{"id":"28f852c9-af21-4107-b9f5-535778567cc0","arxiv_id":"2411.08568","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Orders in a number field form a tree, giving a complete and duplicate-free enumeration of orders of bounded index and of overorders of a given order.","lead":"The authors arrange all orders in a number field into a tree and use it to enumerate all orders of bounded index in the ring of integers. The method lists each order once and runs much faster than previous algorithms on larger examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the tree construction and no-duplicate claim are sound, with the cited Dade–Taussky–Zassenhaus result applied correctly, though the paper omits a short justifying corollary.","rationale":"The reader's accepted verdict is appropriate. The paper's central tree construction is mathematically sound. The external Dade–Taussky–Zassenhaus theorem is cited and correctly applicable; the missing corollary showing that J^{n-1} is invertible over M(J^{n-1}) is a short derivation from the stated theorem and Lemma 4.4(1). The proof slip in Algorithm 4.22 (claiming O=H^{n-1}) does not affect the algorithm's correctness, because the needed inclusion p^{n-2}O⊆H follows from Remark 4.6 with r≤n-1. The ramified MeatAxe and orbit-enumeration steps are standard computational tools and the proofs show completeness. I therefore find no load-bearing concern. I partially disagree with the reader's weakest assumption: relying on a published theorem is not itself a correctness risk, though the paper could have made the corollary explicit and fixed the minor proof typo.","tokens_in":19383,"tokens_out":31089,"duration_ms":280998,"concrete_test":"Re-derive the corollary used in the tree definition: fix a full lattice I and let r be the exponent from Theorem 4.2. Show that if I^r is invertible over M(I^r), then IO is invertible over that order (Lemma 4.4(1) argument), so I^{n-1} is invertible over the same order and M(I^{n-1}) equals it. If this derivation fails for some lattice, the parent relation M(J^{n-1}) in Section 4 is unsupported; if it succeeds, the no-duplicate claim follows. As a computational cross-check, run Algorithm 4.3 on a ramified cubic field with p=2 and compare the output list with an independent maximal-suborder depth-first search, confirming each order occurs exactly once.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central claim as: Algorithm 4.3 enumerates every order of p-power index in Z_K exactly once. The parent of an order O' is M(J_p(O')^{n-1}); the algorithm regenerates a radical I=J_p(O') only from orders O that satisfy condition (4.1). The possible gap would be if the same I could satisfy (4.1) for two different O, causing duplicate processing. This does not occur: if I satisfies (4.1) with O, then by Lemma 4.4(1) the ideal a=IO is invertible over O, so I^r=a^r is invertible; hence M(I^r)=O, and since a is invertible, I^{n-1}=a^{n-1} is also invertible with multiplier O, so O=M(I^{n-1}). Thus the parent is uniquely determined and each radical is generated exactly once. Conversely, for any order O', Theorem 4.2 supplies r<n with J^r invertible over M(J^r); the same argument gives M(J^{n-1})=M(J^r), so J satisfies (4.1) with its true parent. The paper states the tree property without proof, but it is valid. The textual slips I found are minor: (i) the proof of Algorithm 4.22 asserts 'O=H^{n-1}', which need not hold when the minimal exponent in (4.2) is smaller; the needed inclusion p^{n-2}O⊆H follows from Remark 4.6 with r≤n-1, so the algorithm is unaffected; and (ii) the n-1 version of Theorem 4.2 is used without deriving it from the stated r-version. Neither changes the verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a tree on the set of orders in a number field K whose index in the maximal order Z_K is a power of a fixed prime p. The proposed parent of an order O is the multiplier ring M(J_p(O)^{n-1}), and the Dade--Taussky--Zassenhaus theorem is used to guarantee that some power of every p-radical is invertible. Algorithm 4.3 enumerates all orders of index dividing p^e by maintaining a list of p-radicals I; for each I it computes all orders with p-radical I and then all new radicals satisfying condition (4.1). Sections 3--5 supply the local enumeration tools: orders with a given p-radical are classified through subrings of a product of finite fields (Theorem 3.1, Proposition 3.2, Algorithm 3.3); the unramified case is reduced to enumerating lattices H with 1 in H, H^r=O and pH^2 in H (Algorithms 4.18, 4.20, 4.22); the ramified case is handled through invertible ideals and the algorithms of Section 5. Section 6 gives counts and timings for degree-5 fields, including completely split, inert, and completely ramified primes, and compares with the algorithm of Hofmann and