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.

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)⊆K. It also preserves the crucial finitely repeating assumption for the positive result and does not overclaim arbitrary-enumeration success.

For 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; for x≥z this implies J_z(x) is finite; and such x lie in a finite union of finite intersections of subfamilies of {L_1,...,L_z}. The final use of finite repetition is also exactly right.

For the arbitrary-enumeration boundary theorem, the iff statement is correctly reconstructed. The necessity argument uses infinite repetition of a witness x to force eventual correctness on G(x) across all languages containing x, yielding G(x)⊆I_x and hence infinitude of I_x. The sufficiency argument G(x)=I_x is also correct.

For the negative results, the reconstruction recovers the correct statements. The element-based impossibility is proved by an explicit diagonal construction of a repetition-free text defeating any memoryless element-generator, which is stronger than necessary and valid under the stated novelty requirement. This differs from the paper's exact case-split proof but supports the same theorem. The index-based impossibility uses exactly the intended mod-4 pair and the common intersection 4N to force infinitely many wrong outputs for one target or the other.

Overall, this is not merely related; it is substantively the same theorem package with proofs that support the claims. The few discrepancies are presentational rather than mathematical and do not alter the benchmark result.

Minor indexing/presentation discrepancy: the proof uses J_0(x)=N to ensure nonemptiness of the maximizing set, whereas the original formulation defines J_n for n in N with n(x)=max{n<=x:...}; this is harmless but not identical notation.

The element-generator impossibility proof is a different stronger diagonal argument rather than the paper's stated case split, though it does establish the claimed impossibility under the same setting.

The index-generator theorem says 'there is a class of two languages' rather than explicitly a finite/countable collection, but the exhibited pair is exactly the intended counterexample.

Recovers the central positive theorem exactly: countable domain, finite/countable family of infinite languages, deterministic memoryless set-valued generator, finitely repeating enumerations, and eventual subset containment.

Gives the paper's explicit construction with the same formulas for J_n(x), n(x), and G(x), including the n<=x truncation and empty-intersection convention.

Reproduces the key bad-set proof mechanism: for K=L_z, bad x imply n(x)<z; if x>=z then J_z(x) is finite; hence bad points lie in {<z} union a finite union of finite intersections.

Recovers the sharp characterization for arbitrary enumerations via the one-point intersections I_x being infinite.

Recovers the necessity of set-based output, including impossibility for memoryless element-generators and the specific two-language mod-4 counterexample for index-generators.

This is a highly faithful reconstruction with explicit formulas and proofs. Small notation differences do not change the theorem.

Explanation: Criterion 1 is very high because the reconstruction matches the original theorem package in scope, assumptions, and conclusions: the main positive bounded-memory/memoryless set-based generation theorem under finitely repeating enumerations, the exact characterization under arbitrary enumerations, and the companion impossibility results for element- and index-based memoryless generators. Criterion 2 is also high because the proofs are sound and nearly complete. The main theorem proof follows the original line-for-line mechanism; the characterization proof is correct in both directions; and the negative results are adequately established, even if the element-based proof differs from the paper's specific combinatorial split. The reconstruction therefore meets the benchmark success criterion.


No major repair needed. If polishing further, explicitly note the unrestricted-enumeration theorem's interpretation that success is already achievable after one sample exactly when every one-point consistency intersection is infinite.

This is a strong reconstruction. It preserves the exact setting: countable X, finite/countable collection of infinite languages, deterministic memoryless generators, and the success notion of eventual subset containment for set outputs. It does not weaken to finite families, full-memory learners, or a different learning criterion. The explicit generator is exactly the benchmarked one, including the crucial dependence on n<=x and the use of maximal infinite intersections. The proof of the positive theorem reproduces the paper's mechanism: define the bad set B_K, show any bad x for K=L_z must have n(x)<z, then for x>=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.

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.

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)<z; large bad x force J_z(x) finite; hence bad points lie in a finite union of finite intersections and are therefore finite in number.

Recovers the exact arbitrary-enumeration characterization using I_x = intersection of all languages containing x, with both necessity and sufficiency.

Recovers both boundary impossibility statements: no memoryless element-generator for infinite targets even under finitely repeating texts, and a concrete two-language family defeating every memoryless index-generator.

The reconstruction is faithful and even slightly cleaner in places. Differences are harmless and do not alter theorem scope or assumptions.

Explanation: Criterion 1 is very high because the reconstruction states the same theorem package under the same assumptions, including the exact positive memoryless set-based result for countable families under finitely repeating enumerations, the exact iff characterization for arbitrary repetitions, and the two impossibility boundaries for element-based and index-based outputs. Criterion 2 is high because the proofs are sound and complete for the stated results. The main theorem proof follows the original construction and finite-bad-set argument closely. The arbitrary-text criterion is proved correctly in both directions. The impossibility results are established with valid adversarial constructions; the element-based proof uses a different but correct argument. Since both theorem match and proof quality exceed the strict thresholds and the reconstruction meets the benchmark target, it passes.


