the audited paper proof but proves a theorem at least as strong as needed under the same model. No material weakening of quantifiers, assumptions, or success notion occurs. Therefore the reconstruction satisfies the benchmark.", "scored_at": "2026-07-19T07:24:39+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 98}, "trial_id": "inspect-c200-f-200-20260719T072439958682Z-ctx", "_exp_budget": 200, "_exp_condition": "on", "_exp_run": 5} {"accepted": false, "budget_feedback": "Repair the companion boundary theorems, especially Theorem 3.2. Keep the original success notion for index-based output (eventual output language contained in the target, not exact equality), and supply a complete adversarial proof for the element-based impossibility rather than a sketch. The main positive theorem is already strong enough; the benchmark is being lost on the boundary statements/proofs.", "budget_words": 300, "calibration": {"score": 2}, "candidate_id": "c300-c", "context": {"context_papers": [], "has_context": false}, "critique": "The reconstruction does an excellent job on the paper's flagship positive result. It has the right setting (countable domain, finite/countable family of infinite languages), the right finitely repeating notion, and the exact construction J_n(x)=\u2229{L_j:j\u2264n,x\u2208L_j}, n(x)=max{n\u2264x:J_n(x) infinite}, G(x)=J_{n(x)}(x). The proof also mirrors the original: define B_K={x\u2208K:G(x)\u2284K}; show x\u2208B_K implies n(x)=z that J_z(x) is finite, place x into a finite union U_z of finite intersections from {L_1,...,L_z}, conclude B_K finite, then finitely repeating enumerations eventually avoid B_K. This is faithful and essentially complete. \n\n2. Arbitrary repetitions characterization (Theorem 3.1): recovered exactly in content. Sufficiency by choosing G(x) subseteq I_x is correct. Necessity by picking y in G(x)\\I_x and then a target K containing x but not y, followed by an admissible enumeration repeating x infinitely often, is also correct. This matches the target proof idea closely. \n\n3. Index-based impossibility (Theorem 3.2, second bullet): recovered with the same two-language example L1={0,1 mod 4}, L2={0,2 mod 4}. The proof is valid: residues 1 mod 4 force output 1 on L1, residues 2 mod 4 force output 2 on L2, and multiples of 4 cannot eventually satisfy both targets. This is an acceptable proof, even if phrased slightly differently than the target's 'whichever choice occurs infinitely often can be used against the other target' argument. \n\n4. Element-based impossibility (Theorem 3.2, first bullet): theorem statement is correct, but the proof is not. The author begins one recursive construction, then admits it does not finish the job, starts another, abandons it as inconvenient, then asserts a direct recursive argument with a key unsupported claim: that the set of x with g(x) in used-set union {x} 'must be infinite at infinitely many stages, for otherwise cofinitely many unused elements would map injectively into the finite complement of the used set, impossible.' This is not a valid argument as written and does not establish the proposition. The target paper's audit says there is an exhaustive combinatorial case split; this reconstruction does not recover a sound version of it. \n\nBottom line: theorem recovery is strong and paper-specific, but proof recovery is incomplete because one of the benchmark's required companion results lacks a valid proof. Under the strict pass rule, this fails.", "failure_tags": ["proof_gap"], "include_in_aggregate": true, "major_errors": ["The proof of the element-generator impossibility is not rigorous: it contains abandoned constructions, an unsupported infinitary pigeonhole/diagonal claim, and does not actually establish the proposition as stated.", "There is a small indexing inconsistency around starting at 0 versus 1 in the definition of J_n and the justification that n=0 witnesses existence of n(x); this is harmless for the main theorem but imprecise.", "Because one companion theorem lacks a sound proof, the reconstruction does not fully support the entire stated theorem package at benchmark level."], "major_successes": ["Correctly reconstructs the central positive theorem for memoryless set-based generation on finitely repeating enumerations, including the explicit construction via J_n(x), n(x), and G(x).", "Correctly includes the sharp characterization for arbitrary repetitions using the one-point intersections I_x and gives the right necessity/sufficiency idea.", "Correctly includes the two negative boundary statements: impossibility for memoryless element-based generation and failure of memoryless index-based generation for a two-language family.", "Main positive proof follows the paper's mechanism: define bad points B_K, show x in B_K implies n(x)=z force J_z(x) finite and place x in a finite exceptional union U_z."], "mean_score": 3.7, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "The reconstruction is very faithful on theorem content and highly specific to the paper. The only serious issue is proof completeness for Proposition element.", "phase": "measurement", "proof_completeness": "Partial", "proof_correctness": {"rubric_score": 