{"id":"7d05a16b-6355-46aa-97df-4bc16105e6f7","arxiv_id":"2501.13578","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every indecomposable peak P-space of a type A poset is stable under a bilinear-form weight and under an angle stability function from a new geometric model.","lead":"This paper proves that every indecomposable socle-projective representation of a type A poset is stable, using a bilinear form and also a new geometric model with polygons and line segments. It builds a new geometric realization of the category of such representations and connects the bilinear-form stability with the geometric stability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.5's case analysis omits proper subspaces supported on isolated maximal points, so the written proof of Theorem 2.6 does not check all proper subrepresentations.","rationale":"The reader's weakest-assumption pinpoints the omitted S_2^(r) and S_3^(r) computations in Proposition 2.5. My stress-test found a related but distinct and arguably more serious omission: the decomposition into W_X blocks used after equation (2.5) does not include proper subspaces whose support contains isolated maximal points of I, even in the fully written S_1^(r) case. Thus the proof of Proposition 2.5, and hence of Theorem 2.6, does not currently verify the defining inequality θ(W) < 0 for all proper peak P-subspaces. This is a proof gap rather than a demonstrated counterexample: for the explicit S_1^(3) example the missing θ value is negative, and the same is plausible for the other isolated-maximal configurations. The geometric proof in Theorem 4.3 is also terse, but it does not repair the Proposition 2.5 gap because it is presented as an independent route. Given that the main theorems are plausible and supported by worked examples, but the central bilinear-form proof has an incomplete case check, the reader's CONDITIONAL verdict remains appropriate. I do not see grounds to reject the paper outright, nor to accept it unconditionally without the missing computations.","tokens_in":29019,"tokens_out":17206,"duration_ms":159563,"concrete_test":"Enumerate all proper nonempty subsets I of supp U ∩ max P for the sincere type-A families S_1^(r), S_2^(r), and S_3^(r) with r = 2, 3, 4; for each compute θ(W) = b_P(dim U, dim W) - b_P(dim W, dim U) using the displayed incidence matrices and their inverses, including isolated maximal components not captured by any X ⊆ A, and also computing the standard-vector values for S_2^(r) and S_3^(r). If every θ(W) < 0, the gap is repairable; if any θ(W) >= 0, Proposition 2.5 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 2.5 reduces every proper peak P-subspace W = U_{K_I} to blocks W_X, where X is a set of consecutive points in A = {x_i in min P | z_i, z_{i+1} in I}, and then checks θ(W_X) < 0. But this decomposition does not cover maximal points of I that are not adjacent to any element of A. For example, for P = S_1^(3) and I = {z_1, z_3}, one has A = ∅, supp W = {z_1, z_3}, while the sum over all W_X with X ⊆ A is zero. The inequalities in cases (i)-(iii) therefore never evaluate θ on this proper subrepresentation. This is a genuine gap in the only proof of the bilinear-form stability theorem, and it already occurs for the fully displayed family S_1^(r), not only for the delegated S_2^(r) and S_3^(r) computations. The missing cases are likely harmless — for the displayed example θ(e_{z_1}+e_{z_3}) = -2 — but the proof as written does not establish the required strict negativity for every proper peak P-subspace. Since Theorem 2.6 passes directly through Proposition 2.5, the central claim rests on this unfinished case check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the Futorny–Iusenko stability notion from ordinary poset representations to the category P-spr of peak P-spaces, i.e., socle-projective modules over incidence algebras, and proves that for every poset of type A every indecomposable object is stable. The first proof uses a weight derived from the incidence-algebra bilinear form (Proposition 2.5 and Theorem 2.6). The second proof constructs a geometric model: inspired by Schiffler–Serna and the BGMS polygon model, the authors define a category of sp-segments, prove it is equivalent to the category of socle-projective modules (Theorem 3.22), and then define a stability function from the angles of oriented line segments (Theorem 4.3). A final result (Theorem 4.4) connects the geometric stability function to the bilinear-form weights for sincere type A posets.","tokens_in":29258,"tokens_out":14379,"duration_ms":124219,"significance":"If the proofs are completed, the paper gives a genuine extension of stability theory to socle-projective categories for all type A posets, with two complementary mechanisms: explicit bilinear-form weights