{"id":"bd070198-a9ad-4f55-843a-78b0503654a3","arxiv_id":"2507.07022","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For squarefree principal vector-spread Borel ideals, the paper gives the minimal primary decomposition, proves sequential Cohen-Macaulayness, and classifies normal torsionfreeness via the index bounds j_i <= sum_{s<=i} t_s.","lead":"This paper studies squarefree principal vector-spread Borel ideals, a class of monomial ideals in polynomial rings. It describes their minimal primary decomposition, proves they are sequentially Cohen-Macaulay, and classifies when their ordinary and symbolic powers coincide.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 3.1(b)⇒(c) contains a false case step: p2∈{j2,...,jd} fails for v1=x1 x_{t1+1} x_{j2}..., so w∉I^2 is not established as written.","rationale":"I agree with the reader's overall CONDITIONAL verdict, but I locate the most load-bearing concern differently. The reader's weakest assumption is the facet enumeration in Theorem 1.2. I examined that proof: the greedy construction of ℓ_i and the containment of every face in one of the claimed facet forms appears sound, and small examples (e.g., t=(2,1), u=x2x5x8 and t=(3,1), u=x2x5x6) produce exactly the listed facets. No concrete counterexample to Theorem 1.2 emerged. By contrast, the proof of Theorem 3.1(b)⇒(c) contains a demonstrable false assertion in the step that rules out w∈I^2. The monomial v1=x1x4x7 in the reduction case t=(3,1), u=x4x7x8 is a valid generator dividing w with p2=t1+1 not in {j2,j3}, contradicting the proof's claim that p2 must lie in {j2,...,jd}. This is not a typo or a reversed inclusion; it is a missing case in the central argument. The classification may still be true, and in the tested case the intended conclusion w∉I^2 holds by a different counting argument, so the appropriate response is CONDITIONAL rather than REJECT. Since my verdict matches the reader's, I set verdict_should_be to UNCHANGED, but flag that the weakest point is the false proof step, not the primary decomposition.","tokens_in":14851,"tokens_out":33864,"duration_ms":299936,"concrete_test":"Run Macaulay2 on the reduction case t=(3,1), u=x4x7x8 in K[x1,...,x8]: (i) compute associatedPrimes I and check w=x1x2x3x4x7x8 belongs to I^(2); (ii) check w∉I^2, e.g., by noting I^2 is generated in degree 6 and testing membership. If w∉I^2, the intended claim holds but the proof's p2-step is still false, so Theorem 3.1 requires a repaired argument (CONDITIONAL). If w∈I^2, the classification itself is false (REJECT).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.1, implication (b)⇒(c), after localization one may assume ℓ=1 and j1≥t1+1. The argument defines w=(x1...x_{t1+1})(x_{j2}...x_{jd}) and tries to show w∉I^2. If w=v1v2g with v1=x_{p1}...x_{pd}∈G(I), the proof asserts: 'Since p2 ≥ p1 + t1 it follows that p2 ∈ {j2,...,jd}.' This is false. Take d=3, t=(3,1), u=x4x7x8 (so j1=4≥t1+1, j2=7, j3=8). Then w=x1x2x3x4x7x8, and v1=x1x4x7 is a generator: it is t-spread (gaps 3 and 3 ≥ 1) and satisfies the upper bounds p1=1≤4, p2=4≤7, p3=7≤8. It divides w, but p2=4=t1+1∉{7,8}. Thus the claimed deduction fails. The theorem may still be true (direct degree and counting arguments show w∉I^2 in this example), but the proof as written does not cover the case p2=t1+1. Since this step is the only place where the contradiction w∈I^2 is ruled out, the central classification is not fully established by the given argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies squarefree principal vector-spread Borel ideals B_t(u), where t=(t_1,...,t_{d-1}) and u is a t-spread monomial of degree d. Its main results are: Theorem 1.2 gives an explicit minimal primary decomposition in terms of t-spread supports; Theorem 2.1 proves these ideals are sequentially Cohen-Macaulay by showing their Alexander duals are vertex splittable; Theorem 3.1 classifies when such ideals satisfy I^(k)=I^k for all k, by showing this is equivalent to the inequalities j_i ≤ t_1+...