Repair the element-generator impossibility proof. The next candidate should give the paper’s actual exhaustive case split (infinitely many immediately bad points / infinite fibers / infinitely many distinct images) or another fully rigorous adversarial construction that yields infinitely many violations on a finitely repeating enumeration while remaining an admissible enumeration of K. Keep the main theorem and arbitrary-repetition characterization as they are.

Main theorem: very faithful reconstruction. The setup matches the original assumptions: countable domain, finite or countable family of infinite languages, memoryless set-valued generator, finitely repeating enumerations, eventual subset containment. The explicit construction J_n(x)=⋂{L_j:j≤n and x∈L_j}, n(x)=max{n≤x:J_n(x) infinite}, G(x)=J_{n(x)}(x) is exactly the target construction up to 0/1-based indexing. The proof also mirrors the original closely. Defining B_K={x∈K:G(x)⊄K}, showing x∈B_K implies n(x)<z, then for x≥z deducing J_z(x) finite and hence x lies in a finite union U_z of finite intersections of subfamilies of the first z languages, is the right argument. The final use of finite repetition to avoid B_K eventually is correct.

Arbitrary repetitions characterization: theorem statement is correct and essentially equivalent to the original. Sufficiency by choosing an infinite subset S_x⊆I_x is correct. Necessity by repeating x infinitely often in an enumeration of any K containing x is the intended mechanism and is adequately conveyed, though a bit terse.

Index-based impossibility: statement and mod-4 example are correct. The proof that whichever label occurs infinitely often on the common core C=L_1∩L_2 can be exploited by enumerating the other target with infinitely many such c appearing late is sound.

Element-based impossibility: this is where the reconstruction falls short. The original audit indicates a genuine combinatorial proof with an exhaustive case split. The reconstruction instead proposes a stagewise choice x_t∉{previous x_i and previous g(x_i)} and then claims this can be interleaved with an enumeration of all K while still preserving the needed failure property. But choosing x_t outside previous g-values does not by itself force g(x_t) to be old or outside K, and the later assertions ('the adversary can schedule before or at the time it is needed', 'by refining the construction ... we may insist ... for infinitely many t') are unsupported. As written, this proof does not establish that for infinitely many t, g(x_t)∉K\{x_0,…,x_t}. Thus the stated theorem is not properly proved.

Because the benchmark requires recovering not only the main positive theorem but also the precise boundary statements with supporting proof obligations, this unsupported negative proof is a material deficiency. The reconstruction is close, but under strict grading it should not pass.

The proof of the element-based impossibility theorem is not rigorous enough to establish the stated theorem. The diagonal construction is muddled: the admissibility/interleaving claim is not justified, and the key conclusion that g(x_t) is in the seen set or outside K for infinitely many t is asserted rather than proved.

There are indexing inconsistencies (0-based vs 1-based, use of L_0=ℕ) that are mostly harmless for the main theorem but indicate some reconstruction looseness.

The characterization theorem proof is slightly under-argued on necessity: to infer G(x)⊆I_x it relies on infinitely many late occurrences of x in an admissible enumeration, which is fine, but the adversarial construction of such an enumeration is only implicit.

Correctly recovers the central positive theorem for countable families of infinite languages on a countable domain under finitely repeating enumerations, including the explicit construction via J_n(x), n(x), and G(x).

Correctly recovers the arbitrary-repetition characterization using the one-point intersections I_x and the repeat-x adversary argument.

Correctly recovers the two-language mod-4 obstruction for memoryless index-based generators.

Uses essentially the same proof mechanism for the main theorem: define bad points B_K, show badness implies n(x)<z, then for x≥z force x into a finite union U_z of finite intersections, yielding finiteness of B_K.

The reconstruction is very strong on theorem recovery and paper-specific details, but one companion boundary theorem lacks a sound proof at the benchmark’s required strictness.

Explanation: Criterion 1 is high because the reconstructed theorem package matches the target theorem package almost exactly: same model, same finitely-repeating restriction, same explicit construction, same arbitrary-repetition iff characterization, and same negative results for element- and index-based memoryless generation. Minor indexing/presentation differences do not materially change the theorem. Criterion 2 is below pass threshold because although the main theorem proof is essentially complete and faithful, and the arbitrary-repetition and index-based boundary proofs are broadly sound, the element-based impossibility proof does not actually prove the stated result. Since the benchmark targets the central positive result together with precise companion impossibility/characterization statements, that proof gap is significant enough to fail strict proof quality/sufficiency.


Already sufficient. If forced to improve further, tighten small presentation points: state from the outset that all languages are infinite, and phrase the arbitrary-repetition adversarial text more explicitly as an enumeration of K followed by infinite repetition of x after every element has appeared.

Compared with the target theorem, the reconstruction is nearly identical. For Theorem 1.1, it sets X=ℕ, assumes a countable indexing with repetitions allowed, defines J_n(x) by intersecting languages among the first n containing x, defines n(x) as the largest n≤x with J_n(x) infinite, and outputs G(x)=J_{n(x)}(x). It then fixes K=L_z, defines the bad set B_K={x∈K:G(x)⊄K}, proves x∈B_K implies n(x)<z, and for x≥z deduces J_z(x) finite. It packages all finite intersections among {L_1,…,L_z} into U_z and concludes B_K⊆{x<z}∪U_z, hence finite; finitely repeating texts eventually avoid B_K. This is the original proof mechanism. For Theorem 3.1, it states the exact iff condition that I_x:=⋂{L∈𝓛:x∈L} be infinite for every x in the union, proves necessity by eventual repetition of a single x, and proves sufficiency by choosing G(x) as an infinite subset of I_x. This matches the original characterization. For Theorem 3.2, it states no memoryless element-generator works for any infinite target under the freshness criterion, and no general memoryless index-generator exists, already for the two-language family L_1={0,1 mod 4}, L_2={0,2 mod 4}. The element-generator proof follows the intended three-case split from the summary; although reconstructed, it is logically adequate and tailored to the theorem. The index-generator argument uses the common core C=4ℕ and the same pigeonhole/adversarial-target reasoning as the original. There is no harmful weakening of quantifiers, change of model, or substitution of a nearby result. Hence this reconstruction satisfies the benchmark.