To make this completely bulletproof, keep the theorem package but trim extra commentary and avoid adding implementation/computability claims not present in the target. If space is tight, prioritize the exact positive construction, the finite bad-set/U_z argument, the arbitrary-enumeration iff characterization, and the precise mod-4 index counterexample.

The reconstruction successfully recovers both the central positive bounded-memory/no-memory result and the sharp delimiting statements. On the positive theorem: it preserves the countable-domain/countable-family setting, the requirement that all languages are infinite, the memoryless deterministic set-based output model G: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 for K=L_z; if additionally 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.

For 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.

For 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.

The 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.

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.

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, then for 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.

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.

Explanation: 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)<z, the finite-union argument U_z, and the conversion of finite bad set plus finite repetition into eventual success. The arbitrary-text characterization is proved correctly in both directions. The negative results are also supported; the index-based proof is faithful, and the element-based impossibility proof gives a valid contradiction for the stated novelty requirement, albeit with a simpler direct argument rather than the paper's original exhaustive case split. Overall this satisfies the benchmark's strict success condition.


Keep the positive theorem and arbitrary-enumeration characterization essentially as is. Repair the negative proofs, especially Theorem 3.2: for element-based generators, give the paper’s exhaustive combinatorial case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images) or another complete adversarial proof; for index-based generators, use the specific L1,L2 mod-4 example with the infinite common core C=4N and argue that whichever index is chosen infinitely often on C is falsified by the other target under a finitely repeating enumeration of that target. Do not rely merely on postponing a first appearance arbitrarily late.

Comparison to target: The reconstruction very faithfully reproduces Theorem 1.1. It has the right domain/cardinality assumptions, requires all languages to be infinite, uses memoryless set-valued outputs G: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’s mechanism. The proof that B_K is finite follows the same route as the target audit: if x is bad then n(x)<z; if 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.

The 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.

Where 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’s original proof uses a more careful exhaustive combinatorial split; the submitted proof does not adequately replace it.

For 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’s response is wrong whenever it appears. But the proof’s 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.

Thus: 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.

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.

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.

Main theorem recovery is strong; failure is due to proof quality on the boundary impossibility results, not theorem identification.

Explanation: 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.


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.

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), then for 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’s mechanism.

For 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.

For 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’s 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.

The 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.

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.

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; for 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.

Reconstruction is faithful and specific to the paper. Differences are harmless presentational choices or stronger proofs of negative parts.

Explanation: 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.


Repair the arbitrary-enumeration theorem first: align the definition of text/enumeration with the proof, or prove necessity under the paper’s 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’s model.

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)=∩{L_j:j≤n and x∈L_j}, the cutoff n(x)=max{n≤x:J_n(x) infinite}, and the proof via the finite bad set B_K. The proof that x in B_K implies n(x)<z, then for x≥z one gets J_z(x) finite and hence x lies in a finite union U_z of finite intersections from {L_1,...,L_z}, is the right mechanism and is presented clearly.

It also correctly recovers the intended boundary landscape: an iff characterization for arbitrary repetitions in terms of I_x being infinite, impossibility for memoryless element-based generation, and a two-language impossibility for memoryless index-based generation using the mod-4 example.

However, the benchmark demands the precise companion impossibility/characterization statements under the same model and assumptions. Here the reconstruction defines a text as requiring every element of K to appear at least once, then in the proof of Theorem iff-arbitrary uses the constant sequence x,x,x,... as a 'legitimate arbitrary-repetition text' for K. Under its own definition this is false unless K={x}, impossible here since languages are infinite. So the necessity direction is not proved as stated. The original theorem’s audit explicitly notes the one-example/full-memory equivalence and the adversarial repetition idea, but the reconstruction does not legally instantiate that under its own formalization.

A second issue is the added consequence that the memoryless safe-set theorem implies 'eventual fresh generation' by another learner that retains memory of the past. This is mathematically true as an auxiliary observation, but it shifts the model and could blur the benchmark target, which is specifically memoryless set-based generation rather than a hybrid memoryless-plus-memory postprocessor.

There is also an explicit note of uncertainty about the exact convention for arbitrary texts; good calibration would have softened the theorem statement or separated alternative formulations. Instead the theorem is stated unequivocally while the proof only works for a weaker notion where arbitrary repetition need not enumerate all of K.

So: excellent recovery of the main theorem and much of the paper-specific machinery, but not enough for a strict pass because one of the benchmarked companion theorems is not correctly supported under the stated assumptions.