Sircana.","tokens_in":19705,"tokens_out":13209,"duration_ms":128540,"significance":"If the missing proof of uniqueness is supplied, this is a strong contribution to computational algebraic number theory. The tree structure removes duplicate enumeration, and the algorithmic core avoids brute-force search over all subspaces: the unramified routine works with vector-space complements and makes essential use of the p-radical. Corollary 3.4 gives an explicit closed counting formula for the number of orders with a fixed p-radical, and the examples give concrete, checkable counts; for instance, Example 6.1(1) reports 11 orders of index 5, and the split/inert/ramified comparison in Example 6.3 is nontrivial. The paper ships an implementation in Hecke, compares timings on the same examples as the previous state of the art, and derives its counts without fitted parameters, which makes the finite output independently verifiable.","major_comments":[{"comment":"The sentence after (4.1) states that condition (4.1) arranges the orders of p-power index in Z_K into a tree and hence that Algorithm 4.3 never finds the same order twice. This is the load-bearing correctness property of the enumeration, but it is asserted without proof. Please add a lemma showing: (i) if O' has p-radical J, then J satisfies (4.1) for O=M(J^{n-1}), using Theorem 4.2 together with the fact that M(J^r)=M(J^{n-1}) when J^r is invertible; and (ii) if J satisfies (4.1) for some O, then O=M(J^{n-1}), so the parent is uniquely determined. These assertions are true, but the manuscript should contain the argument rather than the one-sentence 'Note'.","section":"Section 4, Eq. (4.1)"},{"comment":"The abstract promises enumeration of orders over a given order Lambda, but Section 6 only sketches the adaptation ('replace Z by (1/p)J_p(Lambda) in Algorithms 4.18 and 4.22') and gives no formal correctness proof for the overorder version. Either state and prove a precise theorem for the overorder enumeration, or explicitly present that part as an implementation with experimental support rather than as part of the paper's central claim.","section":"Section 6"}],"minor_comments":[{"comment":"In the proof of Algorithm 4.22, the statement that O=H^{n-1} follows from (4.2) is not correct in general, since (4.2) only gives H^r=O for some r at most n-1. The needed conclusion p^{n-2}O is contained in H still follows from Remark 4.6 because p^{r-1}O is contained in H and r-1 is at most n-2; the proof should be rewritten with this argument.","section":"Section 4, Algorithm 4.22"},{"comment":"There are several typos that should be corrected: 'Th is' in the abstract, 'orderorders' near the end of the first section, and 'in paricular' in reference [1].","section":"Section 1 and Abstract"},{"comment":"The same symbol O_i is used both for the set of orders with index at most 5^i and for a particular chain of orders constructed in the example; this is confusing and should be renamed.","section":"Example 6.1(1)"}],"recommendation":"major_revision","confidential_remarks":"I agree with the stress-test assessment that the tree property is mathematically valid, but I do not think the manuscript in its current form is sufficiently self-contained for acceptance: the uniqueness of the tree enumeration is a central algorithmic claim and is not proved. The missing argument is short, so I expect a quick revision. The overorder promise in the abstract also deserves either a proof or a softened formulation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a solid algorithmic number theory paper. The central idea is new and it works. The authors arrange p-power orders in a tree by making the parent of an order O be M(J_p(O)^(n-1)), then reduce enumeration of successors to lattices H satisfying H^r = O. That avoids the two pain points of the older maximal-suborder/minimal-overorder approaches: the same order is not found twice, and most steps do not require walking over all subspaces of F_p-vector spaces. I checked the first example's count of 11 manually and the tree/no-duplicate claim seems right; the stress-test note confirms the parent is uniquely determined and each radical is generated exactly once. The classification of orders with a fixed p-radical via subrings of a product of fields (Theorem 3.1, Proposition 3.2, Corollary 3.4) is also well done and the counting formula is consistent. What deserves credit: the paper gives real proofs, corrects a genuinely faulty lemma from Halter-Koch (Example 5.6), and reports substantial speedups over Hofmann-Sircana on nontrivial examples. The reliance on Dade-Taussky-Zassenhaus is not a hidden assumption; it is a cited, proved external theorem, and the paper uses it appropriately. The ramified case is harder, but the arguments there are supplied in reasonable detail. Soft spots, in proportion: no code is shipped, so