Minor presentational deviation: defines languages as arbitrary subsets in preliminaries before later reinstating the standing assumption that every L_i is infinite; harmless but slightly loose.

Necessity proof for arbitrary repetitions uses a text that 'first enumerates all elements of K and then repeats x forever'; this is acceptable because K is countable, but the construction is informally phrased rather than explicitly interleaved as an ω-sequence.

The element-generator impossibility proof is reconstructed rather than verbatim; while sound in substance, the third-case ordering argument is slightly more elaborate than the paper's summary cue.

Recovers the central positive theorem exactly: for every finite/countable family of infinite languages over a countable domain, there is a deterministic memoryless set-generator succeeding on every finitely repeating positive text.

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

Proves the finite-bad-set argument for K=L_z using n(x)<z and the finite union U_z of finite intersections among the first z languages.

Recovers the exact arbitrary-repetition characterization via I_x being infinite for every x in the union.

Includes the sharp negative boundary results for memoryless element-based and index-based generators, including the same two-language mod-4 counterexample.

This is a high-fidelity reconstruction of the theorem package, including the companion impossibility/characterization statements requested by the benchmark.

Explanation: The reconstruction matches the original theorem package extremely closely in statement, assumptions, model, and scope. It preserves the countable domain, countable family of infinite languages, memoryless deterministic set-based output, finitely repeating texts for the positive theorem, and eventual subset containment as the success notion. It also includes the exact boundary theorem under arbitrary repetitions and the negative results for element- and index-based memoryless generation. The proof of the main theorem uses the same explicit J_n/n(x) construction and the same finite exceptional-set argument. The arbitrary-repetition theorem is proved by the same one-point-intersection idea. The negative results are faithful and sufficient, including the specific mod-4 example. Proof quality is strong and essentially complete, with only minor informal phrasing issues, not substantive gaps.


Repair the companion impossibility statements exactly. In particular: (i) define the memoryless index-based success criterion as eventual output of an index whose language is contained in the target, not exact identification; (ii) state the element-based theorem with the original quantifiers—no infinite language is generable by any memoryless element-based generator under finitely repeating texts—and give a proof aligned to that claim. The main positive theorem and arbitrary-repetition characterization are already strong.

Comparison against the target theorem package:

1. Main positive theorem (Theorem 1.1): very faithful. The reconstruction uses the correct setting (countable domain, countable family of infinite languages, memoryless set-valued map), the exact construction J_n(x), n(x), G(x), and the standard proof via the bad set B_K and finite exceptional union U_z. This is essentially the same theorem and proof.

2. Arbitrary-repetition boundary (Theorem 3.1): also very faithful. The iff condition using I_x is correct, and both directions are proved with the same key adversarial repetition idea / safe intersection choice as in the target.

3. Necessity of set-based output — element-based case: this is where the reconstruction drifts. The target theorem states that under finitely repeating enumerations no infinite language is generable by a memoryless element-based generator. The reconstruction instead states a universal impossibility claim of the form: for every g there exists some infinite K and repetition-free text where g fails. While related, this is not the same quantifier structure as the target package's statement. Moreover the proof as written has a problematic first case and does not cleanly establish the claimed theorem in the original formal model.

4. Necessity of set-based output — index-based case: the target success notion is eventual output of an index whose associated language is contained in the target (so eventual exact equality is not required in general, especially when duplicates or proper sublanguages may be present in the indexed family). The reconstruction defines success as eventual exact equality L_{i(x_t)}=K, which is strictly stronger and changes the model. The mod-4 counterexample still obstructs the weaker notion for the displayed two-language family, but because the formal theorem statement is altered, this is not a faithful recovery.

5. Specificity/calibration: the reconstruction is highly paper-specific and even flags uncertainty in the element-based proof details, which is good calibration. But benchmark sufficiency requires exactness on the theorem package, not just strong resemblance.

Net: excellent recovery of the main positive theorem and characterization, but the package fails strict evaluation because the negative companion statements are not reproduced under the original formal success criteria and one proof is not sound enough.

The index-based model is changed: the target requires eventual output of an index whose language is contained in the target, whereas the reconstruction strengthens this to eventual exact equality L_{i(x_t)}=K. This is a non-harmless model change.

The element-based impossibility theorem is misstated: the original boundary says no infinite language is generable by a memoryless element-based generator under finitely repeating enumerations, whereas the reconstruction proves only that no single generator works for all infinite targets and phrases the theorem as a universal impossibility over generators rather than per-language impossibility.

The proof of the element-based impossibility is not sound as written: Case 1 improperly refers to an arbitrary infinite set X_0 and does not establish the claimed universal statement; overall the case split does not match the quantified original theorem precisely.

Because the companion impossibility statements are part of the benchmarked theorem package, these model/quantifier changes prevent a full match even though the central positive theorem is recovered.

Correctly reconstructs the main positive theorem for memoryless set-based generation on finitely repeating texts, including the explicit construction via J_n(x), n(x), and the finite bad-set argument.

Correctly states and proves the arbitrary-text characterization via the one-point intersections I_x.

Correctly captures that set-valued output is essential and gives the specific mod-4 two-language obstruction for index-based generators.