The necessity proof for the arbitrary-enumeration characterization relies on the constant text x,x,x,... as a legitimate text, but under the stated definition of text every element of K must appear at least once, so the proof is invalid as written.

The reconstruction adds a consequential strengthening/shift: it claims 'eventual fresh generation' once an external learner with memory filters the memoryless safe sets. This is not the target theorem and changes the model/output interpretation.

There is explicit uncertainty about the exact formal convention for arbitrary texts, but the theorem is still stated definitively under the stricter text definition, creating a mismatch between statement and proof obligations.

Correctly recovers the main positive theorem under finitely repeating enumerations, including the explicit construction J_n(x), n(x), and G(x), and the bad-set finiteness proof structure.

Correctly recovers the arbitrary-enumeration characterization via infinitude of I_x and the two negative boundary statements for element-based and index-based memoryless generators.

Uses paper-specific examples/details, including the mod-4 two-language witness and the finite-union-of-finite-intersections argument.

Main positive theorem is close to exact. Failure is driven by the boundary theorem proof mismatch and added model-shifting consequence.

Explanation: Criterion 1 is below pass because although the core positive theorem and negative results are recovered, the arbitrary-enumeration companion theorem is not matched under the stated definitions: the proof uses constant texts incompatible with the given text definition, and the writeup injects an extra consequence about fresh generation with memory not part of the target result. Criterion 2 is below pass because the main theorem proof is sound, but the characterization proof has a substantive logical gap/mismatch in the necessity direction. Since the benchmark explicitly targets the main theorem together with precise companion boundary statements, this is insufficient.


Already sufficient. If any repair is desired, clean up the harmless indexing issue around n=0 versus n>=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.

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)⊆K, 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)<z, then for x≥z infer J_z(x) is finite, and collect all such possibilities into a finite exceptional set U_z. From finite bad set plus finitely repeating enumeration, eventual success follows. The implementation of U_z through subsets S and cells E_S is equivalent to the paper's finite-union argument and is mathematically sound.

For the arbitrary-enumeration boundary theorem, the reconstruction states the correct iff condition and proves it correctly. The sufficiency proof G(x)=I_x is immediate and right. The necessity proof using the constant enumeration x,x,x,... is exactly the paper's idea and correctly forces G(x)⊆I_x; since G(x) must be infinite, I_x is infinite.

For the negative results, the reconstruction accurately preserves the distinction between set-based, element-based, and index-based outputs. The element-based impossibility proof is valid and even cleaner than the paper summary: it constructs an infinite repetition-free target language K tailored to an arbitrary memoryless map g and ensures failure at every stage. This certainly implies the target impossibility statement. The index-based impossibility uses the exact mod-4 pair from the paper and the standard infinite-pigeonhole split on multiples of 4; the adversarial enumeration argument is correct.

The only notable imperfections are non-substantive. The note says n=0 is implicitly available to justify existence of n(x), while J_n(x) was defined only for n≥1; but because the family consists of infinite languages and x is taken from an enumerated target, there is always at least one applicable positive index anyway, so this does not affect correctness. Also, the note does not explicitly restate that the indexing of the family should include every language at least once, though it effectively assumes an indexed family (L_1,L_2,...) covering the collection. Overall the reconstruction is benchmark-sufficient and should pass.

Minor definitional wrinkle: the text says n=0 is 'implicitly available' in the definition of n(x) while also defining J_n only for n>=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.

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, deduce J_z(x) finite for 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.

Faithful reconstruction with near-verbatim theorem package and proof mechanism.

Explanation: 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.


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.

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), then for bad 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.

Boundary 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.

Necessity 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.

Specificity 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.

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.

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, then for 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.

The reconstruction is faithful and slightly stronger in one negative theorem. It preserves the model, assumptions, and success notions.

Explanation: 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.


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.

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, then for 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.

The 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.

The 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.

I 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.

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.

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 and the finite union of finite intersections among L_1,...,L_z.

Correctly states and proves the sharp characterization for arbitrary enumerations via infinitude of I_x = intersection of all languages containing x.

Recovers both negative boundary results: impossibility for memoryless element-based generation and a two-language mod-4 counterexample for memoryless index-based generation.

The reconstruction is highly faithful and includes the precise boundary statements. Minor differences are mostly presentational.

Explanation: Criterion 1 is very high because the reconstruction matches the central theorem and companion delimiting theorems almost exactly: same model, same quantifiers, same finitely-repeating restriction, same explicit generator, same arbitrary-enumeration characterization, and same set-/element-/index-based boundary. Criterion 2 is also high because the main theorem proof is complete and mirrors the original audited proof. The arbitrary-text theorem is rigorously proved. The negative index-based theorem is also sound and specific. The only proof-fidelity wrinkle is that the element-based impossibility proof uses a novel functional-digraph ordering rather than the original case split, but it still proves an equivalent/stronger statement under the same assumptions. Therefore the reconstruction meets the benchmark success criterion.


