{"id":"ce412478-3ef0-4bd3-a307-938e84eceac0","arxiv_id":"2411.19682","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives a recursive enumeration algorithm and complete lists of all tame periodicity shadows for sizes n ≤ 6, with full tables in an appendix.","lead":"This paper presents a recursive algorithm that computes every tame periodicity shadow, a type of skew-symmetric matrix encoding quivers of symmetric algebras with periodic simple modules, up to matrix size 6. It provides complete tables for n ≤ 6, including full lists in an appendix, as a data resource for classifying tame periodic algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exhaustive shadow census depends on an omitted PS3 verification; without re-running it, the published counts S(5)=65 and S(6)=516 are not certified.","rationale":"The reader's weakest assumption correctly identifies the central reproducibility gap. The algorithm's recursive generation of basic shades is supported by a genuine correctness proof and the appendix supplies large explicit lists, so the shading generation itself is less vulnerable. The decisive step for the headline census is PS3: the paper explicitly says the nonnegativity check is skipped, yet the counts 65 and 516 are precisely the result of that check. The 1260/1290 disagreement for n=6 shades is a concrete warning sign that the computational output has not been independently audited. My proposed test is a full independent re-run with an exact feasibility solver for each listed triple; it is expensive but finite and would settle both the shadow counts and the shade-count discrepancy. I therefore keep the reader's CONDITIONAL assessment rather than escalating to rejection, because the mathematical framework and the supplied lists may well be correct; what is missing is the verification that makes a computational census paper self-certifying.","tokens_in":196961,"tokens_out":3153,"duration_ms":40342,"concrete_test":"Re-run the complete algorithm in a standard CAS: implement MatricesSatisfyingTPS exactly as written (including the omitted M≺ test via canonical permutation), generate all basic shades for n=5 and n=6, and for each listed triple (A,x,C) in the Appendix solve the feasibility problem: does there exist symmetric C∈M_n(N) with nonzero columns satisfying AC=0, e.g., by linear programming over the parameter cone? Compare the resulting shadow lists and counts with 65/516 and the Appendix, and resolve whether the number of basic 6×6 shades is 1260 or 1290. If the re-run matches all published matrices, the census is confirmed; otherwise the false entries or counts identify the error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that Sections 4-5 and the Appendix list all tame periodicity shadows for n≤6 rests on the PS3 deletion step in Section 2. After listing three sufficient conditions for deleting shades, the authors state that for every remaining triple one can check C∈M_n(N) 'automatically via standard computational environments like Maple, but we skip this for simplicity.' No code, certificates, or Maple output is supplied. The deletion conditions are one-sided: a shade with no all-natural symmetric C could pass all three filters and be counted as a shadow, inflating S(5)/S(6); conversely, a bug in the omitted feasibility check could exclude genuine shadows. The correctness proof in Section 2 proves only that MatricesSatisfyingTPS produces basic shades; it does not prove that the PS3 filtering is complete and sound. The internal inconsistency between the table's 'number of shades' 1260 for n=6 and the statement '|S(6)|=1290 basic shades' (also in the Appendix) makes this gap concrete rather than hypothetical.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is the computational companion to arXiv:2411.17381. It defines periodicity shadows as skew-symmetric integer matrices satisfying PS1–PS3 plus tameness conditions T1–T3, and presents a recursive row-by-row generator, MatricesSatisfyingTPS, which is intended to produce all basic shades (shades modulo row/column permutation and transposition) for a given n. A correctness proof is given for the generator: every shade is eventually generated, and the canonical-representative filter in SetRow removes duplicates up to permutation and sign. The authors then filter shades by the PS3 condition, i.e. existence of a symmetric matrix C with natural, non-zero columns satisfying AC = 0, and list the resulting tame periodicity shadows and essential shadows for n ≤ 6. The announced counts are 5, 12, 65 and 516 tame shadows for n = 3, 4, 5, 6, with 4, 7, 26, 223 essential shadows, and the Appendix contains the full lists of basic shades, tame shadows and non-shadow shades.","tokens_in":197181,"tokens_out":3402,"duration_ms":36757,"significance":"If the enumeration is correct, this is a useful exhaustive census of the small cases that are directly relevant to Gabriel quivers of tame symmetric algebras with periodic simples of period four. The recursive generation scheme is a