Strong recovery of the paper-specific construction and boundary characterization, but the negative output-model theorems are not reconstructed with the right formal success notions/quantifiers.

Explanation: The reconstruction nails Theorem 1.1 and Theorem 3.1 in both statement and proof idea. However, the benchmark explicitly includes the precise companion impossibility/characterization statements delimiting the positive result. On that front, the index-based definition is materially altered from eventual subset containment to eventual exact equality, and the element-based impossibility is weakened/misstated and supported by a flawed proof sketch. These are not harmless reformulations because they change the task and proof obligations. Therefore theorem match falls below the pass threshold. Proof quality also falls below threshold because one of the package's core negative theorems is not rigorously established under the correct formalization.


Little needs repair. If improving further, align the definition of memoryless index-generator with the benchmark's containment-based success notion rather than exact identification, even though the given counterexample still works.

The reconstruction is faithful to the paper's theorem package. On the main positive result, it preserves all essential assumptions: countable domain, every language infinite, finite or countable family, memoryless deterministic set-valued outputs, finitely repeating positive texts, and eventual subset containment. It also reproduces the explicit construction J_n(x), n(x), G(x) rather than merely asserting existence. The proof is the same core proof as in the original: fix K=L_z, define the bad set B_K, show any bad x must satisfy n(x)<z, then for x≥z infer J_z(x) finite and therefore x lies in a finite union U_z of finite intersections among the first z languages; conclude B_K is finite and finitely repeating texts eventually avoid it. This is exactly the benchmarked mechanism.

For the arbitrary-repetition boundary theorem, the reconstruction states the correct iff condition involving I_x and proves it by the same amplification idea: if a single current input x can be repeated forever, then G(x) must be safe for every target containing x, forcing G(x)⊆I_x; since outputs are infinite, I_x must be infinite. Sufficiency is also exactly right by selecting an infinite subset of I_x.

For the negative results, the element-generator theorem is stronger than required and is supported by a solid three-case proof matching the summary cues. The index-generator impossibility uses the correct two-language mod-4 family and the standard argument on the common core C=4ℕ. The one notable mismatch is definitional: the reconstruction defines success for index generators as eventual exact output of the target language index, whereas the benchmark defines success more weakly in terms of eventual containment of the output language in the target. In the provided two-language example there is no proper containment between L1 and L2, so the impossibility argument still establishes the intended boundary statement for that family. Thus this discrepancy is not fatal but is the main reason not to award a perfect theorem-match score.

Overall, this is a benchmark-sufficient recovery: mathematically faithful, specific to the paper, and complete.

The index-based model is stated with success requiring eventual exact output of K (L_{h(x_t)}=K), whereas the original benchmark phrases success more generally as eventual output of a language contained in the target; however for the supplied two-language counterexample these notions coincide, so this does not affect the boundary claim proved.

Minor harmless indexing variation: domain identified with ℕ starting at 0 and uses J_0(x)=ℕ for well-definedness instead of the original's presentation with positive integers and J_1(x) observation.

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

Gives the explicit original construction using J_n(x), n(x)=max{n≤x: J_n(x) infinite}, and G(x)=J_{n(x)}(x), with the same bad-set proof that B_K is finite via {x<z}∪U_z.

Recovers the exact characterization for arbitrary repetitions via I_x being infinite iff memoryless set-generation succeeds on all positive texts.

Recovers both negative boundary statements: impossibility for memoryless element output on any infinite target, and a two-language counterexample for memoryless index output using the mod-4 construction.

Proofs are self-contained, detailed, and track the original mechanisms closely.

Excellent reconstruction; only a slight model-definition drift in the index-based auxiliary notion.

Explanation: Criterion 1 is very high because the reconstruction matches the target theorem package almost exactly: same main theorem, same explicit generator, same characterization under arbitrary repetitions, and same necessity of set-based output with the same mod-4 counterexample. I stop short of 100 because the index-generator success notion is strengthened to exact identification of K rather than eventual output of a language contained in K. This is a genuine definitional change, though it does not materially alter the proved impossibility for the two-language family. Criterion 2 is high because the proofs are complete and sound. The main theorem proof correctly shows B_K finite using the same key implication n(x)<z and the finite exceptional set U_z. The arbitrary-repetition characterization is proved rigorously in both directions. The element-generator impossibility uses the intended three-case combinatorial split with a valid adversarial text construction, and the index-generator impossibility is correct for the chosen family. Because the theorem and proof quality both exceed threshold and the reconstruction meets the benchmark success criterion, it passes.


Repair the element-based impossibility result. State exactly the paper's claim ('no infinite language is generable by a memoryless element-based generator under finitely repeating texts') and provide a valid Case 3 construction that handles cycles/infinite components without the false global topological-ordering claim. Once that boundary theorem is correctly recovered, the package should likely pass.

The reconstruction is very strong on the central constructive theorem. It gets the model right: countable domain, countable family of infinite languages, memoryless deterministic set output, finitely repeating texts, eventual subset containment. The explicit definitions J_n(x)=⋂{L_j:j≤n, x∈L_j}, n(x)=max{n≤x:J_n(x) infinite}, and G(x)=J_{n(x)}(x) are all present. The proof mirrors the paper's audited proof: define B_K={x∈K:G(x)⊄K}; if x is bad then n(x)<z; if also x≥z then J_z(x) is finite; therefore x belongs to a finite union U_z of finite intersections of subfamilies of {L_1,…,L_z}; hence B_K is finite and finitely repeating texts eventually avoid it. This is faithful and essentially complete.