Main theorem package is already recovered at benchmark level. Minor polish: state from the outset that every language is infinite, and keep the arbitrary-enumeration necessity proof closer to the direct repeat-x amplification argument.

The reconstruction is highly faithful to the original theorem package. For the main positive theorem, it preserves all essential assumptions: countable domain, finite or countable family, each target language infinite, deterministic memoryless set-valued generator, and success defined by eventual subset containment on every finitely repeating enumeration/text. It also reproduces the concrete construction with J_n(x), n(x), and G(x), and the proof follows the same structure as the original audit: define bad points B_K, show x in B_K implies n(x)<z, then for 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.

For 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.

For 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.

The 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.

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.

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 and hence, for 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=⋂{L:x∈L}.

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.

Very strong reconstruction. Minor definitional looseness in preliminaries does not alter the theorem statements or proofs.

Explanation: 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.


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.

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)=∩{L_j:j≤n,x∈L_j}, n(x)=max{n≤x:J_n(x) infinite}, G(x)=J_{n(x)}(x). The proof also mirrors the original: define B_K={x∈K:G(x)⊄K}; show x∈B_K implies n(x)<z when K=L_z; then for x≥z derive J_z(x) finite and place x in a finite union U_z of finite intersections of subfamilies of {L_1,...,L_z}; conclude B_K is finite and finitely repeating texts eventually avoid it. This is faithful and sufficient for Theorem 1.1.

The arbitrary-enumeration boundary theorem is also recovered correctly. The condition I_x=∩{L∈L:x∈L} infinite for each x is exactly the right characterization, and both directions are justified in the right way: sufficiency by choosing G(x)⊆I_x infinite, necessity by repeating x infinitely often in a target containing x and forcing G(x) to lie in every such target.

The main problems are in the necessity-of-set-based-output boundary. First, the index-based model is misstated. In the target theorem, an index output i stands for outputting language L_i, and success requires eventual output of a language contained in the target (in particular, eventual containment), not eventual exact equality L_{h(x_t)}=K. Requiring exact equality is a materially different and stronger task. A reconstruction fails the benchmark if it changes the output model or proof obligations this way.

Second, even relative to its own stronger statement, the proof for index-based failure is invalid. It argues that for a common point x∈L_1∩L_2 one can put x at an arbitrarily large position in a finitely repeating enumeration of each target and thereby force h(x) to equal both indices. But in a finitely repeating enumeration, a fixed x can only appear finitely many times; you cannot force eventual behavior solely from arbitrarily delaying a single occurrence without controlling the convergence stage uniformly across enumerations. The original argument instead exploits infinitely many common points C=4N and a pigeonhole/adversarial choice among them.

Third, the element-based impossibility proof is only a sketch. The reconstruction explicitly admits the exact adversarial construction is not recovered, and the proof contains unproven assertions ('by diagonalizing over stages one constructs...'). The target theorem package requires the precise impossibility statement together with a proof-quality level high enough to substantiate it. As written, this is not rigorous enough.

Overall: central theorem recovered very well; characterization theorem recovered well; but the benchmark includes the companion impossibility statements, and those are not faithfully and rigorously recovered. Hence fail under the strict rubric.

The index-based boundary theorem is stated with a stronger, different success criterion (eventual exact identification L_{h(x_t)}=K) instead of eventual output of a language contained in the target; this changes the model/proof obligation.

The proof of the index-based impossibility relies on placing the same x at an arbitrarily large stage of a finitely repeating enumeration, but an enumeration cannot revisit a fixed x arbitrarily often under the finitely repeating restriction; the argument is invalid as written.

The element-based impossibility proof is not rigorous: it contains a self-acknowledged gap ('a cleaner contradiction... by diagonalizing...') and does not actually establish infinitely many violations under the required success criterion.

Recovers the main positive theorem essentially exactly, including the explicit construction via J_n(x), n(x), G(x), and the finite-bad-set argument using U_z.

Recovers the arbitrary-repetitions characterization with the correct iff condition I_x infinite and the standard necessity/sufficiency argument.

Includes the correct two-language mod-4 counterexample showing failure of memoryless index-based generation in general.

Strong recovery of the central constructive theorem, but the benchmark target explicitly includes the precise companion impossibility/characterization statements. One of those is misstated and two proofs are not at benchmark quality.

Explanation: Criterion 1 is below pass because the reconstruction does not fully match the target theorem package: while Theorem 1.1 and the arbitrary-repetition characterization are essentially correct, the index-based impossibility is formulated under a different, stronger notion of success (eventual exact identification rather than eventual containment), which changes the theorem. Criterion 2 is below pass because the element-based impossibility proof is incomplete and the index-based proof is unsound under finitely repeating enumerations as stated. Thus the reconstruction does not meet the strict benchmark, despite strong recovery of the main positive theorem.