2, "score": 72}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is high because the reconstructed statements match the target package essentially exactly: countable domain, countable family of infinite languages, memoryless set-based positive theorem under finitely repeating enumerations, exact characterization under arbitrary repetitions, and the two impossibility results. I scored below 100 only for minor presentation/indexing differences and because the index-generator impossibility is stated for h:N->{1,2} rather than arbitrary index-generators before specializing, though this still captures the required theorem. Criterion 2 is below pass because, while the main theorem and the arbitrary-repetition characterization are proved soundly and the index-based impossibility proof is solid, the element-based impossibility proof is incomplete/invalid as written. Since the benchmark explicitly targets the positive theorem together with the precise companion impossibility/characterization statements, that proof gap prevents sufficiency.", "scored_at": "2026-07-19T08:09:08+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 92}, "trial_id": "inspect-c300-c-300-20260719T080908380233Z", "_exp_budget": 300, "_exp_condition": "off", "_exp_run": 4} {"accepted": false, "budget_feedback": "Repair the element-generator impossibility proof. Give a complete adversarial finitely-repeating enumeration argument (or the original exhaustive case split) showing infinitely many violations for any memoryless g:X\u2192X. The main theorem and arbitrary-repetition characterization are already strong; the missing formal negative proof is the key blocker.", "budget_words": 300, "calibration": {"score": 4}, "candidate_id": "c300-c", "context": {"context_papers": [], "has_context": false}, "critique": "Comparison to target: The reconstructed main theorem is faithful. It keeps the countable domain, countable/finite family of infinite languages, memoryless deterministic set-valued generator, finitely repeating enumerations, and eventual subset containment. The explicit definitions J_n(x)=\u22c2{L_j:j\u2264n and x\u2208L_j}, n(x)=max{n\u2264x:J_n(x) infinite}, and G(x)=J_{n(x)}(x) are recovered. The proof reproduces the original bad-set method: for K=L_z, define B_K={x\u2208K:G(x)\u2284K}; show x\u2208B_K implies n(x)=z that J_z(x) is finite and x lies in finite U_z, conclude B_K finite, then invoke finite repetition. This is paper-faithful.\n\n2. Characterization under arbitrary repetitions (Theorem 3.1): Also recovered faithfully. The if-and-only-if condition that every one-point consistency intersection I_x is infinite is exactly the target statement. Sufficiency via choosing any infinite subset of I_x and necessity via repeating x infinitely often in an admissible enumeration of a target containing x are both correct and aligned with the original proof idea.\n\n3. Index-based impossibility (Theorem 3.2(ii)): Correctly recovered with the exact mod-4 pair. The contradiction argument from shared multiples of 4 and memorylessness is sound.\n\n4. Element-based impossibility (Theorem 3.2(i)): Statement matches the target, but proof quality is insufficient. The target audit says the paper gives an exhaustive combinatorial case split. Here the argument is largely heuristic: it says one can choose x_t so that g(x_t) is already seen or outside K, and if not then after some finite prefix predecessor-chasing yields either a contradiction or an infinite backward chain. But this does not fully handle all possibilities, does not guarantee coverage of all elements of K in a finitely repeating enumeration, and does not rigorously prove that infinitely many failures can be forced for an arbitrary g and arbitrary infinite K. The later 'more direct formulation' is intuition, not proof. Under strict evaluation, this is a significant proof gap.\n\n5. Specificity/calibration: The reconstruction is highly specific to this paper. However, it overstates some contextual facts about an external paper and includes bibliographic matter not needed for theorem recovery. That is mostly harmless but slightly reduces calibration.\n\nBottom line: theorem recovery is excellent, but proof recovery is not sufficient because one of the companion boundary theorems lacks a rigorous proof. Under the benchmark\u2019s strict pass criterion, this should fail.", "failure_tags": ["proof_gap"], "include_in_aggregate": true, "major_errors": ["The proof of the element-generator impossibility is not rigorous enough to support the stated universal impossibility theorem; the predecessor-chasing/backward-chain argument is only a sketch and does not establish existence of a finitely repeating enumeration causing infinitely many failures in all cases.", "There is a minor indexing/definition wobble around n(x) and J_0(x) after identifying X with N={0,1,2,...