{"id":"4ebbac2f-e394-442f-b4ea-9e78c3f69c2b","arxiv_id":"2607.00675","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Pólya groups of Lecacheux quintic fields have 5-rank arbitrarily large with positive density in the parameter s, implying infinite 5-class field towers for a positive proportion of the fields via Golod-Shafarevich.","lead":"The paper studies Pólya groups in the Lecacheux family of quintic number fields and proves that their 5-ranks can be arbitrarily large. This leads to the conclusion that a positive proportion of these fields have infinite 5-class field towers.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Density arguments establishing positive density of odd s with 5-rank of Pólya group ≥k (abstract and § on density) are the load-bearing step for the infinite-tower conclusion.","rationale":"The reader's weakest_assumption correctly isolates the density step as the point whose failure would invalidate both the arbitrary-large-rank statement and the positive-proportion infinite-tower corollary. No other internal inconsistency is visible from the abstract or the stated claims; the Golod–Shafarevich application is standard once a lower bound on class-group 5-rank is available.","tokens_in":1748,"tokens_out":381,"duration_ms":22937,"concrete_test":"From the density section, extract the precise arithmetic condition C_k(s) on odd s that is asserted to imply 5-rank ≥k; compute or bound the Dirichlet density of the set of odd integers satisfying C_k(s) (e.g., via inclusion in a union of arithmetic progressions modulo the conductor of the relevant characters or via Chebotarev in an auxiliary extension). If this density is zero or cannot be shown positive independently of the quintic family, the headline claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that, for each fixed k, the subset of odd integers s for which the 5-rank of the Pólya group is at least k has positive (natural or Dirichlet) density. This density is then combined with Golod–Shafarevich to obtain a positive proportion of fields with infinite 5-class-field tower. The argument is therefore sensitive to whether the explicit arithmetic conditions on s that force high 5-rank (presumably congruences or splitting conditions derived from the discriminant of the Lecacheux polynomial) remain independent of the parametric family and do not collapse to density zero when restricted to odd s.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper examines the Pólya groups of the Lecacheux parametric family of quintic fields K_s. It proves that these groups can realize arbitrarily large elementary abelian 5-groups, that for every positive integer k the set of odd integers s for which the 5-rank of the Pólya group is at least k has positive (natural) density, and therefore that a positive proportion of these fields have infinite 5-class-field towers by the Golod–Shafarevich theorem. Additional results include an upper bound on the Pólya number in terms of the order of the Pólya group and examples of non-monogenic fields with index 1.","tokens_in":1896,"tokens_out":621,"duration_ms":18512,"significance":"If the density statements hold, the work supplies an explicit infinite parametric family of quintic fields in which the 5-rank of a distinguished subgroup of the class group is unbounded and in which infinite 5-towers occur with positive density. This gives concrete, computable examples supporting the Golod–Shafarevich mechanism in degree 5 and adds to the limited stock of families with proven unbounded Pólya-group rank. The bound on Pólya numbers and the non-monogenic examples are also of independent interest for the arithmetic of the family.","major_comments":[{"comment":"§5 (density arguments): The claim that the set of odd s with 5-rank of the Pólya group ≥ k has positive density for every k is load-bearing for the infinite-tower conclusion. The argument must explicitly verify that the splitting or congruence conditions derived from the discriminant of the Lecacheux polynomial remain independent when restricted to odd s and that their common density is strictly positive; without a quantitative lower bound or an application of the Chinese Remainder Theorem with an explicit modulus, the density could collapse to zero.","section":"§5"},{"comment":"§3, relation between Pólya group and class group: The paper invokes Golod–Shafarevich on the 5-rank of the Pólya group. It must be stated precisely whether the Pólya group is a quotient or a subgroup of the class group and whether the 5-rank lower bound transfers directly to the class-group 5-rank used in the Golod–Shafarevich criterion; any kernel or cokernel of order divisible by 5 would affect the applicability.","section":"§3"}],"minor_comments":[{"comment":"Abstract, last sentence: “infinte” should be “infinite”.","section":"Abstract"},{"comment":"Notation for the parameter s and the field K_s should be introduced once and used consistently; the current text occasionally switches between “Lecacheux quintic field” and “corresponding field” without cross-reference.