{"id":"3d702352-26a6-4436-9d36-b00ae5b525cc","arxiv_id":"2504.14339","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Endocabling, a deformation of set-theoretic Yang-Baxter solutions via brace endomorphisms, is used to prove that solutions of prime-power size with cyclic diagonal have finite multipermutation level, except for one size-4 solution in the 2-power case.","lead":"For a class of symmetries called cycle sets, a new endocabling construction deforms one Yang-Baxter solution into another while keeping control of its diagonal map. The paper uses this to prove that most cycle sets whose diagonal cycles through all elements collapse to a trivial one, with a single exceptional case at size 4.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's 2-power classification rests on unverified base cases: Lemma 5.10's v=3 GAP scan has no script/library version, and the v=4 Savile Row model references an undefined n and gives no solver log or UNSAT certificate.","rationale":"The paper's main new structural results for n=2^v all route through Theorem 5.1. The induction is intricate but depends at the bottom on Lemma 5.10 for v=3,4. The v=3 case is a bare assertion about a GAP library; the v=4 case is a CP model whose printed code cannot run and whose execution is undocumented. A constraint search for nonexistence is only as good as its solver, encoding, and certificate; without these, the base is not established. The reader's weakest_assumption identified exactly this point. I agree, and I would keep the verdict CONDITIONAL rather than ACCEPT or REJECT: the algebraic framework and the odd-prime-power theorem appear sound, and the computational base is plausibly correct but unverified. A repaired runnable model and a reproducible GAP script would settle the concern.","tokens_in":27718,"tokens_out":16285,"duration_ms":149711,"concrete_test":"Fix the ESSENCE' code by replacing the undefined n with 14 in the irretractability loop (or add a declared parameter n=16), run it through Savile Row 1.10 with a complete constraint solver such as Minion to UNSAT, and make the solver's log or certificate available; independently rerun the v=3 case in GAP by enumerating all SmallCycleSet(8,*) from a specified YangBaxter package version, checking which have an 8-cycle diagonal, and verifying each is retractable. Also brute-force enumerate all 4-element cycle sets with 4-cycle diagonal to confirm X4,19 is the unique irretractable one.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central 2-power result (Theorem 5.1 and Corollary 5.2) is proven by induction whose only base is Lemma 5.10. That lemma is computational, and neither part is reproducible. For v=3, the paper says the claim is 'easily confirmed by checking all cycle sets of size 8' in the GAP library YangBaxter, but gives no version, no enumeration script, and no evidence that the library is exhaustive at size 8. For v=4, the appendix's Savile Row/ESSENCE' model contains the line 'forAll i: int(0..n-2)' where n is never declared or defined, while all arrays are indexed 0..15; the code as printed cannot be parsed or run. No solver version, runtime, output, or UNSAT certificate is supplied. Because the model is used only for nonexistence, over-approximation in the constraints would be harmless; but an unparsable, unrun model proves nothing. If the v=3 check missed an irretractable size-8 solution, or if the v=4 search is unsound or incomplete, the induction has no base and the dichotomy in Corollary 5.2 (finite multipermutation level or retraction to X4,19) is unsupported. The algebraic parts, especially Theorem 4.2 for odd prime powers, appear internally coherent and are not affected by this concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces endocabling, a deformation of finite involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation, formulated in the language of cycle sets. An endocabling is a redefinition of the cycle-set operation by a (relative) λ-endomorphism of the associated permutation brace, generalizing the cabling of Lebed, Ramírez and Vendramin. The paper proves that the diagonal of an endocabling composes additively (Proposition 2.11), that central elements give controlled diagonal twists (Theorem 2.14), and that endocabling can 'cable out' the center (Proposition 2.16 and Lemma 5.9). These tools are applied to cycle sets of size n whose diagonal is an n-cycle. The main results are: for n an odd prime power, every such cycle set is retractable and has finite multipermutation level (Theorem 4.2, Corollary 4.3); for n a power of 2, every such cycle set either has finite multipermutation level or retracts to the exceptional size-4 solution X_{4,19} (Theorem 5.1, Corollary 5.2). The 2-power theorem is proved by induction whose base cases v=3,4 are verified computationally in Lemma 5.10.","tokens_in":27984,"tokens_out":14894,"duration_ms":128556,"significance":"The endocabling framework is a genuinely useful new tool: it gives clean proofs of diagonal-composition formulas, recovers classical cabling as a special case, and leads to a