{"id":"2f5f2ba7-0034-48f3-9be6-3f20f66fe227","arxiv_id":"2411.16151","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every rational q in (0,1), except reciprocals of integers, whose denominator is odd, the monoid algebra Q[M_q] is atomic.","lead":"This paper proves that certain algebraic rings built from rational exponents are atomic, meaning every element factors into irreducibles. The proof yields the first infinite family of simple one-dimensional rings that are atomic but fail the classical chain condition on principal ideals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2 hinges on a stabilization Claim whose only cited support, Proposition 5.1, leaves the lc(f) in {±1} case unproved; unless that case and the monotonicity of factor counts are supplied, the final ACCP contradiction is not justified.","rationale":"The reader correctly identifies the stabilization Claim as load-bearing. My stress-test goes one level deeper: the Claim's proof is not merely terse; it rests on Proposition 5.1, and the monic case of Proposition 5.1 is explicitly left to a \"similar proof structure\" that the text never supplies. That case is needed not only for monic f but also as the terminal phase of the non-monic case, so it is not an optional simplification. Lemma 4.5, which would control Lambda(b,pi) for all composed divisors b, has a dense and difficult unit case; if the inequality Lambda*(b,P) <= Lambda*(f,P) fails for some b, the bound r^{Lambda*} is unjustified. A secondary but real gap is the use of v_{n(q)} for composite n(q) with no definition; this appears in the final valuation contradiction and should be fixed by defining v_n for composite n, for example as the exponent of n in the numerator, or by switching to a prime divisor of n(q). These are repairable issues and do not give me evidence that the theorem is false, so the reader's CONDITIONAL verdict should stand; I do not see a reason to change it. The computational test on f=x^2+x+2 and r=3 would either find a counterexample to the omitted bound or provide evidence that the omitted case is valid for that instance.","tokens_in":19804,"tokens_out":31825,"duration_ms":303039,"concrete_test":"For a concrete non-exceptional monic f, say f(x)=x^2+x+2, and r=3, use a CAS to compute the number of irreducible factors of f(x^{3^lambda}) in Z[x] for lambda=1,...,8, and independently compute Lambda*(f,{3}) from Definition 4.3 or from the inequalities (4.3)-(4.4). If any lambda gives more than r^{Lambda*(f,{3})} factors, Proposition 5.1's monic case is false. Separately, compute the factorizations in the rings Z[1/3^{ell+gamma} N0] for ell=0 and gamma=1,...,8 and check that the multiset of irreducible factors stabilizes and that the count is monotone; a failure of stabilization would disprove the Claim directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final contradiction in Theorem 5.2 depends on the Claim that an irreducible f in Z[1/r^ell N0] with non-exceptional f(x^{r^ell}) has a factorization in Z[1/r^{ell+gamma} N0] that stabilizes for all gamma at least Gamma. The proof of the Claim is a direct appeal to Proposition 5.1 plus the tower of UFD extensions. Proposition 5.1, however, only proves the bound A(f,pi) in the case lc(f) not in {±1}; the case lc(f) in {±1} is dismissed as a \"similar proof structure\". This omitted case is not cosmetic: it must show that for a monic irreducible non-exceptional f and every lambda, f(x^{r^lambda}) has at most r^{Lambda*(f,pi)} irreducible factors in Z[x]. That requires Lemma 4.5, whose proof, especially Case 2 with subcases 2.1 and 2.2, is very terse and is not independently verified. Furthermore, the step from the uniform bound to \"same factorization\" needs the additional observation that the number of irreducible factors of f in the tower Z[1/r^{ell+gamma} N0] is nondecreasing in gamma; that observation is not stated. If either the monic bound fails, for example by splits occurring after a first lift under a different prime, or the factor count can oscillate, the Claim fails, and with it the contradiction at equations (5.3)-(5.4) is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies atomicity of monoid algebras of exponentially cyclic Puiseux monoids M_q over Q. The main theorem (Theorem 5.2) asserts that Q[M_q] is atomic for every q in ((0,1) ∩ Q) \\ N^{-1}_{>1} with odd denominator, extending