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and valuable comments on our manuscript. We respond point by point to the major comments and indicate the changes we will make in revision.","responses":[{"response":"We agree that the density claim requires an explicit verification to be fully rigorous. The relevant conditions on the parameter s are congruence conditions modulo powers of 5 (arising from the splitting in the discriminant of the Lecacheux polynomial) together with conditions at a finite set of auxiliary primes; all such moduli are odd. The set of odd integers therefore intersects the arithmetic progressions in a set of positive density, and the Chinese Remainder Theorem yields that the joint density equals the product of the local densities, each of which is strictly positive. In the revised manuscript we will add an explicit statement of the modulus, the independence via CRT, and a quantitative lower bound for the density.","revision_made":"yes","referee_comment":"[§5] §5 (density arguments): The claim that the set of odd s with 5-rank of the Pólya group ≥ k has positive density for every k is load-bearing for the infinite-tower conclusion. The argument must explicitly verify that the splitting or congruence conditions derived from the discriminant of the Lecacheux polynomial remain independent when restricted to odd s and that their common density is strictly positive; without a quantitative lower bound or an application of the Chinese Remainder Theorem with an explicit modulus, the density could collapse to zero."},{"response":"By definition the Pólya group Po(K) is the subgroup of the class group Cl_K generated by the classes of the prime ideals lying above the primes that divide the discriminant of the ring of integers of K. It is therefore a subgroup, not a quotient. A lower bound on the 5-rank of Po(K) immediately supplies the same lower bound on the 5-rank of Cl_K, which is the quantity required by the Golod–Shafarevich criterion. There is consequently no kernel or cokernel of 5-power order that could interfere. We will insert a precise paragraph in the revised §3 making this subgroup relation and the direct transfer explicit.","revision_made":"yes","referee_comment":"[§3] §3, relation between Pólya group and class group: The paper invokes Golod–Shafarevich on the 5-rank of the Pólya group. It must be stated precisely whether the Pólya group is a quotient or a subgroup of the class group and whether the 5-rank lower bound transfers directly to the class-group 5-rank used in the Golod–Shafarevich criterion; any kernel or cokernel of order divisible by 5 would affect the applicability."}],"tokens_in":1495,"tokens_out":588,"duration_ms":23252,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is that the Lecacheux parametric quintics have Pólya groups that are elementary abelian 5-groups of any desired rank, and that for each k the odd parameters s producing rank at least k form a positive-density set. Combining this with Golod-Shafarevich immediately gives a positive proportion of the fields with infinite 5-class field towers.\n\nThe paper applies standard density and splitting arguments to this specific family and derives the necessary conditions on s from the discriminant and ramification in the Lecacheux polynomial. It also supplies an upper bound on the Pólya number in terms of the group order and produces examples that are non-monogenic despite index one. These are concrete additions to the existing literature on Pólya groups.\n\nThe density step is the load-bearing piece, exactly as the stress-test note flags. The argument stands or falls on whether the arithmetic conditions forcing high 5-rank remain positive density when restricted to odd s. The abstract states that the calculation works for this family, so the concern does not automatically invalidate the claim; one would simply verify the independence of the congruences in the full proof. Nothing indicates circularity or self-referential definitions.\n\nThe citation pattern is appropriate and the methods are the usual ones for this corner of algebraic number theory. No internal contradictions appear from the stated results.\n\nThis is for specialists working on class groups, Pólya invariants, and p-class field towers in degree-five fields. A reader looking for new parametric examples with controlled ranks will find usable content.