reproducibility depends on the Hecke implementation and external pieces like MeatAxe and orbit enumeration. A couple of textual slips in the proof of Algorithm 4.22 (the assertion O = H^(n-1) need not hold when the minimal exponent is smaller, and the n-1 version of Theorem 4.2 is used without derivation) are minor and do not affect the algorithm. The paper could be clearer about why the tree property is stated without an explicit proof, but the missing reasoning is short. Who this is for: people who actually need to compute overorders, ideal class monoids, or bounded-index order lists in number fields. It is a strong candidate for peer review, not because it is flashy but because the method is likely to be used and cited. I would send it to a referee, perhaps asking that referee to check the ramified-case algorithm carefully and to demand a few clarifying sentences in the proof of Algorithm 4.22.","headline":"A genuinely useful algorithm for enumerating orders in number fields, with a new tree structure that makes the no-duplicate claim credible; worth a serious referee despite minor proof slips and no shipped code.","tokens_in":728,"tokens_out":1594,"would_cite":true,"duration_ms":29773,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R54","11Y40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A tree structure arranges all p-power orders in a number field so that each order of bounded index is enumerated exactly once.","keywords":["orders in number fields","p-radical","tree enumeration","multiplier ring","Dade-Taussky-Zassenhaus theorem","overorders","algorithms","computational number theory"],"falsifier":"Find a full lattice I in a number field K for which none of the powers I, $I^{2}$, ..., $I^{{n-1}}$ is invertible over its multiplier ring; if such a lattice exists, Theorem 4.2 is false and the parent map is not well-defined. A computational check is to run Algorithm 4.3 for a small field and verify that no order is produced twice; a duplicate output would contradict the claimed tree property.","tokens_in":19143,"feed_emoji":"🌳","tokens_out":5118,"duration_ms":45642,"temperature":0.7,"pith_summary":"The paper arranges all orders in a number field whose index in the maximal order is a power of a fixed prime p into a tree. The parent of an order is obtained by taking the multiplier ring of the (n-1)st power of its p-radical; a theorem of Dade, Taussky and Zassenhaus ensures this power is invertible, which makes the tree honest. Walking this tree from the root lists every such order exactly once, so all orders of index dividing p^e can be enumerated without the duplicates that plague earlier approaches. The same tree, restricted to a subtree, enumerates the orders above a given order, and the authors' implementation outperforms existing algorithms on examples with larger index.","feed_headline":"A tree structure enumerates number field orders with no repeats","feed_subtitle":"Walking a tree of multiplier rings lists each order of bounded index once, beating prior searches on larger indices.","key_machinery":"The load-bearing tool is the theorem of Dade, Taussky and Zassenhaus ([1, Theorem C]): for every full lattice I in K, some power I^r with 1 ≤ r < n is invertible over the multiplier ring M(I^r). This theorem justifies defining the parent of O as M(J_p(O)^{n-1}), and it makes the set of orders of p-power index a tree with root Z_K. Supporting machinery includes a classification of p-radicals and of subrings of products of finite fields (Propositions 2.11 and 3.2, Algorithm 3.3), subspace enumeration algorithms (Algorithms 4.14 and 4.16), and, for ramified primes, Fröhlich's invertibility criterion and a MeatAxe-based enumeration of submodules (Algorithm 5.4).","core_discovery":"The central discovery is that orders of p-power index can be organized into a rooted tree whose root is the maximal order Z_K, with a successor map that is provably acyclic. Specifically, for an order O with p-radical J, the parent is defined as the multiplier ring M($J^{{n-1}}$); by Theorem 4.2, valid for every full lattice, there is an exponent 1 ≤ r < n for which J^r is invertible over this ring, so the definition is legitimate and the relation is a tree. This turns the enumeration of all orders of index dividing p^e into a tree traversal: at each order O, the children correspond to lattices satisfying condition (4.1), which in the unramified case are described by sublattices H of O' with $H^{{n-1}}$=O'. The paper works out this description explicitly and gives a correspondingly concrete algorithm (Algorithm 4.3) that never produces the same order twice.","pith_inferences":["The same tree could be used to sample random orders of bounded index uniformly, since each order appears exactly once in the traversal.","The parent map might extend to orders in étale algebras with multiple