The unrestricted-repetition boundary theorem is also faithfully reconstructed. The necessity proof uses the standard adversarial text that eventually repeats x forever, forcing G(x) to be safe for every language containing x. The sufficiency proof chooses any infinite subset of I_x. This matches the original theorem and proof idea closely.

The index-based impossibility is likewise correctly recovered with the same two-language example and the same common-core argument.

The problem is the element-based impossibility theorem. The original benchmark package requires the paper's precise negative statement that no infinite language is generable by a memoryless element-based generator under finitely repeating texts, supported by a rigorous combinatorial case split. The reconstruction states an even stronger form ('for every infinite K there exists a repetition-free text causing infinitely many failures') and gives Cases 1 and 2 plausibly, but Case 3 relies on a dubious graph-ordering assertion. In a functional digraph with cycles, one cannot linearly order all vertices so that every edge points backward; directed cycles obstruct this. The text's suggestion to 'cut the cycle at an arbitrary point and postpone one chosen predecessor' does not show infinitely many failures for the required selected points, because for a cycle some vertex must appear before its image. Without a corrected combinatorial construction, the proof does not establish the theorem as stated.

Because the user instructed that the benchmark target includes the main theorem together with the precise companion impossibility/characterization statements, and because a proof is required that supports the stated theorem package, the reconstruction should fail under the strict rubric. It is close to passing: the central theorem is recovered very well, setting and mechanism are excellent, and only one boundary proof needs repair. But under the benchmark's all-of-package requirement and strict proof-quality threshold, this is insufficient.

The element-based impossibility theorem is overstated and not faithfully proved: it claims a stronger universal statement ('for every infinite K') and its Case 3 scheduling argument is not valid as written for arbitrary functional digraphs with cycles.

Because the element-based negative result is one of the sharp boundary statements in the benchmark package, this proof gap prevents full support for the reconstructed theorem package.

The proof fidelity is not identical in the element-based part: the summary only indicated a rigorous three-case split, while the reconstruction's third case introduces an unsubstantiated graph-ordering lemma that can fail in the presence of cycles.

Minor theorem-match issue: the reconstruction changes some indexing conventions (starts at 0 with J_0(x)=N) and slightly strengthens one negative claim beyond the paper's stated theorem.

Correctly recovers the main positive theorem for countable families of infinite languages over a countable domain under finitely repeating texts, including the explicit construction G(x)=J_{n(x)}(x).

Correctly recovers the arbitrary-repetition characterization via the one-point intersections I_x being infinite.

Correctly recovers the two-language index-based impossibility example with L1={0,1 mod 4}, L2={0,2 mod 4}.

Uses the same finite-bad-set mechanism for the main theorem: show x bad implies n(x)<z, then for x≥z force x into a finite union U_z of finite intersections.

Main positive theorem and Theorem 3.1 are essentially recovered. Failure is driven by the unsupported strengthened proof of the element-based impossibility, which is part of the benchmark target package.

Explanation: Criterion 1 is below pass because the main theorem and characterization theorem are matched closely, but one companion theorem is stated somewhat differently/stronger than the original and not clearly equivalent from the given proof. Criterion 2 is below pass because the proof of the central positive theorem is solid, as is the arbitrary-text characterization and index-based counterexample, but the element-based impossibility proof contains a significant gap in Case 3: the claimed ordering of a general countable out-degree-1 finite-in-degree digraph so that every image precedes its preimage is false without further work due to cycles, and the offered 'cut the cycle' description does not resolve the requirement for all selected u_i simultaneously. Since the benchmark explicitly includes the sharp companion impossibility/characterization statements, this gap is disqualifying.


Already sufficient. If tightening further, the only repair would be to phrase the arbitrary-repetition adversarial text more formally and note explicitly that the domain reduction from an arbitrary countable X to ℕ is harmless.

Compared with the target theorem, the reconstruction preserves all essential mathematical content.

Main theorem: It correctly fixes a countable domain identified with ℕ, a finite or countable family of infinite languages with an enumeration in which every language appears at least once, and defines memoryless set-generation with the correct limit criterion. The explicit construction J_n(x)=⋂{L_j:j≤n and x∈L_j}, n(x)=max{m≤x:J_m(x) infinite}, G(x)=J_{n(x)}(x) is exactly the target construction up to harmless indexing normalization (starting ℕ at 0 and using J_0(x)=ℕ). The proof follows the original structure: define B_K, show x∈B_K implies n(x)<z, then for x≥z obtain J_z(x) finite, then collect all such witnesses into a finite union U_z of finite intersections from the first z languages, concluding B_K is finite and finitely repeating texts eventually avoid it. This is exactly the benchmark proof.

Boundary theorem under arbitrary repetitions: The statement is the correct iff characterization. Sufficiency is exactly the original argument: choose G(x) as an infinite subset of I_x. Necessity uses the same adversarial idea that if some x has finite I_x, repetition of x forces the constant output G(x) to lie in every target containing x, hence in I_x, impossible since G(x) is infinite. The proof's wording about a text that 'first enumerates every element of K at least once and then repeats x forever' is a bit informal because enumerating all elements 'first' over an infinite set requires formal interleaving, but this is a presentation issue rather than a logical gap.

Element-based impossibility: The target only required the theorem and the broad three-case mechanism. The reconstruction supplies a careful three-case proof: infinitely many immediately bad points, an infinite fiber over some y, or all fibers finite so the image is infinite and can be scheduled before preimages. This aligns with the paper summary and proves the required impossibility even on repetition-free texts. Good fidelity.

Index-based impossibility: The reconstruction matches the paper-specific two-language example L1={0,1 mod 4}, L2={0,2 mod 4}, using C=4ℕ and the pigeonhole argument on infinitely many common points. This is exactly right.