+t_i for all i<d. The paper also records corollaries on height, vertex decomposability, and non-finiteness of associated primes.","tokens_in":15209,"tokens_out":17435,"duration_ms":191136,"significance":"If Theorem 3.1 is correct, the paper provides a complete and elegant classification of normally torsionfree ideals in this class, and the explicit primary decomposition in Theorem 1.2 is independently valuable and likely to be useful for further work on vector-spread Borel ideals. The sequential Cohen-Macaulayness result in Theorem 2.1 is also a natural strengthening of known properties of Borel ideals. The proof of Theorem 3.1, however, contains a false intermediate assertion in the implication (b)=> (c), and that implication is the core of the classification. The other parts of the paper appear sound to me; in particular, I did not find a concrete gap in the facet-exhaustion argument of Theorem 1.2, although that proof is terse. The paper also reports computational verification of examples, but the main theorems do not rely on those computations.","major_comments":[{"comment":"The assertion 'Since p2 ≥ p1 + t1 it follows that p2 ∈ {j2,...,jd}' is false in general. For example, take d=3, t=(3,1), and u=x4x7x8 in K[x1,...,x8]. Then the proof's monomial is w=(x1x2x3x4)(x7x8), and v1=x1x4x7 is a generator of B_t(u): it is t-spread and satisfies the upper bounds p1=1≤4, p2=4≤7, p3=7≤8. It divides w, yet p2=4=t1+1 is not in {j2,j3}={7,8}. Thus the claim that p2 belongs to the right-hand factor is not valid, and the proof does not rule out w∈I^2 as written. Since this is the only step that excludes w from I^2, the implication (b)⇒(c) is not established by the given argument. The theorem may still be true—degree and support considerations rule out this particular example—but a corrected argument is required.","section":"§3, proof of Theorem 3.1, (b)⇒(c), around Eq. (3.3)"}],"minor_comments":[{"comment":"The expression 'B′t′(u/j1)' should read 'B′t′(u/x_{j1})', consistent with the notation in Eq. (2.1).","section":"Lemma 2.3, text near Eq. (2.1)"},{"comment":"The proof that every face G is contained in one of the listed facets is rather compressed; in the first case it is not immediately clear which generator v has t-spread support equal to the containing set. I verified that v = x_{ℓ1}⋯x_{ℓk}x_{j_{k+1}}⋯x_{jd} works, so this is a readability issue rather than a mathematical gap, but the generator should be written explicitly.","section":"Theorem 1.2, converse direction"},{"comment":"In the definition of ℓ, the condition 'jp − jp−1 = tp−1' is used for p=1, but j0 is never defined. The range of p should be adjusted or j0 should be declared, to avoid ambiguity.","section":"§3, proof of Theorem 3.1, (c)⇒(a)"},{"comment":"There are several minor typographical slips, such as 'witht-spread Borel generators' in the Introduction; these should be corrected in a final proofread.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The false step in Theorem 3.1 is localized and likely repairable; the rest of the paper appears sound, and the theorem itself is plausible. I would not reject on this basis, but the authors must supply a correct proof of the divisibility claim in (b)⇒(c) before the classification can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely useful paper on squarefree principal vector-spread Borel ideals. The explicit primary decomposition in Theorem 1.2 and the vertex-splitting proof of sequential Cohen-Macaulayness in Section 2 are real contributions, and the classification in Theorem 3.1 plausibly extends [20] beyond the monotone case. But the proof of (b)=>(c) in Theorem 3.1 has a real gap, and the stress-test note is correct.\n\nFor d=3, t=(3,1), u=x4x7x8, take v1=x1x4x7 in G(I). With w=(x1x2x3x4)(x7x8), the proof asserts that since p2>=p1+t1, p2 must lie in {j2,j3}={7,8}. This fails: p2=4=t1+1. The subsequent conclusion that x_{p2}...x_{pd}=x_{j2}...x_{jd} is unsupported. The theorem may well be true, and the gap looks fixable with a degree or counting argument handling p2=t1+1, but as written the central implication is not established.