{"id":"9b83ce0d-6a12-4c03-ace7-9c169bb8f95a","arxiv_id":"2412.15170","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partition-regular complexity-1 arithmetic patterns admit exception-free induced removal: few pattern occurrences imply a small recolouring makes the space pattern-free.","lead":"This paper proves that any partition-regular arithmetic pattern of complexity 1 can be completely eliminated from a finite-field vector space by recolouring only a tiny fraction of the space. It closes a gap left by earlier induced removal lemmas, which either had to leave some pattern instances behind or only worked for translation-invariant patterns.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4 sets D0 = ceil(log_p(epsilon/4r)); for the nontrivial regime epsilon < 4r this is non-positive, so no subspace of that codimension exists and the final epsilon-recolouring bound is not established.","rationale":"The paper's central claim, Theorem 1.5, depends on constructing an epsilon-close recolouring with the strong property (iv) in Lemma 5.4. The D0 sign error is the most load-bearing defect because it breaks the proof at the point where the recolouring cost is bounded: the proof cannot instantiate H0 for small epsilon, and even a charitable reading with D0 = 0 fails the final p^{-D1} <= epsilon/4 inequality by a large factor. This is not merely a cosmetic typo; without a corrected threshold, the constructed colouring may differ from phi on more than an epsilon proportion of F_p^n, so the removal guarantee is unproven. I agree with the reader's weakest_assumption on this point. The reader also flags two further issues (the 2r versus 2^r colour count in Lemma 5.3 and an off-by-m factor in the non-regular solution count), but those are secondary: they affect constants and intermediate existence arguments, whereas D0 directly controls the claimed epsilon-close recolouring. The concern is repair-grade rather than evidence of falsity: replacing D0 by ceil(log_p(4/epsilon)) is a natural correction, and the surrounding machinery (subcoset selection, counting lemma, Rado-theoretic argument) is coherent. Therefore the appropriate disposition remains CONDITIONAL, and the reader's verdict is unchanged.","tokens_in":17123,"tokens_out":7910,"duration_ms":71140,"concrete_test":"Re-run Lemma 5.4 with D0 replaced by ceil(log_p(4/epsilon)): instantiate Proposition 5.1 and Lemma 5.3 with this corrected codimension and verify that the final inequality (3epsilon/4 + p^{-D1})|F_p^n| <= epsilon|F_p^n| and properties (iii) and (iv) all hold with the same or adjusted constants. A smaller, fully explicit check is to set p = 2, r = 2, epsilon = 1/100 and substitute the paper's D0 = ceil(log_2(1/800)) = -9 into the proof; observe that no subspace has codimension -9, and if the proof is run with D0 = 0 then p^{-D1} = 1, so the claimed cost bound fails because 1 > 1/400. If the corrected threshold is used and all downstream inequalities survive, the sign error is a repairable defect; if not, the epsilon-approximation guarantee itself is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 5.4, the proof defines D0 = ceil(log_p(epsilon/4r)) and takes H0 of codimension D0. For every nontrivial choice of parameters (epsilon < 4r, including the paper's own illustrations with small epsilon), this number is non-positive, so the proof cannot even be instantiated. If one ignores this and starts from D0 = 0, the final line '(3epsilon/4 + p^{-D1})|F_p^n| <= epsilon|F_p^n|' requires p^{-D1} <= epsilon/4, equivalently D1 >= ceil(log_p(4/epsilon)). The proof only guarantees D1 >= D0 = ceil(log_p(epsilon/4r)), which is far smaller. For a concrete example, take p = 2, r = 2, epsilon = 1/100: the written D0 is ceil(log_2(1/800)) = -9, while the needed lower bound is D1 >= ceil(log_2(400)) = 9. With D1 = 0, p^{-D1} = 1, which exceeds epsilon/4 = 1/400 by a factor of 400, so the recolouring cost bound can fail massively. The entire epsilon-approximation guarantee of Theorem 1.5 rests on this parameter, so the central claim is not proved as written. The error appears repairable by taking D0 = ceil(log_p(4/epsilon)) (or a similar positive threshold) and checking that the subsequent uses of Proposition 5.1 and Lemma 5.3 still go through, but this repair is exactly what the current text omits.