{"id":"6f00405f-92fa-4ba4-95ff-dbe82de80947","arxiv_id":"2511.22905","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sufficiently large polynomial set with asymptotically trivial multiplicative energy yields a complex Gaussian limit for Steinhaus random multiplicative sums; short intervals, few prime factors, shifted primes, and rough polynomials satisfy the criterion.","lead":"This paper gives a general condition on sets of degree-N polynomials over F_q[t] under which sums of Steinhaus random multiplicative functions converge to a complex normal distribution, and verifies it for four natural families. It is the function-field analogue of a 2023 criterion of Soundararajan and Xu, with new estimates for smooth and rough polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Shifted-prime CLT (§9) rests on a false Selberg-sieve local density for primes dividing the shift Z; Theorem 2.3 is not established as written.","rationale":"The core criterion, Theorem 1.1, appears well supported: the martingale decomposition and Lemma 5.2 give the required control on the blocks, and the energy condition (1.4) is exactly what the fourth-moment calculation needs. The applications are where the proof is most exposed. The shifted-prime proof is the least secure: the Selberg sieve is used to bound the G,H sums in (9.1), but its local-density hypothesis fails for primes dividing the shift Z. This is not a stylistic concern—it is a false equality used to set the sieve remainders R_D to zero. Consequently, the bound (9.3), and hence Theorem 2.3, is not established as written. The reader identified precisely this issue, and I concur. The paper should remain conditionally accepted pending a repaired shifted-prime argument; there is no evidence in this pass that the main theorem or the other applications collapse. A concrete counterexample to the claimed local density is given in the test above, and a successful repair would need to rework the sieve with the corrected densities and verify the final o(q^{2N}/N^2) bound.","tokens_in":30602,"tokens_out":30834,"duration_ms":258757,"concrete_test":"Compute X_D explicitly for q=3, Z=t, A=t^r+1, B=t^r+2, D=t, r=N−d≥1. Since Z≡0 mod t and A(0),B(0)≠0, the condition t | (GA−Z)(GB−Z) is equivalent to G(0)=0, so |X_D|=q^{d-1}; the paper's density formula predicts 2q^{d-1}. This contradiction confirms the sieve remainder R_D is nonzero. Then verify whether a corrected local density—setting g(P)=|P| for P|Z with P∤AB, and handling P|Z∩AB separately—still yields the required o(q^{2N}/N^2) in the Selberg upper bound after summing over A,B. If yes, Theorem 2.3 can be repaired; if not, the shifted-prime result fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §9, after defining X_D = {G∈M_d : D | (GA−Z)(GB−Z)}, the paper claims for each irreducible P|D that the congruence GA≡Z or GB≡Z mod P leaves exactly 2 residue classes when P∤AB(A−B), and exactly 1 class when P|AB(A−B). This is false when P|Z. If P|Z and P∤AB, both congruences reduce to G≡0, so the union is a single class; if P|Z and P|A, the first congruence holds for all G. In either case |X_D| ≠ q^d/g(D) with the stated g, so the remainder R_D in Lemma 3.2 is not zero and the Selberg step leading to (9.3) is unjustified. Since (9.3) is the key estimate for the off-diagonal energy, Theorem 2.3 lacks support as written. A concrete failure: take q=3, Z=t, A=t^r+1, B=t^r+2 with r=N−d≥1 and D=t. Then |X_D|=q^{d-1}, while the paper's formula gives q^d/g(t)=2q^{d-1}.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a function-field analogue of the Soundararajan–Xu criterion: if A⊂M_N has size |A| ≫ q^N exp(-(1/3)√(N log q)) and contains a subset S with |S|=(1+o(1))|A| and asymptotically trivial multiplicative energy, then normalized sums of Steinhaus random multiplicative functions over A converge in distribution to CN(0,1). The proof uses a refined prime-based martingale filtration and a new estimate for smooth polynomials in short intervals. The criterion is applied to four families: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. Auxiliary results include an explicit Hildebrand-type inequality, a function-field Shiu theorem, and a short-interval Chebyshev bound.","tokens_in":30911,"tokens_out":21673,"duration_ms":170325,"significance":"If correct, Theorem 1.1 is a valuable and flexible criterion, extending a recent number-field theorem to F_q[t] and yielding four new CLTs. The paper is largely self-contained and provides explicit parameter-free estimates (Proposition 2.5, Theorem 2.6, Lemma 2.7) that should be of independent interest. The martingale argument for the main criterion is coherent. However, the shifted-prime application currently rests on a false Selberg-sieve local-density assertion, so Theorem 2.3 is not established as written. The remaining applications appear coherent and carefully argued.","major_comments":[{"comment":"The moment computations are written with the wrong orthogonality relation. The displayed fourth moment is Σ E[f(F1)f(F2)f(G1)f(G2)] without conjugates, and the text says the expectation vanishes unless F1G1=F2G2. For |Z_P|^4 the correct expression contains \\(\\overline{f(G_1)}\\,\\overline{f(G_2)}\\), and the non-vanishing condition is