}; the theorem can be repaired, but the presentation is not fully clean.", "Some contextual/citation material introduces extraneous claims about a follow-up paper/prior paper not part of the target theorem package."], "major_successes": ["Correctly recovers the central positive theorem for countable families of infinite languages over a countable domain under finitely repeating enumerations.", "Includes the explicit construction via J_n(x), n(x), and G(x), and uses the same finite-bad-set proof strategy with U_z.", "Correctly recovers the sharp characterization for arbitrary repetitions via infinitude of I_x.", "Correctly includes the two impossibility boundaries: element-based impossibility and the mod-4 two-language counterexample for index-based generators."], "mean_score": 3.5, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "The reconstruction is very close on theorem content and proof architecture for the positive theorem and arbitrary-repetition characterization, but fails the strict benchmark because one boundary proof is insufficiently substantiated.", "phase": "measurement", "proof_completeness": "Partial", "proof_correctness": {"rubric_score": 2, "score": 78}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is high because the reconstructed theorem package matches the target almost exactly: same model, same finitely repeating restriction, same explicit generator, same arbitrary-repetition iff condition, and same two impossibility statements. Minor deductions only for allowing languages to be 'simply subsets' in preliminaries before later restoring infinitude, and for small indexing/presentation drift. Criterion 2 is below pass because while Parts (1), (2), and (4) are essentially sound, Part (3) does not rigorously prove the no-memoryless-element-generator theorem. Since the benchmark explicitly targets the central positive result together with the precise companion impossibility/characterization statements, this proof gap matters. Thus sufficient=false and overall pass=false under the strict rubric.", "scored_at": "2026-07-19T08:18:36+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 92}, "trial_id": "inspect-c300-c-300-20260719T081836064973Z-ctx", "_exp_budget": 300, "_exp_condition": "on", "_exp_run": 2} {"accepted": false, "budget_feedback": "Repair the negative element-generator theorem with a rigorous adversarial proof matching the paper\u2019s combinatorial case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images). The main positive theorem and the other boundary results are already essentially recovered; the decisive missing piece is a valid proof of Theorem 3.2(i).", "budget_words": 300, "calibration": {"score": 3}, "candidate_id": "c300-c", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "Comparison to target: (1) Main theorem 1.1: faithfully reconstructed. The setup (countable domain, countable family of infinite languages, memoryless set-based generator, finitely repeating enumerations, eventual subset containment) is correct. The explicit construction using J_n(x), n(x), G(x) is correct, and the proof follows the original mechanism: define B_K, show x in B_K implies n(x)=z derive J_z(x) finite and hence x in the finite union U_z, concluding B_K is finite and finitely repeating streams eventually avoid it. This is the same proof strategy and essentially complete. (2) Theorem 3.1 characterization under arbitrary repetitions: also faithful. Sufficiency by choosing an infinite subset of I_x is correct; necessity using a target containing x and an enumeration repeating x infinitely often is correct. This matches the original idea closely. (3) Theorem 3.2(ii) index-based impossibility: faithful and rigorous. The mod-4 example is exactly the paper\u2019s example, and the contradiction from a repetition-free enumeration of the common subset is valid. (4) Theorem 3.2(i) element-based impossibility: statement is correct, but proof is the weak point. The initial recursive construction chooses x_t avoiding previous outputs g(x_1),...,g(x_{t-1}); that alone does not force failure at time t, since success concerns whether g(x_t) is a fresh element of K relative to current history. The subsequent prose acknowledges this by saying the construction may be strengthened, then gives a vague alternative ('if not, follow the orbit map... one obtains an infinite chain...') without a rigorous contradiction. The original audit indicates the paper has an exhaustive combinatorial split; the reconstruction does not reproduce such a valid argument. Hence the theorem package is not fully substantiated. Overall: theorem recovery is strong and highly specific, proof fidelity is mostly the same, but proof completeness is only partial because one required companion theorem lacks a sound proof.", "failure_tags": ["proof_gap"], "include_in_aggregate": true, "major_errors": ["The proof of the element-generator impossibility is not sound as written: it becomes speculative ('if not... one obtains an infinite chain', 'either way') and does not actually establish the theorem rigorously.", "There is an unnecessary change in the setup by defining languages initially as arbitrary subsets before later restricting to infinite languages; harmless in context but slightly sloppy.", "The reconstruction includes citation/context material and presentation extras not part of the theorem package, though this does not affect theorem content."], "major_successes": ["Correctly