{"id":"8cd7405d-4017-400c-8705-519779fd8f41","arxiv_id":"2607.23795","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every infinite B_h-set with h even satisfies liminf A(n)/(n/log n)^{1/h} at most (pi/log 2 * Gamma(1+h/2)^2 / Gamma(1+1/h)^h)^{1/h}.","lead":"The paper proves an explicit upper bound on how thick an infinite B_h-set (a generalized Sidon set) can be, for every even h. The bound improves a 35-year-old finiteness result by giving a concrete constant built from pi, log 2, and Gamma values.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Theorem 1's explicit constant is only as secure as Lemma 6, which is stated but never proved in this manuscript and is imported from Part I — itself an unrefereed preprint; its constant sqrt(8c/log 2) propagates unchanged into the final Gamma expression.","rationale":"The reader identified Lemma 6 as the weakest assumption, and my independent pass lands in the same place: it is the only load-bearing statement in the paper that is invoked rather than proved, it sits on an unrefereed companion preprint, and its numerical constant is exactly what is at stake in a paper whose entire contribution is an explicit constant. I add two sharpenings the reader did not state: the change of block parameter M between Parts I and II means Lemma 6 must hold under its general hypotheses, not just Part I's setting; and a naive reconstruction of the energy-to-liminf optimization fails to produce the log 2 factor, which localizes the risk to a specific step the reader of this manuscript cannot inspect. Everything else — Lemmas 7-11, the Dirichlet integral, the limiting algebra, and the h = 2 consistency check against Part I's Sidon result — verifies cleanly, so this is a one-point failure mode, not a diffuse one. The reader's CONDITIONAL / MODERATE assessment already prices in exactly this gap, so I recommend no verdict change: CONDITIONAL remains correct, and the condition is precisely \"supply or verify Lemma 6 with its stated constant under hypotheses (i)-(iii).\" The author's own closing note that Part I's structural weaknesses carry over reinforces rather than changes this assessment.","tokens_in":11668,"tokens_out":7833,"duration_ms":152071,"concrete_test":"Extract the proof of the energy-to-liminf lemma from Part I [12] and re-run its final optimization step with M = N/log^3 N (rather than Part I's M = N/log N) to confirm the output is exactly sqrt(8c/log 2) under hypotheses (i)-(iii) as stated. As a sharp sub-check: for the extremal block profile F_l = lambda * d/dℓ sqrt(ℓN/log(ℓN)), compute whether the argument's lower bound on (1/N) sum binom(F_l,2) is lambda^2 (log 2)/8 + o(1) or only lambda^2/8 + o(1). If only the latter, Theorem 1's constant must be multiplied by (log 2)^{-1/h} and Corollary 2's by (log 2)^{1/h}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I re-checked the internal machinery and it is largely sound: the pair-averaging in Lemma 11 (blocks of width N, at most N-d offsets containing a pair of difference d), the disjoint-pair injectivity P_0(d) ≤ 1 via Lemma 3, the regularity-free ratio A(W)^r/A(N)^{2r} → 0 with W = N^2/log^3 N, the Dirichlet integral evaluation in Lemma 8 (m^{1/2} Γ(1/h)^k / Γ(3/2) = 2Γ(1/h)^k √m/√π, since k/h = 1/2), and the final algebra all verify, and h = 2 correctly recovers Part I's sqrt(4/log 2) Sidon constant. The single point where the theorem can break is Lemma 6: the manuscript verifies its hypotheses (i)-(iii) in §4 but gives no proof, and the constant sqrt(8c/log 2) it outputs is exactly what becomes the 1/log 2 inside Theorem 1. Two specific vulnerabilities: (a) Part I used M = N/log N; here M = N/log^3 N, so one must confirm Part I's proof works under the general hypotheses (i)-(iii) and not only for its own M; (b) a straightforward Cauchy/extremal-profile computation (F_l ∝ d/dℓ sqrt(ℓN/log ℓN) ∝ 1/√ℓ) yields only sum F_l^2 ≈ λ²N/4, i.e. λ ≤ sqrt(8c) with no log 2 — so the log 2 factor must arise from a specific, non-reproduced argument in Part I, and if that step is off, every constant in Theorem 1 and Corollary 2 shifts by a factor (log 2)^{∓1/h}.