{"id":"4c8c8eb7-f407-41a9-b5b4-c3e3a6552735","arxiv_id":"2607.08516","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Row-span-constrained LCL properties let random code ensembles inherit the near-optimal proximity gaps of subspace-design codes without the exponential-in-ℓ parameter loss of prior proxy reductions.","lead":"This paper improves proximity-gap parameters for random linear codes, random Reed-Solomon codes, and Gallager LDPC codes so they match the near-optimal bounds previously known only for subspace-design codes. The improvement comes from a black-box transfer that casts curve-decodability as a row-span-constrained local property, removing an exponential loss in degree that earlier proxy arguments incurred.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s contribution is precisely the enlargement of the LCL framework that lets curve-decodability sit inside it without the W-decoding proxy that produced the (ℓ/η^{2})^O(ℓ) loss. All subsequent steps (threshold theorems for the three random ensembles, the size bound on F, and the final parameter plug-in via Corollary 2.11) are standard first-moment / union-bound arguments that follow once that casting is accepted. The only place the argument could fail is if GG25’s Theorem 2.15 itself is false for some natural subspace-design family; that is an external algebraic claim already accepted by the community and correctly treated as a black box. Because the present manuscript neither re-proves nor strengthens that claim, a stress-test of this paper cannot raise it as an internal load-bearing flaw. The concrete verification suggested above would still be a useful sanity check on the new potential-function bookkeeping, but a negative outcome would indicate only a technical slip, not a collapse of the central transfer. Hence the reader’s ACCEPT / high-confidence / low-correctness-risk verdict stands unchanged.","tokens_in":34144,"tokens_out":659,"duration_ms":7186,"concrete_test":"Independently re-derive the potential-function identity (3.6) and the maximal-quotient Lemma 3.14 for a concrete small instance (r=2, a concrete curve-free U of dimension 2) and verify that the resulting RF lower bound of Corollary 4.5 still holds when the same parameters are plugged into the original (non-row-span) LCL threshold of LMS25; if the two thresholds diverge by more than η the new constraint is not faithfully captured.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a clean black-box transfer: once curve-decodability is cast as a row-span-constrained a-local LCL family F (Theorem 4.4) whose threshold satisfies RF ≥ 1-δ-η (Corollary 4.5/4.6), the existing low-rate thresholds for RLC/RRS/RLDPC (Corollaries 3.19, 3.21, 3.26) immediately give the improved proximity-gap parameters that match the subspace-design bound of GG25. The only external algebraic input is GG25’s Theorem 2.15 (curve-decodability of τ-subspace-design codes). That statement is used exactly as a black box; the present paper never re-proves it and correctly advertises the dependence. No internal inconsistency, hidden parameter loss, or circularity appears in the new LCL machinery or the size bound |F| ≤ qa+a^{2} 2an. The reader’s weakest-assumption note is therefore accurate but does not constitute a correctness risk for this manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper improves proximity-gap (and stronger CA/MCA) parameters for three random ensembles—random linear codes, Reed–Solomon codes with random evaluation points, and Gallager LDPC codes—so that they match the near-optimal bounds previously obtained only for subspace-design codes. The key technical step is an extension of the LCL framework of LMS25/BCDZ25 to a row-span-constrained version (Section 3). With this machinery the authors cast non-curve-decodability exactly as membership in an explicit a-local family F (Theorem 4.4), prove that the associated threshold satisfies RF ≥ 1-δ-η (Corollaries 4.5–4.6), and then invoke the existing low-rate thresholds for the three ensembles (Corollaries 3.19, 3.21, 3.26) together with known list-decoding results. The final statements (Theorems 5.3, 5.6, 5.10) therefore replace the earlier (ℓ/η^{2})^O(ℓ) term by an additive O(ℓ^{2}/η^{3}) term, matching the subspace-design bound of GG25. The argument is advertised as a black-box transfer: any future improvement of curve-decodability for subspace-design codes automatically upgrades the random ensembles.","tokens_in":34458,"tokens_out":869,"duration_ms":7917,"significance":"The