{"id":"c09fe4ea-60fa-439c-8dc1-dbadc68b940b","arxiv_id":"2607.10431","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"lim inf (F(n)-f(n,n))/(n log n) ≥ 1/e, so some intervals of length c n log n contain no distinct multiples of 1..n for any fixed c<1/e.","lead":"The paper proves that the longest gap without a full set of distinct multiples of 1 through n is at least roughly (1/e) n log n. This strengthens a recent result of van Doorn and shows that the Erdős–Pomerance smooth-number obstruction already forces gaps on the n log n scale once the starting point is moved off the diagonal.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the Hildebrand–Tenenbaum ratio estimate as the sole nontrivial analytic ingredient and correctly judges that its error is under control. The remainder of the argument is elementary counting and elementary optimization of three real parameters; both are fully rigorous and self-contained. No stronger load-bearing concern exists. The manuscript therefore continues to merit ACCEPT with high confidence.","tokens_in":15069,"tokens_out":449,"duration_ms":4424,"concrete_test":"Independently recompute the three applications of (3.2) that produce (4.2), (4.3) and (4.4) for a concrete admissible triple (e.g., d=1, a=2/5, b=1/2 of Corollary 4.2) using only the statements of [9, Thms 2–3] and the elementary expansions (3.6); verify that the coefficient of Ψ(n,y)/ℓ remains strictly positive for all large n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Lemma 2.1 (smooth obstruction) plus the three-parameter construction of Proposition 4.1, optimized in Lemma 5.1/Section 5 to the constant 1/e. The only analytic input that could threaten the sign of (2.1) is the Hildebrand–Tenenbaum local ratio (3.2) under the specialization of Lemma 3.1. That specialization is a direct, standard consequence of the cited theorems [9, Thms 2(i) and 3] once y = d log n and x lies in a fixed multiple of [n, n log n]; the relative error is O(ℓ/L) = o(1/ℓ), strictly smaller than the main-term gap of size ≍1/ℓ produced by condition (4.1). No hidden uniformity failure, circularity, or gap appears in the derivation. The constant 1/e is optimal inside the family used (Remark 5.2), and the comparison with van Doorn’s transfer is accurate.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the maximal gap F(n) for systems of distinct multiples of 1,...,n in an interval of length h, and the diagonal quantity f(n,n). Building on Erdős–Pomerance and van Doorn’s lower bound of order n log n / log log n for F(n)−f(n,n), it proves liminf (F(n)−f(n,n))/(n log n) ≥ 1/e. The argument applies the classical smooth-number obstruction (Lemma 2.1) at off-diagonal starting points m ≍ a n log n with smoothness threshold y = d log n, compares the two sides of the obstruction inequality via Hildebrand–Tenenbaum local ratio estimates (specialized in Lemma 3.1), and optimizes the free parameters (d,a,b) in Proposition 4.1 / Lemma 5.1 to obtain the constant 1/e. Corollaries strengthen van Doorn’s scale, and Section 6 compares the construction with diagonal transfer, recovers the Erdős–Pomerance diagonal lower bound, and discusses the upper-bound problem via Hall’s theorem.","tokens_in":15314,"tokens_out":854,"duration_ms":8764,"significance":"The result advances a long-standing Erdős problem (Bloom #711) by moving the lower bound for F(n)−f(n,n) from the n log n / log log n scale to the n log n scale, with an explicit constant 1/e that is optimal inside the three-parameter family used. The method is elementary once the Hildebrand–Tenenbaum estimates are granted, and the paper carefully situates the new bound relative to van Doorn’s transfer inequality and the original diagonal obstruction. The discussion of Hall violators and the partial n^{1+o(1)} upper bound at polynomial heights (Proposition 6.2) is a useful contribution to the remaining open upper-bound question. Strengths include clean error tracking, an explicit admissible triple giving a concrete 1/10 coefficient, and transparent comparison with prior work.","major_comments":[],"minor_comments":[{"comment":"In the proof of Corollary 1.2 the constant 1/(1754 