\n\nWhat is solid: the primary decomposition is the main new structural result, and the facet analysis, though intricate, is plausible. The vertex splittability argument in Section 2 is a clean extension of known machinery. The authors are also honest about the limitation of [20] and explicitly say where their proof diverges. Citation practice is normal; self-citations are to independent prior results.\n\nSoft spots beyond the gap: Section 2 has typos and some reversed inclusions in the displays of Lemma 2.3 and equation (2.2). These look superficial but need cleanup. The proof of Theorem 1.2 is a long case analysis; I did not find a counterexample, but it is hard to check quickly.\n\nBottom line: this paper deserves a serious referee and likely acceptance after the (b)=>(c) proof is repaired. I would not desk-reject it, and I would bring it to a reading group as a useful example of a plausible classification with a genuinely fixable proof gap.","headline":"Solid structural results with a genuine, fixable gap in the main classification proof.","tokens_in":15747,"tokens_out":2480,"would_cite":true,"duration_ms":23867,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13B25","13F20","13H10","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for a squarefree principal vector-spread Borel ideal, symbolic powers equal ordinary powers exactly when the generator's indices satisfy $j_i \\le t_1+\\cdots+t_i$ for all $i$, through an explicit primary decomposition…","keywords":["squarefree monomial ideals","vector-spread Borel ideals","primary decomposition","sequentially Cohen-Macaulay","symbolic powers","normally torsionfree","Alexander duality","t-spread support"],"falsifier":"Enumerate the facets of the Stanley-Reisner complex for an example such as $t=(2,2,1)$, $u=x_3x_6x_{10}x_{13}$ in $K[x_1,\\ldots,x_{13}]$, and compare the list with the description in Theorem 1.2; any missing or spurious facet would falsify the primary decomposition. To test Theorem 3.1 directly, compute the ordinary square and the second symbolic power of an ideal satisfying $j_i \\le \\sum_{s=1}^i t_s$ for all $i$, for instance $t=(3,2,1)$, $u=x_4x_9x_{13}x_{15}$ in $K[x_1,\\ldots,x_{15}]$; if they differ, the classification collapses.","tokens_in":14657,"feed_emoji":"⚖️","tokens_out":17438,"duration_ms":144089,"temperature":0.7,"pith_summary":"This paper investigates squarefree principal vector-spread Borel ideals: monomial ideals generated by a single monomial whose successive variable indices are separated by prescribed gaps $t_1,\\ldots,t_{d-1}$. The authors compute the minimal primary decomposition of such an ideal in closed form, show that every member of the class is sequentially Cohen-Macaulay — it admits a filtration whose successive quotients are Cohen-Macaulay with increasing dimension — and use those two structural results to classify precisely when ordinary powers coincide with symbolic powers. The classification is a finite list of integer inequalities: $j_i \\le t_1+\\cdots+t_i$ for $i=1,\\ldots,d-1$, where the $j_i$ are the indices of the generator. The upshot is a complete, checkable structural picture of this class of ideals, with symbolic power equality reduced to simple arithmetic.","feed_headline":"One inequality classifies symbolic vs ordinary powers","feed_subtitle":"One short inequality on the generator decides whether symbolic and ordinary powers match.","key_machinery":"The t-spread support of a monomial $v = x_{i_1}\\cdots x_{i_\\ell}$ is the union of intervals $[i_s, i_s + t_s - 1]$; these sets are the building blocks for the facets of the Stanley-Reisner complex whose ideal is $B_t(u)$. The load-bearing mechanism is the complete description of these facets (equation (1.1)): every facet is either the t-spread support of a minimal generator or a set $\\operatorname{supp}_t(x_{\\ell_1}\\cdots x_{\\ell_{s-1}}x_{j_s}) \\cup [j_s+1,n]$. From that description flow the minimal primary decomposition, the vertex-splitting decomposition of the Alexander dual ideal in Lemma 2.3, and the enumeration of associated primes used throughout. Here vertex splitting means a recursive construction of a monomial ideal as $x_i I_1 + I_2$ with $I_2 \\subseteq I_1$, a property that guarantees the linear-resolution behavior needed for the sequential