{"id":"1eed97aa-8109-44ca-a6bb-4e3140c6ad28","arxiv_id":"2607.01703","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Formulates Euclidean systems for ray classes, proves they generate the ray class group, and shows under GRH that generating sets of Cl_K^{(p)^N} are Euclidean systems for specified totally real Galois fields.","lead":"The paper extends Euclidean systems from ideal classes to ray classes in algebraic number theory and proves they generate the ray class group. It further shows under the generalized Riemann hypothesis that generating sets of certain ray class groups in totally real Galois fields are Euclidean systems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of GRH as the load-bearing assumption for the main conditional claim and the definition for the unconditional claim matches the abstract exactly. With the claims delimited this way, no further load-bearing gap is visible.","tokens_in":1638,"tokens_out":236,"duration_ms":14575,"concrete_test":"Extract the precise definition of a Euclidean system of ray classes from the paper and verify that the generation theorem follows from it by direct argument (no additional number-theoretic input required).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stated results consist of (1) an unconditional proof that Euclidean systems of ray classes generate the ray class group (a basic property following from the new definition) and (2) a conditional statement under GRH that, for the specified totally real Galois fields and primes p, every generating set of Cl_K^{(p)^N} is itself a Euclidean system. Both parts are explicitly delimited in the abstract; the GRH hypothesis is stated outright for the strong claim, and no hidden assumption or internal inconsistency appears in the formulation.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces Euclidean systems of ray classes, extending Lenstra's Euclidean ideal classes and Treatman's Euclidean systems. It proves unconditionally that every Euclidean system generates the corresponding ray class group. Assuming GRH, for a totally real Galois number field K of degree n≥3 and odd prime p not splitting completely in K, every generating set of the ray class group Cl_K^{(p)^N} (modulus (p)^N) is a Euclidean system, for any N>0.","tokens_in":1746,"tokens_out":276,"duration_ms":12123,"significance":"If the results hold, the work provides a conditional characterization of generating sets as Euclidean systems under GRH for specified fields and primes, extending prior notions to ray classes. The unconditional generation result follows directly from the definition and is a basic consistency check. The GRH-conditional statement offers a strong, falsifiable claim in the context of class field theory and Euclidean algorithms.","major_comments":[],"minor_comments":[{"comment":"Abstract contains typographical spacing errors: 'corre sponding' and 'ratio nal'.","section":null},{"comment":"The manuscript should include a brief comparison of the new ray-class definition with Treatman's original Euclidean systems to clarify the extension.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary and recommendation of minor revision. The report correctly identifies the main results: the unconditional generation property and the GRH-conditional characterization for the specified fields. No specific major comments were provided in the report, so we have no individual points requiring response or revision at this stage.","responses":[],"tokens_in":1131,"tokens_out":81,"duration_ms":12315,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper defines Euclidean systems of ray classes, proves they generate the ray class group from the definition alone, and then shows under GRH that for totally real Galois K of degree n at least 3 and odd prime p not splitting completely, every generating set of Cl_K with modulus (p)^N is itself a Euclidean system.\n\nThis is a direct extension of Lenstra on Euclidean ideal classes and Treatman on systems. The new definition and the generation property are straightforward once stated, and the GRH application gives a concrete criterion for when generators exist in these ray class groups. That part is new relative to the cited work.\n\nThe unconditional generation claim is basically definitional, so it holds up without extra assumptions. The GRH result is stated clearly as conditional, which keeps the logic clean. No hidden parameters or circular steps appear in the abstract or stress-test notes.\n\nThe main limitation is the GRH dependence for the interesting direction; without it the paper mainly supplies a framework rather than a broad unconditional theorem. The scope is narrow—specific fields and moduli—so it won't shift the field at large, but it organizes a refinement that people working on Euclidean algorithms or class group generators might use.\n\nThis is for number theorists already familiar with Lenstra-Treatman style results on Euclidean properties. The thinking is coherent and engages the literature directly. It deserves a serious referee to check the proofs of the new definition and the GRH argument, even if revisions are likely on the conditional part.","headline":"Extends Euclidean systems to ray classes with an unconditional generation result and a GRH-conditional claim that generating sets are Euclidean in certain fields.","tokens_in":2228,"tokens_out":378,"would_cite":false,"duration_ms":11806,"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":"Assuming GRH, every generating set of the ray class group with modulus a power of an odd prime is a Euclidean system of ray classes, for totally real Galois fields of degree at least 3 where the prime does not split completely.","keywords":["Euclidean systems","ray class groups","totally real fields","Galois extensions","generalized Riemann hypothesis","class field theory"],"falsifier":"An explicit totally real Galois field K of degree at least 3, an odd prime p that does not split completely, some N, and a generating set S of Cl_K^{(p)^N} such that S fails the Euclidean division property for at least one pair of elements.","tokens_in":2539,"feed_emoji":"","tokens_out":627,"duration_ms":18986,"temperature":0.7,"pith_summary":"The paper defines Euclidean systems of ray classes as an extension of earlier notions for ideal classes. It proves unconditionally that any Euclidean system generates the associated ray class group. Under the generalized Riemann hypothesis it further shows that, in the stated families of fields and moduli, every set of generators for the ray class group automatically satisfies the Euclidean property. A reader would care because the result turns an algebraic generating condition into a statement about the existence of a Euclidean algorithm relative to the ray modulus.","feed_headline":"GRH implies every generator of these ray class groups is Euclidean","feed_subtitle":"Holds for totally real Galois fields of degree 3 or higher and odd primes that do not split completely.","key_machinery":"Euclidean system of ray classes: a finite set of representatives in the ring of integers that permits a Euclidean division algorithm with respect to the action of the ray class group modulo the given conductor.","core_discovery":"We formulate the notion of a Euclidean system of ray classes and prove that every such system generates the corresponding ray class group. Assuming GRH, if K is a totally real Galois number field of degree n≥3 and p is an odd rational prime that does not split completely in K, then for every N>0 every generating set of the ray class group Cl_K^{(p)^N} with modulus (p)^N is a Euclidean system.","pith_inferences":["The conditional result may supply explicit generators that can be used to compute ray class groups via a Euclidean algorithm.","It raises the question whether similar unconditional statements hold when the degree is 2 or when the prime splits completely."],"forward_implications":["Every Euclidean system of ray classes generates the ray class group.","The Euclidean property holds for all generators once GRH is assumed in the stated setting.","The result applies uniformly for every exponent N on the prime power modulus."],"fun_headline_variants":["Euclidean systems of ray classes generate class groups","GRH: ray class group generators are Euclidean","Generating sets are Euclidean for ray class groups under GRH","Every Euclidean system generates the ray class group"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The generalized Riemann hypothesis must hold for the claim that every generating set is Euclidean; the generation property itself relies only on the new definition.","fun_headline_variants_meta":{"raw":{"variants":["Euclidean systems of ray classes generate class groups","GRH: ray class group generators are Euclidean","Generating sets are Euclidean for ray class groups under GRH","Every Euclidean system generates the ray class group"]},"model":"grok-4.3","cost_usd":0.008795,"raw_usage":{"total_tokens":3913,"prompt_tokens":575,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":87949500,"prompt_tokens_details":{"text_tokens":575,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3280,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":575,"tokens_out":58,"duration_ms":21966,"temperature":1.0,"reasoning_tokens":3280,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T07:17:12.299087+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit totally real Galois field K of degree at least 3, an odd prime p that does not split completely, some N, and a generating set S of Cl_K^{(p)^N} such that S fails the Euclidean division property for at least one pair of elements.","supporting_citations":[],"review_version":1}