genuine algorithmic contribution, and the proof in Section 2 that the generator produces every shade up to permutation and sign is plausible and covers the orbit deduplication. The displayed triples (A, x, C) provide explicit nullspace data and symmetric solution matrices for every essential shadow, which is valuable for further structural study. However, the paper stops short of certifying the final PS3 filtering step and contains a direct numerical inconsistency in the central counts, so the census as published is not yet fully verifiable.","major_comments":[{"comment":"The completeness of the published census for n = 5 and n = 6 rests on the PS3 verification, but this verification is explicitly not carried out in the paper. After listing three sufficient deletion conditions, the authors state that for each remaining triple one can check C ∈ M_n(N) 'automatically via standard computational environments like Maple, but we skip this for simplicity' (Section 2, end). No code, Maple output, or certificates are supplied, and the deletion rules are only sufficient conditions for failure. Therefore the claim that Sections 4–5 and the Appendix list all tame periodicity shadows S(5) and S(6) is not certified by the manuscript; a bug in the omitted feasibility check could either include non-shadows or exclude genuine shadows.","section":"Section 2 (PS3 filtering)"},{"comment":"There is a direct numerical inconsistency in the central data. The table in Section 3 gives 1260 basic shades for n = 6, while Section 5 states |S(6)| = 1290 basic shades, and the Appendix repeats the figure 1290 and lists items (1)–(1290). Since these counts are the main quantitative output of the paper, the discrepancy must be resolved; if 1290 is the correct count, the table entry 1260 is wrong, and if 1260 is correct, the Appendix and Section 5 overcount.","section":"Section 3 table vs Section 5 and Appendix"},{"comment":"The correctness proof for MatricesSatisfyingTPS depends on the test M = M≺ for canonical representatives, but the paper states that this test 'can be efficiently verified by a recursive procedure (we omit the details)' and no implementation is given. Together with the omitted PS3 check, the algorithm as presented is therefore not fully reproducible from the text, although the high-level correctness argument for the recursive enumeration is otherwise coherent.","section":"Section 2, SetRow line 8"}],"minor_comments":[{"comment":"There are several typographical errors that should be corrected, including 'cosiderations' in the abstract and 'partial oreder' in Section 2, as well as the entry '−0' in the description of ComposeRow.","section":"Abstract and Section 2"},{"comment":"The name 'BIA/suppress lkowski' appears corrupted, and 'Skowyrski' is used inconsistently with 'Skowroński' in the references; the LaTeX source appears to have a suppression artifact that must be fixed before publication.","section":"Author and address lines"},{"comment":"Given the extreme length of the lists for n = 6, the authors should provide machine-readable data (e.g. an ancillary file) so that readers can verify the counts and reuse the matrices; this also mitigates the omitted-certificate problem.","section":"Appendix data"},{"comment":"The hand-drawn quiver diagrams for n = 4 are very hard to read; vector graphics or a structured description would be preferable.","section":"Section 3 quiver diagrams"}],"recommendation":"major_revision","confidential_remarks":"I would condition acceptance on the authors either supplying verifiable code or certificates for the PS3 feasibility check and the M = M≺ test, or making those checks fully explicit in the paper, and on correcting the 1260/1290 inconsistency. The topic is appropriate for the journal, and the algorithmic framework is a solid basis, but without these fixes the central census claim is not externally verifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real new content is the recursive row-by-row generator with a correctness proof for basic shades, and the first complete tables for n=5 and n=6, including the 26 and 223 essential shadows. That is useful, concrete material for anyone trying to classify tame symmetric algebras with period-four simples or algebras of generalized quaternion type. The generator proof is sound in structure: the total order on matrices, the M≺ and (-M)≺ orbit filtering, and the argument that every shade appears up to permutation and sign are all coherent. The Appendix, with full shade lists, is a substantial data dump and appears organized. The soft spots are real but localized. The paper explicitly skips the PS3 check for the surviving triples: verifying C in M_n(N) is described as doable in Maple, then not done. No code, certificates, or output files are provided. For a computational census whose main claim is exhaustiveness, that is a material gap. A second, smaller gap is the omitted recursive detail for testing M=M≺. And there is a worrying count discrepancy: Section 3's table says 1260 