Cohen-Macaulay conclusion. The proof machinery for the classification also includes monomial localization and the a-restriction criterion, which reduce the symbolic-power equality to a smaller ideal after removing one index of the generator.","core_discovery":"The central claim is Theorem 3.1: for a squarefree principal vector-spread Borel ideal $I = B_t(u)$, the properties 'normally torsionfree', '$I^{(k)} = I^k$ for all $k \\ge 1$', and the inequalities $j_i \\le \\sum_{s=1}^i t_s$ for $i = 1,\\ldots,d-1$ are equivalent. The proof rests on Theorem 1.2, which describes the minimal primary decomposition as the intersection of primes $P_{[n]\\setminus G}$ over two explicitly listed families of subsets, and on Theorem 2.1, which establishes sequential Cohen-Macaulayness. To prove Theorem 2.1 the authors show the Alexander dual ideal $I^\\vee$ — the squarefree monomial ideal generated by the facet complements of the Stanley-Reisner complex — admits the vertex-splitting decomposition $x_1 B'_t(u)^{\\vee} + B'_{t'}(u/x_{j_1})^{\\vee}$, allowing induction on degree and number of variables. To prove Theorem 3.1 they combine a monomial-localization reduction with the a-restriction criterion for symbolic powers. The paper claims these results hold for every vector $t_1,\\ldots,t_{d-1} \\ge 1$, extending previously known cases such as $d = 2$.","pith_inferences":["The contradiction argument in the proof targets the second symbolic power, so in this class the full equality of symbolic and ordinary powers may already be forced by the single equality of the ordinary square and the second symbolic power; a reader could test whether condition (c) is equivalent to that single equality.","The vertex-splitting recursion used here is limited to squarefree ideals, but the same recursion suggests a route into the non-squarefree case that the paper leaves open, provided the t-spread support construction is replaced by a suitable non-squarefree analogue.","The analytic spread analysis in Proposition 3.6 implies that under condition (c) the maximal ideal is never an associated prime of any power; examining the depths of the quotients $S/I^k$ might reveal stronger homological stabilization than the paper states.","Because the class is parameterized by an arbitrary spread vector and a generator, it gives a two-parameter family of monomial ideals with a closed-form normal-torsion-freeness answer, which could serve as a test bed for more general criteria."],"forward_implications":["Deciding normal torsion-freeness for this class requires only checking finitely many integer inequalities on the generator's indices; no primary decomposition is needed in practice.","Every squarefree principal vector-spread Borel ideal is sequentially Cohen-Macaulay, and its Stanley-Reisner complex is vertex decomposable, so the class inherits the homological and combinatorial consequences of those properties.","The explicit primary decomposition gives a closed-form list of associated primes and the height of the ideal, which is the smallest index of the generator.","The classification covers arbitrary spread vectors, subsuming the known one-dimensional and squarefree strongly stable cases.","The induction pattern used in the proof — remove one index of the generator and pass to a smaller spread vector — provides a recursive certificate for the equality of powers."],"supporting_citations":[{"why":"Supplies the equivalence between normal torsion-freeness and equality of ordinary and symbolic powers, and the Alexander-dual criterion used for sequential Cohen-Macaulayness.","marker":"[15]"},{"why":"Gives the classification for the case d = 2 that the paper extends to arbitrary d, and is the starting point for the (b) implies (c) direction.","marker":"[20]"},{"why":"Provides the a-restriction criterion (Theorem 3.2) that reduces symbolic-power equality to smaller ideals in the proof of (c) implies (a).","marker":"[21]"},{"why":"Provides vertex splittability theory and the Alexander-dual splitting lemma used to prove sequential Cohen-Macaulayness.","marker":"[18]"},{"why":"Establishes linear resolutions for vector-spread