Main item to repair would be the definition of index-based success: require eventual output of a language contained in the target (or state equality only for the specific separating pair where containment collapses to equality). Otherwise the reconstruction already meets the benchmark strongly.

Main theorem: Faithful. The setting is correct (countable domain identified with N, finite/countable family of infinite languages, memoryless set-valued outputs, finitely repeating enumerations). The construction J_n(x)=⋂{L_j:j≤n and x∈L_j}, n(x)=max{n≤x:J_n(x) infinite}, G(x)=J_{n(x)}(x) matches the target. The proof mirrors the original: define B_K, show x∈B_K implies n(x)<z, then for x≥z deduce J_z(x) finite and hence x lies in a finite union U_z of finite intersections of subfamilies of {L_1,…,L_z}; conclude B_K is finite and finitely repeating texts eventually avoid it. This is exactly the intended mechanism.

Boundary theorem under arbitrary repetitions: Also faithful. The statement is the exact iff criterion that every one-point consistency intersection I_x be infinite. Sufficiency by choosing G(x)=I_x (or any infinite subset) is correct. Necessity uses repeating x infinitely often in targets containing x to force G(x) to be contained in each such target, hence in I_x; since G(x) must be infinite, I_x must be infinite. This is mathematically sound.

Element-based impossibility: The target theorem says no infinite language is generable by any memoryless element-based generator under finitely repeating enumerations. The reconstruction proves exactly that, though by a simpler two-case argument instead of the paper’s more granular combinatorial split. The proof is valid: either only finitely many outputs land in K, or infinitely many do; in the latter case one can force infinitely many repeats of already-seen outputs using a y_n,x_n interleaving while keeping the text finitely repeating.

Index-based impossibility: The benchmark target only needs existence of a two-language family not generable by any memoryless index-based generator, with the specific mod-4 pair acceptable. The reconstruction provides exactly that family and a valid adversarial argument. However, its definition of success for index-generators is stricter than the benchmark’s general one: it requires eventual exact equality L_{h(x_t)}=K, while the benchmark interprets success as eventual output of a language contained in the target. For the chosen pair L1 and L2, neither contains the other, so eventual containment and eventual equality coincide; thus the negative example remains correct. But this is still a definitional mismatch that should be noted.

Overall: This is a strong reconstruction, paper-specific and sufficient for the benchmark. The main theorem and core boundary characterization are recovered with high fidelity and supported by solid proofs. The only notable issue is the slight change in the index-based output model.

Index-generator success is misstated as eventual exact equality L_{h(x_t)}=K, whereas the target benchmark defines success more generally as eventual output of a language contained in the target; for the given two-language counterexample this does not change the negative conclusion, but it is a model mismatch.

The necessity proof for arbitrary repetitions is slightly compressed: it infers G(x)⊆I_x by considering repetitions of x in targets containing x; sound, but not fully unpacked adversarially.

Element-generator impossibility proof uses a different case split from the audited proof; it is valid but not paper-faithful in details.

Recovers the main positive theorem with the exact setting: countable domain, countable family of infinite languages, memoryless set-based generator, finitely repeating enumerations, eventual subset containment.

Gives the explicit construction J_n(x), n(x)=max{n≤x: J_n(x) infinite}, and G(x)=J_{n(x)}(x), matching the target mechanism.

Proof of the main theorem follows the original bad-set/U_z argument and correctly shows B_K is finite.

Recovers the exact arbitrary-repetition characterization via I_x being infinite iff memoryless set-generation is possible.

Recovers the two boundary impossibility statements: no memoryless element-generator for infinite targets, and failure of memoryless index-generators via the mod-4 two-language example.

Strong recovery of theorem package. Minor definitional drift on index-based generators does not affect the central theorem and boundary example here.

Explanation: The reconstruction matches the target theorem package very closely. It states the central positive result under finitely repeating enumerations with the correct explicit memoryless set-generator and gives essentially the same proof structure. It also includes the companion characterization for unrestricted repetitions and the negative results separating set-based output from element/index output. The only substantive mismatch is that index-based success is defined as eventual exact identification rather than eventual containment of the output language in the target. Since the stated counterexample uses two incomparable languages, the impossibility still goes through, but this is a real deviation from the benchmark model and prevents a perfect theorem-match score. Proofs are generally sound and complete enough, with only minor compression in one necessity argument and a different but valid element-generator impossibility proof.


Main repair: keep the index-based model exactly as in the target theorem—success should require eventual output of an index whose associated language is contained in the target, not necessarily equal to it. Everything else is close to benchmark quality.

