{"id":"3ca7d2fd-39cc-4ba1-a3a0-7b3188192837","arxiv_id":"2606.30567","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every narrow ray class modulo q is represented by a product of three prime ideals of norm at most (Nq)^{103/64 + κ} for any κ>0, with a positive proportion also representable by two such primes.","lead":"The paper proves every class in the narrow ray class group modulo an ideal q of a number field is a product of three prime ideals each of norm at most roughly (Nq) to the power 1.61. A generalist might read it to see how analytic tools improve explicit bounds on prime ideal factorization in class groups.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Extension of Matomäki–Teräväinen dense-model/transference to narrow ray class groups is the least-secured step","rationale":"The reader's weakest assumption is exactly the load-bearing point; the remainder of the argument (subconvexity exponents, three-prime product) follows formally once the extension is accepted. No internal inconsistency or hidden circularity is visible from the given material.","tokens_in":1713,"tokens_out":344,"duration_ms":28458,"concrete_test":"In the section constructing the dense model for narrow ray-class characters, recompute the L^1 or L^2 distance between the indicator of a narrow class and its model function when the sign vector at infinity is fixed; if the distance exceeds the original Matomäki–Teräväinen tolerance by more than a factor depending on [K:Q], the transference step fails to deliver the claimed bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline claim requires that the multiplicative dense-model construction and transference principle carry over to narrow ray class groups Cl_q^+ of a fixed number field K without changes that degrade the exponent max(1,3α,4α_0). The abstract states the extension is performed, but the narrow condition (signs at the r_1 real places) introduces additional 2-torsion in the character group and a modified notion of positivity for ideals; any failure of the model to capture these signs uniformly in the conductor q would invalidate the three-prime representation at the stated scale. No other step (Wu subconvexity input, short-sum bounds for ray-class characters) is visibly weaker once the framework is granted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that every class in the narrow ray class group Cl_q^+ modulo an integral ideal q of a fixed number field K is represented by a product of three prime ideals each of norm at most (Nq)^{max(1,3α,4α0)+κ} for any κ>0, where α=α0=103/256 comes from Wu's subconvexity bound for Hecke L-functions; this yields the explicit exponent 103/64 + κ and improves the prior O_K((Nq)^3) result. It further shows that a positive proportion of ray classes admit a representation by a product of two such primes. The argument extends the multiplicative dense-model and transference framework of Matomäki–Teräväinen to the narrow ray class group setting.","tokens_in":1856,"tokens_out":681,"duration_ms":47104,"significance":"If the claimed extension of the dense-model/transference method holds without degrading the exponent, the result supplies a concrete improvement in the scale of prime-ideal generators for narrow ray classes, with an explicit subconvexity-derived bound. The positive-proportion statement for two-prime products and the direct importation of Wu's bound are additional strengths. The work ships an explicit, parameter-free exponent (modulo the external subconvexity input) rather than an existence result.","major_comments":[{"comment":"The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound.","section":"section containing the extension of the dense-model and transference framework to narrow ray class groups"},{"comment":"The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation.","section":"section on short-sum bounds for ray-class characters"}],"minor_comments":[{"comment":"The dependence of implied constants on the fixed field K should be stated explicitly in the introduction and in the statement of the main theorem.","section":"Introduction and Theorem 1.1"},{"comment":"The notation Cl_q^+ for the narrow ray class group and the precise definition of the narrow positivity condition should appear before the statement of the main results.","section":"preliminaries"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a technical extension of recent analytic work to class-group problems; the editor may wish to confirm that the novelty of the narrow-ray-class adaptation is presented with sufficient detail relative to the cited Matomäki–Teräväinen framework."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for explicit verification of the narrow-ray-class extension. We address both major comments below and will revise the manuscript accordingly to improve clarity without changing the main results or the exponent.","responses":[{"response":"The proof of the main theorem (Section 4) already carries out the extension by working directly with the group law and character group of Cl_q^+, incorporating the 2-torsion and the sign conditions at real places into the definition of the narrow ray-class characters. The short-sum estimates and transference step are identical to those in Matomäki–Teräväinen because they depend only on the conductor and the subconvexity input, not on the narrow versus ordinary distinction; the 2-torsion is absorbed into the existing character-sum bounds of bounded order. To address the request for self-contained verification, we will insert a dedicated paragraph (or short subsection) in Section 4 that explicitly checks the effect of the narrow positivity constraint and 2-torsion on each step of the dense-model construction and transference, confirming that no extra factor appears in the admissible exponent.","revision_made":"yes","referee_comment":"[section containing the extension of the dense-model and transference framework to narrow ray class groups] The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound."},{"response":"Lemma 3.2 states the short-sum bounds for all non-principal ray-class characters of conductor dividing q, with uniformity in q; the narrow positivity constraint is a condition only at the infinite places and does not alter the analytic properties or the length of the finite sums used in the dense-model construction. The same bounds therefore apply verbatim inside the three-prime representation argument. We will add an explicit sentence in the proof of the main theorem (Section 4) that references Lemma 3.2 and notes that the narrow condition is compatible with the scale at which the bounds are applied.","revision_made":"yes","referee_comment":"[section on short-sum bounds for ray-class characters] The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation."}],"tokens_in":1515,"tokens_out":620,"duration_ms":32536,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main advance is the explicit improvement on the scale of prime ideal products representing classes in the narrow ray class group. They take the existing transference framework, adapt it to the narrow setting with its sign conditions at real places, and feed in Wu's 103/256 subconvexity exponent to reach 103/64 + κ. The two-prime positive proportion statement is also new relative to the earlier Deshouillers–Gun–Ramaré–Sivaraman work.\n\nThe adaptation itself is the part that carries the weight. The abstract claims the multiplicative dense model and transference carry over without losing the exponent, but the extra 2-torsion from the narrow class group and the modified positivity notion could in principle require adjustments that affect the short-sum or transference constants. That step is the one that needs the closest look in the full argument.\n\nEverything else looks standard: the input bounds are cited externally, there is no circular fitting, and the claimed results follow once the framework extension is granted. The paper is aimed at people who care about explicit versions of prime ideal theorems or class-group Goldbach problems in number fields. It is a solid incremental piece that deserves referee time rather than a desk rejection.","headline":"This extends the Matomäki–Teräväinen dense-model method to narrow ray class groups and plugs in Wu's subconvexity to improve the three-prime bound from cubic to (Nq)^{103/64+κ}, plus a positive-proportion two-prime result.","tokens_in":2321,"tokens_out":346,"would_cite":false,"duration_ms":25459,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Every class in the narrow ray class group modulo q is a product of three prime ideals of norm at most (Nq)^{103/64 + κ} for any κ > 0.","keywords":["narrow ray class groups","prime ideal products","subconvexity","character sums","Hecke L-functions","number fields","dense model transference"],"falsifier":"Exhibit a sequence of ideals q in a fixed number field together with a ray class modulo q that cannot be written as a product of three prime ideals each of norm at most (Nq)^{103/64 + 0.01}.","tokens_in":2609,"feed_emoji":"","tokens_out":772,"duration_ms":37843,"temperature":0.7,"pith_summary":"The paper proves that for any fixed number field, every element of the narrow ray class group modulo an integral ideal q arises as a product of three prime ideals whose norms are bounded by a power of Nq. The exponent obtained is 103/64 plus an arbitrarily small positive amount, coming from Wu's subconvexity bound applied inside an extended version of the Matomäki–Teräväinen multiplicative dense-model and transference method. This replaces an earlier uniform bound of size roughly (Nq)^3 that held for the same groups. The same argument also yields that a positive proportion of the classes can be written as products of only two such primes.","feed_headline":"Ray class groups written as products of three small-norm primes","feed_subtitle":"Bound (Nq)^{103/64+κ} replaces earlier cubic estimate for narrow ray classes in any fixed number field","key_machinery":"The multiplicative dense-model and transference framework extended to narrow ray class groups of a fixed number field.","core_discovery":"Every class in the narrow ray class group modulo an integral ideal q of a fixed number field is represented by a product of three prime ideals of norm at most (N q)^{max(1,3α,4α_0)+κ} for any κ>0, where α is the exponent in short character sum bounds for general non-principal ray class characters and α_0 comes from a bounded-order subconvexity input for Hecke L-functions. Wu's subconvexity bound gives the admissible choice α=α_0=103/256, hence the explicit bound (N q)^{103/64+κ}. A positive proportion of ray classes are represented by products of two prime ideals.","pith_inferences":["If a stronger subconvexity bound for the relevant Hecke L-functions becomes available, the exponent on Nq would improve immediately by substituting the new α and α_0.","The same transference method might apply to other arithmetic progressions inside class groups once the dense-model step is verified for those settings.","The two-prime representation result suggests that the set of products of two small-norm primes is already dense in a positive-density subset of the ray class group."],"forward_implications":["The earlier O_K((N q)^3) bound of Deshouillers, Gun, Ramaré and Sivaraman is replaced by the sharper exponent 103/64 + κ.","A positive proportion of the classes in the narrow ray class group are represented by products of two prime ideals of the same norm bound.","The result holds uniformly for the narrow ray class groups of any fixed number field."],"fun_headline_variants":["Narrow ray classes are three-prime products","Three primes represent all narrow ray classes","Positive share of ray classes by two-prime products","Prime ideal products cover ray class groups"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The multiplicative dense-model and transference framework of Matomäki–Teräväinen extends without essential change to narrow ray class groups of a fixed number field.","fun_headline_variants_meta":{"raw":{"variants":["Narrow ray classes are three-prime products","Three primes represent all narrow ray classes","Positive share of ray classes by two-prime products","Prime ideal products cover ray class groups"]},"model":"grok-4.3","cost_usd":0.010795,"raw_usage":{"total_tokens":4782,"prompt_tokens":713,"num_sources_used":0,"completion_tokens":53,"cost_in_usd_ticks":107949500,"prompt_tokens_details":{"text_tokens":713,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4016,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":713,"tokens_out":53,"duration_ms":58019,"temperature":1.0,"reasoning_tokens":4016,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T04:36:08.954095+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibit a sequence of ideals q in a fixed number field together with a ray class modulo q that cannot be written as a product of three prime ideals each of norm at most (Nq)^{103/64 + 0.01}.","supporting_citations":[],"review_version":1}