Overall, this is sufficient for the benchmark and should pass. The tiny deviations are stylistic or indexing-related and do not alter the theorem, setting, or proof obligations.

Minor indexing/presentation deviation: defines J_0(x)=ℕ implicitly by working over ℕ={0,1,2,...} rather than the original positive integers; harmless but not identical.

In the arbitrary-repetition necessity proof, the text says 'first enumerates every element of K at least once and then repeats x forever'; strictly this is only a conceptual schedule over an infinite sequence, though the intended meaning is clear and can be formalized by interleaving an enumeration with infinitely many later x's.

Element-generator proof is reconstructed in more detail than the seed and may not match the paper line-for-line, though it proves the stated impossibility.

Recovers the central positive theorem exactly: every finite/countable family of infinite languages over a countable domain admits a deterministic memoryless set-valued generator for all finitely repeating texts.

Gives the explicit construction with J_n(x), n(x)=max{m≤x: J_m(x) infinite}, and G(x)=J_{n(x)}(x).

Uses the same finite-bad-set proof: for K=L_z, shows bad points have n(x)<z and, for x≥z, force J_z(x) finite, hence lie in a finite union of finite intersections.

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

Recovers both necessity results delimiting the theorem: impossibility for memoryless element output and a two-language impossibility for memoryless index output.

The negative index-based example matches the original modular-language construction.

This is a high-fidelity reconstruction with essentially all benchmark-required components present.

Explanation: The reconstruction matches the target theorem package almost exactly in model, assumptions, quantifiers, and conclusions. It states the positive result for finite or countable families of infinite languages on a countable domain, with memoryless set-valued outputs and success defined by eventual subset containment on every finitely repeating positive text. It also recovers the explicit generator G(x)=J_{n(x)}(x) and the same proof mechanism via finiteness of the bad set B_K. The companion characterization for arbitrary repetitions is exactly the original iff condition involving I_x, and the two impossibility statements for element-based and index-based memoryless generation are also present with correct scope. Proof quality is strong: the main theorem proof is complete and rigorous, the boundary theorem is sound, and the negative results are adequately justified. Small presentational differences and one slightly informal adversarial-text description do not affect validity.


Repair the element-generator impossibility proof, especially the third-case selection argument. Give a fully explicit construction of a repetition-free text causing infinitely many failures under the 'infinitely many distinct images' branch, rather than a heuristic graph/path sketch.

The reconstruction is impressively faithful on the central positive result. It preserves the exact model: countable domain, finite or countable family of infinite languages, deterministic memoryless set-valued generator, finitely repeating texts, and eventual subset containment. It also gives the same concrete construction J_n(x)=\cap\{L_j:j\le n, x\in L_j\}, n(x)=\max\{n\le x:J_n(x)\text{ infinite}\}, and G(x)=J_{n(x)}(x). The proof for Theorem 1.1 follows the original mechanism closely: define bad points B_K, show x\in B_K implies n(x)<z, then for x\ge z derive J_z(x) finite, encode such points into the finite union U_z of finite intersections among L_1,...,L_z, and conclude B_K is finite. This is the right proof and it is presented well.

The boundary theorem for arbitrary repetitions is also essentially exact. The iff condition involving I_x=\cap\{L\in\mathcal L:x\in L\} being infinite is correctly recovered, and the proof idea is the same: if I_x were finite then infinitely repeating x amplifies any mistake forever; conversely choosing G(x)\subseteq I_x gives correctness on every round. There is a minor presentation issue in the necessity proof because 'enumerate every element of K once and then repeat x forever' is not literally a standard sequence specification unless one means an enumeration of K in which x appears again infinitely often after all other elements have appeared. But this is a harmless expository blemish, not a theorem mismatch.

The index-generator impossibility result is recovered exactly, with the correct two-language mod-4 example and the right adversarial use of the infinite intersection C=4\mathbb N.

The main weakness is the element-generator impossibility proof. The original answer key says the paper has a rigorous exhaustive case split. The reconstruction reproduces the three-case architecture, and Cases 1 and 2 are fine. But Case 3 is not a valid proof: it offers several incompatible sketches and claims ('there is some value y ... with infinitely many as-yet-unchosen preimages', 'or more simply', 'König-style infinite-path reasoning', 'this can be achieved') without actually establishing the needed recursive construction. Since this theorem is part of the required target package, that gap materially lowers proof quality. The author even notes this uncertainty explicitly, which is good calibration, but it still means the proof package is only partial.

Therefore the reconstruction is theorem-faithful but does not meet the benchmark's strict proof-quality bar for the full package.

The proof of the element-generator impossibility theorem is not fully rigorous: the third case contains a handwavy reconstruction ('König-style', 'more simply', 'this can be achieved') without a valid complete combinatorial argument.

In the arbitrary-repetition necessity proof, the text described as 'first enumerating every element of K at least once in some order, and then repeating x forever' is not literally a well-formed sequence construction; the intended idea is clear but the presentation is sloppy.

Because one companion theorem proof is incomplete, the overall theorem package is not fully substantiated at the benchmark's strict proof-quality threshold.

Accurately reconstructs the main positive theorem for memoryless set-based generation on finitely repeating texts, including the explicit construction J_n(x), n(x), and G(x)=J_{n(x)}(x).

Correctly proves the finite bad-set argument B_K\subseteq\{x<z\}\cup U_z and deduces eventual success on finitely repeating enumerations.

Correctly recovers the exact boundary theorem for arbitrary repetitions via the pointwise intersections I_x and the repeat-a-single-point necessity argument.

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

Good uncertainty calibration: it flags the element-generator proof’s third case as the least certain reconstruction.

Main theorem and arbitrary-repetition characterization are recovered very faithfully. Failure is due to strict proof-quality requirements for the full theorem package, not theorem mismatch.

