{"id":"48d47753-e485-404f-b897-451cba7f5496","arxiv_id":"2607.16022","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A singularity-theoretic 'stable hierarchy' on Artin algebras is shown to be computable from automorphism-group data and to extend the classical degeneration order beyond fixed rank.","lead":"The authors introduce a new partial order on finite-dimensional complex algebras, the stable hierarchy, based on singularities of stable maps rather than Hilbert schemes. They prove it can be computed from symmetry data via Thom polynomials and that, for fixed dimension, it agrees with the classical degeneration order from deformation theory.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithmic claim depends on unproven interpolation theorems (4.4/4.5) and external [TPP] symmetry data; if either is defective, the computed posets in Figures 4–6 are unsupported even though the structural Theorem 7.1 remains valid.","rationale":"The reader's weakest_assumption is the status of the interpolation theorems, and I agree. The strongest claim is not a single monolithic theorem; the proven structural part (Theorem 7.1 and Theorem 9.1/Corollary 9.2) is in reasonable shape, while the advertised algorithmic computation is only as sound as Theorem 4.5 plus the [TPP] data. The paper's own text disclaims a proof of Theorem 4.5, and the l>100 consistency check is also not documented. I do not see an internal contradiction in the main structural proofs; the issue is a missing proof and external reliance. Hence the verdict should remain CONDITIONAL: it should be stated as conditional on a proof or independent verification of the interpolation theorem and the symmetry data. There is no new reason to reject, but also no reason to accept unconditionally. Independent support exists in the derived prototype geometry and in the Poincaré-series consistency check, which is real evidence, but it does not replace the missing proof.","tokens_in":30158,"tokens_out":10282,"duration_ms":102125,"concrete_test":"Take a nontrivial multisingularity with an independently known Thom polynomial—e.g. the triple point locus A_0^3 in relative dimension 1. Compute Th^T_{A_0^3} by an independent geometric method (Kazarian/Ohmoto formulas), then run the interpolation system of Theorem 4.5 using the [TPP] stabilizer weights for the relevant prototypes and check the unique solution equals that polynomial. Agreement would show the unproven theorem and the data are correct in a nontrivial case; disagreement would pinpoint a concrete failure in the computational engine.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim has two parts: a structural equivalence (Theorem 7.1: η≤lζ iff Th^T_ζ(θ^l_η)≠0 in C*-equivariant cohomology, plus Theorem 9.1/Corollary 9.2 comparing with the Hilb order) and the advertised algorithmic determination of the poset from symmetry data. The structural part is argued from Mather theory and the appendix. The algorithmic part, however, rests on Theorems 4.4–4.5, the multisingularity interpolation method. The text explicitly does not prove these theorems: after Theorem 4.5 it states the theorem 'was known, or at least widely believed, by experts... In concrete applications a formal proof is not required' and refers to [KR25, Thm. 8.2] for a generalization. That is not a proof. If the interpolation conditions do not uniquely characterize Th^T_η in this setting, or if the stabilizer weights listed on [TPP] are wrong or incomplete, every computed evaluation in Section 7 and every edge in Figures 4–6 could be incorrect. Theorem 7.1's equivalence would be unharmed, but the paper's first main result—that the hierarchy is determined algorithmically by symmetry data—would not be established. This is a gap in support, not a demonstrated contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a singularity-theoretic degeneration order ('stable hierarchy') on finite-dimensional commutative Artin algebras, defined via the singularity loci of stable maps rather than via Hilbert-scheme deformations. The first main claim is that, in Mather nice dimensions, this hierarchy is characterized by evaluations of Thom polynomials of multisingularities, and can therefore be determined algorithmically from symmetry data (automorphism groups / torus weights). The second main claim is that, for algebras of fixed rank, the stable hierarchy coincides with the classical degeneration order on Hilbert schemes of points. The paper also contains an appendix formalizing the stratification of Hilbert schemes by algebra type, and it presents computed Hasse diagrams (Figures 4–6) for low-dimensional algebras.","tokens_in":30549,"tokens_out":6366,"duration_ms":69163,"significance":"If the computational engine is made fully rigorous, this is a valuable unification: it connects Mather's classification of stable singularities with the deformation theory of Artin algebras, and it gives a genuinely new method for computing degeneration posets, including comparisons across different ranks. The structural results—especially Theorem 7.1 and Theorem 9.1/Corollary 9.2—are significant and, as far as I can check, argued carefully. The appendix's representable-functor