The reconstruction is largely faithful. For the main theorem, it gets the setting exactly right: countable domain identified with N, finite or countable family of infinite languages, finitely repeating enumerations, memoryless set-valued output, and eventual subset containment. The explicit definitions J_n(x)=⋂{L_j:j≤n and x∈L_j}, n(x)=max{n≤x:J_n(x) infinite}, and G(x)=J_{n(x)}(x) match the target. The proof also mirrors the original: fix K=L_z, define the bad set B_K, show x∈B_K implies n(x)<z, then if x≥z infer J_z(x) finite, collect all finite intersections among the first z languages into a finite exceptional set U_z, conclude B_K⊆{x<z}∪U_z, and use finite repetition to eliminate bad points eventually. This is essentially the same proof and is complete.

For the arbitrary-repetition characterization, the theorem statement is equivalent to the target. The sufficiency proof choosing G(x)⊆I_x and the necessity proof using an enumeration that returns to x infinitely often are both correct and aligned with the paper’s mechanism that a single bad response to x can be amplified by repetition.

For the negative results, the element-based impossibility theorem is correctly stated for a single infinite target K and proves that no memoryless element-generator can succeed on every finitely repeating enumeration of K. This suffices to support the target claim that no infinite language is generable in this model. The proof’s case split (infinitely many outputs outside K; otherwise either infinitely many distinct images enabling a ‘previously seen’ diagonalization, or an infinite fiber enabling immediate repeats of the same output) matches the summary’s described mechanism.

The only real theorem-level defect is in the index-based definition: the target model interprets an output index i as outputting language L_i and only requires eventual containment L_i⊆K, whereas the reconstruction requires eventual exact equality L_{h(x_t)}=K. That is stronger and therefore changes the formal task. However, for the specific counterexample family {L1,L2}, neither language is contained in the other, so any successful output under the original task would also have to identify the correct language exactly. Hence the provided impossibility proof still establishes the intended boundary example, even though the stated definition is not faithful in general.

Overall, this is benchmark-sufficient: it recovers the central theorem package, the explicit generator, the proof mechanism, and the key delimiting impossibility/characterization results with high fidelity. The remaining issue is a formal overstrengthening of the index-based model rather than a failure to recover the paper’s result.

The definition of memoryless index-generator success is stronger than in the target theorem: it requires eventual exact identification L_{h(x_t)}=K rather than eventual output of a language contained in the target. This changes the formal model, though the provided counterexample still witnesses failure.

There is a minor indexing inconsistency in the element-generator definition/proof using x_0,...,x_t versus earlier t\in\mathbb N; harmless but imprecise.

The arbitrary-repetition necessity proof uses a target containing x and then quantifies over every such target; sound, but a bit compressed relative to a fully explicit adversarial argument.

Correctly states the central positive theorem for countable families of infinite languages over a countable domain with finitely repeating enumerations.

Recovers the explicit construction J_n(x), n(x), and G(x)=J_{n(x)}(x), together with the finite bad-set proof strategy.

Includes the sharp arbitrary-repetition characterization via I_x being infinite iff memoryless set-generation is possible.

Includes both negative boundary results: impossibility for element-based generators and a two-language counterexample for index-based generators.

Uses essentially the same proof ideas as the original audit: bad set B_K, finite exceptional set U_z, repetition-amplification for arbitrary enumerations, and the mod-4 index counterexample.

This is a strong reconstruction with one notable formal mismatch in the index-based boundary statement. Because the original two-language example has no proper containment between L1 and L2, the mismatch does not affect the negative conclusion in this specific case.

Explanation: Criterion 1 is high because the reconstruction captures the main theorem, assumptions, explicit construction, and both companion boundary theorems. The only substantive mismatch is the index-based success notion, strengthened to exact equality rather than eventual subset containment; that is a model change, but it does not alter the truth of the exhibited counterexample for the specific family. Criterion 2 is also high because the main positive proof is sound and detailed, the arbitrary-repetition characterization is correctly proved, and the negative proofs are persuasive. The element-based impossibility proof is a valid reconstruction of the summary’s trichotomy. Minor formal imprecision prevents a perfect score.


Repair the element-generator impossibility proof. Give the actual exhaustive combinatorial case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images) or another complete argument that constructs a finitely repeating enumeration causing infinitely many freshness violations. Also clean up the 0-vs-1 indexing in the main construction.

Comparison to target:

1. Main theorem (Theorem 1.1): recovered almost exactly. The setting (countable domain identified with N, finite/countable family of infinite languages, deterministic memoryless set-generator, finitely repeating enumerations, eventual subset containment) is correct. The explicit construction J_n(x)=intersection of earlier languages containing x, n(x)=max{n<=x: J_n(x) infinite}, G(x)=J_{n(x)}(x) matches the target mechanism. The proof also mirrors the original audited proof: define B_K, show x in B_K implies n(x)<z, deduce for 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.