Explanation: Criterion 1 is high because the reconstruction matches the target theorem package almost exactly: same countable-domain/countable-family setting, same finitely repeating assumption, same explicit generator, same unrestricted characterization, and same element/index impossibility boundaries. Criterion 2 falls below pass because although the main theorem proof and characterization proof are essentially correct, the element-generator impossibility proof is not fully rigorous in its hardest case and explicitly admits reconstruction uncertainty. Under the strict rubric, the stated theorem package is only partially proved, so sufficient=false and overall pass=false.


Repair the index-based model exactly: define success as eventual output of an indexed language that is an infinite subset of the target (equivalently eventually contained in the target), not eventual exact equality to the target. Then restate the impossibility theorem under that exact criterion. Everything else is essentially benchmark-ready.

The reconstruction recovers almost the entire theorem package faithfully. The core setting is correct: countable domain identified with N, finite or countable family of infinite languages, memoryless set-generator G:X->[X]^infty, positive texts/enumerations, finitely repeating condition, and eventual subset containment. The main theorem is essentially identical to Theorem 1.1. The explicit construction J_n(x)=intersection of early languages containing x, n(x)=max{n<=x:J_n(x) infinite}, G(x)=J_{n(x)}(x) is exactly the right mechanism. The proof also tracks the original proof very closely: define B_K, show x in B_K implies n(x)<z, then for x>=z derive J_z(x) finite, package bad x into a finite union U_z of finite intersections of subfamilies of {L_1,...,L_z}, conclude B_K finite, and finish using finite repetition. This is a faithful recovery both mathematically and mechanistically.

The arbitrary-repetition boundary theorem is also reconstructed essentially exactly. The necessity direction uses the standard adversary that enumerates all of K and then repeats x forever, forcing G(x) to be contained in every language containing x, hence in I_x, so I_x must be infinite because G(x) is infinite. The sufficiency direction chooses any infinite subset of I_x. This matches the target theorem and proof idea.

The element-based impossibility is also faithful. The success condition is correctly stated as eventual output of a fresh element of K, and the proof follows the intended three-case split: infinitely many immediately bad points, else an infinite fiber, else infinitely many distinct images among the good points. In each case a repetition-free text forces infinitely many failures. This is specific and correct.

The only serious issue is the index-based model. The original target defines an index output as naming one of the languages in the collection, with success requiring eventual output of a language whose set is contained in the target; exact equality is not required in general. The reconstruction instead defines success as eventual exact equality L_{h(x_t)}=K at every step. That is a stricter and different proof obligation, and the benchmark instructions explicitly disallow changing the output model or proof obligations. The proposition proved remains true under the original criterion for the specific family chosen, because neither L_1 subseteq L_2 nor L_2 subseteq L_1, and outputs outside {1,2} are also wrong if the family is just these two languages. But the theorem statement itself is not the same model as the target statement.

Because the benchmark targets not just the main positive theorem but also the precise companion impossibility/characterization boundary statements, this mismatch prevents a full pass despite the otherwise excellent recovery. The proof quality remains high because the reconstruction is rigorous for its own statements. The proof fidelity is 'Same' since the main construction and negative arguments use the same key ideas as the original. Completeness is 'Full' since complete proofs are given for all stated results.

The index-based success criterion is changed: the reconstruction requires eventual exact equality L_{h(x_t)}=K, whereas the target only requires eventual output of a language contained in the target. This alters the model/obligation even though the provided counterexample still happens to work.

Because of that altered definition, the companion impossibility statement is not stated under exactly the same output model as the original theorem package.

Minor presentational deviation: the proof of the main theorem uses J_0(x)=N rather than the original audit's J_1 nonemptiness route, which is harmless but not identical.

Correctly reconstructs the central positive theorem for memoryless set-based generation on countable families of infinite languages under finitely repeating texts, including the explicit construction via J_n(x), n(x), and the finite bad-set argument.

Correctly reconstructs the exact characterization under arbitrary repetitions using the one-point consistency intersections I_x and the repeat-x adversary argument.

Correctly includes the necessity of set-based output: impossibility for memoryless element-generators and a two-language counterexample for memoryless index-generators.

Proof ideas and structure closely match the target, especially the U_z finite exceptional-set construction and the three-case element-generator impossibility proof.

Reconstruction is very strong and paper-specific, but strict benchmarking requires exact preservation of the companion boundary statements and output model definitions.

Explanation: Criterion 1 is below pass threshold because the reconstruction changes one of the target theorem package's stated models: for index-based generators it requires eventual exact identification of K, while the original only requires eventual output of an indexed language contained in K. Since the benchmark explicitly includes the companion impossibility/characterization statements as part of the target package and forbids changing proof obligations or output models, this is a substantive mismatch. Aside from that, the main theorem and arbitrary-repetition characterization are essentially exact, and the index counterexample would also imply the weaker original impossibility in this specific two-language family. Criterion 2 is high because the proofs given are complete and logically sound for the theorems as stated; the main positive proof, boundary characterization, element-generator impossibility, and index-generator counterexample are all well argued.


No major repair needed. If compressing further, preserve the discriminator that arbitrary repetitions require the exact pointwise-intersection condition I_x infinite, and keep the mod-4 two-language index-generator counterexample.

The main theorem is recovered with very high fidelity. The statement preserves all essential assumptions: X is countable and identified with N; the family is finite or countable; every language is infinite; success is for every finitely repeating positive text; and the learner is memoryless and set-based. The construction J_n(x), n(x), G(x) is the same one as in the target theorem. The proof mirrors the original line-by-line: define B_K, show bad points have n(x)<z, deduce for x≥z that J_z(x) is finite, place such x into a finite exceptional set U_z built from finite intersections among L_1,...,L_z, conclude B_K is finite, and then use finite repetition to show eventual avoidance.