treatment of the Hilbert strata is a useful contribution. However, the advertised algorithmic determination rests on an interpolation theorem that the paper explicitly does not prove, and on external symmetry/weight data that is not shipped; these are load-bearing gaps.","major_comments":[{"comment":"Theorem 4.5 is the computational engine: all computed Thom polynomials and therefore the evaluations in Section 7 and the edges in Figures 4–6 depend on its uniqueness claim. Yet the text states after the theorem: 'This theorem was known, or at least widely believed, by experts... In concrete applications a formal proof is not required.' This is not acceptable for a rigorous proof of the paper's first main result. Either include a proof of Theorem 4.5 (and Theorem 4.4) under precisely stated hypotheses, or state the exact theorem from [KR25] and verify that the hypotheses of that theorem are satisfied in the present setting. The current deferral leaves the algorithmic claim unsupported.","section":"§4.3, Theorem 4.5"},{"comment":"The computations depend on the completeness and correctness of the symmetry/weight data listed on the external website [TPP]. This data is not reproduced or archived with the paper. The consistency check of Theorem 5.2 is only reported for l=0,...,100, and for l>100 the paper says 'with an extra algebraic argument (not detailed here)'. Since incorrect or incomplete weights would change every Thom polynomial evaluation and hence every edge in Figures 4–6, the paper should make the data available in a citable/archived form and give a complete verification, or clearly state this as an assumption rather than a proven fact.","section":"§5.2 and §7"},{"comment":"The extent of the computational claim should be stated more precisely. Figure 5 explicitly leaves some covering relations unresolved ('To decide what arrows point into vertices D51 and D52 we would need more computing power'), and Figure 6 excludes C58,...,C5,11. This is compatible with an algorithmic claim, but the abstract and introduction should not imply that the full posets for dimensions up to 6/7 have been computed. The distinction between 'the hierarchy is algorithmically determinable' and 'we have determined the hierarchy in these ranges' should be made explicit.","section":"§7, Figures 5–6"}],"minor_comments":[{"comment":"In the exponential equation in Theorem 4.5, the left-hand sum appears to use the variable t_ζ where t_η is intended; as written, the expression is hard to parse.","section":"§4.3, Theorem 4.5"},{"comment":"The letter A is used both for the right-left group Diff(C^m,0)×Diff(C^{m+l},0) in Section 3 and for an arbitrary algebra throughout; this can be confusing, especially in Section 5.1 and the appendix.","section":"Notation"},{"comment":"The dependence on the website [TPP] is heavier than a typical reference. Please provide a stable version/archive link or a data appendix, especially since the paper's own verification is incomplete for large l.","section":"References"},{"comment":"The captions and text should clarify which parts of the diagrams are proven computations, which are conjectural for all l, and which are incomplete due to computing-power limits. Currently the phrase 'conjecturally for all l' is used but its precise scope is not defined.","section":"Figures 4–6"}],"recommendation":"major_revision","confidential_remarks":"The structural contributions are substantial, but the algorithmic main result is not yet fully supported because Theorem 4.5 is unproved and the external symmetry data is not shipped. This is fixable within the manuscript's scope: the authors can prove or precisely import Theorem 4.5, and can archive the data. I would not reject, but I would not accept until these load-bearing points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nIf you read one paper this month on the interface of singularity theory and commutative algebra, this is it. The stable hierarchy — a partial order on all Artin algebras, allowing varying rank — is a genuinely new object, and Theorem 9.1, that for fixed rank it agrees with the classical degeneration order from the Hilbert scheme, is an important bridge. The paper does a careful job of turning folklore statements about Hilbert scheme strata into proven propositions (Section 9, Appendix 10), and I believe the structural theorems are solid. Theorem 7.1, linking the hierarchy to non-vanishing of Thom polynomial evaluations, is plausible and proved from known results.\n\nThe soft spot is exactly where the reader and stress-test put their fingers. The advertised algorithmic determination of the posets (Figures 4–6) rests on Theorems 4.4 and 4.5, the interpolation method. Those theorems are stated and then promptly asserted to be 'known, or at least widely believed, by experts' and 'a formal proof is not required'. That is not satisfactory for a paper whose first main result is that the hierarchy is computable from symmetry data. The uniqueness of the solution to those interpolation conditions is the load-bearing unsupported step. If the interpolation theorem is wrong, or the symmetry data from [TPP] is incomplete, every edge in the figures could be wrong, even though the equivalence theorems would survive.