{"id":"40b777bc-fdae-4634-8296-76eae859ac41","arxiv_id":"2607.22669","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"No positive quadratic monogenic semiring, and no semiring whose defining algebraic number has composite primitive constant term, is a bi-UFS; new bi-HFS semirings are also constructed.","lead":"This paper proves that several natural families of 'bi-unique-factorization' semirings cannot exist beyond the standard nonnegative integers, and it constructs new examples for the relaxed half-factorial version. It is progress on the Bi-UF Positive Conjecture, a niche but active question at the intersection of factorization theory and semiring theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6 leaves the atomicity of B in the m=0, even n>2 subcase unjustified; without a patch, the main theorem is not fully established.","rationale":"Theorem 3.6 is the paper's main new result and Corollary 3.7 depends on it. The only unpatched step I can find is the B subcase; all other cases in the case split are justified by Propositions 3.4 and 3.5. This gap is internal and concrete, unlike the imported Theorem 1.1, which is a cited result with no evidence against it. The gap is likely fixable by the outlined argument, so it does not invalidate the conjecture; it means the manuscript should be revised before acceptance. The abstract's missing 'positive' and the final-sentence typo in Theorem 3.8 are editorial and not load-bearing.","tokens_in":16994,"tokens_out":38055,"duration_ms":370955,"concrete_test":"Complete the missing subcase: for even nonsquare n>2, let B=2√n+(n+4)/2. (i) For n≡2 mod4, use ac+nbd=(n+4)/2 with bd=0 and ad+bc=2 to show the integer factor cannot be 2, so B is irreducible. (ii) For n≡0 mod4, write B=2(√n+(n+4)/4) and verify √n+(n+4)/4 is irreducible by Proposition 3.4 (n∤(n+4)/4). If both checks pass, re-verify the distinctness of the two factorizations in (3.6) for e.g. n=8,12,20; if either fails, Theorem 3.6 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3.6, Case 2, subcase m=0 with even nonsquare n>2, the proof of atomicity of B is incomplete. From B=(a+b√n)(c+d√n), coefficient comparison gives ad+bc=2 and ac+nbd=(n+4)/2. The text shows bd>0 impossible and then asserts that, after swapping, one factor is a positive integer dividing 2, so B is either irreducible or twice an irreducible. This does not follow as written: if the integer factor is 2, then B=2(√n+(n+4)/4); one must prove that √n+(n+4)/4 is irreducible (e.g., via Proposition 3.4) and that the case n≡2 mod4 is impossible from the constant term. Without this, the right-hand side of (3.6) is not shown to produce a factorization into irreducibles distinct from (√n+2)^2. Since n ranges over an infinite family, the main theorem is not fully proved until this step is supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies complex semirings, focusing on monogenic semidomains S_α = N_0[α]. Its central claims are: (1) Theorem 3.6 shows that if α is a positive quadratic algebraic number for which the additive monoid (S_α,+) is factorial, then S_α is not multiplicatively factorial; hence no positive quadratic monogenic semidomain is a bi-UFS (Corollary 3.7). (2) Theorem 3.8 gives a second class: if α is a positive non-rational algebraic number whose primitive minimal polynomial has composite constant term, then S_α is not factorial. (3) Section 4 proves a structure theorem for semidomains whose additive monoid is a finite-rank free commutative monoid and uses it to formulate a Bi-UF Complex Conjecture. (4) Section 5 proves that N_0 is the only positive rational bi-HFS and constructs two families of bi-HFS semidomains distinct from N_0, one via pullbacks and one via Laurent polynomials.","tokens_in":17266,"tokens_out":19219,"duration_ms":170796,"significance":"If the proofs are completed, the paper gives substantial progress on the Bi-UF Positive Conjecture: it settles the conjecture for the natural class of quadratic monogenic semidomains and identifies another infinite class of monogenic semidomains that satisfy the conjecture. The structural theorem in Section 4 usefully narrows the scope of the extended complex conjecture, and the constructions in Section 5 enrich the known examples of bi-HFS semidomains. The paper is clearly written and builds carefully on the published characterization in [14]; I do not see circularity in that