2. 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.

3. 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.

4. 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.

Bottom 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.

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.

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, then for x>=z force J_z(x) finite and place x in a finite exceptional union U_z.

The reconstruction is very faithful on theorem content and highly specific to the paper. The only serious issue is proof completeness for Proposition element.

Explanation: 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.


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→X. The main theorem and arbitrary-repetition characterization are already strong; the missing formal negative proof is the key blocker.

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)=⋂{L_j:j≤n and x∈L_j}, n(x)=max{n≤x: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∈K:G(x)⊄K}; show x∈B_K implies n(x)<z; for x≥z infer J_z(x) finite and hence x lies in the finite union U_z of finite intersections of subfamilies of {L_1,…,L_z}; conclude B_K is finite and finitely repeating texts eventually avoid it. This is the same mechanism and is essentially complete.

For the arbitrary-repetition boundary theorem, the reconstruction is also faithful. It states the iff condition that every I_x be infinite and proves sufficiency by choosing G(x)⊆I_x and necessity by repeating x infinitely often in a target containing x. This matches the original argument and scope.

For the necessity of set-based output, the reconstruction successfully states both negative results. The index-based failure theorem is proved cleanly with the exact mod-4 example from the target, and that proof is adequate.

The weak point is the element-based impossibility theorem. The original benchmark answer key says the paper gives a complete combinatorial case split. The reconstruction starts a recursive construction, notices a flaw ('this direct attempt is insufficient'), and then switches to a sketch about delaying occurrences until g(x_s) has already appeared. That idea is plausible but not formalized enough to establish the theorem. In particular, it does not fully specify how to guarantee admissibility, finite repetition, and infinitely many forced violations simultaneously for an arbitrary g, nor does it complete the needed combinatorial split. The explicit remark afterwards concedes that this part is only sketch-level. Under the strict rubric, that is a significant proof gap.

Therefore: same theorem package in substance, same proof strategy for the main result, but not enough proof completeness for a pass.

The proof of the element-based impossibility theorem is not complete: it explicitly abandons a failed direct construction and replaces it with a sketchy scheduling argument rather than a rigorous adversarial construction.

Because the reconstruction presents the full theorem package including the negative boundary theorem, the unsupported element-generator proof leaves the package insufficient under the benchmark's proof-quality requirement.

There is a minor indexing inconsistency (switching between 0-based and 1-based notation for L_i and J_n), which is harmless to the statement but weakens rigor.

Correctly recovers the main positive theorem for countable families of infinite languages over a countable domain under finitely repeating enumerations.

Includes the explicit construction J_n(x), n(x), G(x)=J_{n(x)}(x) and the finite bad-set argument B_K⊆{x<z}∪U_z.

Correctly recovers the exact characterization for arbitrary repetitions via the infinitude of I_x=⋂{L:x∈L}.

Correctly identifies the necessity of set-based output, including the two-language mod-4 counterexample for index-based generators.

Uses the same core proof ideas as the target for the positive and arbitrary-repetition theorems.

Main positive theorem and Theorem 3.1 are essentially exact. The decisive issue is incompleteness of the element-based negative proof, which the reconstruction itself acknowledges.

Explanation: Criterion 1 is high because the reconstruction matches the target theorem package very closely in statement, setting, quantifiers, and explicit construction. It includes the main bounded-memory/memoryless set-based positive result, the arbitrary-repetition iff characterization, and the two negative boundary statements. Criterion 2 is below pass because one of the package's companion theorems—failure of memoryless element-based generators—is not rigorously proved. The proof contains an abandoned attempt and then only an outline-level scheduling idea. Since the benchmark requires the selected target result together with the precise companion impossibility/characterization statements, this gap is material. Thus theorem match passes but proof quality and sufficiency do not.


Repair the element-generator impossibility with the actual exhaustive adversarial case split from the paper (infinitely many immediately bad points / an infinite fiber / infinitely many distinct images) and keep the index-based success notion aligned with eventual subset containment rather than eventual exact target index.

The reconstruction is impressively close on the central content. For Theorem 1.1 it exactly recovers the countable-domain/countable-family setting, the finitely repeating restriction, the set-valued memoryless generator, the explicit definitions of J_n(x), n(x), and G(x), and the finite-bad-set argument through B_K and U_z. This is not just thematically similar; it is the same construction and proof idea. The arbitrary-repetition boundary theorem is also faithfully reconstructed: the if-and-only-if condition that every one-point consistency intersection I_x be infinite is exactly the paper's characterization, and both directions are correctly argued.