The boundary theorem for arbitrary repetitions is also correctly recovered as an exact iff statement. The sufficiency direction is immediate and correct by choosing an infinite H_x⊆I_x. The necessity direction is proved using a text that first enumerates K and then repeats x forever; this correctly forces eventual correctness on the repeated symbol and therefore implies G(x)⊆K for every K containing x, so G(x)⊆I_x and I_x must be infinite. This is equivalent to the original argument.

The element-generator impossibility is stronger than necessary in presentation (it shows failure already on repetition-free texts for any infinite K), but this is compatible with the target theorem and proved by the same exhaustive three-case split over bad points, infinite fibers, and infinitely many distinct images. The index-generator impossibility uses exactly the intended two-language mod-4 example and the same adversarial reasoning on the common core C=4N.

Specificity is excellent: the explicit mod-4 languages, the finite-union U_z mechanism, and the pointwise-core characterization are all paper-specific details, not generic field knowledge. Calibration is also strong: the reconstruction does not overclaim beyond the paper's setting, and it explicitly preserves the necessity of set-based outputs and of the finitely repeating restriction.

Overall, this is a benchmark-successful reconstruction.

Minor presentational deviation: introduces J_0(x)=X to justify existence of n(x), whereas the target formulation used n≤x and relied on x≥1 / J_1(x) or the empty-intersection convention; this is harmless.

The necessity proof for arbitrary repetitions uses a text that enumerates K once and then repeats x forever; this is valid but phrased slightly differently from the target's more adversarial formulation.

Recovers the central main theorem exactly: every countable family of infinite languages over a countable domain has a deterministic memoryless set-based generator for finitely repeating texts/enumerations.

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 same finite bad-set proof: for K=L_z, bad points satisfy n(x)<z, and for x≥z one gets J_z(x) finite, hence x lies in a finite union U_z of finite intersections of subfamilies of {L_1,...,L_z}.

Includes the exact characterization under arbitrary repetitions using I_x=⋂{L:x∈L}, with both directions proved.

Includes both necessity-of-set-output impossibility results: no memoryless element-generator on infinite languages, and a two-language counterexample for index-generators using the mod-4 construction.

This is a faithful reconstruction of the theorem package, with essentially the same model, quantifiers, construction, and delimiting impossibility results.

Explanation: The reconstruction matches the original benchmark target almost exactly. It keeps the same setting (countable domain, finite/countable family, each language infinite), the same success notion (eventual subset containment on every finitely repeating positive text), and the same explicit memoryless set-valued generator. It also recovers the two key companion results: the iff characterization for arbitrary repetitions via I_x, and the impossibility of memoryless element-based generation together with a two-language impossibility for memoryless index-based generation. The proofs are sound and substantially identical to the audited original arguments. Minor notation changes such as adding J_0(x)=X are harmless and do not alter assumptions or conclusions. Because both theorem match and proof quality clear the strict thresholds, this reconstruction passes.


No major repair needed. If tightening further, keep the theorem package focused on the exact success criterion and explicitly note that the unrestricted characterization is an iff for memoryless set-based generation under arbitrary repetitions.

The reconstruction is a strong pass. On theorem content, it fully captures the central positive bounded-memory result and the precise delimiting impossibility/characterization statements. It does not weaken countable to finite, does not replace set-based output by a different model, and does not alter the limiting success notion. It explicitly states finitely repeating texts for the main theorem and the iff characterization for arbitrary repetitions. It also includes the necessity of set-based output through element-based and index-based impossibility theorems.

For Theorem 1.1, the recovered construction is essentially identical to the original: define J_n(x) as the intersection of early languages containing x, define n(x) as the largest n≤x leaving an infinite intersection, and output G(x)=J_{n(x)}(x). The proof fixes K=L_z, defines the bad set B_K, shows badness implies n(x)<z), then for bad x≥z derives finiteness of J_z(x), and finally places all such x into a finite union U_z of finite intersections of subfamilies of {L_1,...,L_z}. This is exactly the target mechanism. The conclusion that finitely repeating texts eventually avoid the finite bad set is also exactly right.

For Theorem 3.1, the reconstruction states the correct necessary and sufficient condition I_x infinite for all x in the union. The necessity proof via enumerating the target once and then repeating x forever is the intended adversarial idea. The sufficiency proof choosing G(x)⊆I_x is immediate and correct. This matches the target characterization.

For Theorem 3.2, the reconstruction recovers both negative results. The element-generator impossibility uses the same trichotomy: infinitely many immediately bad/self-mapped points; else an infinite fiber to a single y; else finitely many bad points and all fibers finite, forcing infinitely many distinct images and allowing an ordering where images appear before preimages. The index-generator impossibility uses the same pair L_1={0,1 mod 4}, L_2={0,2 mod 4}, their infinite intersection 4N, and the pigeonhole argument on whichever index appears infinitely often on the intersection. These are paper-specific details, not generic substitutes.

The only noticeable deviations are cosmetic: the reconstruction uses N starting at 0 and introduces J_0(x)=X to ensure n(x) is defined, whereas the audited theorem phrased things with positive integers and J_1(x). This is harmless and equivalent. There is also a small expository overstatement in the arbitrary-repetition necessity argument where the conclusion 'G(x)⊆K for every language K containing x' is reached after discussing one chosen K, but the intended universal statement is obviously obtained by rerunning the same argument for each such K. No substantive theorem mismatch or proof defect is present.

Overall, this reconstruction more than meets the benchmark: it recovers the target theorem package faithfully, with the same setting, same quantifiers, same construction, and same proof architecture.