dependence. However, a key step in the proof of Theorem 3.6 is not fully justified, and since that theorem is the basis of Corollary 3.7, the main claim is not yet established as written.","major_comments":[{"comment":"The proof that B = 2√n + (n+4)/2 is irreducible or twice an irreducible is not derived. From a factorization B=(a+b√n)(c+d√n), coefficient comparison gives ad+bc=2 and ac+nbd=(n+4)/2. Ruling out bd>0 only shows that, after swapping, one factor is a positive integer dividing 2. If that integer is 1, the factorization is trivial; if it is 2, one must prove that the cofactor √n + (n+4)/4 is actually an element of S_α (which requires (n+4)/4 to be an integer, i.e., n≡0 mod 4), and that this cofactor is irreducible, e.g. via Proposition 3.4. For n≡2 mod 4 the factor 2 cannot occur, so B is irreducible, but this also needs an argument. Without this step, the right-hand side of (3.6) is not shown to be a factorization into irreducibles distinct from (√n+2)^2. Since n ranges over an infinite family, this gap is load-bearing.","section":"§3.1, Theorem 3.6, Case 2 (m=0, even n>2)"},{"comment":"The same type of gap occurs for C = 2α + r - 2m_1 + 1. From ad+bc+2m_1bd=2 and m_1>1, the paper correctly gets bd=0 and ad=2, so after swapping a=1 or a=2. It then concludes 'C is either irreducible or 2(α+ℓ)' without justification. If a=2, the cofactor has constant term (r-2m_1+1)/2, which must be an integer; the parity condition is not discussed. If a=1, the factorization is by a unit, but one still has to argue that reducibility of C would force a factorization with a=2 and positive b,d. These details are needed to ensure that splitting C on the right-hand side of (3.7) yields a genuine factorization into irreducibles distinct from (α+k)^2.","section":"§3.1, Theorem 3.6, Case 2 (m even positive)"}],"minor_comments":[{"comment":"The last sentence reads 'Since every atom in a factorial monoid is prime, so S_α is factorial.' The intended conclusion is that S_α is not factorial. Please correct this typo; the surrounding argument clearly supports the negation.","section":"Theorem 3.8, final paragraph"},{"comment":"The phrase 'we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem' is grammatically awkward; suggest 'motivated by a structural theorem'.","section":"Abstract"},{"comment":"There is a typo 'becase' in the sentence 'which is not possible becase α is not rational.' Also, in the proof of the claim, the induction step would be easier to follow if the sign condition on S_k were stated explicitly as an invariant, since it is used later to conclude b_k ∈ N_0.","section":"Theorem 3.8, proof"}],"recommendation":"major_revision","confidential_remarks":"The main results are likely correct and the gaps in Theorem 3.6 appear patchable with a short additional argument; I do not think rejection is warranted. The referee report focuses on the incomplete justification of atomicity in the even m=0 and m>0 subcases. The authors should also double-check the final sentence of Theorem 3.8 and the parity issue in the m>0 case. The dependence on Theorem 1.1 from [14] is a legitimate external characterization, not a circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this paper proves two substantial new chunks about the Bi-UF Positive Conjecture—no positive quadratic monogenic semidomain is a bi-UFS, and any positive non-rational algebraic α whose primitive minimal polynomial has composite constant term gives a non-factorial S_α. Both are real progress. The paper also contains a clean finite-rank structure theorem and two neat bi-HFS constructions, including a Laurent-polynomial example that generalizes the known pullback example. The core is almost certainly correct, and the authors engage seriously with the existing literature.\n\nThe main strategy in Theorem 3.6 is sensible: additive factoriality forces α² = mα + n, so you can compare coefficients and engineer explicit non-unique factorizations. The proof works for most cases. The one real soft spot is exactly what the stress-test note flags: in the m = 0, even n > 2 subcase, they assert B = 2√n + (n+4)/2 is either irreducible or twice an irreducible without proving the cofactor irreducible. That is a genuine gap as written. It is also a very local one. If a factor is the integer 2, the other factor is √n + (n+4)/4; when n ≡ 2 mod 4 that element is not even in the semiring, and when n ≡ 0 mod 4 Proposition 3.4 applies directly because n does not divide (n+4)/4 and α² ≠ α + k. So this is a patchable omission, not a fatal flaw.