\n\nThat said, I want to be clear: this is a gap in support, not a demonstrated contradiction. The interpolation method is well-established in the Thom polynomial literature, and the paper is honest about what it has not proven. The fix is straightforward: either prove Theorem 4.5 (or give a precise reference for a proof), or make the code and the stability data available so the computations are reproducible. The Poincaré series consistency check is a nice idea, but the 'extra algebraic argument not detailed here' for l>100 is another omitted detail that should be supplied.\n\nA minor point: some entries in the Hasse diagrams are left unresolved (e.g., the arrows into D51 and D52). That is fine, but it undercuts the completeness claim for dimension 6; the paper should clearly label which parts of the poset are complete and which are conditional on more computing power.\n\nBottom line: this is a serious paper with a real new theorem. It deserves a careful referee, with the understanding that the computational engine is not yet fully proven and the reproducibility is not yet there. I would bring it to our reading group and cite the structural results.\n\nRecommendation: send to peer review, with the request that the authors either prove or precisely reference the interpolation theorem and make the data/code available. If they do that, this could be a very nice addition to the literature.","headline":"Strong new bridge between singularity theory and deformation theory, but the advertised algorithmic computation rests on an unproven interpolation theorem and unsupplied data.","tokens_in":30951,"tokens_out":3329,"would_cite":true,"duration_ms":36349,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C05","14B05","32S20","58K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the degeneration order on Artin algebras is decided by a single equivariant characteristic-class evaluation, and that this order agrees with the classical Hilbert-scheme degeneration order for fixed rank.","keywords":["Artin algebras","degenerations","stable hierarchy","multisingularities","Thom polynomials","equivariant cohomology","Hilbert scheme","singularity theory"],"falsifier":"Take any adjacent pair in the dimension-6 poset that the algorithm produces (for instance, an arrow into D51 or D52, which the paper leaves open due to computing limits) and determine the relation by classical deformation theory: write down an explicit one-parameter family of algebras over C[[t]] realizing the alleged degeneration, or verify directly that Hilb_A(X) ⊂ closure(Hilb_B(X)) fails. If the algorithm's arrow contradicts the deformation-theoretic containment, Theorem 7.1's evaluation criterion is refuted; if the open arrows into D51/D52 resolve in the predicted direction, the conjectur","tokens_in":30094,"feed_emoji":"🧮","tokens_out":10317,"duration_ms":106049,"temperature":0.7,"pith_summary":"This paper attempts to show that the degeneration poset of finite-dimensional commutative Artin algebras—which algebra can deform into which—can be read off from symmetry data alone, without exploring Hilbert schemes. The authors define a singularity-theoretic 'stable hierarchy': for a fixed relative dimension, one multisingularity is below another if its target singularity locus is contained in the closure of the other's, for every stable map. The first main result (Theorem 7.1) says that, for Mather multisingularities, this relation is equivalent to a single characteristic-class evaluation: the target Thom polynomial of the upper singularity, evaluated on the prototype of the lower one in C*-equivariant cohomology, must be nonzero; the interpolation method then computes these polynomials from the automorphism-group weights of the algebras, making the hierarchy algorithmic. The second main result (Corollary 9.2) identifies the stable hierarchy, for algebras of fixed rank, with the classical degeneration order defined by closures of strata in the Hilbert scheme. If both claims hold, degeneration posets in the Mather range are determined by linear algebra, and singularity theory and deformation theory describe the same partial order.","feed_headline":"Symmetry data decides which algebras deform into which","feed_subtitle":"A singularity-theoretic hierarchy matches classical Hilbert-scheme degenerations and turns the poset into computation.","key_machinery":"The machine is the pair consisting of prototypes and target Thom polynomials. For a multisingularity η and a relative dimension l, the prototype θ^l_η is a universal stable map germ realizing η; its existence turns containment of singularity strata into a local question at the origin. The target Thom polynomial Th^T_ζ is a universal equivariant characteristic class whose fundamental class on the target of any stable map is the (weighted) singularity locus of ζ; substitution into it is governed by torus weights. The interpolation theorems (4.4, 4.5) assert that Th^T_ζ is uniquely determined by its evaluations on prototypes—conditions (2) and (3) are linear equations in the polynomial's coeffi","core_discovery":"On the paper's own terms: for Mather multisingularities