shades for n=6, while the text in Section 5 and the Appendix say |S(6)|=1290. I did not find a decisive derivation error in the mathematics, and the reduction to essential shadows is plausibly imported from Part I and [3]. The citation pattern is appropriate; the authors flag their dependence. The paper deserves referee time, but only with a request for code/data or certificates for the final PS3 filtering, and a fix of the count mismatch. As it stands, the complete lists should not be accepted as certified. For a specialist in self-injective algebras the data is still worth having, and the appendix is a useful reference. I would not cite it in a proof without re-running the computation.","headline":"A useful computational census with a real reproducibility gap: the PS3 filter is explicitly skipped and the n=6 shade count is inconsistent (1260 vs 1290), so the exhaustive lists are not yet certified.","tokens_in":197632,"tokens_out":3489,"would_cite":false,"duration_ms":30728,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E16","16D50","16E20","16G20","16Z05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a recursive row-by-row algorithm enumerates all tame periodicity shadows up to size 6, with counts 5, 12, 65, and 516.","keywords":["periodicity shadows","tame algebras","symmetric algebras","periodic modules","generalized quaternion type","Gabriel quiver","skew-symmetric matrices","recursive enumeration"],"falsifier":"Run an independent implementation of MatricesSatisfyingTPS(n) for n = 3, 4, 5, 6 together with a certified linear-inequality solver for PS3, and compare the outputs with the published lists: the claim stands only if the counts 5, 12, 65, 516, the essential counts 4, 7, 26, 223, and every listed matrix (A, x, C) match exactly, with all C entries nonnegative and no zero columns.","tokens_in":196775,"feed_emoji":"🧮","tokens_out":5850,"duration_ms":48002,"temperature":0.7,"pith_summary":"The paper claims that a recursive algorithm, which builds skew-symmetric matrices row by row and keeps only canonical representatives under row/column permutation and sign reversal, generates every basic shade of a given size. Filtering those shades by the existence of a symmetric natural-coefficient solution to AC = 0 then yields exactly the tame periodicity shadows. The authors assert that for n = 3, 4, 5, 6 the counts of basic tame periodicity shadows are 5, 12, 65, and 516, with the essential shadows (those that can belong to a tame symmetric algebra with period-4 simples) numbering 4, 7, 26, and 223. The point is to make the combinatorial classification of Gabriel quivers of small tame symmetric algebras computationally explicit and complete.","feed_headline":"All tame periodicity shadows up to size 6 are now listed","feed_subtitle":"Recursive enumeration yields 5, 12, 65, and 516 shadows for n=3-6, and 4, 7, 26, 223 essential ones.","key_machinery":"The central object is a periodicity shadow: a singular skew-symmetric integer matrix A whose rows do not have nonzero entries of one sign and for which some symmetric matrix C with natural entries and nonzero columns satisfies $AC = 0$, subject to the tame entry bounds T1-T3. The machinery that makes enumeration possible is the total order $\\preceq$ on matrices, comparing entries lexicographically after rows and columns are listed in a fixed order. For each orbit under simultaneous row/column permutation the algorithm keeps the minimal representative $A^{\\prec}$, and since $A^T = -A$ has the same nullspace, it also identifies a matrix with its negative; line 8 of SetRow returns $\\{M\\}$ only when $M$ is its own canonical representative and $M \\preceq (-M)^{\\prec}$. This canonical filter runs inside a recursion that builds matrices row by row, collapsing the search space to the 5, 12, 138, 1290 basic shades for $n = 3, 4, 5, 6$. PS3 is then decided from the parametrised nullspace of $A$.","core_discovery":"The central claim is that the recursive procedure MatricesSatisfyingTPS(n), whose core is the function SetRow, generates one representative from each orbit of shades under simultaneous row/column permutation, with opposites identified. The filter at line 8 of SetRow keeps a matrix M only when M = M^prec and M is no larger than the canonical representative of -M, so every generated matrix is the minimal representative of its orbit. For each surviving shade A, the paper solves AC = 0 with C = C^T and tests whether a generic nullspace vector admits a symmetric natural-coefficient solution; triples whose vector has a zero entry, opposite parameter entries, or a nonnegative linear relation are deleted, because each such configuration forces every solution to have a zero column or a negative entry. The paper concludes that the remaining matrices form the complete set of basic tame periodicity shadows, and that the tables in Sections 3-5 and the Appendix are complete for n <= 6.","pith_inferences":["If the PS3 filter is independently reimplemented and verified, the same counts would certify