e) is described as aesthetic; a short remark that any fixed positive constant less than 1/e works would make the dependence clearer for readers who only skim that proof.","section":null},{"comment":"Section 6.1 optimizes the transfer parameter κ and obtains the same constant 1/e on the weaker n log n / log log n scale. A one-sentence pointer that this numerical coincidence is left unexplained (as already noted later in the section) would help readers who stop after the transfer calculation.","section":null},{"comment":"In Lemma 3.1 the uniformity statement over compact subsets of (0,∞) for d,c1,c2 is correct but slightly stronger than needed for the fixed-parameter applications that follow; a parenthetical that fixed d,a,b suffice would reduce notational overhead.","section":null},{"comment":"Typographical consistency: the manuscript mixes “log log n” and “ℓ” freely; once ℓ is introduced it would be cleaner to use it uniformly in the asymptotic displays of Sections 4–5.","section":null},{"comment":"Reference [15] is cited as INTEGERS 2026; if the final pagination is known it should be inserted, otherwise the arXiv identifier already given is adequate.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, self-contained, and of clear interest for the Erdős-problems community. I see no load-bearing technical gap; the only analytic input is a standard specialization of Hildebrand–Tenenbaum, which the author tracks carefully. Fit for a strong number-theory journal is good. The LLM-assistance acknowledgment is unusually frank but does not affect the mathematical content."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new fact is simple: liminf (F(n) - f(n,n))/(n log n) ≥ 1/e. That moves the known lower bound for F(n) from van Doorn’s n log n / log log n scale onto the pure n log n scale, and it is optimal inside the three-parameter family the author uses.\n\nWhat is actually new is the off-diagonal placement. The obstruction itself is the classical Erdős–Pomerance counting argument (Lemma 2.1). The author runs it at m ≈ a n log n with smoothness y = d log n, feeds in the Hildebrand–Tenenbaum local ratios, and optimizes the free parameters (d,a,b). The comparison with van Doorn’s diagonal-transfer inequality is accurate: under the present diagonal estimates the transfer cannot reach n log n, so the improvement is genuine. Section 6 also recovers the classical diagonal lower bound with the same constant 2/√e and gives a clean Hall-theoretic reading of the upper-bound problem, including the n^{1+o(1)} bound at every polynomial height.\n\nThe soft spots are minor and already flagged by the author. The constant 1/e is only the supremum of the three-parameter family (Remark 5.2); it is not claimed to be absolute. The method cannot produce intervals longer than ≈ n log n because the obstruction forces E < n y. The only analytic input that could threaten the sign of the counting inequality is the relative error in the Hildebrand–Tenenbaum ratio under the specialization of Lemma 3.1; that error is O(ℓ/L) = o(1/ℓ), strictly smaller than the main-term gap of size ≈ 1/ℓ, so the sign is controlled. No circularity or hidden uniformity failure appears.\n\nThis is a short, self-contained analytic-number-theory note for people who care about Erdős matching problems or smooth-number applications. The citations are appropriate, the proofs are elementary once the saddle-point estimates are granted, and the result is a real quantitative strengthening of a known conjecture. It deserves a serious referee and should be accepted after ordinary polishing.","headline":"Clean off-diagonal smooth obstruction that lifts the lower bound for F(n) from n log n / log log n to (1/e) n log n; the math holds.","tokens_in":15938,"tokens_out":602,"would_cite":true,"duration_ms":6253,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N25","11B75"],"pacs":[],"model":"grok-4.5","headline":"Some intervals of length about (1/e) n log n contain no complete set of distinct multiples of 1 through n.","keywords":["distinct multiples","smooth numbers","Erdős–Pomerance obstruction","Hildebrand–Tenenbaum estimates","interval matchings","Hall's