and a geometric angle stability function. The categorical equivalence in Theorem 3.22 is an independently interesting structural result, and the paper rightly credits the prior geometric work of Schiffler–Serna and BGMS. The bilinear-form construction is explicit and algorithmic, and the geometric model gives a uniform reason why all indecomposables are stable. The significance is currently conditional, because several load-bearing computations and inequalities are delegated or asserted rather than demonstrated.","major_comments":[{"comment":"The case analysis reduces a proper peak P-subspace W = U_{K_I} to a sum over sets X of consecutive points in A = {x_i in min P | z_i, z_{i+1} in I}, and verifies θ(W_X) < 0 for each X. This does not cover proper subspaces supported on maximal points that are not adjacent through A. For P = S_1^(3) and I = {z_1, z_3}, one has A = ∅ and supp W = {z_1, z_3}, so the sum over the W_X is empty and no inequality is checked. The missing case is harmless in this example because θ(e_{z_1}+e_{z_3}) = -2, but the written proof of Theorem 2.6 therefore does not establish strict negativity for every proper peak P-subspace, and the gap already occurs in the fully displayed family S_1^(r). The proof needs an explicit treatment of isolated maximal points of I or a different decomposition that accounts for them.","section":"§2, Proposition 2.5, proof after eq. (2.5)"},{"comment":"The formulas for the families S_2^(r) and S_3^(r) are delegated: the text states that 'the same arguments can be used' without displaying the analogous computations with the inverse incidence matrices, the endpoint corrections, or the resulting signs. These two families have different boundary conventions (the extra points x_r, x_0, and x_{r+1}), and the subsequent case analysis in items (ii) and (iii) depends on exactly those endpoint terms. Since Theorem 2.6 relies on Proposition 2.5 for all sincere type A posets, the omitted S_2^(r) and S_3^(r) computations are load-bearing and should be written out.","section":"§2, Proposition 2.5, eqs. (2.3)–(2.4)"},{"comment":"The proof asserts that after partitioning I into consecutive subsets I_1, ..., I_s, each summand Φ_n(M^{I_j}) has argument below θ(M), and 'hence' the argument of the sum is below θ(M). This implication is not valid without an additional sector argument: vectors whose arguments are individually smaller than θ(M) can sum to a vector whose argument is larger than θ(M). The proof must show that all vectors Φ_n(M^{I_j}) lie in a common half-plane bounded by the line of angle θ(M), or otherwise bound the argument of the sum directly. Because Theorem 4.3 is one of the two central stability theorems, this assertion cannot remain as it is.","section":"§4, Theorem 4.3, proof"},{"comment":"The proof claims that for every m ≥ 1 the vectors w+η and κ+η have positive entries. This is false for m = 1: in case (i) of (4.2), the entry of w+η at an interior maximal point z_i with 1 < i < r equals zero, and the entry of κ+η at every minimal point x_j equals zero. Concretely, for P = S_1^(3) and M the simple representation supported at the middle maximal point z_2, one obtains Z_1(M) = 2i, whose argument is π/2, i.e., on the boundary of the half-plane H in Definition 4.1. Thus the central-charge condition of Definition 4.1 fails for m = 1. The statement should be changed to m sufficiently large, or a separate treatment of the boundary cases should be supplied.","section":"§4, Theorem 4.4 and eqs. (4.2)–(4.3)"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors, including 'objet', 'dence', 'subpace', 'asocciated', 'Auslander-Reiter', and inconsistent hyphenation of 'peak P-space'; the paper should be carefully copyedited.","section":"General"},{"comment":"The lemma speaks of 'proper subsets' of supp U ∩ max P, which includes the empty subset; the corresponding admissible subspace K = 0 gives the zero subrepresentation, for which θ(U_0) = 0, not < 0. The stability arguments should explicitly restrict to nonempty proper subsets or otherwise exclude the zero subrepresentation.","section":"§2, Lemma 2.1"},{"comment":"In the density part of the proof the text refers to 'Lemma 3.10 (b)' and 'Theorem 4.4 [21]'; with the numbering in this preprint, Lemma 3.10 has no part (b) and the intended statement in the current paper is not clear. The cross-reference should be corrected.","section":"§3, Theorem 3.22, proof"},{"comment":"The supports of the proper subrepresentations N and L are stated as {1} and {3,4}; this appears inconsistent with Lemma 2.2 for the displayed poset, where the two proper subsets of the relevant maximal set {3,5} give supports {3,2} and {5,4}. The example should be checked and the supports corrected.","section":"§4, Example 4.5"},{"comment":"The notation overloads P(Q) and P(QF): the same symbol is used for a polygon, a set of non-frozen line segments, and a category of sp-segments. A consistent notation or an index of symbols would greatly improve readability.