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a complete induced arithmetic removal lemma for partition-regular patterns of complexity 1 over F_p^n: whenever an r-colouring has pattern density at most delta, one can recolour an epsilon-fraction of the space so that no instance of the pattern remains, including instances involving 0. This would extend the Fox--Tidor--Zhao induced removal lemma, which necessarily leaves non-generic instances for non-partition-regular patterns. The proof uses a two-stage subcoset selection scheme, a quantitative Rado theorem over finite fields, a counting lemma for complexity-1 systems, and a recolouring lemma constructed from regular subcosets.","tokens_in":17425,"tokens_out":21733,"duration_ms":168150,"significance":"If the proof is completed, the result is a natural and significant strengthening of the Fox--Tidor--Zhao theorem: it gives fully pattern-free recolourings for all partition-regular complexity-1 patterns, not just recolourings with possible non-generic leftovers. The multiple-subcoset idea is actually interesting and adapts the Alon--Fischer--Krivelevich--Szegedy approach to the arithmetic setting. The paper also contains a self-contained proof of the needed counting lemma and a useful discussion of quantitative bounds, including the polynomial true-complexity bounds of Manners. However, as written, the central proof contains several concrete parameter errors, and the main theorem is not established by the current text. The errors appear repairable, but they are load-bearing.","major_comments":[{"comment":"The proof defines D0 = ceil(log_p(epsilon/4r)) and takes H0 of codimension D0. For every nontrivial choice of parameters with epsilon < 4r, this D0 is non-positive, so no such subspace exists and the construction cannot be instantiated. Even if one ignores this and starts from D0 = 0, property (i) only gives D1 >= log_p(epsilon/4r), while the final recolouring estimate requires p^{-D1} <= epsilon/4, i.e. D1 >= log_p(4/epsilon). These two lower bounds differ substantially. For example, with p=2, r=2 and epsilon=1/100, the written lower bound is about -9 while the needed lower bound is about 9. Consequently, Lemma 5.4 does not establish that phi' differs from phi in at most epsilon|F_p^n| places, and the epsilon-approximation guarantee of Theorem 1.5 is unproved. The repair is to set D0 = ceil(log_p(4/epsilon)) and then check that the subsequent applications of Proposition 5.1 and Lemma 5.3 still go through; as written, this load-bearing parameter is wrong.","section":"Section 5, Lemma 5.4 (first paragraph and final bound)"},{"comment":"Theorem 5.5 applies Lemma 5.4 with epsilon' = min(epsilon, epsilon_count(delta')), where delta' = (epsilon/8r)^m/(2m). Lemma 5.4(iv) then gives alpha_i >= epsilon'/8r, not epsilon/8r. The displayed lower bound, however, uses (epsilon/8r)^m and claims a contradiction with delta < (8 r p^{C_rcl} epsilon^{-1})^{-m}/2. If epsilon' is strictly smaller than epsilon, which is the typical situation because epsilon_count(delta') tends to 0 with delta', the product (epsilon'/8r)^m can be smaller than the error term m delta' = (epsilon/8r)^m/2, in which case the counting lemma gives no positive lower bound at all. Thus the contradiction argument in Theorem 5.5 is not justified as written. The proof needs a coherent parameter choice that keeps the recolouring threshold used in Lemma 5.4 compatible with the counting error delta', for example by choosing delta' in terms of the actual lower bound on alpha_i, or by applying Lemma 5.4 with the original epsilon and choosing delta' so that both epsilon_count(delta') >= epsilon p^{-C_zreg}/8r and m delta' < (epsilon/8r)^m/2 hold.","section":"Section 5, Theorem 5.5 (proof, final lower bound)"},{"comment":"There are two quantitative gaps in the proof of Lemma 5.3. First, the auxiliary colouring psi takes values in P([r]), so it has up to 2^r colours; Theorem 4.2 should be applied with 2^r colours, not with 2r colours as written. Second, the number of monochromatic solutions that fail property (ii) is at most m epsilon' |F_p^D|^{m-l}, not epsilon' |F_p^D|^{m-l}, since a solution fails if any one of its m coordinates lies in a non-regular coset. With the current choice epsilon' <= crado(p,2r,m)/2, the bound m epsilon' |F_p^D|^{m-l} can exceed the guaranteed number crado |F_p^D|^{m-l} of monochromatic solutions when m > 1. The proof therefore does not establish the