η, ζ occurring for relative dimension l, η ≤_l ζ holds exactly when the target Thom polynomial Th^T_ζ evaluates nonzero on the prototype θ^l_η in C*-equivariant cohomology (Theorem 7.1). The interpolation method then computes Th^T_ζ from the weights of positive C*-actions stabilizing prototypes, i.e. from automorphism groups of the algebras, so the hierarchy becomes algorithmic. Second: for algebras of fixed rank, the limit stable hierarchy ≤_∞ equals the classical degeneration order ≤_Hilb from Hilbert-scheme stratum closures (Theorem 9.1, Corollary 9.2). Thus the two notions agree where both are defined, and the singularity-theoretic","pith_inferences":["If the folklore interpolation theorem is given a full proof, the same pipeline extends to dimension 7 and beyond, where continuous moduli first appear; the 'dressing' phenomenon in the diagrams—monosingularities that refuse to split until they first transform into another monosingularity—would then be a systematic feature of the poset, deserving an invariant.","The equality with the Hilbert-scheme order suggests a translation dictionary: dimensions and tangent spaces of Hilbert-scheme strata can be read from singularity codimensions and derivation spaces, so the Thom-polynomial evaluation matrix could serve as a fast oracle for component structure of Hilbert schemes of points in higher dimensions.","The unresolved dependence of ≤_l on l for algebras of unequal dimension (Question 6.11) is the main structural gap; a positive answer would make the stable hierarchy a single global poset on all Artin algebras, and the paper's Proposition 6.8 plus Corollary 9.11 already settle the equal-rank case.","Because the criterion is a single nonzero evaluation, the full poset can be precomputed once for the Mather lists and reused, turning each future classification of small algebras into a lookup rather than a new deformation-theoretic search."],"forward_implications":["Degeneration posets in the Mather range are computable by linear algebra: the paper implements this for all multisingularities of algebra dimension ≤5 and for monosingularities of dimension 6, giving explicit Hasse diagrams and elementary-splitting tables.","The stable hierarchy extends the Hilbert-scheme degeneration order beyond fixed rank; for example, any quotient Q/(s) of a local algebra Q by an element of the maximal ideal satisfies Q ≤_{l+1} Q/(s), so the two hierarchies compare algebras of different dimensions.","For fixed rank, the two theories become one: singularity computations determine Hilbert-scheme closures, and deformation-theoretic facts such as openness of the Gorenstein property translate into statements about the stable hierarchy (Corollary 9.13).","Corollary 9.12 supplies a closed formula tcodim_l(η) = n·l + dim Der(A,A) for the target codimension of the singularity attached to an algebra of rank n, tying singularity invariants to algebra derivations.","The Poincaré-series identity of Theorem 5.2 gives, for each l, a numerical consistency check on the classification lists of small-dimensional algebras; the authors report computer verification for l=0,...,100 and an algebraic argument beyond."],"fun_headline_variants":["Automorphism groups determine degeneration hierarchy","Singularity theory computes algebra degeneration order","New hierarchy equals Hilbert scheme degenerations","Degeneration poset from symmetry data, algorithmically"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the interpolation theorems (4.4 and 4.5) genuinely compute Thom polynomials from symmetry data: Theorem 4.5 is explicitly left as folklore ('This theorem was known, or at least widely believed, by experts... a formal proof is not required'), and the symmetry data are taken from a registry whose completeness is checked numerically rather than proven; if either fails, the nonvanishing evaluations that define the hierarchy in Theorem 7.1(3) could","fun_headline_variants_meta":{"raw":{"variants":["Automorphism groups determine degeneration hierarchy","Singularity theory computes algebra degeneration order","New hierarchy equals Hilbert scheme degenerations","Degeneration poset from symmetry data, algorithmically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1294,"prompt_tokens":736,"completion_tokens":558,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":504}},"tokens_in":480,"tokens_out":558,"duration_ms":6372,"temperature":1.0,"reasoning_tokens":504,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:44:28.614628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any adjacent pair in the dimension-6 poset that the algorithm produces (for instance, an arrow into D51 or D52, which the paper leaves open due to computing limits) and determine the relation by classical deformation theory: write down an explicit one-parameter family of algebras over C[[t]] realizing the alleged degeneration, or verify directly that Hilb_A(X) ⊂ closure(Hilb_B(X)) fails. If the algorithm's arrow contradicts the deformation-theoretic containment, Theorem 7.1's evaluation criterion is refuted; if the open arrows into D51/D52 resolve in the predicted direction, the conjectur","supporting_citations":[],"review_version":1}