{"id":"a5089439-91e7-47b7-8f8e-37dac9f57020","arxiv_id":"2607.08129","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For fixed retention probability θ, the process log L_⌊nt⌋ obeys a functional LDP with geometric-mark entropy rate, an MDP with the CLT Gaussian RKHS rate, and a Strassen LIL with that unit ball as cluster set.","lead":"The paper proves functional large-deviation, moderate-deviation, and Strassen LIL theorems for the log of the LCM of a random Bernoulli subset of {1,...,n}. It completes the fixed-probability picture beyond the known LLN and CLT for this classical random-arithmetic object.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim (functional LDP/MDP/LIL for the rescaled log-LCM process via geometric-mark entropy contraction and the associated RKHS) rests on a transparent reduction: discard primes ≤√n and higher powers (Lemma 5.1), obtain independent geometric marks for the remaining primes, convert prime sums to integrals by the prime-number theorem (Lemma 6.1), then apply Gärtner–Ellis plus equicontinuity. The only quantitative error that must be controlled is the O(√n) remainder; classical Chebyshev bounds already make it negligible on every scale appearing in Theorems 2.1, 2.3 and 2.5. The reader correctly isolates this step as the weakest link, yet the link holds under the stated hypotheses. Numerical diagnostics (Figs. 1–4, Table 1) further corroborate the endpoint formulas. Consequently the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":30848,"tokens_out":558,"duration_ms":5518,"concrete_test":"Recompute the endpoint LDP rate I_end_θ(y) from the explicit Legendre transform (2.9)–(2.11) for θ=1/2 at y=0.75 and compare with the finite-n large-prime Legendre transform I_LP_θ,n(y) at n=10^5 (already plotted in Fig. 2); if the absolute difference exceeds 0.005 the arithmetic-reduction error is not negligible on the LDP scale.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption flag (Lemma 5.1 O(√n) remainder) is correctly identified but does not threaten the central claims. The bound is elementary Chebyshev (ϑ(x)≤C x), not a deep PNT error term: the discarded contribution is at most ϑ(√n)+∑_{r≥2}ϑ(n^{1/r})≤C√n uniformly in t. Relative to LDP speed n/log n the error is O((log n)/√n)→0; relative to MDP/LIL denominators b_n√(n log n) and √(n log n log log n) it is o(1) under the stated regimes. The subsequent independent geometric-mark representation, Gärtner–Ellis finite-dimensional limits, exponential equicontinuity (monotonicity for LDP; Appendix A maximal inequality for MDP/LIL), and RKHS identification are standard and internally consistent. No hidden circularity or scale pollution appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the process t → log L_⌊nt⌋ for a Bernoulli random subset A_n of {1,…,n} with fixed retention probability θ ∈ (0,1). After the von Mangoldt reduction to large primes and an independent geometric-mark representation, the authors prove three functional limit theorems: an LDP for the polygonal processes of n^{-1} log L_⌊nt⌋ in C[0,1] with speed n/log n and good rate I_θ given by the entropy contraction of geometric marks (Theorem 2.1); an MDP for the centered and moderately scaled processes with speed b_n^{2} and rate ½∥f∥_{H_θ}^{2} (Theorem 2.3); and a Strassen-type LIL whose almost-sure cluster set is the unit ball of the same RKHS (Theorem 2.5). Endpoint and continuous-linear-functional consequences are derived by contraction. The proofs follow a standard route: finite-dimensional Gärtner–Ellis limits, exponential equicontinuity (monotonicity for the LDP; a local maximal inequality for the MDP/LIL), and identification of the projective rates with the entropy and RKHS contractions.","tokens_in":31093,"tokens_out":887,"duration_ms":7921,"significance":"The work completes the fixed-θ asymptotic picture for random LCMs beyond the existing LLN and functional CLT of Alsmeyer–Kabluchko–Marynych. The rate functions are derived from first principles (relative entropy of geometric marks; the covariance kernel of the already-established Gaussian limit) rather than fitted, and the three theorems sit cleanly on the same arithmetic reduction. The explicit endpoint formulas, the local quadratic expansion of the LDP rate, the variational form of the MDP rate, and the numerical checks of finite-n Legendre transforms and importance-sampling tails are useful additions. The results are of genuine interest in probabilistic number theory and large-deviation theory for arithmetic functionals.","major_comments":[],"minor_comments":[{"comment":"The title of the manuscript and the running head contain a typographical space in “MUL TIPLES” / “COMMON MUL TIPLES”; this should be corrected before publication.","section":null},{"comment":"Section 3 (Numerical simulations) is helpful, but the figure captions and Table 1 would be clearer if the precise values of n, θ, bin half-width, and number of tilted replications were repeated in every caption rather than only in the surrounding text.","section":null},{"comment":"Remark 2.7 points to a companion working paper [24] on varying θ_n. A one-sentence pointer to the precise regimes (sparse / nearly complete) already treated there would help the reader place the fixed-θ results.","section":null},{"comment":"In Appendix B the dual Euler equation is presented as supplementary. A brief remark that it is not used in the main proofs (already stated) could be moved into the introduction of the appendix so that readers who skip the appendix do not wonder whether the main rate identification depends on it.","section":null},{"comment":"A few minor notational inconsistencies appear: m(x) is defined both as ⌊1/x⌋ and used as a running index; the same letter appears for the number of grid cells in the equicontinuity arguments. Distinct letters would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that fits a probability journal with interest in number-theoretic applications. The self-citation to the companion working paper [24] is appropriate and does not create a circularity problem for the fixed-θ results under review. I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes the fixed-θ story for random LCMs. Earlier work stopped at LLN and functional CLT; here they get the full functional LDP (speed n/log n, rate = entropy contraction of geometric marks), MDP (Gaussian quadratic / RKHS), and Strassen LIL (cluster set = unit ball of that RKHS). That is the real news.