existence of a monochromatic solution with all cosets regular. Both issues are local and repairable, for instance by using 2^r colours and taking epsilon' <= crado/(4m), but as written Lemma 5.3 is not fully proved.","section":"Section 5, Lemma 5.3 (proof, regularity of monochromatic solutions)"}],"minor_comments":[{"comment":"In the statement of Proposition 5.1, the bound in part (i) is written as D1 <= D2 <= Creg(epsilon, zeta, r, N, d), but N is not defined; this should presumably be r, as in the rest of the statement. There is also a typo in Theorem 2.3, where 'regularity lemm' should be 'regularity lemma'.","section":"Section 5, Proposition 5.1"},{"comment":"In the recolouring step for u != 0, the text says that recolouring low-density colour classes changes at most epsilon/4r + zeta <= epsilon/2r proportion of H1+u. This is the per-colour bound; the total proportion over all r colours is at most sum of the low densities, which is at most r(epsilon/4r + zeta) = epsilon/2. The subsequent line 'zeta + r epsilon/2r <= 3epsilon/4' shows that this is what was intended, but the phrasing should be clarified to avoid an apparent off-by-r error.","section":"Section 5, Lemma 5.4 (recolouring cost accounting)"},{"comment":"In Example 1.6, the claim that there are at most delta |F_p^n|^3 instances is off by a factor of |F_p^n|: the number of l-tuples is |F_p^n|^4, and fixing one variable in a set of size at most delta|F_p^n|/4 gives at most delta|F_p^n|^4/4 instances. The intended statement is that the density is at most delta/4, not that the number of instances is at most delta |F_p^n|^3.","section":"Section 1, Example 1.6"}],"recommendation":"major_revision","confidential_remarks":"The errors appear localized and repairable, so I recommend major revision rather than rejection. The D0 issue in Lemma 5.4 and the epsilon'/epsilon mismatch in Theorem 5.5 both need to be fixed in the proof itself, not merely in the discussion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main theorem is new and likely true, but the written proof has a parameter bug in Lemma 5.4 that breaks the epsilon-recolouring estimate. I would send it to peer review, with the request that the referee focus on that lemma.\n\nWhat's genuinely good: Fox-Tidor-Zhao explicitly left non-generic instances, and the translation-invariant result does not cover all partition-regular complexity-1 patterns. Closing that gap is a real contribution. The multiple-subcoset selection scheme is new, and the exposition in Section 3 explains clearly why the zero coset forces the change. The counting lemma in Appendix A is proved independently and looks correct. The paper relies honestly on external results — Green's regularity lemma, Serra-Vena's Rado theorem, FTZ's subcoset selection, Manners's bounds — with no circularity. Citation pattern is fine; the one self-reference to forthcoming work is used only for the discussion of bounds, not as a load-bearing step.\n\nThe soft spots, in order:\n\n1. Lemma 5.4 sets D0 = ceil(log_p(epsilon/4r)). For any non-trivial epsilon (epsilon < 4r) this is non-positive, so there is no subspace of that codimension. The proof cannot be instantiated as written. If you ignore that and start with D0 = 0, the final inequality requires p^{-D1} <= epsilon/4, i.e. D1 >= log_p(4/epsilon), whereas the proof only guarantees D1 >= log_p(epsilon/4r). For p=2, r=2, epsilon=1/100 this is the difference between D1 >= 9 and D1 >= -9. The entire epsilon-approximation guarantee rests on this line. The repair looks straightforward — take D0 = ceil(log_p(4/epsilon)) and check the subsequent uses of Proposition 5.1 and Lemma 5.3 still go through — but that repair is absent.\n\n2. Lemma 5.3 applies Rado's theorem with 2r colours, but the auxiliary colouring actually maps into subsets of [r], so it uses 2^r colours. This only affects constants.\n\n3. There is an off-by-m factor in the count of non-regular solutions in the same lemma. Again cosmetic.\n\nThese are repair-grade defects, not signs of a confused argument. The core idea is sound and the theorem is probably true. The paper will be useful to people working on arithmetic removal, property testing, and finite-field Ramsey theory, and it would make a good reading-group paper because the gap is instructive.