recovers the central positive theorem for countable families of infinite languages over a countable domain under finitely repeating enumerations.", "Includes the explicit construction G(x)=J_{n(x)}(x) with the right definitions of J_n(x) and n(x), and the core bad-set/U_z proof that B_K is finite.", "Correctly recovers the arbitrary-repetition characterization via the one-point intersections I_x being infinite.", "Correctly recovers the memoryless index-generator impossibility with the mod-4 two-language example.", "States the right output/success notions for set-, element-, and index-based generators."], "mean_score": 3.5, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Fails strict benchmark because Criterion 2 is below threshold: the theorem package is mostly recovered, but one companion boundary proof is materially unsupported.", "phase": "measurement", "proof_completeness": "Partial", "proof_correctness": {"rubric_score": 2, "score": 78}, "proof_fidelity": "Same", "reasoning": "The reconstruction matches the target theorem package very closely on statements: the main theorem, explicit generator, arbitrary-repetition iff characterization, and index-generator impossibility are all essentially exact. That yields a high theorem-match score. However, the benchmark asks for the central positive bounded-memory generation result together with the precise companion impossibility/characterization statements. One of those companion statements\u2014the impossibility for memoryless element-based generators\u2014does not receive a correct proof. The proof begins with a construction that does not imply failure, then switches to an informal diagonal/case discussion without a complete argument. Because the stated theorem package includes this boundary theorem, proof quality must be reduced materially. The positive theorem proof and the arbitrary-repetition characterization are sound and close to the original, but the overall proof package is only partial, not complete enough for a pass under the strict thresholds.", "scored_at": "2026-07-19T08:22:02+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 92}, "trial_id": "inspect-c300-c-300-20260719T082202502499Z-ctx", "_exp_budget": 300, "_exp_condition": "on", "_exp_run": 3} {"accepted": true, "budget_feedback": "For a perfect reconstruction, strip irrelevant citation/context to the prior paper and preserve the paper-specific theorem packaging more tightly: explicitly state the unrestricted characterization as an iff theorem and the index-based impossibility as 'there exists a two-language family not generable by any memoryless index-generator.'", "budget_words": 300, "calibration": {"score": 3}, "candidate_id": "c300-c", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "The central theorem is recovered faithfully. The setting is correct: countable domain identified with N, countable family of infinite languages, memoryless deterministic set-valued generator, admissible versus finitely repeating enumerations, and eventual subset containment as the success notion. The explicit construction J_n(x), n(x), G(x) is exactly the one from the original theorem. The proof mirrors the source argument: define the bad set B_K, show any bad x must satisfy n(x)=z deduce J_z(x) is finite, package all such finite intersections among the first z languages into a finite set U_z, and conclude B_K is finite. The final use of finite repetition is exactly right.\n\nThe unrestricted-enumeration boundary theorem is also correctly reconstructed. The sufficiency direction chooses G(x)=I_x, and the necessity direction uses an enumeration repeating x infinitely often to force G(x) to lie in every target containing x, hence in I_x; because G(x) must be infinite, I_x must be infinite. This is the same mechanism as the original.\n\nThe negative results are sufficiently faithful. For element-based generators, the reconstruction gives a direct adversarial enumeration that defeats any g by presenting g(x) earlier whenever it is a viable unseen target element. This is a different presentation from the audited proof's case split but proves the stated impossibility under the same model. For index-based generators, the same mod-4 pair is used, and the argument that one of the two indices is chosen on infinitely many common points, which then defeats the opposite target, is correct.\n\nWeaknesses are minor. The writeup includes paper-context references not necessary for the benchmark and a bit of extra framing. Also, theorem numbering and paper-specific packaging are slightly loosened. But mathematically the reconstruction matches the target result and its boundaries, with a sound proof.", "failure_tags": [], "include_in_aggregate": true, "major_errors": ["Minor presentation drift: cites an external prior paper and frames some intuition relative to it, which is not needed for the target theorem package.", "The arbitrary-repetition result is labeled as a proposition rather than Theorem 3.1, and the index impossibility is stated as existence of a two-language family rather than explicitly emphasizing 'not generable by any memoryless index-based generator'\u2014though the proof establishes