the recent result for q = 3/4. The strategy is to reduce atomicity of Q[M_q] to atomicity of Z[M_q] via Proposition 2.1, use the almost ACCP of M_q to reduce to ACCP-supported polynomials, and then prove that every ACCP-supported polynomial in Z[M_q] satisfies the ACCP. The proof introduces splitting sequences, bounds their lengths through algebraic number theory (Lemma 4.4 and Lemma 4.5), and then uses a stabilization claim inside Theorem 5.2 to rule out an infinite ascending chain of principal ideals. The paper concludes with a contradiction involving cyclotomic factorizations and p-adic-type valuations.","tokens_in":20149,"tokens_out":6552,"duration_ms":62678,"significance":"If the proof is completed, the result provides the first infinite family of one-dimensional integral domains that are atomic but fail the ACCP, making substantial progress on Gotti's conjecture. The splitting-sequence machinery and the reduction to Z[M_q] are natural and potentially reusable tools for further cases of the conjecture. The paper contains no fitted parameters or empirical constants, and the overall proof architecture is coherent: it reduces the problem to concrete polynomial-factorization bounds. However, several load-bearing technical steps are currently either omitted or too terse to verify, so the main theorem is not yet fully established as written.","major_comments":[{"comment":"The proof of inequality (4.4) in Case 2 is too terse to be checkable. In particular, the construction of the sequence S, the assertion that the exponents a_1,...,a_m can be chosen 'maximally,' and the contradictions in Subcases 2.1 and 2.2 are not fully argued; Subcase 2.2 appears to rely on an ordering of the primes in P that is not introduced. This lemma is load-bearing because it supplies the uniform bound for monic irreducible polynomials that Proposition 5.1 uses in the case lc(f) ∈ {±1}.","section":"Section 4, Lemma 4.5"},{"comment":"The proof of Proposition 5.1 explicitly treats only the case lc(f) ∉ {±1} and dismisses the case lc(f) ∈ {±1} with the statement that 'a similar proof structure may be used.' Since the monic bound r^{Λ*(f,π)} is exactly what the stabilization Claim in Theorem 5.2 invokes for non-exceptional irreducible polynomials, this case must be written out in full rather than omitted.","section":"Section 5, Proposition 5.1"},{"comment":"The symbol v_{n(q)} is used for the valuation of an lcm and later of the exponent c, but Section 2.1 only defines p-adic valuations v_p for primes p. The final contradiction depends on comparing these valuations, so either v_m must be defined for composite m or the argument must be rewritten with an explicit prime divisor of n(q). In addition, the displayed equality (5.3) needs a proof explaining how the cyclotomic factorization and the condition gcd(n(q), r) = 1 imply the claimed equality of valuations.","section":"Section 5, equations (5.3)-(5.4)"},{"comment":"The step from the uniform divisor bound in Proposition 5.1 to the statement that f has 'the same factorization into irreducibles' for all sufficiently large γ is not fully justified. One needs to prove both that the number of irreducible factors of f in the tower Z[1/r^{ℓ+γ} N0] is nondecreasing in γ and that stabilization of that number, together with uniqueness of factorizations in these UFDs, forces the factorization itself to stabilize. Neither observation is stated or proved.","section":"Theorem 5.2, stabilization Claim"}],"minor_comments":[{"comment":"The notation /llbracketa, b/rrbracket is used for discrete intervals with unusual delimiters; please ensure it is rendered consistently as a discrete interval in the published version.","section":"Section 2.1"},{"comment":"The expression F := f(x) ∘ (g(x^{3^{n-1}}))_{n∈N} is ambiguous about the starting index and the intended sequence of polynomials; please clarify the notation.","section":"Example 3.4(3)"},{"comment":"The paragraph beginning 'Let P′(x) := f(x^p)/P(x)' appears to be part of the proof but is not labeled as such; it should be moved into the proof environment.","section":"Lemma 2.5"},{"comment":"In Definition 3.3, Nspl(F) is described as the 'number of splits at the beginning' of F but is then defined as a minimal index; the wording should be aligned with the formal definition.","section":"Definition 3.