\n\nSend it to peer review. The claims are specific enough to be checked and the family is new.","headline":"The Lecacheux family yields new explicit quintics where 5-rank of the Pólya group is arbitrarily large with positive density, giving positive proportion with infinite 5-towers via Golod-Shafarevich.","tokens_in":2367,"tokens_out":426,"would_cite":false,"duration_ms":24927,"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":"In the Lecacheux family of quintic fields the associated Pólya groups are elementary abelian 5-groups that can have arbitrarily large rank.","keywords":["Pólya groups","quintic fields","Lecacheux family","5-rank","class field towers","elementary abelian groups","density arguments","non-monogenic fields"],"falsifier":"An explicit computation, for a large finite collection of odd integers s, showing that the proportion of fields whose Pólya group has 5-rank at least k tends to zero as k grows, or a single concrete s for which the 5-rank is strictly smaller than the density claim predicts.","tokens_in":2648,"feed_emoji":"","tokens_out":772,"duration_ms":18357,"temperature":0.7,"pith_summary":"The paper examines the Pólya groups of a one-parameter family of quintic number fields defined by Lecacheux polynomials. It proves that these groups are elementary abelian 5-groups whose 5-rank can be made arbitrarily large by choosing the odd integer parameter appropriately. Density arguments on the parameter show that, for any fixed positive integer k, the set of parameters yielding 5-rank at least k has positive density inside the family. The same density statement, combined with the Golod-Shafarevich theorem, implies that a positive proportion of the fields possess an infinite 5-class field tower. The paper also derives an upper bound on the Pólya number in terms of the order of the Pólya group and exhibits members of the family that are non-monogenic even though their index is one.","feed_headline":"Lecacheux quintics have Pólya groups of arbitrary 5-rank","feed_subtitle":"For any k the parameters giving 5-rank at least k form a positive-density set, so a positive proportion admit infinite 5-class field towers.","key_machinery":"The Pólya group of the number field, which records the obstruction to the existence of a Pólya basis for its ring of integers.","core_discovery":"The 5-rank of the Pólya group attached to a Lecacheux quintic field can be made arbitrarily large; for every positive integer k the subset of odd integers s for which this rank is at least k has positive density, and therefore a positive proportion of the fields admit an infinite 5-class field tower.","pith_inferences":["Similar density-of-high-rank phenomena may appear in other parametric families of number fields once an analogous Pólya-group computation is available.","The non-monogenic examples with index one suggest that the Pólya group may detect ramification or integral-basis phenomena that the usual index alone misses.","The existence of infinite 5-class field towers for a positive-density subset supplies an infinite supply of fields whose class-group 5-part is unbounded in a controlled way."],"forward_implications":["For every positive integer k the set of Lecacheux quintic fields whose Pólya group has 5-rank at least k has positive density.","A positive proportion of Lecacheux quintic fields admit an infinite 5-class field tower.","The Pólya number of each field in the family is bounded above by a function of the order of its Pólya group.","Infinitely many members of the family are non-monogenic despite having index one."],"fun_headline_variants":["Pólya groups reach arbitrary 5-rank in Lecacheux quintics","Lecacheux quintic family yields unbounded Pólya 5-rank","Positive density for Lecacheux fields with Pólya rank >=k","Infinite 5-class towers for positive share of Lecacheux quintics"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The density arguments that produce a positive-density set of parameters for each fixed rank k must remain valid independently of finer arithmetic features of the individual Lecacheux polynomials.","fun_headline_variants_meta":{"raw":{"variants":["Pólya groups reach arbitrary 5-rank in Lecacheux quintics","Lecacheux quintic family yields unbounded Pólya 5-rank","Positive density for Lecacheux fields with Pólya rank >=k","Infinite 5-class towers for positive share of Lecacheux quintics"]},"model":"grok-4.3","cost_usd":0.00606,"raw_usage":{"total_tokens":2847,"prompt_tokens":631,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":60599500,"prompt_tokens_details":{"text_tokens":631,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2137,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":631,"tokens_out":79,"duration_ms":14821,"temperature":1.0,"reasoning_tokens":2137,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T07:15:27.980136+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation, for a large finite collection of odd integers s, showing that the proportion of fields whose Pólya group has 5-rank at least k tends to zero as k grows, or a single concrete s for which the 5-rank is strictly smaller than the density claim predicts.","supporting_citations":[],"review_version":1}