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a clear, valuable goal: it extends Futorny–Iusenko stability from one-peak posets to all socle-projective categories of type A, and it adds a new geometric model via sp-segments. If the main theorem is right, it completes a natural story, and the geometric equivalence in Section 3 is a genuine new construction, not just a repackaging of BGMS. The authors position it honestly relative to [7] and [21].\n\nThe soft spots are in the proofs, and they are real. The stress-test note is correct. In Proposition 2.5, the reduction to consecutive blocks X ⊆ A does not cover proper peak P-subspaces whose support contains isolated maximal points. Example: P = S1^(3), I = {z1,z3}. Then A = ∅, so the W_X sum is empty and the proof never evaluates θ on this subspace. Direct computation gives θ = −2, so the theorem survives this example, but the proof as written does not establish it. This gap occurs in the fully displayed S1 family, not only in the delegated S2/S3 computations.\n\nI also have trouble with the displayed formulas (2.2)–(2.4). They appear to conflate the dimension vector of the auxiliary W_j with that of the sincere U. Computing directly from the incidence matrix for S1^(3) gives θ(e_z1) = −1, θ(e_z2) = −2, θ(e_x1) = 2, which do not match what the formulas suggest. The final estimates in cases (i)–(iii) seem to use the correct numbers, so the intended argument is probably salvageable, but the text is internally inconsistent.\n\nSection 3 is better. The category of sp-segments and the functor Ω are carefully defined, and the proof of the equivalence, though long, is mostly checkable. The weak spot is Theorem 4.3: the proof is only about a page, and the central angle inequality is asserted rather than shown. The claim that M^I decomposes as a direct sum of indecomposable M^{I_j} needs justification, and the statement that each Φ_n(M^{I_j}) is an sp-segment at some translate needs detail. This is a moderate gap, less serious than the Section 2 issue.\n\nBottom line: the paper is for representation theorists working on poset stability and geometric models. It deserves a serious referee; the results are plausible and the geometric model is a real contribution. But the proof of the main bilinear-form theorem is not complete, and Theorem 4.3 is too sketchy. I would send it out, with the expectation of a major revision.","headline":"Plausible and interesting results, but the proof of the main bilinear-form stability theorem has a real gap (missing isolated maximal-point cases) and the geometric stability proof is too sketchy; deserves review but needs major revision.","tokens_in":29807,"tokens_out":15395,"would_cite":false,"duration_ms":120796,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every indecomposable peak P-space over a type A poset is stable—for a weight from the incidence matrix and for an angle from a polygon model.","keywords":["stability","socle-projective representations","peak posets","posets of type A","bilinear form","geometric model","line segments","Auslander-Reiten quiver"],"falsifier":"Compute the weight θ(W_X) = b_P(dim U, dim W_X) - b_P(dim W_X, dim U) directly for the sincere peak P-spaces of $S_2^{{(r)}}$ and $S_3^{{(r)}}$ using the incidence matrices displayed in Proposition 2.5, for r = 2 and r = 3. If any proper peak P-subspace W_X of the sincere space has θ(W_X) ≥ 0, Theorem 2.6 is false; a negative value on every W_X would confirm the missing step.","tokens_in":28798,"feed_emoji":"📐","tokens_out":10952,"duration_ms":92412,"temperature":0.7,"pith_summary":"The paper extends a stability notion for representations of posets to the wider category of socle-projective representations, which in this setting are called peak P-spaces: systems of finite-dimensional vector spaces indexed by the points of a poset, one subspace for each point inside an ambient space built from the maximal points. The main result is that when the poset has type A, every indecomposable peak P-space is stable for some weight. This is established twice, once with an explicit weight derived from the poset's incidence matrix and once with a stability function that measures the angle of an oriented line segment in a polygon model. The paper also gives a new geometric realization of the whole category as a category of such line segments and shows the two stability constructions agree in explicit form. If the result is right, the stability machinery previously available for one-peak poset representations now covers all type A socle-projective categories, with weights that can be written