F1F2=G1G2, matching (1.4) and condition (ii) of Theorem 5.3. The same inconsistency appears in the second-moment-squared argument. As written, the claimed bound on Σ E|Z_P|^4 and hence the proof of Theorem 1.1 does not follow. This is straightforward to repair, but it must be corrected systematically throughout the proof.","section":"§5.3, proof of Theorem 5.3"}],"minor_comments":[{"comment":"After the local-density correction, the condition 'P∤AB(A−B)' should be replaced by a condition involving Z as well, and the role of Z in the Pollack divisor M=AB(B−A)Z should be stated before the local-density claim, not only in the denominator computation.","section":"§9"},{"comment":"The two occurrences of 'F1G1=F2G2' should read 'F1F2=G1G2'.","section":"§5.3"},{"comment":"In the final line of case (a), 'This verifies (iii)' appears to mean condition (ii) of Theorem 5.3; condition (iii) was already verified above.","section":"§7, proof of Theorem 2.1(a)"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth publishing after the §9 shifted-prime sieve gap is repaired and the §5.3 moment expressions are corrected. The main criterion and the other three applications appear sound and interesting. The use of [2] is as a lemma, not as the target theorem, so I see no circularity issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a meaningful extension of the Soundararajan–Xu CLT criterion to F_q[t], and most of it holds up. But the shifted-primes application (Theorem 2.3) has a genuine gap in its Selberg sieve step, so that theorem is not established as written.\n\nWhat's actually new: the refined prime-wise martingale filtration (Lemma 5.2) lets them weaken the density condition from q^N/N to q^N exp(-(1/3)√(N log q)), and the auxiliary estimates — Proposition 2.5, Theorem 2.6, Lemma 2.7 — look genuinely useful. The main theorem and the short-interval, few-prime-factor, and rough-polynomial applications are coherent and carefully argued. This is not a routine translation; the filtration is a real technical improvement.\n\nThe soft spot is in §9. The local density claim for the shifted-prime sieve is false for primes dividing the shift Z. The paper says that for each P|D, the congruence (GA−Z)(GB−Z) ≡ 0 mod P leaves 2 residue classes when P ∤ AB(A−B) and 1 when P | AB(A−B). That ignores P | Z. If P | Z and P ∤ AB, both congruences force G ≡ 0 mod P — one class, not two. If P | Z and P | A, the first congruence holds for every G — all classes, not one. Concrete failure: q=3, Z=t, A=t^r+1, B=t^r+2, D=t. Then |X_D| = q^{d−1}, whereas the paper's formula with g(t)=|t|/2 gives 2q^{d−1}. So the remainder R_D in Lemma 3.2 is not zero, and (9.3) is unjustified. Since (9.3) drives the bound on Ξ, Theorem 2.3 lacks support. This looks fixable — split off primes dividing Z or use a different sifting set — but as written it is a real gap. Minor: in §7 the line 'using our choice m=(1+ε)N' should read m=(1+ε) log N; harmless typo.\n\nThe main theorem itself holds up on inspection. The citation pattern is fine; [2] has overlapping authors but is used as a lemma, not as the target result.\n\nBottom line: worth a serious referee. The paper deserves revision, not rejection. Send it to peer review with a request to repair the shifted-prime sieve argument; the rest of the package can be evaluated in the same round.","headline":"Solid function-field extension of Soundararajan–Xu with a real but likely repairable gap in the shifted-primes section.","tokens_in":31358,"tokens_out":4498,"would_cite":true,"duration_ms":36774,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11T55","60F05","11K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a subset of polynomials over F_q[t] whose multiplicative energy is asymptotically trivial has Gaussian-distributed Steinhaus random multiplicative sums, and derives four central limit theorems from it.","keywords":["random multiplicative functions","function fields","central limit theorem","multiplicative energy","short intervals","shifted primes","rough polynomials","sieve methods"],"falsifier":"Take q=3, N=2, Z=t, A=t, B=1, and the degree-one prime P=t. Write X={ (GA−Z)(GB−Z) : G monic linear }. For every G∈M_1, P divides GA−Z = t(G−1), so all 3 elements of X are divisible by P. The sieve local density used in the proof of Theorem 2.3 would predict only |X|/|P| = 1 such G. This direct count shows the shifted-prime bound (9.3) is not supported as written.","tokens_in":30523,"feed_emoji":"🎲","tokens_out":11577,"duration_ms":94662,"temperature":0.7,"pith_summary":"The paper's goal is to say when a randomly weighted sum over a set of monic polynomials of fixed degree over F_q[t] has a Gaussian limit. The main theorem gives a sufficient condition: if a set A is not too small and contains a nearly full subset S whose multiplicative energy is asymptotically trivial — meaning almost all solutions of F1F2=G1G2 come from swapping the two factors — then the normalized Steinhaus random multiplicative sum over A converges in distribution to the standard complex normal distribution CN(0,1). A sympathetic reader should care because this transfers a known integer-number-theory criterion to the function-field setting, where far fewer examples were known. The