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript proves that for every even positive integer h and every B_h-set A of nonnegative integers, liminf A(n)/(n/log n)^{1/h} is bounded by the explicit constant (π/log 2 · Γ(1+h/2)²/Γ(1+1/h)^h)^{1/h}. This is the first explicit constant for this liminf for general even h; Chen (1993) had shown finiteness without a constant, and the h=2 case recovers the √(4/log 2) bound from the author's Part I. The proof is by contradiction: assuming A(x) ≥ τ(x/log x)^{1/h} with τ above the claimed constant, a new Lemma 8 lower-bounds liminf S(m)/√(m/log m) for the k-fold sumset S = kA (k = h/2) via a lattice-point count and a Dirichlet integral; Lemma 11 produces a shift t* whose block counts satisfy Σ C(F_ℓ,2) ≤ N/2 + o(N) by averaging over shifts and stratifying difference pairs by |V_s ⋒ V_s'|, using a quantitative form of Jia's Lemma 9; and Lemma 6 (imported from Part I) converts the block-energy bound into liminf S(m)/√(m/log m) ≤ √(4/log 2). Comparing the two bounds on the same liminf yields the contradiction.","tokens_in":12109,"tokens_out":11647,"duration_ms":386915,"significance":"If correct, this is a solid contribution: the first explicit, parameter-free constant for the thickness of infinite B_h-sets for all even h, improving on 35-year-old finiteness results, and it eliminates Jia's growth-regularity hypothesis A(N²) = O(A(N)²) by separating the block count M from the block width N (M = N/log³ N) — a genuine technical advance over Erdős–Helm–Jia–Chen. The constant is fully explicit and falsifiable, the h=2 specialization correctly recovers Part I's √(4/log 2), and the asymptote h/(2e) with minimum at h=4 is a clean, checkable statement. The argument is short and, apart from one imported lemma, self-contained and verifiable line by line; I checked the Dirichlet integral evaluation, the Γ(1+1/h) = (1/h)Γ(1/h) reconciliation, the final algebra, and Corollary 2 (including the factor h from log n ≈ (1/h) log x), all of which are correct. The one soft point is that the constant's log 2 factor enters entirely through Lemma 6, which is stated but not proved here.","major_comments":[{"comment":"Lemma 6 is stated with hypotheses (i)–(iii) but no proof, and no precise citation (not even a theorem number) into Part I [12], which is listed only as 'Preprint.' This is load-bearing: the entire explicit content of Theorem 1 and Corollary 2 inherits the factor √(8c/log 2), and the log 2 — the only transcendental input beyond the Dirichlet integral — is produced nowhere in this manuscript. A naive Cauchy/extremal-profile computation with F_ℓ ∝ 1/√ℓ yields only Σ F_ℓ² ≲ λ²N/4, i.e. √(8c) with no log 2, so the constant depends on a specific argument in Part I that the reader cannot audit. Moreover, Part I used M = N/log N while this paper needs M = ⌊N/log³ N⌋; the manuscript asserts that hypotheses (i)–(iii) suffice but gives no indication that Part I's proof runs under these general hypotheses rather than for its specific M. Please either include a proof of Lemma 6 (an appendix is fine)","section":"§3, Lemma 6; invoked in §4"},{"comment":"The left-hand side of the displayed chain changes from Σ_{ℓ=1}^{M} C(F_ℓ,2) in (7) to Σ_{ℓ=1}^{M−1} C(F_ℓ,2) in (8), and the proof concludes with the M−1 sum, while the lemma's statement — and the application to Lemma 6 in §4, which needs exactly the blocks ℓ = 1,…,M — concerns the M sum. Since C(F_M,2) ≥ 0, an upper bound on the M−1 sum does not imply one on the M sum, so as written the proof does not establish the stated lemma. The repair is immediate (rerun the argument with M+1 in place of M, or restate the lemma with M−1 and note Lemma 6's hypotheses are asymptotic and insensitive to this shift), and no constant is affected, but the mismatch must be fixed.","section":"§3, proof of Lemma 11, Eqs. (7)–(10)"}],"minor_comments":[{"comment":"In the three-line chain ending the proof, the middle relation is printed as '≤' but must be '≥': (1 − 1/log u_0)^k ≥ 1 − k/log u_0 ≥ 1 − k/log log m since u_0 ≥ log m. As printed the chain reads N*(m) ≥ · ≤ ·, which is confusing.","section":"§3, proof of Lemma 8, final display"},{"comment":"The hypothesis allows τ ≥ 0, but the proof's ratio A(W)^r/A(N)^{2r} → 0 requires τ > 0. State τ > 0. Also 'let k := h/2' begins with a lowercase letter after a period.","section":"§3, Lemma 11 statement"},{"comment":"'B(N) = o(√N log N)' is ambiguous between o(√(N log N)) and o(√N · log N); in §4 it is verified as O(N^{1/2}) = o(√(N log N)). Please disambiguate with parentheses.","section":"§3, Lemma 6, hypothesis (iii)"},{"comment":"Typos: 'with substantials detours' and 'artifical manner.'","section":"§1.2"},{"comment":"[12] (Part I) is cited only as 'Preprint'; please add an arXiv identifier and, given that Lemma 6 and the weighted Cauchy inequality are imported from it, precise internal pointers (section/theorem numbers).","section":"References"},{"comment":"The display lacks a terminal period. It would also help the reader to note in one line where the factor h comes from (log a_n ≈ (1/h) log(a_n/(log a_n)^{...}) inversion), since it is not the naive reciprocal of Theorem 1's constant.","section":"§1, Corollary 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a sequel whose single external ingredient (Lemma 6) rests on the author's own unrefereed Part I; the paper cannot be fully verified until that proof is public and confirmed to apply at M = N/log³ N. I do not doubt the result — the internal machinery checks out in detail and h=2 reproduces the known constant — but the explicit constant is the paper's entire point, so I have recommended major revision rather than minor. The author discloses substantial AI assistance in writing and checking; the disclosure is transparent and I detected no artifacts of it in the mathematics, but the editor may wish to confirm the journal's policy."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this turns Chen’s 1993 finiteness theorem into a concrete numerical bound for even h, with a Gamma/pi/log-2 expression that recovers the known Sidon case when h=2. That is real progress inside a narrow classical line.\n\nWhat is new is the constant itself, the Dirichlet-integral lower bound on the k-fold sumset (Lemma 8), and the flexible block width M=N/log^3 N that lets the author drop Jia’s regularity hypothesis. The architecture is the usual one—assume the set is denser than the constant, lower-bound the energy of S=kA via the integral, upper-bound the same energy by averaging differences over blocks, then compare—but the execution is careful. Multiset language, the injectivity for disjoint pairs, and the ratio A(W)^r/A(N)^{2r}\\to0 all check out. Algebra of the final constant is consistent; the h=2 reduction matches Part I.\n\nThe soft spot is exactly the one the stress-test flags: Lemma 6 (the block-energy conversion that produces the sqrt(8c/log 2) factor) is stated with hypotheses but not proved here. It is imported from the author’s own unrefeered Part I. The present manuscript verifies that its M satisfies the listed hypotheses, yet the log-2 itself is not re-derived. If that step in Part I is off by a constant, every number in Theorem 1 and Corollary 2 shifts. That is a genuine dependence, not a fatal hole; it is ordinary for a Part II, and the author is transparent about carrying structural weaknesses forward. Minor indexing/sign slips and “routine” asymptotics are present but do not look load-bearing.\n\nCitation pattern is appropriate (Erdős–Stöhr, Jia, Chen, Cilleruelo, Green). No circularity, no free parameters, no data issues.\n\nThis is for people who already care about quantitative Sidon/B_h problems. A serious referee in additive combinatorics should see it; the central claim is plausible and most of the derivation is on the page. I would send it to peer review, with the explicit request that Lemma 6 be either reproduced or tightly referenced to a stable source. Engage if you work on these sets; otherwise file it as a clean incremental benchmark.","headline":"Explicit constant for even-order infinite B_h thickness, cleanly upgrading Chen, but the load-bearing energy-to-liminf step lives in an unrefeered Part I.","tokens_in":13394,"tokens_out":591,"would_cite":false,"duration_ms":16800,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B10","11B83","11B05"],"pacs":[],"model":"grok-4.5","headline":"Every infinite even-order generalized Sidon set is thinner than a sharp multiple of (n/log n) to the power 1/h.","keywords":["B_h-sets","Sidon sets","thickness","sumsets","liminf bounds","additive bases","even order"],"falsifier":"Exhibit an infinite B_h-set (h even) whose counting function satisfies A(n) > C (n/log n)^{1/h} for