\n\nThere are also several editorial problems that should be fixed before publication: the abstract says 'quadratic algebraic number' where the theorem needs 'positive quadratic'; the multiplicative monoid is written as (S\\{1},·) instead of (S\\{0},·); there is a grammatical 'by motivated by' in the abstract; and the last sentence of Theorem 3.8 says 'so S_α is factorial' when it must mean 'not factorial.' These are easy.\n\nOn the broader picture, the dependence on Theorem 1.1 from [14] is acceptable—it is a published characterization theorem by Correa-Morris and the first author, and the new results do not reduce to its statement. The structural reduction in Section 4 is a useful way to narrow the extended conjecture, and the bi-HFS proofs are sound as far as I can tell.\n\nWho is this for: anyone working on factorization theory of semirings or the Bi-UF/Bi-HF conjectures. It deserves a serious referee. I would send it to peer review, with a request to patch the m=0 subcase and clean up the abstract and typos. A revised version would be a solid contribution.","headline":"Solid, genuinely new results on the Bi-UF Positive Conjecture, but Theorem 3.6 has a local proof gap in one subcase and the abstract overclaims 'quadratic' for 'positive quadratic.'","tokens_in":17728,"tokens_out":11130,"would_cite":true,"duration_ms":100115,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16Y60","13F15","13A05","11R09","13G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every positive quadratic algebraic number alpha, the monogenic semiring N0[alpha] cannot be both additively and multiplicatively factorial.","keywords":["Bi-UF Positive Conjecture","monogenic semiring","unique factorization monoid","half-factorial monoid","semidomain","quadratic algebraic number","atomic factorization","complex semiring"],"falsifier":"Find one positive quadratic algebraic number alpha for which S_alpha is both additively and multiplicatively factorial; the paper's Theorem 3.6 says none exists. Concretely, for alpha satisfying alpha^2 = m alpha + n, test the explicit candidates in the proof — for odd m the element (alpha + k)^2 with k = (r-m)/2, r = m^2 + 4n, and for even positive m the element (alpha + k)^2 with k = m1^2 - m1 + n, m1 = m/2 — and check whether the displayed factorizations actually give two distinct atomic factorizations; any failure would expose a gap.","tokens_in":16901,"feed_emoji":"🔢","tokens_out":4516,"duration_ms":41626,"temperature":0.7,"pith_summary":"The paper proves that the Bi-UF Positive Conjecture — which says that the nonnegative integers are the only positive semiring whose additive and multiplicative monoids both have unique factorization — holds for every quadratic monogenic semiring. It does so by assuming additive factoriality and deriving explicit non-unique multiplicative factorizations from the coefficient identities forced by alpha^2 = m alpha + n. The paper also shows that if the constant term of the primitive minimal polynomial of a positive non-rational algebraic generator is composite, the semiring is not factorial, giving another class satisfying the conjecture. It extends the conjecture from positive to complex semirings and narrows the search to finite N0-spans of algebraic numbers. In the relaxed half-factorial setting, it proves that N0 is the only positive rational bi-HFS and constructs two families of bi-HFS examples distinct from N0.","feed_headline":"No quadratic monogenic semiring is a bi-UFS","feed_subtitle":"Quadratic generators never give a bi-UFS; the only positive rational bi-HFS is N0.","key_machinery":"The load-bearing tool is a characterization (Theorem 1.1) stating that for a positive algebraic number alpha, the additive monoid of S_alpha is a UFM exactly when its atoms are the powers 1, alpha, ..., alpha^{d-1}. In the quadratic case this reduces every element to a unique expression c + d alpha with c,d in N0, and the relation alpha^2 = m alpha + n converts any multiplicative product into two coefficient equations. These equations are used to certify atoms and to manufacture explicit non-unique factorizations. For the non-quadratic Theorem 3.8, the primitive minimal polynomial and Gauss's lemma control divisibility, showing that alpha is an atom