down.","feed_headline":"All type A peak spaces are stable, two ways","feed_subtitle":"Bilinear-form weights and polygon angles both certify every indecomposable socle-projective representation.","key_machinery":"The argument rests on four pieces. First, proper peak P-subspaces: in type A every indecomposable peak space has one-dimensional subspaces, and Lemma 2.1 identifies each proper peak P-subspace with a proper subset I of the maximal points in its support, with support I^△ \\ (I^c)^△; this turns the stability check into finitely many support computations. Second, the bilinear form b_P(α,β) = α·$C_P^{{-1}}$·$β^{{tr}}$ associated with the poset P through its incidence matrix C_P; the stability weight is the antisymmetrization of this form against the dimension vector of U. Third, the category of sp-segments: inside the polygon of a type A Dynkin quiver, line segments that are not frozen by the added alien arrows and not swept into a certain forbidden set form a subcategory of line segments; the paper constructs a functor from this category to the socle-projective modules and proves it is an equivalence. Fourth, the angle stability function: each sp-segment is an oriented vector in the plane, so its direction defines a central charge and a phase φ(M) = (1/π) arg(Z[M]); the geometric proof shows that every proper subrepresentation decomposes into consecutive blocks whose angle vectors add to a smaller phase, giving stability.","core_discovery":"The central claim is Theorem 2.6: for any poset P of type A, every indecomposable peak P-space U is θ-stable with a weight θ built from the bilinear form of P. Specifically, θ(W) = b_P(dim U, dim W) − b_P(dim W, dim U), where b_P is defined from the incidence matrix of P and its inverse. The proof reduces to sincere objects: every indecomposable is the image under a subposet-induced lift functor of a sincere peak space on a sincere peak-subposet, and stability is preserved by that lift. For the three sincere families of type A posets, the paper checks that every proper peak P-subspace W_X has θ(W_X) < 0, using the fact that such subspaces are indexed by subsets of the maximal points in the support. The geometric half proves Theorem 4.3: in the line-segment model, the oriented segment representing an indecomposable M gives a central charge Z([M]) and a stability function φ(M) = (1/π)arg Z([M]); every indecomposable socle-projective representation is φ-stable because every proper subrepresentation is a direct sum of consecutive blocks whose angle vectors sum to a strictly smaller angle than the original. Section 4 closes the circle by writing the bilinear-form weight as θ = κ(M)w − w(M)κ and giving explicit φ_m stability functions on the sincere families.","pith_inferences":["The reduction to sincere objects plus a classification of sincere posets is a template: if a larger class of posets of finite prinjective type has a sincere-poset classification, the same two proofs would go through whenever the bilinear-form computation and the line-segment model are available.","The angle stability suggests a comparison question the paper does not ask: as the polygon vertices move, the phases of indecomposables vary, so one could look for walls in polygon-coordinate space where two indecomposables exchange order and stability changes, giving a concrete stability-space picture for the category.","A direct symbolic check of the S_2^{(r)} and S_3^{(r)} weight computations, which the paper leaves to 'the same arguments', would settle the only unexhibited step in Theorem 2.6; if a counterexample appeared there, Theorem 4.3 would still provide stability through the geometric route."],"forward_implications":["In a type A socle-projective category every indecomposable object lies in the stable locus for an explicit weight, so θ-stability is a structural feature rather than a rare condition.","The categorical equivalence identifies the Auslander–Reiten quiver of mod_sp kP with the translation quiver of sp-segments, giving a complete combinatorial description of irreducible morphisms and mesh relations.","Lemma 1.4 turns each θ-stable object into a µ-stable object for every positive slope function κ, so the Harder–Narasimhan and Jordan–Hölder filtrations exist for all peak P-spaces of type A.","The angle-based central charge lands in the strict right half-plane and attaches a real phase to every indecomposable, connecting the additive-category stability with the usual complex-stability language.","The explicit φ_m functions of Theorem 4.4 give a one-parameter family of stability functions on sincere type A posets, all of which certify stability of every indecomposable."],"supporting_citations":[{"why":"Introduces stability for poset representations and the bilinear-form weight that this paper extends to peak