exactly that.", "The necessity proof for arbitrary repetitions could have explicitly stated that the same fixed G(x) must work simultaneously for every target containing x; this is nevertheless clearly implied and correctly used."], "major_successes": ["Recovers the central positive theorem exactly: every finite/countable family of infinite languages over a countable domain is memorylessly set-generable under finitely repeating enumerations.", "Includes the explicit original construction with J_n(x), n(x)=max{n<=x : J_n(x) infinite}, and G(x)=J_{n(x)}(x).", "Proof of the main theorem follows the same bad-set/U_z argument and correctly shows B_K is finite, then uses finite repetition to conclude eventual success.", "Recovers the exact characterization for arbitrary repetitions via the one-point intersections I_x being infinite.", "Recovers both negative boundary statements: impossibility for memoryless element-based generation and failure of memoryless index-based generation for the mod-4 two-language family."], "mean_score": 3.83, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "The reconstruction is highly faithful and comfortably meets the benchmark. Any deviations are cosmetic rather than mathematical.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 93}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is very high because the reconstruction matches the target theorem package almost exactly: same model, same quantifiers, same finitely repeating condition, same set-based success notion, same explicit generator, same arbitrary-repetition characterization, and same element/index impossibility boundaries. Criterion 2 is also high because the main proof is complete and sound, and the supporting results are proved correctly. The element-generator impossibility is not the exact combinatorial case split described in the audit, but it proves the same theorem under the same assumptions, so that is acceptable. The reconstruction therefore satisfies the strict pass criterion.", "scored_at": "2026-07-19T08:25:50+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 4, "score": 98}, "trial_id": "inspect-c300-c-300-20260719T082550850279Z-ctx", "_exp_budget": 300, "_exp_condition": "on", "_exp_run": 4} {"accepted": false, "budget_feedback": "Repair the element-generator impossibility proof. The next candidate should give the paper\u2019s actual exhaustive case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images) or another fully rigorous adversarial construction that yields infinitely many violations on a finitely repeating enumeration while remaining an admissible enumeration of K. Keep the main theorem and arbitrary-repetition characterization as they are.", "budget_words": 300, "calibration": {"score": 3}, "candidate_id": "c300-c", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "Main theorem: very faithful reconstruction. The setup matches the original assumptions: countable domain, finite or countable family of infinite languages, memoryless set-valued generator, finitely repeating enumerations, eventual subset containment. The explicit construction J_n(x)=\u22c2{L_j:j\u2264n and x\u2208L_j}, n(x)=max{n\u2264x:J_n(x) infinite}, G(x)=J_{n(x)}(x) is exactly the target construction up to 0/1-based indexing. The proof also mirrors the original closely. Defining B_K={x\u2208K:G(x)\u2284K}, showing x\u2208B_K implies n(x)[X]^infty, positive texts/enumerations, finitely repeating condition, and eventual subset containment. The main theorem is essentially identical to Theorem 1.1. The explicit construction J_n(x)=intersection of early languages containing x, n(x)=max{n<=x:J_n(x) infinite}, G(x)=J_{n(x)}(x) is exactly the right mechanism. The proof also tracks the original proof very closely: define B_K, show x in B_K implies n(x)=z derive J_z(x) finite, package bad x into a finite union U_z of finite intersections of subfamilies of {L_1,...,L_z}, conclude B_K finite, and finish using finite repetition. This is a faithful recovery both mathematically and mechanistically.\n\nThe arbitrary-repetition boundary theorem is also reconstructed essentially exactly. The necessity direction uses the standard adversary that enumerates all of K and then repeats x forever, forcing G(x) to be contained in every language containing x, hence in I_x, so I_x must be infinite because G(x) is infinite. The sufficiency direction chooses any infinite subset of I_x. This matches the target theorem and proof idea.\n\nThe element-based impossibility is also faithful. The success condition is correctly stated as eventual output of a fresh element of K, and the proof follows the intended three-case split: infinitely many immediately bad points, else an infinite fiber, else infinitely many distinct images among the good points. In each case a repetition-free text forces infinitely many failures. This is specific and correct.