\n\nRecommendation: serious referee, yes. Ask the referee to verify the repaired parameter choice in Lemma 5.4 and the colour-count in Lemma 5.3.","headline":"New and likely true, but a fixable parameter bug in Lemma 5.4 currently blocks the proof of the main theorem; worth a serious referee and a focused revision.","tokens_in":17974,"tokens_out":5371,"would_cite":false,"duration_ms":47498,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","05D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every partition-regular pattern of complexity 1 over a finite field can be completely eliminated from a colouring by recolouring only a tiny fraction of the space.","keywords":["arithmetic removal lemma","induced removal","partition-regular patterns","true complexity 1","finite fields","Rado's theorem","arithmetic regularity lemma","zero coset"],"falsifier":"Take $p=2$, $r=2$, $\\epsilon=2^{-10}$. Then Lemma 5.4 sets $D_0 = \\lceil \\log_2(\\epsilon/8)\\rceil = -13$, so $D_1$ need not be large and $p^{-D_1}$ can exceed $\\epsilon/4$. Computing this inequality would show the proof does not establish the claimed recolouring bound.","tokens_in":16877,"feed_emoji":"🧩","tokens_out":8928,"duration_ms":50607,"temperature":0.7,"pith_summary":"The paper proves that if a pattern of linear equations over a finite field is partition-regular and has true complexity 1, then any colouring of $\\mathbb{F}_p^n$ with very few instances of that pattern can be recoloured on a tiny fraction of the points to eliminate every instance, including those involving the zero vector. Earlier induced removal results for such patterns always left a small residue of 'non-generic' instances behind; the author shows that partition-regularity supplies enough flexibility to remove all of them. The proof works by selecting not just one but several regular subcosets inside each large coset, using a quantitative form of Rado's theorem to ensure that the chosen subcosets still solve the pattern's equations. If the theorem is right, it closes the gap between the induced removal lemmas for general complexity-1 patterns and the stronger complete-removal results known for translation-invariant patterns.","feed_headline":"Full removal proved for every partition-regular complexity-1 pattern","feed_subtitle":"Extends induced removal lemmas so no pattern instances, even at zero, survive after recolouring a tiny fraction of the space.","key_machinery":"The proof's central device is a two-stage subcoset selection. A strong regularity lemma first partitions $\\mathbb{F}_p^n$ into cosets of a coarse subspace $H_1$ and then picks regular cosets of a finer subspace $H_2$ inside each $H_1$-coset, as in [7]. The novel step, Lemma 5.3, selects several cosets $H_3+z_1, \\dots, H_3+z_m$ inside the zero coset of $H_2$ at once, with the translations $z_j$ solving the pattern's equation $A z = 0$; this is achieved by applying a quantitative Rado theorem over finite fields to an auxiliary colouring of the coset space. Because a whole consistent family of regular subcosets is chosen, the recolouring can be decided by the colour densities on these representatives without any special handling of the zero coset; partition-regularity guarantees that such a family exists.","core_discovery":"The central claim (Theorem 1.5) is: for every $\\epsilon>0$, integer $r$, and every partition-regular arithmetic pattern $H$ of complexity 1, there is a $\\delta > 0$ such that any $r$-colouring of $\\mathbb{F}_p^n$ with $H$-instance density at most $\\delta$ can be recoloured on at most an $\\epsilon$-fraction of $\\mathbb{F}_p^n$ to become completely $H$-free. Unlike the earlier induced removal lemma [7], the recolouring removes all instances, including those containing 0; a non-partition-regular pattern can force a monochromatic solution at 0 in every recolouring, so partition-regularity is exactly the condition that makes complete removal possible.","pith_inferences":["A corrected parameter choice in the recolouring lemma (taking the coarse-subspace codimension large enough that the unchanged fraction is bounded by $\\epsilon/4$) would repair the proof's approximation bound and leave the main theorem intact.","The same multiple-subcoset strategy is the natural candidate for extending complete removal to partition-regular patterns of higher true complexity, although the author notes that the required rank control is not yet available.","If the theorem survives scrutiny, it implies that