the completeness of the published tables; until then the counts 65 and 516 are best read as computational claims awaiting independent confirmation.","The canonical-representative ordering used here is a general principle: any class of matrices closed under simultaneous permutation and negation can be enumerated by row-by-row generation with the same filtering, so the technique may transfer to other nullspace-defined combinatorial classes.","The listed essential shadows for n <= 6 give a testbed for conjectures about n >= 7 stability: one could check whether every larger shadow decomposes into smaller ones, which would make the apparent stability at n >= 7 a consequence of the small cases.","Because the paper leaves the natural-solution check to external software, a natural extension is to replace it with an explicit linear-inequality certificate per shadow, turning the tables into machine-checkable proofs."],"forward_implications":["For n <= 6, every Gabriel quiver of a tame symmetric algebra with simple modules of period 4 is, modulo loops and 2-cycles, one of the essential shadows listed in Sections 3-5.","Any algebra whose periodicity shadow fails the essentiality conditions PS4 or PS5 is wild, so the search for tame algebras can be restricted to the essential lists.","For n = 3 and 4 the sets of shades and shadows coincide, while for n = 5 and 6 the appendix lists the 73 and 774 shades that are not tame periodicity shadows, respectively.","The algorithm terminates for n = 7 and n = 8 only with impractical runtime, and n >= 9 is intractable; the published tables therefore cover exactly the feasible range n <= 6.","From the complete lists one can read off the generic symmetric Cartan-matrix solutions C for every shadow, which is the data needed to reconstruct candidate algebras."],"supporting_citations":[{"why":"Defines periodicity shadows and proves the identity that the signed adjacency matrix of a symmetric algebra with period-4 simples annihilates its Cartan matrix; the algorithm and lists implement this definition.","marker":"[6]"},{"why":"Supplies the tame/wild dichotomy for algebras of generalized quaternion type and the arguments behind the essentiality conditions PS4 and PS5 used to cut the lists.","marker":"[3]"},{"why":"Provides the standard background on symmetric and Frobenius algebras, idempotents, projective modules, and Cartan matrices that motivates the shadow conditions.","marker":"[4]"}],"fun_headline_variants":["Tame periodicity shadows enumerated through size 6","All tame periodicity shadows up to size 6 computed","Tame shadows enumerated: 5,12,65,516 for n=3-6","Algorithmic census of tame periodicity shadows up to n=6"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The completeness of the published shadow lists depends on two unverified implementation details: the automatic check that a symmetric matrix with positive integer entries sits in the nullspace, and the omitted recursive procedure that tests whether a matrix is its own canonical representative under permutations; a bug in either would make the counts 5, 12, 65, 516 incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Tame periodicity shadows enumerated through size 6","All tame periodicity shadows up to size 6 computed","Tame shadows enumerated: 5,12,65,516 for n=3-6","Algorithmic census of tame periodicity shadows up to n=6"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001224,"raw_usage":{"total_tokens":4980,"prompt_tokens":843,"completion_tokens":4137,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":4061}},"tokens_in":459,"tokens_out":4137,"duration_ms":22584,"temperature":1.0,"reasoning_tokens":4061,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:56:59.643950+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent implementation of MatricesSatisfyingTPS(n) for n = 3, 4, 5, 6 together with a certified linear-inequality solver for PS3, and compare the outputs with the published lists: the claim stands only if the counts 5, 12, 65, 516, the essential counts 4, 7, 26, 223, and every listed matrix (A, x, C) match exactly, with all C entries nonnegative and no zero columns.","supporting_citations":[{"cited_title":"Periodicity shadows I: A new approach to combinatorics of periodic algebras","cited_arxiv_id":"2411.17381","evidence_quote":"Defines periodicity shadows and proves the identity that the signed adjacency matrix of a symmetric algebra with period-4 simples annihilates its Cartan matrix; the algorithm and lists implement this definition."},{"cited_title":"Erdmann, A","cited_arxiv_id":null,"evidence_quote":"Supplies the tame/wild dichotomy for algebras of generalized quaternion type and the arguments behind the essentiality conditions PS4 and PS5 used to cut the lists."},{"cited_title":"Skowro\\' n ski, K","cited_arxiv_id":null,"evidence_quote":"Provides the standard background on symmetric and Frobenius algebras, idempotents, projective modules, and Cartan matrices that motivates the shadow conditions."}],"review_version":1}