\n\nThe only serious issue is the index-based model. The original target defines an index output as naming one of the languages in the collection, with success requiring eventual output of a language whose set is contained in the target; exact equality is not required in general. The reconstruction instead defines success as eventual exact equality L_{h(x_t)}=K at every step. That is a stricter and different proof obligation, and the benchmark instructions explicitly disallow changing the output model or proof obligations. The proposition proved remains true under the original criterion for the specific family chosen, because neither L_1 subseteq L_2 nor L_2 subseteq L_1, and outputs outside {1,2} are also wrong if the family is just these two languages. But the theorem statement itself is not the same model as the target statement.\n\nBecause the benchmark targets not just the main positive theorem but also the precise companion impossibility/characterization boundary statements, this mismatch prevents a full pass despite the otherwise excellent recovery. The proof quality remains high because the reconstruction is rigorous for its own statements. The proof fidelity is 'Same' since the main construction and negative arguments use the same key ideas as the original. Completeness is 'Full' since complete proofs are given for all stated results.", "failure_tags": ["theorem_mismatch"], "include_in_aggregate": true, "major_errors": ["The index-based success criterion is changed: the reconstruction requires eventual exact equality L_{h(x_t)}=K, whereas the target only requires eventual output of a language contained in the target. This alters the model/obligation even though the provided counterexample still happens to work.", "Because of that altered definition, the companion impossibility statement is not stated under exactly the same output model as the original theorem package.", "Minor presentational deviation: the proof of the main theorem uses J_0(x)=N rather than the original audit's J_1 nonemptiness route, which is harmless but not identical."], "major_successes": ["Correctly reconstructs the central positive theorem for memoryless set-based generation on countable families of infinite languages under finitely repeating texts, including the explicit construction via J_n(x), n(x), and the finite bad-set argument.", "Correctly reconstructs the exact characterization under arbitrary repetitions using the one-point consistency intersections I_x and the repeat-x adversary argument.", "Correctly includes the necessity of set-based output: impossibility for memoryless element-generators and a two-language counterexample for memoryless index-generators.", "Proof ideas and structure closely match the target, especially the U_z finite exceptional-set construction and the three-case element-generator impossibility proof."], "mean_score": 3.7, "mechanism": {"score": 4}, "models": {"analyzer": "openai/azure/gpt-5.4", "reproducer": "openai/azure/gpt-5.4"}, "notes": "Reconstruction is very strong and paper-specific, but strict benchmarking requires exact preservation of the companion boundary statements and output model definitions.", "phase": "measurement", "proof_completeness": "Full", "proof_correctness": {"rubric_score": 4, "score": 92}, "proof_fidelity": "Same", "reasoning": "Criterion 1 is below pass threshold because the reconstruction changes one of the target theorem package's stated models: for index-based generators it requires eventual exact identification of K, while the original only requires eventual output of an indexed language contained in K. Since the benchmark explicitly includes the companion impossibility/characterization statements as part of the target package and forbids changing proof obligations or output models, this is a substantive mismatch. Aside from that, the main theorem and arbitrary-repetition characterization are essentially exact, and the index counterexample would also imply the weaker original impossibility in this specific two-language family. Criterion 2 is high because the proofs given are complete and logically sound for the theorems as stated; the main positive proof, boundary characterization, element-generator impossibility, and index-generator counterexample are all well argued.", "scored_at": "2026-07-19T09:53:52+00:00", "setting": {"score": 4}, "specificity": {"score": 4}, "theorem_equivalence": {"rubric_score": 3, "score": 88}, "trial_id": "inspect-c400-d-400-20260719T095352583222Z-ctx", "_exp_budget": 400, "_exp_condition": "on", "_exp_run": 2} {"accepted": true, "budget_feedback": "No major repair needed. If compressing further, preserve the discriminator that arbitrary repetitions require the exact pointwise-intersection condition I_x infinite, and keep the mod-4 two-language index-generator counterexample.", "budget_words": 400, "calibration": {"score": 4}, "candidate_id": "c400-d", "context": {"context_papers": ["2404.06757"], "has_context": true}, "critique": "The main theorem is recovered with very high fidelity. The statement preserves all essential assumptions: X is countable and identified with N; the family is finite or countable; every language is infinite; success is for every finitely repeating positive text; and the learner is memoryless and set-based. The construction J_n(x), n(x), G(x) is the same one as in the target theorem. The proof mirrors the original line-by-line: define B_K, show bad points have n(x) n(x)