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript reports a genuine advance, but the monic case in Proposition 5.1 and the valuation issue in Theorem 5.2 must be repaired before the main theorem can be considered proved; the omitted details appear to be local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves Q[M_q] is atomic for every q in Gotti's set S with odd denominator. That is a genuine step: before this, only q=3/4 was known. If the proof holds, it gives the first infinite family of one-dimensional atomic monoid algebras failing the ACCP, and it makes the odd-denominator half of Gotti's conjecture a single technique away. The framework is inherited from Bu et al., but the generalization of splitting sequences and the algebraic number theory bounds on splitting sequence lengths are new and are the right tools.\n\nThe overall architecture is sound. Atomicity of Z[M_q] reduces to showing ACCP-supported polynomials satisfy the ACCP, and the contradiction at equations (5.3)–(5.4) is well motivated. There is no circularity: the cited results (almost ACCP for exponentially cyclic monoids, Capelli, Guersenzvaig) are independent of the target theorem.\n\nThe soft spots are real but, as far as I can tell, patchable. Proposition 5.1 states a bound for lc(f) in {\\pm 1} but does not prove it; ‘a similar proof structure’ is not a proof, and this case is load-bearing because the stabilization Claim in Theorem 5.2 leans on it. Lemma 4.5, which the monic case needs, is very compressed, especially subcases 2.1 and 2.2; I could not fully verify them on a first pass. The stabilization Claim also silently uses the fact that the number of irreducible factors of f in the tower Z[1/r^{\\ell+\\gamma}N_0] is nondecreasing in \\gamma; that is true because each ring is a subring of the next, but it should be said. And v_{n(q)} in equations (5.3)–(5.4) is undefined when n(q) is composite; the fix is to pick a prime p | n(q) and use v_p, relying on gcd(n(q), r)=1. None of these looks fatal. They are omitted details in a technically dense argument, not a broken strategy.\n\nWho is this for? People working on factorization theory, Puiseux monoids, and atomic domains. It deserves a serious referee. I would send it out and ask the author to expand Prop 5.1's monic case, define the valuation used (or switch to v_p), and add the monotonicity observation plus more detail in Lemma 4.5. With those additions the proof should be checkable.","headline":"Real progress on Gotti's conjecture: the odd-denominator case is reduced to a handful of fillable gaps, not a broken strategy.","tokens_in":20627,"tokens_out":4105,"would_cite":true,"duration_ms":37380,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F15","13A05","20M25","13B22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Odd-denominator monoid algebras are atomic, without ACCP","keywords":["monoid algebra","atomicity","ascending chain condition on principal ideals","Puiseux monoid","exponentially cyclic monoid","splitting sequence","cyclotomic polynomial","factorization"],"falsifier":"Find an irreducible polynomial $f$ in $\\mathbb{Z}[x]$ outside $\\{\\pm x\\} \\cup \\{\\pm \\Phi_y(x) : y \\in \\mathbb{N}\\}$ for which the number of irreducible divisors of $f(x^{r^\\lambda})$ in $\\mathbb{Z}[x]$ grows faster than the bound $A(f,\\pi)$ of Proposition 5.1 as $\\lambda$ grows, or exhibit an ACCP-supported non-monomial in $\\mathbb{Z}[M_q]$ with $q$ of odd denominator that has a strictly ascending chain of principal ideals that never stabilizes. Either observation would disprove the claim that $\\mathbb{Q}[M_q]$ is atomic for this class.","tokens_in":19610,"feed_emoji":"🧮","tokens_out":4814,"duration_ms":38744,"temperature":0.7,"pith_summary":"This paper proves that the monoid algebra $\\mathbb{Q}[M_q]$ is atomic for every rational $q \\in ((0,1) \\cap \\mathbb{Q}) \\setminus \\mathbb{N}^{-1}_{>1}$ whose denominator is odd. Atomic means every nonunit element factors into irreducibles, the building blocks of multiplication. The result matters because these algebras are one-dimensional integral domains, and atomicity is known to coexist with failure of the ascending chain condition on principal ideals (ACCP) only in rare, usually technical, examples. The theorem supplies an infinite family of such domains and progresses toward Gotti's conjecture that atomicity ascends from the exponentially cyclic