{"id":"ba60d870-6294-478f-83fa-0ffed8935a21","arxiv_id":"2608.06681","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An n-divisor arithmetic progression of sub-square-root logarithmic height must have length at least roughly n^{3/4}/sqrt(log n), which disproves the arithmetic-progression version of the Strong Divisor Conjecture at (1/3,1/3).","lead":"This paper proves a lower bound on the length of any arithmetic progression that contains a multiple of every integer up to n, and it uses that bound to refute a conjecture proposed by Umans and Wang. The result eliminates one proposed route toward faster factorization algorithms, while leaving the stronger higher-rank conjecture open.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"No significant mathematical issue found. The proof is complete and checkable: I verified the incidence lemma, the prime-band count, the removal of primes dividing c, the congruence degree bound, and the optimization at a=2/3, b=1. The reader's weakest assumption, namely dependence on the prime number theorem, is real but not a correctness risk; PNT is standard and the argument only needs its asymptotic form. The paper scopes the result correctly, explicitly leaving the higher-rank Strong Divisor Conjecture open, and the Codex-assisted discovery is disclosed without affecting the mathematical content. The only minor expositional point is that u,c∈Z>0 is stated before the singleton case with c=0, but that case is handled separately and does not affect the proof. Therefore the ACCEPT verdict should remain unchanged.","tokens_in":4241,"tokens_out":19127,"duration_ms":194835,"concrete_test":"Independently re-derive Lemma 3.1 with a=2/3 and b=1, checking the boundary case: for distinct primes p,q≤√n one has pq≤n; and from L≤(2/3)^2 n it follows that 0<|i−j|≤L−1<L≤(2/3)^2 n≤pq, so pq∤(i−j). If this check passes, the endpoint choice b=1 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central argument. Theorem 1.1 follows from Lemma 3.1: the singleton case is eliminated by the prime number theorem; for L≥2, the set P excludes primes dividing c so each prime selects exactly one residue class of indices; the block-intersection argument uses only L≤a^2 n≤pq and remains valid at the endpoint b=1; properness of every block follows from the log-product estimate. The only external input is the prime number theorem, which is standard and used only asymptotically; even a Chebyshev-class estimate would preserve the contradiction structure, with different constants. The scope restrictions in Remarks 4.2 and 4.3 are accurate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an unconditional lower bound on the length L of any arithmetic progression of positive integers that is n-divisor (every integer 1 through n divides at least one term) and has height H with log H = o(sqrt n). The bound is L >= (sqrt(8/27) - o(1)) n^{3/4} / sqrt(log n). As a consequence, the arithmetic-progression version of the Strong (alpha, beta)-Divisor Conjecture of Umans and Wang is false whenever alpha < 1/2 and beta < 3/8, including the proposed (1/3, 1/3) point even under n^{o(1)} exponent-level relaxations. The proof combines a bounded-degree linear-space lemma (Lemma 2.1) with a construction of blocks from primes in a fixed band below sqrt n (Lemma 3.1). The only external analytic input is the prime number theorem.","tokens_in":4325,"tokens_out":12349,"duration_ms":102907,"significance":"If correct, the result is a clean and decisive refutation of the one-dimensional arithmetic-progression variant of the Umans-Wang conjecture over a substantial parameter range. The proof is fully written and self-contained; the combinatorial core is elementary and transparent. The paper is careful to state that the higher-rank Strong Divisor Conjecture is not affected. The PNT is the only external input, and the argument does not rely on numerical computation, fitted parameters, or any circular assumption. This should be of interest to number theorists and theoretical computer scientists working on divisor covers and factorization algorithms.","major_comments":[],"minor_comments":[{"comment":"The disclosure that the proof was discovered in an OpenAI Codex run is unusual for a mathematics paper; the authors should verify that this presentation conforms to the journal's policies on AI-assisted work and should consider moving it to the acknowledgments.","section":"Section 1 (Discovery using Codex)"},{"comment":"The phrase \"If the progression has only one distinct term\" covers both the case L=1 and the case c=0; the subsequent reduction to L>=2 and c>=1 is correct, but the two cases could be stated explicitly for readability.","section":"Section 3, proof of Lemma 3.1"},{"comment":"The phrase \"relaxed by n^{o(1)} at the exponent level\" is clear from context, but it could be made explicit by writing L_n <= n^{2 beta + o(1)} and H_n <= exp(n^{alpha + o(1)}) directly after the corollary statement.","section":"Section 4, Corollary 1.2"},{"comment":"The reference to \"Proposition 3.4 and Section 6\" of the Umans-Wang paper would be more helpful with a precise statement or a page/equation number, since these items are used to frame the conjecture being refuted.","section":"Introduction"}],"recommendation":"accept","confidential_remarks":"This is a mathematically sound and well-scoped refutation of the arithmetic-progression version of the Umans-Wang conjecture. The only unusual feature is the prominent AI-discovery disclosure, which is a policy matter for the editor rather than a substantive mathematical issue; the proof itself is fully human-verifiable from the text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know: this paper proves a real lower bound and kills the one-dimensional AP version at (1/3,1/3) without circularity. The argument is elementary and verifiable, and the authors are explicit that the higher-rank Strong Divisor Conjecture is untouched.