every n larger than some n0, where C is strictly larger than the constant stated in Theorem 1.","tokens_in":13048,"feed_emoji":"🔢","tokens_out":899,"duration_ms":15493,"temperature":0.7,"pith_summary":"A B_h-set is a set of nonnegative integers in which every sum of h elements, written in nondecreasing order, is unique. For ordinary Sidon sets (h=2) it has long been known that the counting function cannot stay as large as a constant times sqrt(n/log n) forever. This paper supplies an explicit constant that works for every even h: the liminf of A(n) divided by (n/log n)^{1/h} is at most a concrete expression built from pi, log 2 and gamma values. The bound improves earlier qualitative finiteness results and recovers the author's earlier Sidon-set constant when h=2. A sympathetic reader cares because the constant is sharp enough to be compared with known constructions and because the argument isolates exactly where the even-order hypothesis is used.","feed_headline":"Even-order Sidon sets can't stay thicker than this constant","feed_subtitle":"An explicit liminf bound on how dense infinite B_h-sets may remain, for every even h","key_machinery":"Control of the squared block counts of the k-fold sumset S=kA (k=h/2) via a multiset difference stratification and Jia-type counting of Phi tuples, then conversion of that energy bound into a liminf upper bound by a weighted block-energy lemma.","core_discovery":"For every even positive integer h and every B_h-set A, the liminf as n tends to infinity of A(n) divided by the h-th root of n/log n is at most (pi/log 2 times Gamma(1+h/2)^2 over Gamma(1+1/h)^h) raised to the power 1/h.","pith_inferences":["Because every B_h-set is automatically a B_{h-1}-set, the even-order bound already gives a (weaker) thickness statement for odd order by reduction, though the paper treats that as artificial.","The asymptotic of the constant is h/(2e); numerical plots in the paper show a surprising minimum at h=4, suggesting that thickness is hardest to force precisely at order 4.","If a matching construction ever reaches the same constant, the liminf would be settled exactly for even h."],"forward_implications":["The corresponding limsup lower bound on the n-th term a_n is at least (log 2/pi) times h Gamma(1+1/h)^h over Gamma(1+h/2)^2, times n^h log n.","The same constant specializes, when h=2, to the Sidon-set thickness bound previously obtained by the author.","Any future improvement of the block-energy constant c=1/2 immediately improves the explicit thickness constant for every even h.","The method does not yet yield an analogous explicit liminf for odd h, leaving Jia's conjecture open."],"fun_headline_variants":["Even h B_h-sets obey this explicit liminf density cap","Infinite even-order B_h-sets stay thinner than this constant","Liminf bound caps thickness of every even-h B_h-set","Even-order generalized Sidon sets hit this density ceiling","B_h-sets for even h cannot exceed this liminf root bound"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The conversion from a uniform upper bound on the sum of binomial block counts into an explicit liminf constant is taken as already proved in an earlier paper and is not re-established here.","fun_headline_variants_meta":{"raw":{"variants":["Even h B_h-sets obey this explicit liminf density cap","Infinite even-order B_h-sets stay thinner than this constant","Liminf bound caps thickness of every even-h B_h-set","Even-order generalized Sidon sets hit this density ceiling","B_h-sets for even h cannot exceed this liminf root bound"]},"model":"grok-4.5","effort":"low","cost_usd":0.003933,"raw_usage":{"total_tokens":1202,"prompt_tokens":711,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":39328000,"prompt_tokens_details":{"text_tokens":711,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":412,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":711,"tokens_out":79,"duration_ms":5777,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T12:09:46.094583+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit an infinite B_h-set (h even) whose counting function satisfies A(n) > C (n/log n)^{1/h} for every n larger than some n0, where C is strictly larger than the constant stated in Theorem 1.","supporting_citations":[],"review_version":1}