quantitative improvement is load-bearing for applications that use large curve degree ℓ (e.g., WHIR, Blaze, Bolt). The black-box character of the transfer is a genuine methodological contribution: once curve-decodability is recognized as a row-span-constrained LCL property, every subsequent advance on subspace-design codes immediately yields the same advance for the three random ensembles. The re-derivation of the LCL threshold and transfer theorems in the constrained setting (Section 3) and the clean equivalence for Gallager codes (Appendix A) are carefully executed and should be reusable. The paper therefore both solves a concrete parameter gap left open by GG25 and supplies a reusable technical tool.","major_comments":[],"minor_comments":[{"comment":"In Definition 3.3 the phrase “row-span constrained r-local LCL family” is introduced; a short parenthetical reminder that the ordinary LCL notion of LMS25 is recovered by restricting U to L_dist would help readers who skip the introduction.","section":null},{"comment":"Corollary 4.6 chooses the concrete constants a = ⌈100ℓ^{2}/η^{3}⌉ and b = ⌈4ℓ/η+ℓ⌉. A one-sentence remark that any sufficiently large absolute constants work (and that the 100 can be reduced) would make the dependence clearer.","section":null},{"comment":"The factor-of-two loss in the list size for Gallager codes (Theorem 5.7 / Remark 5.8) is correctly flagged; it would be useful to state explicitly that the same loss already appears in the list-decoding transfer of MRRZ+21 and is not introduced by the new LCL machinery.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “row-span constrained” vs. “row-constrained” in the introduction; occasional missing spaces after commas in displayed equations). A light copy-edit pass would remove them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, high-quality follow-up to GG25 that removes a genuine parameter obstacle for random ensembles. I see no correctness risk and no novelty-disclosure issue. Fit for a top theory / coding-theory venue is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does exactly what the abstract claims: it removes the (ℓ/η^{2})^O(ℓ) blow-up that GG25 paid when transferring proximity gaps from subspace-design codes to random linear, random RS, and Gallager LDPC ensembles, so the random ensembles now match the near-optimal εq ≥ nℓ(1-R)/η + O(ℓ^{2}/η^{3}) bound.\n\nWhat is new is the technical device. They enlarge the LCL framework of LMS25/BCDZ25 to a row-span-constrained version (pairs (V,U) instead of just profiles V). Curve-decodability then becomes exactly the property of avoiding an explicit a-local family F whose size is only q^{a+a^{2}}2^{an}; the row-span constraint encodes the “no b columns on a degree-ℓ codeword curve” condition that vanilla LCL could not capture. Once that is done, the existing low-rate thresholds for the three ensembles (and the subspace-design transfer) apply black-box. The re-derivation of the potential/threshold machinery under the extra constraint is careful and the size bound on F is tight enough that the union bounds go through under the same q,n assumptions already needed for list-decoding.\n\nSoft spots are minor and correctly advertised. Everything still rests on GG25’s algebraic statement that \tau-subspace-design codes are already curve-decodable; if that fails for some natural family the transfer collapses, but the present paper never pretends otherwise. The LDPC list-size carries a factor-of-two slack that can be improved, and the alphabet/size lower bounds remain exponential in poly(ℓ/η), which is the usual price of these random-code arguments. None of that undercuts the main claim.\n\nMath, citations, and black-box structure look solid. This is for people who care about concrete SNARK/IOP parameters or about local-property frameworks for codes. I would send it to referees; it is a genuine, usable improvement inside its subfield.","headline":"Clean black-box fix that removes the exponential-ℓ loss for random ensembles and matches the subspace-design proximity-gap bound.","tokens_in":35033,"tokens_out":554,"would_cite":true,"duration_ms":8062,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","68Q25","94B65"],"pacs":[],"model":"grok-4.5","headline":"Random codes match subspace-design codes for near-optimal proximity gaps by casting curve-decoding as a row-span constrained local property.","keywords":["proximity gaps","curve-decodability","row-span constrained