Minor notational/conventional deviation: indexing starts at 0 and uses J_0(x)=X to guarantee existence of n(x), whereas the audited target phrasing used positive integers and noted J_1(x) is infinite. This is harmless.

The necessity proof for the arbitrary-repetition characterization says 'for every language K containing x' after constructing a text for a chosen K; strictly, one argues separately for each such K. The intended conclusion is still valid.

Some exposition references the prior paper and motivation not needed for the target theorem, but this does not alter the mathematics.

Recovers the central positive theorem exactly: for every finite/countable family of infinite languages over a countable domain, there exists a deterministic memoryless set-based generator succeeding on every finitely repeating positive text.

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

Reproduces the original proof structure for Theorem 1.1 via the bad set B_K, the implication x in B_K ⇒ n(x)<z, the finite exceptional union U_z, and finite avoidance under finitely repeating texts.

Recovers the exact arbitrary-repetition characterization via infinitude of I_x, with both necessity and sufficiency arguments.

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

Uses essentially the same specific mod-4 construction for the index-based impossibility and the same three-case mechanism for the element-based impossibility.

Faithful reconstruction of the theorem package with high specificity and correct proof ideas. Differences are cosmetic rather than mathematical.

Explanation: Criterion 1 is very high because the reconstruction matches the target theorem package almost exactly: same model (memoryless deterministic set-based generator), same assumptions (countable domain, finite/countable family of infinite languages), same finitely repeating condition, same explicit construction, and same two companion boundary results. Criterion 2 is also high because the proofs are complete and sound. The positive theorem proof follows the audited argument line by line. The arbitrary-repetition theorem proves both directions correctly. The element-based impossibility gives the correct exhaustive case split and adversarial text constructions. The index-based impossibility uses the exact two-language mod-4 example and diagonalizes correctly. The few differences are harmless conventions or slight expository imprecision, not proof flaws.


Repair the proof of the element-generator impossibility result, especially Case 3. Give a clean construction of a repetition-free text from infinitely many distinct image pairs that guarantees infinitely many source points appear after their image. Also tighten the index-generator argument so it explicitly treats arbitrary indices in the enumeration, not just outputs 1 and 2.

The reconstruction does an excellent job recovering the paper's actual theorem package rather than drifting to a nearby bounded-memory claim. The main positive theorem is reproduced essentially verbatim in substance: same model, same success notion, same finitely-repeating restriction, same explicit J_n/n(x)/G construction, and the same finite-bad-set argument. The arbitrary-repetition boundary theorem is also recovered exactly, including the standard necessity argument via a text that eventually repeats a shared point forever and the sufficiency construction G(x)⊆I_x. The index-based impossibility is also very close to the original, using the correct mod-4 pair.

The decisive weakness is proof support for the element-based impossibility theorem. The original audit says the paper has a rigorous exhaustive case split. Here Cases 1 and 2 are fine, but Case 3 first constructs infinitely many successful positions rather than failures, recognizes that this is the wrong direction, and then replaces the needed argument by an informal scheduling discussion ('topologically order', 'interleave components', 'a simpler way to package this final step...'). That is not a complete proof. Because the benchmark asks for the central positive result together with the precise companion impossibility/characterization statements that delimit it, this gap prevents a full pass. Proof fidelity is nevertheless 'Same' because the reconstructed proof uses the same core mechanisms and proof skeleton as the original; completeness is 'Partial' because most major parts are present but one boundary proof remains materially incomplete.

The proof of the element-generator impossibility theorem is not rigorous in Case 3: it explicitly reaches the opposite conclusion for the constructed sequence, then proposes an informal/revisional scheduling argument that is not fully established.

Part (iii) of the theorem is stated generally ('No memoryless element-generator succeeds on all finitely repeating positive texts even for a single infinite target language'), but the supplied proof contains a material gap in the hardest case, so the stated theorem is not fully supported.

The index-generator proof is somewhat informal in asserting 'only relevant safe outputs are the indices naming L1 or L2 themselves'; while the core counterexample idea is right, the proof should more explicitly handle arbitrary extra languages/indices in the ambient enumeration.

Correctly reconstructs the main positive theorem for memoryless set-based generation on countable families of infinite languages under finitely repeating texts, including the explicit construction G(x)=J_{n(x)}(x).

Recovers the finite-bad-set proof structure via B_K, the implication x in B_K => n(x)<z, the finite union U_z of finite intersections, and eventual avoidance of B_K on finitely repeating texts.

Correctly states and proves the exact characterization for arbitrary repetitions using the one-point intersections I_x.

Correctly includes the impossibility boundary for memoryless element-based and index-based outputs, with the same two-language mod-4 counterexample for index generators.

Appropriately flags uncertainty about the least recoverable part of the element-generator impossibility proof rather than overstating certainty.

The reconstruction is theorem-faithful and highly specific, but benchmark pass fails because proof quality is below threshold due to the acknowledged Case 3 gap.

Explanation: Criterion 1 is high because the reconstructed theorem package matches the original central result and companion boundary statements under the same model and assumptions: countable domain, countable family of infinite languages, memoryless set-based outputs, finitely repeating texts for the positive theorem, exact iff condition for arbitrary repetitions, and necessity of set-based output via element/index impossibility. Minor presentational differences do not change scope. Criterion 2 is below pass because although the main theorem and arbitrary-repetition characterization are proved correctly and quite faithfully, the element-generator impossibility proof is incomplete in its hardest case. The author explicitly notes that the scheduling step is the least recoverable part and does not provide a rigorous derivation. Since this impossibility statement is part of the target theorem package, the proof package is only partial overall.