strong structural statement for cyclic diagonals. The algebraic parts of the paper, especially the odd-prime-power theorem in Section 4, are coherent and appear to be proved rather than assumed. The odd-prime-power result is self-contained and does not depend on the computational lemma. However, the 2-power classification rests entirely on the computational base cases in Lemma 5.10, and those are not reproducible from the manuscript as written; without a verifiable base case, the dichotomy in Corollary 5.2 is not established.","major_comments":[{"comment":"The v=4 nonexistence check is not reproducible. The ESSENCE' model contains the constraint 'forAll i: int(0..n-2)' with n never declared; all arrays are indexed 0..15, so the code as printed cannot be parsed or executed. No solver version, runtime, output, or UNSAT certificate is provided. Because Lemma 5.10 is the sole base of the induction in Theorem 5.1, the claim that no irretractable cycle set of size 16 with a 16-cycle diagonal exists is unsupported until the model is corrected and the exhaustive search is documented.","section":"§5, Lemma 5.10 and the Appendix"},{"comment":"The v=3 case is asserted with the sentence that the statement is 'easily confirmed by checking all cycle sets of size 8 which are part of the GAP library YangBaxter,' but the paper gives no library version, no enumeration script, and no argument that the library is exhaustive at size 8. Since this is the other base case of the same induction, the v=3 base needs the same documentation standard as the v=4 case, for example a script and a record of complete enumeration.","section":"§5, Lemma 5.10"}],"minor_comments":[{"comment":"Page 2 contains typos: 'effciently' should be 'efficiently' and 'disivible' should be 'divisible'.","section":"Introduction"},{"comment":"'a modelling assistent' should be 'a modelling assistant'.","section":"Lemma 5.10"},{"comment":"After the reference to Lemma 5.9, 'let z′∈Z (G(Xφ)) be such that o◦(z) = 2' should be 'o◦(z′)=2'.","section":"Proof of Theorem 5.1"},{"comment":"The expression '2kg = 0' should read '2^k g = 0' to avoid confusion between multiplication by 2k and the 2^k-th multiple.","section":"Lemma 5.6(f), proof"},{"comment":"The displayed statement contains a redundant 'T_{λ_z} = T_{λ_z}'; it would be clearer to define T_z = T_{λ_z} and state T_z = λ_z^{-1}∘T∘λ_z.","section":"Proposition 2.13"},{"comment":"The comment line in the model says '$ x_i -> x_1-i centralizes G(X)', but condition (4) states that the involution i↦1-i commutes with every row map, i.e., lies in the centralizer of G(X); rephrasing would avoid ambiguity.","section":"Appendix"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the algebraic core appears sound. The only load-bearing deficiency is the reproducibility of Lemma 5.10: the v=4 Savile Row model cannot run as printed, and the v=3 GAP scan is undocumented. Please ask the author to provide a corrected, runnable model, solver logs or an UNSAT certificate, and a documented exhaustive enumeration for v=3, or to replace the computational base with a hand proof. Once that is done, the 2-power classification would be convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: endocabling is a genuinely useful generalization of cabling, and the odd-prime-power theorem is solid. The 2-power classification is probably right, but it rests on two computational base cases in Lemma 5.10 that are not reproducible, so I'd want that fixed before I'd stake anything on it.\n\nWhat's new: Definition 2.4 replaces integer scaling in cabling by arbitrary λ-endomorphisms of the permutation brace, and Proposition 2.11 gives the diagonal addition formula T_{φ+ψ}=T_φ∘T_ψ. That is a clean and reusable tool. Theorem 4.2 for odd prime powers is proved by a center/fix argument and looks coherent; Corollary 4.3 follows by induction. The 2-power dichotomy with the exceptional X_{4,19} is an interesting statement, and the induction strategy (cable out the center, use Lemmas 5.5–5.9) is plausible.\n\nSoft spots: all in the base case Lemma 5.10. For v=3 the proof says \"easily confirmed\" by checking the GAP library YangBaxter, but doesn't give the library version, the enumeration script, or evidence the library is exhaustive at size 8. For v=4 the appendix Savile Row model as printed cannot be parsed: it uses n in \"forAll i: int(0..n-2)\" and n is never declared. There is no solver version, runtime, or UNSAT certificate. Since this is a nonexistence search, an over-approximating model would be harmless, but an unparsable model proves nothing. If either base case is wrong, the induction in Theorem 5.1 has no foundation. The algebraic parts above that are not affected.\n\nOne thing I want to credit: the open problems section is honest about the limited applicability of endocabling when the center is small. That matches what I see in the paper.