\n\nWhat is new is the fixed-band incidence argument. Taking primes in [a sqrt(n), b sqrt(n)], semiprime divisibility forces pair coverage, the difference (i-j)c bounds intersections, and the bounded-degree linear-space lemma yields L on the order of n^{3/4}/sqrt(log n). The optimization a=2/3, b=1 gives the 8/27 constant. I checked the steps: properness of blocks uses the prime number theorem to get log product ~ x, and the degree bound uses one residue class per prime. It holds together. The singleton case is handled by lcm(1..n) having log ~ n, far above o(sqrt n). No hidden assumptions, no fitted parameters.\n\nThe soft spots are mostly scope. The conjecture being refuted is informally named rather than a numbered conjecture in [4], so the headline is slightly softer than 'conjecture refuted'—it is the AP version that falls. The endpoint b=1 is legitimate because pq <= n still holds for distinct primes <= sqrt n. The case log H = O(sqrt n) and beta = 3/8 remain open, and the constant is not claimed optimal. Those are honest limitations, not flaws. The proof leans on PNT; a Chebyshev bound would change constants but the contradiction structure survives. The Codex disclosure is transparent and irrelevant to the mathematics.\n\nI have no serious objection. The citations check out; the only self-reference is to the Moonshot harness in the acknowledgments, which is not used as evidence.\n\nWho this is for: people working on divisor-cover conjectures and anyone interested in elementary incidence proofs. It deserves a serious referee; I would send it out. I would cite it if I worked on n-divisor sets.","headline":"A clean, unconditional refutation of the informal AP version of the Umans–Wang divisor conjecture; the proof is short, checkable, and the scope limits are stated honestly.","tokens_in":4835,"tokens_out":1629,"would_cite":true,"duration_ms":15231,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B25","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"An arithmetic progression whose largest term is not too large must have length at least sqrt(8/27) n^(3/4)/sqrt(log n), up to lower-order terms, to contain a multiple of every integer up to n.","keywords":["n-divisor set","arithmetic progression","divisor cover","prime number theorem","finite linear space","lower bound","Strong Divisor Conjecture","semiprime divisibility"],"falsifier":"Find an unbounded sequence of $n$ and $n$-divisor arithmetic progressions with $\\log H = o(\\sqrt n)$ and length $L \\le (\\sqrt{8/27} - \\varepsilon) n^{3/4}/\\sqrt{\\log n}$ for some fixed $\\varepsilon>0$; any such example would disprove the theorem. A finite search for small $n$ comparing minimal progression length against the claimed bound would give an immediate sanity test, though only an infinite sequence would be decisive.","tokens_in":4051,"feed_emoji":"🔢","tokens_out":19168,"duration_ms":138817,"temperature":0.7,"pith_summary":"The paper proves an unconditional lower bound on the length of any arithmetic progression whose entries include a multiple of every integer from 1 through $n$. Under the condition that the progression's largest term $H$ satisfies $\\log H = o(\\sqrt n)$, the length $L$ must be at least $(\\sqrt{8/27} - o(1)) n^{3/4}/\\sqrt{\\log n}$. This rules out the proposed 'arithmetic-progression version' of the Strong Divisor Conjecture at the $(\\alpha,\\beta)=(1/3,1/3)$ exponent point, even when the length and height bounds are relaxed by $n^{o(1)}$. The proof is elementary apart from the prime number theorem, and it applies only to one-dimensional progressions, not to the higher-rank conjecture.","feed_headline":"n-divisor progressions need length ~0.544 n^(3/4)/sqrt(log n)","feed_subtitle":"The (1/3,1/3) arithmetic-progression case of the Strong Divisor Conjecture is false, even with sub-polynomial slack.","key_machinery":"The argument converts semiprime divisibility into a finite incidence structure. Take the primes $p$ in a fixed band $a\\sqrt n \\le p \\le b\\sqrt n$ that do not divide the progression's step $c$; each progression term $u+ic$ yields a block consisting of the primes in the band that divide it. The $n$-divisor property ensures every pair of distinct band primes lies in some block (their product $pq \\le b^2 n \\le n$), while the small height makes every block proper and the small length limits each prime to about $L/(a\\sqrt n)$ blocks. A bounded-degree