paper then checks the condition in four settings: short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. Along the way it establishes uniform estimates for smooth and rough polynomials in short intervals and a short-interval bound for multiplicative functions that are likely useful independently.","feed_headline":"Gaussian limits proven for random multiplicative sums over polynomial rings","feed_subtitle":"A single energy condition yields CLTs for short intervals, few prime factors, shifted primes, and rough polynomials.","key_machinery":"The multiplicative energy E×(S) and a refined martingale filtration. The paper orders monic irreducible polynomials by degree and defines, for each prime P, the fiber S_P of polynomials in S whose ≺-maximal prime factor is P. Because the random value f(P) is independent of the sigma-algebra generated by earlier primes, the normalized sums over these fibers form a martingale difference sequence. A new bound on smooth polynomials in short intervals shows each fiber is sparse, which permits the size assumption |A| ≫ q^N exp(−(1/3)√(N log q)). The energy condition then forces the error terms in the martingale CLT to vanish.","core_discovery":"The central claim is Theorem 1.1: for any family of subsets A of monic degree-N polynomials over F_q[t] with |A| ⩾ q^N exp(−(1/3)√(N log q)), if there is S⊆A with |S|=(1+o(1))|A| and multiplicative energy E×(S)=|{(F1,F2,G1,G2)∈S^4 : F1F2=G1G2}|=(2+o(1))|S|^2 as q^N→∞, then (1/√|A|)∑_{F∈A} f(F) converges to CN(0,1) for a Steinhaus random multiplicative function f. The proof realizes the partial sum as a martingale difference sequence indexed by primes, using a refined filtration by the maximal prime factor under a degree-respecting ordering, and then applies a martingale central limit theorem; the energy condition controls the fourth-moment and variance-fluctuation terms. The paper derives fo","pith_inferences":["The energy criterion is transferable: any future family of polynomials with provably trivial multiplicative energy inherits the Gaussian conclusion without reworking the martingale proof.","The new short-interval estimates for smooth and rough polynomials may sharpen other function-field distribution questions, such as prime polynomial counts or divisor sums in intervals.","One could test the boundary of the short-interval CLT by computing the multiplicative energy of intervals with h between N/2 and N; the paper's bound is near-optimal in the fixed-degree, large-q limit."],"forward_implications":["If Theorem 1.1 is correct, any subset of monic degree-N polynomials satisfying the size and trivial-energy conditions automatically has a Gaussian limit for Steinhaus random multiplicative sums.","The short-interval application gives a CLT when q^h→∞ with q^{h+1}=o(q^N/N), and when h→∞ with q^{h+1}=o(q^N/N^c) for c>2 log 2−1.","The few-prime-factors application covers polynomials with k=o(log N) irreducible factors, matching the known integer range.","The shifted-prime and rough-polynomial applications give CLTs for sets of primes shifted by a fixed polynomial and for polynomials with all prime factors of degree exceeding z, with z≫√N.","The auxiliary estimates — smooth polynomials in short intervals, a short-interval bound for multiplicative functions, and a Chebyshev-type bound for rough polynomials — are uniform in q and N and may be used independently."],"fun_headline_variants":["Energy criterion proves Gaussian limits for polynomial sums","CLT for polynomial sums: one energy condition, four cases","Gaussian limits for random multiplicative sums over F_q[t]","Random multiplicative sums over F_q[t] approach normal law","One energy bound, four CLTs for polynomial random sums"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The shifted-prime application assumes that in the sieve, the number of G for which D divides (GA−Z)(GB−Z) is exactly q^d/g(D) with g(P)=|P|/2 or |P| according as P divides AB(A−B), but this local density is not correct when P divides the shift Z and exactly one of A,B, so the resulting bound (9.3) is not established as written.","fun_headline_variants_meta":{"raw":{"variants":["Energy criterion proves Gaussian limits for polynomial sums","CLT for polynomial sums: one energy condition, four cases","Gaussian limits for random multiplicative sums over F_q[t]","Random multiplicative sums over F_q[t] approach normal law","One energy bound, four CLTs for polynomial random sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2641,"prompt_tokens":708,"completion_tokens":1933,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":1853}},"tokens_in":452,"tokens_out":1933,"duration_ms":14286,"temperature":1.0,"reasoning_tokens":1853,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:37:54.475415+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take q=3, N=2, Z=t, A=t, B=1, and the degree-one prime P=t. Write X={ (GA−Z)(GB−Z) : G monic linear }. For every G∈M_1, P divides GA−Z = t(G−1), so all 3 elements of X are divisible by P. The sieve local density used in the proof of Theorem 2.3 would predict only |X|/|P| = 1 such G. This direct count shows the shifted-prime bound (9.3) is not supported as written.","supporting_citations":[],"review_version":1}