Puiseux monoid $M_q$ to its monoid algebra over $\\mathbb{Q}$. The proof reduces the task to controlling splitting sequences of composed polynomials $f(x^r)$ and showing that factorizations stabilize as the exponent lattice is refined.","feed_headline":"Odd-denominator monoid algebras are atomic, without ACCP","feed_subtitle":"Every such q gives a one-dimensional atomic domain that still fails the ascending chain condition.","key_machinery":"The paper's central object is the splitting sequence: starting from an irreducible polynomial $f$ in $\\mathbb{Z}[x]$ and an exponent sequence $(e_n)$, each next term is an irreducible divisor of $f_{n-1}(x^{e_n})$; the binary string records whether each step is a 'lift' or a 'split.' The key estimate is Proposition 5.1, which bounds the number of irreducible divisors of $f$ in $\\mathbb{Z}[\\frac{1}{r^\\lambda}\\mathbb{N}_0]$ by a finite quantity $A(f,\\pi)$ depending only on $f$, the prime set $\\pi$ of $r$, and the leading coefficient of $f$. This bound is obtained by combining Capelli's Lemma, a degree-counting argument, and a new invariant $\\Lambda^*(f,P)$ that measures how many initial splits a splitting sequence can have; Lemma 4.5 transfers the bound from $f$ to every polynomial in its composed divisor set. The uniform bound is what forces factorization to stabilize as the exponent lattice is refined, which is the mechanism that excludes an infinite ascending chain of principal ideals.","core_discovery":"The central claim is Theorem 5.2: for every $q \\in ((0,1) \\cap \\mathbb{Q}) \\setminus \\mathbb{N}^{-1}_{>1}$ with odd denominator, the monoid algebra $\\mathbb{Q}[M_q]$ is atomic. Since $M_q$ is an atomic Puiseux monoid that fails the ACCP, and ACCP ascends from a monoid to its monoid algebra, $\\mathbb{Q}[M_q]$ is a one-dimensional integral domain that is atomic but does not satisfy the ACCP. The paper establishes this by showing that every ACCP-supported polynomial in $\\mathbb{Z}[M_q]$ satisfies the ACCP, hence is atomic, and by reducing atomicity of $\\mathbb{Q}[M_q]$ to atomicity of $\\mathbb{Z}[M_q]$ through a coefficient-clearing argument. The load-bearing technical step is a stabilization claim: for any irreducible $f$ in $\\mathbb{Z}[\\frac{1}{r^\\ell}\\mathbb{N}_0]$ avoiding $\\pm x$ and $\\pm$ cyclotomic polynomials, there is a threshold $\\Gamma$ such that $f$ has the same factorization into irreducibles in $\\mathbb{Z}[\\frac{1}{r^{\\ell+\\gamma}}\\mathbb{N}_0]$ for every $\\gamma \\geq \\Gamma$. This claim, derived from a uniform bound on the number of irreducible divisors in these subalgebras, rules out infinite strictly ascending chains of principal ideals.","pith_inferences":["A natural testable extension is to remove the odd-denominator restriction: if the same stabilization claim can be proved with $r$ even, Gotti's conjecture would follow in full, since the apparent obstruction is the parity of the exponent in the cyclotomic factorization steps.","The uniform divisor bound suggests that the refined algebras $\\mathbb{Z}[\\frac{1}{r^\\lambda}\\mathbb{N}_0]$ form a chain whose atomic structure stabilizes entrywise; one could investigate whether the stabilized factorization sets define an invariant of the limit algebra, or of the original $\\mathbb{Q}[M_q]$.","The methods may transfer to other rank-one monoid algebras built from monotone or multi-geometric Puiseux monoids, where analogous splitting-sequence bounds could tame infinite ascending chains of principal ideals."],"forward_implications":["For every odd-denominator $q$ in the parameter set, $\\mathbb{Q}[M_q]$ is a one-dimensional atomic domain that fails the ACCP, so atomicity and the ACCP are genuinely independent even in Krull dimension one.","Gotti's conjecture is now settled for an infinite family of parameters, with only the even-denominator cases left open.","The splitting-sequence machinery, especially Proposition 5.1, gives uniform divisor bounds in $\\mathbb{Z}[\\frac{1}{r^\\lambda}\\mathbb{N}_0]$ that can be reused to study ascent of finite factorization and bounded factorization properties to monoid algebras.","Since $\\mathbb{Q}[M_q]$ is atomic, every nonunit element has at least one factorization into irreducibles, making the factorization theory of these one-dimensional domains explicitly accessible.","The atomicity conclusion implies that the failure of