linear-space lemma (Lemma 2.1) then forces the number $v$ of such primes to satisfy $v \\le \\Delta(\\Delta-1)+1$, and the prime number theorem supplies $v$ and the log-product of the band primes. Optimizing $a=2/3$, $b=1$ gives the constant $\\sqrt{8/27}$.","core_discovery":"The central claim is that short arithmetic progressions cannot cover $1,\\dots,n$ when the logarithm of their largest term is much smaller than $\\sqrt n$. Formally, any $n$-divisor progression with $\\log H = o(\\sqrt n)$ has $L \\ge (\\sqrt{8/27} - o(1)) n^{3/4}/\\sqrt{\\log n}$. A direct corollary excludes every exponent pair $\\alpha<1/2$, $\\beta<3/8$, including the originally proposed $(1/3,1/3)$, even under $n^{o(1)}$ slack. The obstruction is sharp at the level of the argument: the constant $\\sqrt{8/27}$ arises from optimizing the band $a=2/3$, $b=1$ below $\\sqrt n$ and is not claimed to be optimal.","pith_inferences":["The same bounded-degree incidence argument may transfer to other structured covering sets, but the one-dimensional step identity $(u+ic)-(u+jc)=(i-j)c$ carries the proof; for rank-two progressions the difference becomes vector-valued and the pairwise-intersection argument does not follow.","A natural next test is to attempt constructions of $n$-divisor progressions near the new lower bound; if such constructions exist, the constant $\\sqrt{8/27}$ would be optimal up to the $o(1)$ term.","The role of the prime number theorem is only to count primes in a fixed band below $\\sqrt n$; replacing it with explicit Chebyshev-type bounds would turn the asymptotic lower bound into an explicit finite-$n$ inequality, at the cost of worse constants.","The qualitative phenomenon here is that semiprime divisibility alone forces a super-polynomial separation between length and height; this suggests similar trade-offs may hold for other divisor-cover structures, but that is an extrapolation beyond the paper."],"forward_implications":["The arithmetic-progression version of the Strong Divisor Conjecture is false at $(\\alpha,\\beta)=(1/3,1/3)$, so the proposed sufficient statement does not hold.","The failure persists: both the length bound and the height bound can be relaxed by a factor $n^{o(1)}$ at the exponent level before the contradiction disappears.","Any $n$-divisor progression with $\\log H = o(\\sqrt n)$ must have length at least $(\\sqrt{8/27} - o(1)) n^{3/4}/\\sqrt{\\log n}$, a quantitative obstruction independent of the conjecture.","The theorem says nothing when $\\beta=3/8$ or when $\\log H = \\Theta(\\sqrt n)$; these boundary cases remain open.","The higher-rank Strong Divisor Conjecture is untouched by this result."],"supporting_citations":[{"why":"States the Strong (α,β)-Divisor Conjecture and its arithmetic-progression version, which is the conjecture refuted by the theorem.","marker":"[4]"},{"why":"Supplies the prime number theorem estimates for π(x) and ϑ(x) that count the band primes and their log-product.","marker":"[2]"},{"why":"Presents the classical finite-linear-space estimate that motivates Lemma 2.1; the paper gives a self-contained proof of its bounded-degree variant.","marker":"[1]"}],"fun_headline_variants":["Arithmetic-progression divisor conjecture refuted at (1/3,1/3)","n-divisor APs need length >= 0.544 n^(3/4)/sqrt(log n)","Umans–Wang AP conjecture fails even with sub-polynomial slack"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's quantitative conclusion rests on the prime number theorem for the count and log-product of primes in a fixed band just below $\\sqrt n$; with only weaker prime-count estimates, the constant and possibly the whole proper-block argument would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Arithmetic-progression divisor conjecture refuted at (1/3,1/3)","n-divisor APs need length >= 0.544 n^(3/4)/sqrt(log n)","Umans–Wang AP conjecture fails even with sub-polynomial slack"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002025,"raw_usage":{"total_tokens":7924,"prompt_tokens":1008,"completion_tokens":6916,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":6845}},"tokens_in":624,"tokens_out":6916,"duration_ms":48652,"temperature":1.0,"reasoning_tokens":6845,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:33:06.527447+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an unbounded sequence of $n$ and $n$-divisor arithmetic progressions with $\\log H = o(\\sqrt n)$ and length $L \\le (\\sqrt{8/27} - \\varepsilon) n^{3/4}/\\sqrt{\\log n}$ for some fixed $\\varepsilon>0$; any such example would disprove the theorem. A finite search for small $n$ comparing minimal progression length against the claimed bound would give an immediate sanity test, though only an infinite sequence would be decisive.","supporting_citations":[{"cited_title":"Umans and S","cited_arxiv_id":null,"evidence_quote":"States the Strong (α,β)-Divisor Conjecture and its arithmetic-progression version, which is the conjecture refuted by the theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the classical finite-linear-space estimate that motivates Lemma 2.1; the paper gives a self-contained proof of its bounded-degree variant."}],"review_version":1}