\n\nWhat they do well is the arithmetic reduction. After discarding primes ≤ √n and higher powers (O(√n) by elementary Chebyshev, uniformly in t), the large primes give independent geometric marks. Everything else is standard but carefully executed: Gärtner–Ellis for finite-dimensional limits, monotonicity for LDP equicontinuity, a local maximal inequality (Appendix A) for the centered MDP/LIL, then projective identification with the entropy contraction and the RKHS of the known CLT covariance. Rate-function properties (convexity, domain, unique zero, local quadratic expansion) are worked out cleanly, and the numerical checks on the endpoint Legendre transforms and tilted rare-event probabilities line up with the theory.\n\nThe soft spot the reader flagged—Lemma 5.1’s O(√n) remainder—is real but not load-bearing. Relative to n/log n it is o(1); under the MDP/LIL normalizations it is also negligible. No scale pollution, no circularity. The companion note on varying θ is correctly left aside. Self-citations are to their own prior CLT/LLN and are not used as crutches.\n\nThis is for people who work on probabilistic number theory or functional large deviations for arithmetic processes. It is not a new paradigm, but it is honest mid-subfield progress with reusable templates. I would send it to referees without hesitation; a pure-probability journal should take it seriously. Worth reading if you care about the model or about entropy/RKHS contractions for triangular arrays of threshold processes.","headline":"Solid completion of the fixed-θ random-LCM program: functional LDP/MDP/LIL with clean entropy and RKHS rates, proofs that hold up.","tokens_in":31720,"tokens_out":532,"would_cite":true,"duration_ms":5991,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60F17","11N37","60G50"],"pacs":[],"model":"grok-4.5","headline":"Random LCMs obey functional large deviations, moderate deviations, and a Strassen LIL whose rates come from geometric-mark entropy and a Gaussian RKHS.","keywords":["least common multiple","random set of integers","functional large deviations","moderate deviations","law of the iterated logarithm","geometric marks","reproducing kernel Hilbert space","prime number theorem"],"falsifier":"Compute the exact difference log Ln − Sn(1) for large n and check whether it stays O(√n); if the ratio of that difference to the LDP, MDP or LIL scale fails to tend to zero, the reduction that underlies all three theorems collapses.","tokens_in":31743,"feed_emoji":"√","tokens_out":735,"duration_ms":7054,"temperature":0.7,"pith_summary":"When each integer up to n is kept independently with fixed probability θ, the least common multiple Ln of the retained set grows like exp(cθ n). This paper upgrades that growth law to pathwise large-deviation, moderate-deviation, and iterated-logarithm statements for the whole process t ↦ log L⌊nt⌋. After discarding an O(√n) arithmetic remainder, the process reduces to a triangular array of independent geometric marks on large primes; the large-deviation rate is the contraction of relative entropy of those marks, while the moderate-deviation rate and the almost-sure LIL cluster set are the quadratic form and unit ball of the reproducing-kernel Hilbert space of the already-known Gaussian limit. The results therefore complete the fixed-θ picture by giving precise exponential and almost-sure fluctuation scales for a classical arithmetic functional of a random set.","feed_headline":"Random LCMs obey full pathwise large-deviation and LIL laws","feed_subtitle":"Entropy of geometric marks and a Gaussian RKHS fix the rates for the whole process log L⌊nt⌋","key_machinery":"The von Mangoldt identity reduces log LCM to a sum of prime indicators; discarding primes ≤ √n leaves independent geometric marks Gn,p on the large primes, which convert (via the prime-number theorem) into an entropy-contracted path measure for large deviations and into the covariance kernel Cθ for moderate deviations and the LIL.","core_discovery":"The polygonal interpolations of n⁻¹ log L⌊nt⌋ satisfy a large-deviation principle in C[0,1] with speed n/log n and good rate equal to the entropy contraction of geometric marks; after centering and moderate scaling they satisfy an MDP whose rate is half the squared RKHS norm of the Gaussian covariance of the CLT; and under the LIL normalization their almost-sure cluster set is exactly the unit ball of that RKHS.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Functional LDP and Strassen LIL for the random LCM process","Pathwise large deviations and LIL for log of random LCMs","Entropy contraction rates the full path of random LCM logs","Random LCM logs obey LDP, MDP and RKHS unit-ball LIL","Process log L⌊nt⌋ satisfies functional LDP MDP and LIL"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The claim that all primes up to the square root of n and all higher prime powers contribute only an O(square-root-n) error that is negligible on every fluctuation scale used later.","fun_headline_variants_meta":{"raw":{"variants":["Functional LDP and Strassen LIL for the random LCM process","Pathwise large deviations and LIL for log of random LCMs","Entropy contraction rates the full path of random LCM logs","Random LCM logs obey LDP, MDP and RKHS unit-ball LIL","Process log L⌊nt⌋ satisfies functional LDP MDP and LIL"]},"model":"grok-4.5","effort":"low","cost_usd":0.00492,"raw_usage":{"total_tokens":1312,"prompt_tokens":690,"num_sources_used":0,"completion_tokens":97,"cost_in_usd_ticks":49200000,"prompt_tokens_details":{"text_tokens":690,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":525,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":690,"tokens_out":97,"duration_ms":5059,"temperature":1.0,"reasoning_tokens":525,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T12:36:02.464625+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the exact difference log Ln − Sn(1) for large n and check whether it stays O(√n); if the ratio of that difference to the LDP, MDP or LIL scale fails to tend to zero, the reduction that underlies all three theorems collapses.","supporting_citations":[],"review_version":1}