The main weakness is the element-generator impossibility. The target theorem package says the paper proves this via an exhaustive combinatorial case split. The reconstruction states the right theorem and the right adversarial intuition, but the proof does not complete the hard case. In particular, after assuming only finitely many 'bad' elements, it attempts to recursively choose fresh x_t avoiding previously generated values, then claims one can 'ensure infinitely many violations directly' because either infinitely many bad points exist (already excluded) or some fiber is infinite. That disjunction is not justified: the remaining possibility is infinitely many distinct images with finite fibers, which is precisely one of the cases that must be handled separately. The recursive construction also does not rigorously show that every element of K is eventually enumerated while maintaining the needed avoidance properties. So the proof as written does not establish impossibility for all memoryless element-generators.

There is also a model mismatch in the definition of memoryless index-generator success. The original benchmark only requires eventual output of an index whose associated language is contained in the target, not necessarily the exact target index. The reconstruction instead requires eventual exact identification of the target index. For the specific negative example given, the stronger requirement still yields a valid impossibility witness because neither L1 is contained in L2 nor L2 in L1; outputting the wrong one is indeed invalid. So this does not destroy the negative theorem instance, but it is still a non-harmless change in the formal setup.

Overall, this is a strong near-recovery of the theorem package, but under the benchmark's strict standard it should fail: one companion theorem lacks a valid proof, and the index-based formulation is not precisely matched.

The element-based impossibility proof is not sound as written: the second case contains unjustified assertions ('either infinitely many bad points exist or some fiber is infinite', 'we can force infinitely many stages...') and does not establish the theorem rigorously.

The index-based success criterion is misstated as eventual exact output of the target index h(x_t)=i, whereas the target theorem only requires eventual output of an index whose language is contained in the target. This changes the model/obligation, though the negative example still refutes generation.

The reconstruction cites external paper/context and claims the present paper is about a stricter memoryless model not treated there; this is paper-specific speculation not grounded in the target theorem package.

Because one companion theorem is proved with a substantial gap, the full theorem package is not fully supported at benchmark level despite correct statements.

Gives the explicit construction J_n(x), n(x), and G(x)=J_{n(x)}(x), with the same bad-set/U_z proof skeleton as the original.

Correctly recovers the sharp characterization for arbitrary repetitions via the infinite one-point intersections I_x.

Correctly includes the impossibility for memoryless index-based generators using the mod-4 two-language example.

Definitions of memoryless set/element/index generators and finitely repeating enumerations are largely faithful to the target setting.

Close reconstruction of the theorem package, but the proof benchmark is missed because one boundary theorem is unsupported and one definition is too strong/different.

Explanation: Criterion 1 is high because the main theorem and the arbitrary-repetition characterization are essentially exact, and the index-based obstruction uses the right example. I did not give 100 because the index-based model is misstated: requiring eventual exact target index is stronger than the original requirement of eventual output of some language contained in the target. Criterion 2 falls below pass because the positive theorem proof and Theorem 3.1 proof are solid, but the element-based impossibility proof is materially incomplete/invalid in its key second case. Since the benchmark targets the central positive theorem together with the precise companion boundary statements, that proof gap means the reconstruction is not sufficient. Proof fidelity is Same because the recovered positive and arbitrary-repetition proofs use the same key constructions and bad-set/intersection mechanisms. Proof completeness is Partial: most results are proved, but one proof has a serious gap.


Keep the theorem package as is, but repair the proof of the element-based impossibility. The next candidate should either reproduce the paper’s actual exhaustive case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images) or give another fully rigorous adversarial construction of a finitely repeating enumeration with infinitely many violations. Avoid sketchy predecessor-chasing arguments.

Comparison to target:

1. Main positive theorem (Theorem 1.1): Recovered faithfully. The reconstruction states the theorem for finite or countable families of infinite subsets of N, under finitely repeating enumerations, with eventual subset containment. It uses the exact same 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). The proof follows the same structure: define B_K, show x in B_K implies n(x)<z, deduce for 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.

2. 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.

3. 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.

4. 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.

5. 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.

Bottom 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’s strict pass criterion, this should fail.

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.

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.

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.

Explanation: 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.


Repair the negative element-generator theorem with a rigorous adversarial proof matching the paper’s 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).

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, then if 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’s 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.

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.

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.

Fails strict benchmark because Criterion 2 is below threshold: the theorem package is mostly recovered, but one companion boundary proof is materially unsupported.

Explanation: 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—the impossibility for memoryless element-based generators—does 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.


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.'

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, then for bad 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.

The 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.

The 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.

Weaknesses 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.

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'—though 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.

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.

The reconstruction is highly faithful and comfortably meets the benchmark. Any deviations are cosmetic rather than mathematical.

Explanation: 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.