LCL","random linear codes","random Reed-Solomon codes","Gallager LDPC","subspace design codes","correlated agreement"],"falsifier":"Exhibit a concrete subspace-design code (for example a folded Reed-Solomon code) whose curve-decodability radius is strictly worse than the bound claimed in the source theorem, or produce a random linear code of rate R that fails the stated proximity-gap condition for large ℓ.","tokens_in":35087,"feed_emoji":"📐","tokens_out":665,"duration_ms":6539,"temperature":0.7,"pith_summary":"Proximity gaps let a verifier test whether every point on a low-degree curve is close to a code by sampling only a few points; they are central to interactive proofs and succinct arguments. Earlier work already gave near-optimal gaps (distance approaching the Singleton bound) for algebraic subspace-design codes, but the same parameters for random linear codes, random Reed-Solomon codes, and Gallager LDPC codes suffered an exponential blow-up in the degree. This paper removes that blow-up: with high probability the random ensembles achieve exactly the same quantitative bounds as the algebraic codes. The method is black-box transfer: any future improvement for subspace-design codes immediately upgrades the random ensembles. The technical step is to enlarge the existing local-property framework so that the global “no codeword curve through b nearby points” constraint can be encoded as a restriction on the row-span of a witness matrix; once curve-decodability itself becomes a local property, the known transfer theorems apply directly and the exponential loss disappears.","feed_headline":"Random codes match algebraic codes for near-optimal proximity gaps","feed_subtitle":"A row-span constraint turns curve-decoding into a local property and removes the exponential loss","key_machinery":"Row-span constrained LCL properties: an r-local family of pairs (V,U) in which V supplies the usual coordinate-wise linear constraints while U further restricts the row-span of any witness matrix. Curve-decodability is shown to be exactly such a property, so the existing threshold theorems for random ensembles and for subspace-design codes transfer verbatim.","core_discovery":"With high probability a random linear code, a random Reed-Solomon code, or a Gallager LDPC code of rate R exhibits (ℓ,1-R-2η,ε) proximity gaps (and the stronger correlated-agreement and mutual-correlated-agreement properties) whenever εq is at least nℓ(1-R)/η plus an additive O(ℓ²/η³) term, matching the best known bound for subspace-design codes and eliminating the previous (ℓ/η²)^O(ℓ) overhead.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Random codes match subspace designs for near-optimal proximity gaps","Row-span LCL transfers optimal gaps to random linear and LDPC codes","Curve-decoding as constrained LCL closes gap for random ensembles","Random RS and Gallager codes hit algebraic proximity-gap bounds","Black-box lift: subspace progress yields random-code proximity gains"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument treats as a black box the claim that every good subspace-design code is already curve-decodable with the near-optimal parameters; if that algebraic statement fails, the transfer to random ensembles collapses.","fun_headline_variants_meta":{"raw":{"variants":["Random codes match subspace designs for near-optimal proximity gaps","Row-span LCL transfers optimal gaps to random linear and LDPC codes","Curve-decoding as constrained LCL closes gap for random ensembles","Random RS and Gallager codes hit algebraic proximity-gap bounds","Black-box lift: subspace progress yields random-code proximity gains"]},"model":"grok-4.5","effort":"low","cost_usd":0.007836,"raw_usage":{"total_tokens":1972,"prompt_tokens":951,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":78360000,"prompt_tokens_details":{"text_tokens":951,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":949,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":951,"tokens_out":72,"duration_ms":8861,"temperature":1.0,"reasoning_tokens":949,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T06:06:21.031127+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete subspace-design code (for example a folded Reed-Solomon code) whose curve-decodability radius is strictly worse than the bound claimed in the source theorem, or produce a random linear code of rate R that fails the stated proximity-gap condition for large ℓ.","supporting_citations":[],"review_version":1}