partition-regularity is the exact dividing line between induced removal lemmas that leave a zero-residue and those that do not, at least in the complexity-1 range."],"forward_implications":["Complete removal holds for all partition-regular patterns of complexity 1, including non-translation-invariant patterns such as rainbow solutions to $x+y+z=w$.","Partition-regularity is exactly the dividing line: for a non-partition-regular pattern, some instances (e.g. at 0) must survive any recolouring, as shown by the paper's Example 1.4.","Corollary 1.7: a subset $A$ of $\\mathbb{F}_p^n$ with $A+A$ almost disjoint from the complement is $\\epsilon$-close to a subspace.","The theorem extends to removing any finite collection of partition-regular complexity-1 patterns simultaneously."],"supporting_citations":[{"why":"supplies the base induced removal lemma for complexity-1 patterns and the subcoset selection technique that the proof refines.","marker":"[7]"},{"why":"gives the complete-removal result for translation-invariant patterns and the crucial idea of exploiting translation flexibility.","marker":"[3]"},{"why":"provides the quantitative Rado theorem over finite fields used in Lemma 5.3 to find consistent families of regular subcosets.","marker":"[19]"},{"why":"states the arithmetic regularity lemma used to build the honest partition of the space.","marker":"[13]"},{"why":"yields polynomial bounds for the counting-lemma parameter, giving the final quantitative dependence of $\\delta$.","marker":"[17]"},{"why":"demonstrates the multiple-exception strategy in graph removal that motivates the multiple-subcoset selection.","marker":"[1]"}],"fun_headline_variants":["Complete removal for all partition-regular patterns","Zero tolerance: all pattern instances eliminated","Every partition-regular pattern fully removed after recolor","Beyond induced removal: full elimination for partition-regular"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's guarantee that the recolouring changes only a small fraction of the space depends on the coarse subspace being chosen large enough; the bound stated in Lemma 5.4 does not, as written, ensure this for small epsilon.","fun_headline_variants_meta":{"raw":{"variants":["Complete removal for all partition-regular patterns","Zero tolerance: all pattern instances eliminated","Every partition-regular pattern fully removed after recolor","Beyond induced removal: full elimination for partition-regular"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00096,"raw_usage":{"total_tokens":4032,"prompt_tokens":829,"completion_tokens":3203,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":3145}},"tokens_in":445,"tokens_out":3203,"duration_ms":21636,"temperature":1.0,"reasoning_tokens":3145,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:35:00.868677+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $p=2$, $r=2$, $\\epsilon=2^{-10}$. Then Lemma 5.4 sets $D_0 = \\lceil \\log_2(\\epsilon/8)\\rceil = -13$, so $D_1$ need not be large and $p^{-D_1}$ can exceed $\\epsilon/4$. Computing this inequality would show the proof does not establish the claimed recolouring bound.","supporting_citations":[{"cited_title":"Induced arithmetic removal: complexity 1 patterns over finite fields","cited_arxiv_id":null,"evidence_quote":"supplies the base induced removal lemma for complexity-1 patterns and the subcoset selection technique that the proof refines."},{"cited_title":"Every locally characterized affine-invariant property is testable","cited_arxiv_id":null,"evidence_quote":"gives the complete-removal result for translation-invariant patterns and the crucial idea of exploiting translation flexibility."},{"cited_title":"On the number of monochromatic solutions of integer linear systems on abelian groups","cited_arxiv_id":null,"evidence_quote":"provides the quantitative Rado theorem over finite fields used in Lemma 5.3 to find consistent families of regular subcosets."},{"cited_title":"A Szemer´ edi-type regularity lemma in abelian groups, with applications","cited_arxiv_id":null,"evidence_quote":"states the arithmetic regularity lemma used to build the honest partition of the space."},{"cited_title":"Efficient testing of large graphs","cited_arxiv_id":null,"evidence_quote":"demonstrates the multiple-exception strategy in graph removal that motivates the multiple-subcoset selection."}],"review_version":1}