theorem","F(n)","saddle-point"],"falsifier":"Either exhibit, for infinitely many n, a complete system of distinct multiples of 1..n inside every interval of length (1/e - ε) n log n that begins near height a n log n for admissible a, or produce a matching upper bound F(n) ≤ (1/e + o(1)) n log n that would force equality of the liminf.","tokens_in":15970,"feed_emoji":"🔢","tokens_out":801,"duration_ms":6693,"temperature":0.7,"pith_summary":"The paper studies how long an interval of integers can be while still missing a complete system of distinct multiples of 1, 2, ..., n. The quantity F(n) records the worst-case length of such a gap. Earlier work already showed that F(n) grows faster than the diagonal gap f(n,n), but only by a factor of order n log n / log log n. This paper improves the lower bound to the full scale n log n: the liminf of (F(n) - f(n,n))/(n log n) is at least 1/e. Equivalently, for every fixed c smaller than 1/e and all large n, there exists an interval of length c n log n that contains no such system. The argument works by placing the starting point of the interval at height roughly a n log n and applying a classical smooth-number counting obstruction of Erdős and Pomerance, now controlled by local saddle-point estimates of Hildebrand and Tenenbaum. The constant 1/e is optimal inside the three-parameter family used in the construction.","feed_headline":"Gaps of size (1/e)n log n lack multiples of 1..n","feed_subtitle":"A smooth-number count at height n log n forces the longer empty intervals","key_machinery":"The Erdős–Pomerance smooth-number obstruction (Lemma 2.1): if the number of y-smooth indices in (E/y, n] exceeds the number of y-smooth integers in the target window (m, E], then no complete system of distinct multiples can fit inside that window. Local Hildebrand–Tenenbaum ratio estimates turn the comparison into an explicit inequality among three free parameters that is optimized to produce the constant 1/e.","core_discovery":"The paper proves that liminf (F(n) - f(n,n))/(n log n) is at least 1/e. Consequently, for every fixed c < 1/e and all sufficiently large n, some interval of length c n log n contains no system of pairwise distinct multiples of the integers 1 through n. Because f(n,n) is o(n log n), this is equivalent to the lower bound F(n) ≥ (1/e - o(1)) n log n.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Some intervals of length c n log n lack distinct multiples of 1..n","Liminf (F(n)-f(n,n))/(n log n) is at least 1/e","Gaps near (1/e)n log n can miss distinct multiples of 1..n","Smooth-number count forces F(n)-f(n,n) ≥ (1/e)n log n in liminf","No distinct 1..n multiples in some intervals shorter than (1/e)n log n"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument needs the Hildebrand–Tenenbaum local ratio for the count of smooth numbers to have a relative error smaller than the main-term difference of size 1 over log log n, throughout the height range from n to n log n.","fun_headline_variants_meta":{"raw":{"variants":["Some intervals of length c n log n lack distinct multiples of 1..n","Liminf (F(n)-f(n,n))/(n log n) is at least 1/e","Gaps near (1/e)n log n can miss distinct multiples of 1..n","Smooth-number count forces F(n)-f(n,n) ≥ (1/e)n log n in liminf","No distinct 1..n multiples in some intervals shorter than (1/e)n log n"]},"model":"grok-4.5","effort":"low","cost_usd":0.00801,"raw_usage":{"total_tokens":1933,"prompt_tokens":849,"num_sources_used":0,"completion_tokens":135,"cost_in_usd_ticks":80100000,"prompt_tokens_details":{"text_tokens":849,"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":849,"tokens_out":135,"duration_ms":8439,"temperature":1.0,"reasoning_tokens":949,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T11:46:29.472197+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Either exhibit, for infinitely many n, a complete system of distinct multiples of 1..n inside every interval of length (1/e - ε) n log n that begins near height a n log n for admissible a, or produce a matching upper bound F(n) ≤ (1/e + o(1)) n log n that would force equality of the liminf.","supporting_citations":[],"review_version":1}