Borel ideals, enabling the linear-relation-graph formula for analytic spread used in Proposition 3.6.","marker":"[11]"},{"why":"Shows the Rees algebra of a vertex splittable ideal is Cohen-Macaulay and that Ass(I^k) ascends, used in Corollary 3.5 to exclude the maximal ideal.","marker":"[19]"},{"why":"Gives the scaling lemma for symbolic powers of a monomial times an ideal, used in Lemma 3.3 to reduce to the minimal support case.","marker":"[13]"},{"why":"Proves principal vector-spread Borel ideals are vertex splittable, used in Corollary 3.5.","marker":"[4]"},{"why":"Provides the formula for analytic spread of a monomial ideal via its linear relation graph, used in the proof of Proposition 3.6.","marker":"[7]"}],"fun_headline_variants":["A single inequality pins down symbolic vs ordinary powers","Squarefree Borel ideals: equality of powers decided by one inequality","When do symbolic and ordinary powers match? One inequality tells all","Borel ideals: one check for symbolic-ordinary power agreement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the list of sets in Theorem 1.2 exhausts the facets of the Stanley-Reisner complex of $B_t(u)$ is the load-bearing step; if some facet were missed, the primary decomposition, the Alexander dual computation, and the final classification would not follow.","fun_headline_variants_meta":{"raw":{"variants":["A single inequality pins down symbolic vs ordinary powers","Squarefree Borel ideals: equality of powers decided by one inequality","When do symbolic and ordinary powers match? One inequality tells all","Borel ideals: one check for symbolic-ordinary power agreement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1249,"prompt_tokens":849,"completion_tokens":400,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":331}},"tokens_in":465,"tokens_out":400,"duration_ms":4854,"temperature":1.0,"reasoning_tokens":331,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:50:30.388784+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate the facets of the Stanley-Reisner complex for an example such as $t=(2,2,1)$, $u=x_3x_6x_{10}x_{13}$ in $K[x_1,\\ldots,x_{13}]$, and compare the list with the description in Theorem 1.2; any missing or spurious facet would falsify the primary decomposition. To test Theorem 3.1 directly, compute the ordinary square and the second symbolic power of an ideal satisfying $j_i \\le \\sum_{s=1}^i t_s$ for all $i$, for instance $t=(3,2,1)$, $u=x_4x_9x_{13}x_{15}$ in $K[x_1,\\ldots,x_{15}]$; if they differ, the classification collapses.","supporting_citations":[{"cited_title":"Sayedsadeghi, M","cited_arxiv_id":null,"evidence_quote":"Provides the a-restriction criterion (Theorem 3.2) that reduces symbolic-power equality to smaller ideals in the proof of (c) implies (a)."},{"cited_title":"Moradi, F","cited_arxiv_id":null,"evidence_quote":"Provides vertex splittability theory and the Alexander-dual splitting lemma used to prove sequential Cohen-Macaulayness."},{"cited_title":"Ficarra, Vector-spread monomial ideals and Eliahou–Kervaire type resolutions","cited_arxiv_id":null,"evidence_quote":"Establishes linear resolutions for vector-spread Borel ideals, enabling the linear-relation-graph formula for analytic spread used in Proposition 3.6."},{"cited_title":"Normal Rees algebras arising from vertex decomposable simplicial complexes","cited_arxiv_id":"2311.15135","evidence_quote":"Shows the Rees algebra of a vertex splittable ideal is Cohen-Macaulay and that Ass(I^k) ascends, used in Corollary 3.5 to exclude the maximal ideal."},{"cited_title":"Symbolic powers of polymatroidal ideals","cited_arxiv_id":"2502.19998","evidence_quote":"Gives the scaling lemma for symbolic powers of a monomial times an ideal, used in Lemma 3.3 to reduce to the minimal support case."},{"cited_title":"Crupi, A","cited_arxiv_id":null,"evidence_quote":"Proves principal vector-spread Borel ideals are vertex splittable, used in Corollary 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the formula for analytic spread of a monomial ideal via its linear relation graph, used in the proof of Proposition 3.6."}],"review_version":1}