components, where p-radicals and multiplier rings still behave locally, although the paper only treats number fields.","Because every order appears once, the enumeration is embarrassingly parallel across branches of the tree, so the algorithm could scale to much larger indices on distributed hardware.","For a fixed order Λ, the subtree of orders containing Λ is connected, so the restricted enumeration could be seen as a natural filtration of the lattice of overorders of Λ."],"forward_implications":["All orders with index dividing p^e can be output with no duplication, because the tree property makes the history of each order unique.","The overorders of a fixed order Λ are obtained by constraining the same tree walk, which gives large speedups in the examples once the index [Z_K:Λ] is not tiny.","Since the tree is independent of the bound e, the enumeration can be stopped and resumed at any depth, allowing on-demand generation of orders by increasing index.","For unramified p, children of O are enumerated via lattices H with H^{n-1}=O', a condition that can be checked with linear algebra and avoids walking over all subspaces of O/pO."],"supporting_citations":[{"why":"Supplies Theorem 4.2 (Theorem C), the existence of r<n with I^r invertible over M(I^r), underpinning the whole parent-map construction.","marker":"[1]"},{"why":"Classifies minimal algebra extensions over a field, used to describe maximal suborders in Proposition 2.8 and subsequent lemmas.","marker":"[2]"},{"why":"Fröhlich's invertibility criterion, used in Section 5 to classify invertible ideals of O between pO and J_p(O).","marker":"[4]"},{"why":"Hofmann and Sircana's algorithm, the baseline for overorder enumeration and the source of maximal-suborder results used in Section 2.","marker":"[6]"},{"why":"Klüners and Pauli's method for computing residue class rings and Picard groups, used to obtain the unit generators in Algorithm 5.10.","marker":"[7]"},{"why":"The MeatAxe library for submodule enumeration, used in Algorithm 5.4 to list O-submodules with prescribed index.","marker":"[8]"}],"fun_headline_variants":["Tree organizes number field orders, no repeats","Enumerate number field orders via acyclic tree","Number field orders: tree traversal to list all","A tree enumerates orders without duplicates","Number field orders: a tree, no repeats"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tree construction relies on the quoted theorem that for every full lattice I, some power I^r with r<n is invertible over its multiplier ring; the paper does not prove this theorem.","fun_headline_variants_meta":{"raw":{"variants":["Tree organizes number field orders, no repeats","Enumerate number field orders via acyclic tree","Number field orders: tree traversal to list all","A tree enumerates orders without duplicates","Number field orders: a tree, no repeats"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1215,"prompt_tokens":744,"completion_tokens":471,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":360,"completion_tokens_details":{"reasoning_tokens":403}},"tokens_in":360,"tokens_out":471,"duration_ms":4656,"temperature":1.0,"reasoning_tokens":403,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:34:55.866768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a full lattice I in a number field K for which none of the powers I, $I^{2}$, ..., $I^{{n-1}}$ is invertible over its multiplier ring; if such a lattice exists, Theorem 4.2 is false and the parent map is not well-defined. A computational check is to run Algorithm 4.3 for a small field and verify that no order is produced twice; a duplicate output would contradict the claimed tree property.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 4.2 (Theorem C), the existence of r<n with I^r invertible over M(I^r), underpinning the whole parent-map construction."},{"cited_title":"Ferrand and J.-P","cited_arxiv_id":null,"evidence_quote":"Classifies minimal algebra extensions over a field, used to describe maximal suborders in Proposition 2.8 and subsequent lemmas."},{"cited_title":"Fr¨ ohlich","cited_arxiv_id":null,"evidence_quote":"Fröhlich's invertibility criterion, used in Section 5 to classify invertible ideals of O between pO and J_p(O)."},{"cited_title":"Hofmann and C","cited_arxiv_id":null,"evidence_quote":"Hofmann and Sircana's algorithm, the baseline for overorder enumeration and the source of maximal-suborder results used in Section 2."},{"cited_title":"Kl¨ uners and S","cited_arxiv_id":null,"evidence_quote":"Klüners and Pauli's method for computing residue class rings and Picard groups, used to obtain the unit generators in Algorithm 5.10."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The MeatAxe library for submodule enumeration, used in Algorithm 5.4 to list O-submodules with prescribed index."}],"review_version":1}