that fails to be prime.","core_discovery":"The central theorem (Theorem 3.6) states: if alpha is a positive quadratic algebraic number such that the additive monoid of S_alpha = N0[alpha] is a unique factorization monoid, then the multiplicative monoid of S_alpha is not factorial. Hence no positive quadratic monogenic semidomain is a bi-UFS (Corollary 3.7). The proof splits into cases by the parity of the linear coefficient m in the minimal polynomial x^2 - m x - n and produces, in each case, two distinct factorizations of one element into atoms using identities such as (alpha + k)^2 = r(alpha + ...) and (alpha+k)^2 = r(2alpha + ...). A second theorem (Theorem 3.8) shows that if w_alpha(0), the constant term of the primitive integer","pith_inferences":["The coefficient-comparison technique used for quadratics may extend to higher-degree monogenic semirings, where additive factoriality gives a basis {1, alpha, ..., alpha^{d-1}} and multiplication by alpha yields a linear recurrence; analogous identities could yield non-unique factorizations in degree d.","The paper's Theorem 3.8 suggests a testable heuristic: for algebraic integers, composite field norm should imply failure of multiplicative factoriality, so bi-UFS candidates must have prime norm; this could be checked against known classes.","The Laurent-polynomial construction indicates that the bi-HF condition is far less restrictive than bi-UF; a broader classification of bi-HFS may be possible along block-monoid lines.","The structural theorem (Theorem 4.2) effectively converts the complex conjecture into a question about subrings of number fields, which may be approachable by existing results on arithmetic of orders."],"forward_implications":["The Bi-UF Positive Conjecture is now known to hold for every quadratic monogenic semidomain; the open cases are higher-degree generators.","Any bi-UFS with reduced, finite-rank additive monoid must be isomorphic to a finite N0-span of algebraic numbers in a number field (Theorem 4.2), so proving the complex conjecture reduces to ruling out factoriality in those spans.","If the additive monoid has rank 1, the only bi-UFS is N0 (Corollary 4.3).","The half-factorial analogue is strictly weaker: bi-HFS examples exist beyond N0, including Laurent-polynomial semirings, while positive rational bi-HFS still force N0.","The composite-constant-term theorem gives a purely arithmetic obstruction: any positive algebraic generator whose primitive polynomial has composite constant term generates a non-factorial monogenic semiring."],"fun_headline_variants":["Quadratic monogenic semirings never are bi-UFS","Bi-UFS impossible for quadratic monogenic semirings","Proof: quadratic monogenic semirings fail bi-UFS","Quadratic monogenic semirings lack the bi-UFS property"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole quadratic argument rests on the imported characterization that additive factoriality of S_alpha forces the additive atoms to be exactly the powers 1, alpha, ..., alpha^{d-1}; if that characterization fails for some positive quadratic alpha, the coefficient identities and the non-unique factorizations built from them no longer apply.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic monogenic semirings never are bi-UFS","Bi-UFS impossible for quadratic monogenic semirings","Proof: quadratic monogenic semirings fail bi-UFS","Quadratic monogenic semirings lack the bi-UFS property"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":3023,"prompt_tokens":840,"completion_tokens":2183,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":2112}},"tokens_in":584,"tokens_out":2183,"duration_ms":16811,"temperature":1.0,"reasoning_tokens":2112,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:38:38.997577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one positive quadratic algebraic number alpha for which S_alpha is both additively and multiplicatively factorial; the paper's Theorem 3.6 says none exists. Concretely, for alpha satisfying alpha^2 = m alpha + n, test the explicit candidates in the proof — for odd m the element (alpha + k)^2 with k = (r-m)/2, r = m^2 + 4n, and for even positive m the element (alpha + k)^2 with k = m1^2 - m1 + n, m1 = m/2 — and check whether the displayed factorizations actually give two distinct atomic factorizations; any failure would expose a gap.","supporting_citations":[],"review_version":1}