the ACCP in this setting does not obstruct the existence of irreducible factorizations, clarifying the boundary between the two conditions."],"supporting_citations":[{"why":"Proved that $\\mathbb{Q}[M_{3/4}]$ is atomic, introduced splitting sequences, and supplied the proof template that Theorem 5.2 adapts.","marker":"[6]"},{"why":"Posed Gotti's conjecture and established the atomicity and ACCP background for exponentially cyclic Puiseux monoids.","marker":"[18]"},{"why":"Introduced atomicity and the original claim that atomic domains satisfy the ACCP, which this paper's examples disprove.","marker":"[11]"},{"why":"Provided the irreducibility criteria for composed polynomials $f(x^r)$ that underlie the leading-coefficient and divisor-count bounds in Lemmas 2.5 and 2.6.","marker":"[25]"},{"why":"Gave the first atomic domain failing the ACCP, providing the historical context and motivating the search for simpler one-dimensional examples.","marker":"[24]"},{"why":"Proved that exponentially cyclic monoids satisfy the almost ACCP and supplied earlier monoid algebra counterexamples, giving the almost-ACCP input used in the proof of Theorem 5.2.","marker":"[20]"},{"why":"Provides the transfer theorem that irreducibility in $\\mathbb{Z}[M_q]$ implies irreducibility in $\\mathbb{Q}[M_q]$, used in Proposition 2.1.","marker":"[17]"}],"fun_headline_variants":["Odd-denominator monoid algebras atomic without ACCP","Atomic one-dimensional domains from odd-denominator monoids","Odd-denominator Puiseux algebras: atomic, no ACCP","Atomicity ascends for every odd-denominator q","All odd-denominator cases confirm atomicity conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the stabilization claim: for each irreducible polynomial $f$ in the smaller algebra, the number of ways it factors cannot increase without bound as the allowed exponent denominators are enlarged; if that claim fails, an infinite strictly ascending chain of principal ideals could survive and the main theorem would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Odd-denominator monoid algebras atomic without ACCP","Atomic one-dimensional domains from odd-denominator monoids","Odd-denominator Puiseux algebras: atomic, no ACCP","Atomicity ascends for every odd-denominator q","All odd-denominator cases confirm atomicity conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1410,"prompt_tokens":1041,"completion_tokens":369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":286}},"tokens_in":657,"tokens_out":369,"duration_ms":3887,"temperature":1.0,"reasoning_tokens":286,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:29:38.755103+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an irreducible polynomial $f$ in $\\mathbb{Z}[x]$ outside $\\{\\pm x\\} \\cup \\{\\pm \\Phi_y(x) : y \\in \\mathbb{N}\\}$ for which the number of irreducible divisors of $f(x^{r^\\lambda})$ in $\\mathbb{Z}[x]$ grows faster than the bound $A(f,\\pi)$ of Proposition 5.1 as $\\lambda$ grows, or exhibit an ACCP-supported non-monomial in $\\mathbb{Z}[M_q]$ with $q$ of odd denominator that has a strictly ascending chain of principal ideals that never stabilizes. Either observation would disprove the claim that $\\mathbb{Q}[M_q]$ is atomic for this class.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved that $\\mathbb{Q}[M_{3/4}]$ is atomic, introduced splitting sequences, and supplied the proof template that Theorem 5.2 adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced atomicity and the original claim that atomic domains satisfy the ACCP, which this paper's examples disprove."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provided the irreducibility criteria for composed polynomials $f(x^r)$ that underlie the leading-coefficient and divisor-count bounds in Lemmas 2.5 and 2.6."},{"cited_title":"Gotti and B","cited_arxiv_id":null,"evidence_quote":"Proved that exponentially cyclic monoids satisfy the almost ACCP and supplied earlier monoid algebra counterexamples, giving the almost-ACCP input used in the proof of Theorem 5.2."},{"cited_title":"Gotti, Irreducibility and Factorizations in Monoid Rings , Numerical Semigroups 40 (2020) 129–139","cited_arxiv_id":null,"evidence_quote":"Provides the transfer theorem that irreducibility in $\\mathbb{Z}[M_q]$ implies irreducibility in $\\mathbb{Q}[M_q]$, used in Proposition 2.1."}],"review_version":1}