P-spaces.","marker":"[7]"},{"why":"Provides the geometric realization and alien-arrow description of type A socle-projective categories that Section 3 refines.","marker":"[21]"},{"why":"Defines peak P-spaces and supplies the structural facts (unique decomposition, kernels, extensions, AR sequences) the stability theory uses.","marker":"[26]"},{"why":"Classifies sincere one-peak posets of finite prinjective type, the base case of the sincere-poset list.","marker":"[11]"},{"why":"Classifies sincere two-peak posets of finite prinjective type, part of the classification behind Table 1.","marker":"[12]"},{"why":"Classifies indecomposable sincere prinjective modules over multipeak sincere posets with at least four maximal elements, used for the sincere families with large r.","marker":"[13]"},{"why":"Classifies sincere three-maximal-element posets of finite prinjective type and their sincere representations, completing Table 1.","marker":"[14]"},{"why":"Defines the subposet-induced functor T_S that lifts sincere stability to all indecomposables.","marker":"[5]"},{"why":"Supplies the polygon model, category of line segments, pivot morphisms, and angle stability used in the geometric half of the paper.","marker":"[1]"},{"why":"Gives the slope-based stability formalism for abelian categories that the paper adapts to the non-abelian additive setting.","marker":"[19]"}],"fun_headline_variants":["All type A peak spaces stable, via bilinear forms and geometry","Every type A indecomposable socle-projective is stable","Two distinct proofs all type A peak spaces are stable","Bilinear and geometric stability coincide for type A posets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the bilinear-form weight is negative on every proper subspace is written out only for the sincere family $S_1^{{(r)}}$; for the other two sincere families it says the same computation works without displaying it, so the stability theorem for all type A posets rests on those unstated computations being correct.","fun_headline_variants_meta":{"raw":{"variants":["All type A peak spaces stable, via bilinear forms and geometry","Every type A indecomposable socle-projective is stable","Two distinct proofs all type A peak spaces are stable","Bilinear and geometric stability coincide for type A posets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000702,"raw_usage":{"total_tokens":3208,"prompt_tokens":1022,"completion_tokens":2186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":638,"tokens_out":2186,"duration_ms":15115,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:47:58.925197+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the weight θ(W_X) = b_P(dim U, dim W_X) - b_P(dim W_X, dim U) directly for the sincere peak P-spaces of $S_2^{{(r)}}$ and $S_3^{{(r)}}$ using the incidence matrices displayed in Proposition 2.5, for r = 2 and r = 3. If any proper peak P-subspace W_X of the sincere space has θ(W_X) ≥ 0, Theorem 2.6 is false; a negative value on every W_X would confirm the missing step.","supporting_citations":[{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Introduces stability for poset representations and the bilinear-form weight that this paper extends to peak P-spaces."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Provides the geometric realization and alien-arrow description of type A socle-projective categories that Section 3 refines."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Defines peak P-spaces and supplies the structural facts (unique decomposition, kernels, extensions, AR sequences) the stability theory uses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies sincere one-peak posets of finite prinjective type, the base case of the sincere-poset list."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies sincere two-peak posets of finite prinjective type, part of the classification behind Table 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies indecomposable sincere prinjective modules over multipeak sincere posets with at least four maximal elements, used for the sincere families with large r."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies sincere three-maximal-element posets of finite prinjective type and their sincere representations, completing Table 1."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Defines the subposet-induced functor T_S that lifts sincere stability to all indecomposables."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the polygon model, category of line segments, pivot morphisms, and angle stability used in the geometric half of the paper."},{"cited_title":"Algebra 197 (1997), no","cited_arxiv_id":null,"evidence_quote":"Gives the slope-based stability formalism for abelian categories that the paper adapts to the non-abelian additive setting."}],"review_version":1}