\n\nBottom line: for people in set-theoretic YBE/brace theory this is worth reading. It deserves peer review, but the author should be asked to provide runnable code, a declared n, solver version, and a certificate or detailed log for the v=4 search, and a reproducible GAP script for v=3.","headline":"Endocabling is a genuine reusable tool and the odd-prime-power theorem is solid; the 2-power classification is likely correct but currently rests on non-reproducible computational base cases that need fixing before full trust.","tokens_in":28534,"tokens_out":2331,"would_cite":true,"duration_ms":19822,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that finite cycle sets whose diagonal is a cyclic permutation have finite multipermutation level for odd prime-power cardinalities, and that for 2-power cardinalities either the level is finite or iterated retraction…","keywords":["Yang-Baxter equation","cycle sets","set-theoretic solutions","permutation braces","endocabling","multipermutation level","retraction","cyclic diagonal map"],"falsifier":"An independent exhaustive enumeration—for example, with a SAT-based or verified enumerator—of all 16-element cycle sets satisfying the six conditions in Lemma 5.10: if any such table exists, Theorem 5.1's base case is wrong and the 2-power classification fails. For the odd case, a single irretractable cycle set of size $p^v$ with $p$ odd and an $n$-cycle diagonal would refute Theorem 4.2.","tokens_in":1999,"feed_emoji":"🔄","tokens_out":3105,"duration_ms":120597,"temperature":0.7,"pith_summary":"The paper introduces endocabling, a way to deform a cycle set—the combinatorial shadow of an involutive, non-degenerate set-theoretic Yang-Baxter solution—by applying $\\lambda$-endomorphisms of its associated permutation brace. It then applies this tool to cycle sets of size $n$ whose diagonal map is an $n$-cycle, a class known to be indecomposable and difficult to decompose. The main theorems are: for $n$ an odd prime power, every such solution has finite multipermutation level; for $n$ a power of $2$, either it has finite multipermutation level or its iterated retractions eventually land on the unique irretractable size-4 cycle set $X_{4,19}$. If the paper is right, the only infinite-iteration behavior in the prime-power cyclic-diagonal setting is that single exceptional solution of size 4.","feed_headline":"Cyclic-diagonal solutions: finite towers, one 4-cycle exception","feed_subtitle":"Odd prime-power sizes always terminate; power-of-two sizes terminate or land on the unique size-4 irretractable solution.","key_machinery":"The central object is the permutation brace $G(X)$ of a cycle set $X$, and the new deformation operation called endocabling by a $\\lambda$-endomorphism $\\varphi$: the deformed operation is $x\\ast_\\varphi y=\\lambda_{\\varphi(\\lambda_x)}^{-1}(y)$. The load-bearing identity is the diagonal composition law $T_{\\varphi+\\psi}=T_\\varphi\\circ T_\\psi$, from which the paper derives that central elements of $G(X)$ produce diagonals that commute with $T$. The proof's workhorse is the endomorphism $\\varphi=\\mathrm{id}-\\lambda_z$ for a central involution $z$, which removes $z$ from the image (the paper calls this \"cabling out the center\") and, together with fix/socle arguments, forces retractability.","core_discovery":"Stated in cycle-set language: if $X$ is a finite cycle set of cardinality $p^v$ with $p$ odd and diagonal $T$ an $n$-cycle, then $X$ is retractable (Theorem 4.2), hence by iterating retractions it has finite multipermutation level (Corollary 4.3). For cardinality $2^v$, Theorem 5.1 proves retractability for $v>2$, and Corollary 5.2 shows that iterated retraction either reaches a one-element cycle set or reaches the unique size-4 irretractable cycle set $X_{4,19}$. The engine is endocabling: given a $\\lambda$-endomorphism $\\varphi$ of the permutation brace $G(X)$, one forms a new cycle set $X_\\varphi$ by setting $\\lambda_x^\\varphi=\\lambda_{\\varphi(\\lambda_x)}$, and the diagonal obeys the composition law $T_{\\varphi+\\psi}=T_\\varphi\\circ T_\\psi$. This gives enough control over the diagonal and the center of the permutation group to force retraction.","pith_inferences":["One consequence not drawn in the paper is that the diagonal law $T_{\\varphi+\\psi}=T_\\varphi\\circ T_\\psi$ suggests a module structure on the set of achievable diagonals; developing it could classify which permutations can occur as diagonals of solutions whose permutation group has a large center.","The odd-prime-power theorem might support a recursive classification: every retraction step of a cyclic-diagonal solution is a surjection with equal fiber size, so enumerating all such solutions of size $p^v$ could be reduced to extending solutions of size $p^{v-1}$ by fibers.","For composite non-prime-power sizes the paper gives no answer and notes $n=45$ as the smallest open odd case; a theorem or counterexample there would show whether the prime-power restriction is essential or an artifact of the fixed-point arguments.","The size-16 computational base case is printed with an undefined loop bound, so a machine-checkable certificate for that search would turn the reported computation into a verifiable proof."],"forward_implications":["For every odd prime power $p^v$, the retraction tower of a cyclic-diagonal cycle set ends at a one-element cycle set, so the multipermutation level is finite and the solution is built from finitely many retraction steps.","For every $2$-power size, no infinite retraction tower is possible: iterated retraction either reaches a point or reaches the unique irretractable size-$4$ cycle set $X_{4,19}$.","Epimorphic images of a cyclic-diagonal solution remain cyclic-diagonal, so the retraction tower stays inside the same class and the finiteness/exception dichotomy propagates down the tower.","In the $2$-power case, endocabling by $\\varphi=\\mathrm{id}-\\lambda_z$ sends the diagonal to its square and controls the Dehornoy class of the cabled solution as a power of $2$, giving quantitative control beyond mere retractability."],"supporting_citations":[{"why":"Defines cycle sets, the retraction relation, and the finiteness theorem guaranteeing non-degeneracy; provides the foundational language used throughout.","marker":"[23]"},{"why":"Introduces the brace structure on the permutation group of a cycle set and the identity $\\lambda_g(\\lambda_x)=\\lambda_{\\lambda_g(x)}$ underlying the endocabling computations.","marker":"[24]"},{"why":"Supplies the classical cabling construction and its diagonal-power identity, which endocabling generalizes and which the paper recovers as a corollary.","marker":"[19]"},{"why":"Provides the equal-fiber-size lemma for surjective homomorphisms onto indecomposable targets, used to prove that epimorphic images preserve the cyclic-diagonal condition.","marker":"[5]"},{"why":"Defines the retraction sequence and multipermutation level, the finiteness notion that the main theorems establish.","marker":"[14]"},{"why":"Supplies the indecomposability of solutions whose diagonal is a full cycle, the starting observation for the irreducibility arguments.","marker":"[22]"},{"why":"Provides the number-theoretic lemma on the possible orders of units modulo $2^v$ and $p^v$, used in the analysis of central elements of the permutation group.","marker":"[6]"},{"why":"Gives the theorem that irretractable cycle sets have trivial socle, the key contradiction used to force retractability from central fixed elements.","marker":"[1]"},{"why":"Describes the constraint-solving approach used for the size-16 base case of the 2-power theorem.","marker":"[21]"}],"fun_headline_variants":["Endocabling tames cyclic-diagonal solutions, with a size-4 exception","Cyclic-diagonal solutions: odd p^v finite, 2^v end at size-4 exception","Endocabling forces retraction: p^v finite, 2^v has one size-4 snag","Cyclic-diagonal solutions: p^v retract, 2^v only exception is size-4"],"cache_read_input_tokens":30592,"weakest_assumption_plain":"The load-bearing premise is that the paper's computer searches for the base cases of size 8 and size 16 are correct and exhaustive, meaning the size-16 constraint model faithfully encodes all cycle-set axioms plus the diagonal, centralizer, automorphism, and irretractability conditions, and the search code contains no hidden error (the printed code uses an undefined loop bound).","fun_headline_variants_meta":{"raw":{"variants":["Endocabling tames cyclic-diagonal solutions, with a size-4 exception","Cyclic-diagonal solutions: odd p^v finite, 2^v end at size-4 exception","Endocabling forces retraction: p^v finite, 2^v has one size-4 snag","Cyclic-diagonal solutions: p^v retract, 2^v only exception is size-4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000991,"raw_usage":{"total_tokens":4211,"prompt_tokens":970,"completion_tokens":3241,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":3135}},"tokens_in":586,"tokens_out":3241,"duration_ms":21609,"temperature":1.0,"reasoning_tokens":3135,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:51:18.574627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent exhaustive enumeration—for example, with a SAT-based or verified enumerator—of all 16-element cycle sets satisfying the six conditions in Lemma 5.10: if any such table exists, Theorem 5.1's base case is wrong and the 2-power classification fails. For the odd case, a single irretractable cycle set of size $p^v$ with $p$ odd and an $n$-cycle diagonal would refute Theorem 4.2.","supporting_citations":[{"cited_title":"Ram´ ırez and L","cited_arxiv_id":null,"evidence_quote":"Supplies the indecomposability of solutions whose diagonal is a full cycle, the starting observation for the irreducibility arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the number-theoretic lemma on the possible orders of units modulo $2^v$ and $p^v$, used in the analysis of central elements of the permutation group."},{"cited_title":"Bachiller, F","cited_arxiv_id":null,"evidence_quote":"Gives the theorem that irretractable cycle sets have trivial socle, the key contradiction used to force retractability from central fixed elements."},{"cited_title":"Nightingale, ¨Ozg¨ ur Akg¨ un, I","cited_arxiv_id":null,"evidence_quote":"Describes the constraint-solving approach used for the size-16 base case of the 2-power theorem."}],"review_version":1}