{"id":"d8be1f73-4a00-4c2e-8c22-24ef5a111df0","arxiv_id":"1908.08371","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For Latin squares in bipartite min-max-plus systems, the paper gives an eigenvalue-first power algorithm and calls it efficient, but the proof that the first-cycle vector is an eigenvector is invalid and no benchmark is provided.","lead":"A short paper proposes an iterative algorithm for eigenvectors of Latin squares in bipartite min-max-plus systems by first fixing the eigenvalue with a known formula. The claimed efficiency gain is illustrated on one example, but the main proof rests on an invalid min-max exchange and the promised numerical comparison is absent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's proof relies on an unjustified min-max interchange; Algorithm 2's step-5 eigenvector claim and the efficiency comparison are therefore unsupported.","rationale":"The reader's weakest assumption points to the exact place where the proof of Theorem 2 fails: the min-plus block does not commute with the componentwise maximum used to define v. This is load-bearing because Theorem 2 is what lets Algorithm 2 stop at step 5; without it, the algorithm falls back to step 7, which is essentially the restart loop of the older Algorithm 3, and the stated efficiency gain is unsupported. The paper contains no Latin-square-specific lemma or additional argument establishing the missing equality, and the worked Example 3 verifies only one instance. It also claims a numerical comparison in Remark 4 but reports no data, so the efficiency claim has no independent experimental support. I do not see an ad hominem issue or a purely stylistic objection; the concern is internal to the proof. If a computational search over small Latin squares found no counterexample, the objection would reduce to a proof gap rather than a false theorem, but as written the paper still does not establish its central claim. Hence the reader's REJECT verdict stands.","tokens_in":10645,"tokens_out":28137,"duration_ms":270820,"concrete_test":"Enumerate all order-3 and order-4 Latin squares in the n_epsilon/n_tau encoding of Section 3; for each, run Algorithm 2 from several initial vectors and at step 5 test whether N(v)=v. If any run gives N(v)>v, Theorem 2 is false in the claimed Latin-square setting and the efficiency claim collapses. If all runs pass, symbolically re-derive the w-block equality min_i max_l (b_ji+u_i(l)) = max_l min_i (b_ji+u_i(l)) for those orbits; if it cannot be derived, the proof gap remains even though the algorithm may be repairable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion is Theorem 2: if x*(r)=x*(s), then v=x*(s)⊕...⊕x*(r-1) satisfies M(v)=λ⊗v. The proof collapses N(⊕_l x*(l)) into ⊕_l N(x*(l)). This is correct for the u-block because Aλ⊗ distributes over max, but for the w-block it requires Bλ⊗'(⊕_l u*(l)) = ⊕_l (Bλ⊗'u*(l)), i.e. min_i max_l (b_ji+u_i(l)) = max_l min_i (b_ji+u_i(l)). In general only max_l min_i ≤ min_i max_l holds, and strict gaps are easy to construct: for the 2x2 array [[0,10],[9,1]], min_i max_l = 9 while max_l min_i = 1. The proof gives no Latin-square-specific argument that equality holds on the orbit produced by Algorithm 2. If equality fails, the vector v from step 5 has N(v)>v, so step 7's restart is needed and the claimed efficiency advantage of Algorithm 2 over Algorithm 3 disappears. Theorem 3 and Lemma 2 only show N(v)≥v; they do not establish the needed fixed point without an additional eventual-periodicity assumption. Remark 4 asserts a numerical comparison but no experimental data is reported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies eigenproblems for bipartite min-max-plus systems of the form u(l+1)=A⊗w(l), w(l+1)=B⊗'u(l), specializing to the case where A and B are Latin squares. It imports the eigenvalue formula λ=(max(A)+min(B))/2 from earlier work and proposes Algorithm 2, which iterates the shifted map N until a cycle x*(r)=x*(s) is found, forms v as the componentwise maximum over one cycle, and asserts that M(v)=λ⊗v. If this check fails, the algorithm restarts from v and iterates until a fixed point. The paper compares the algorithm with the power algorithm (Algorithm 3) on a single 4×4 example and claims greater efficiency.","tokens_in":10970,"tokens_out":6328,"duration_ms":61469,"significance":"If the central claim were correct, the paper would provide a simple cycle-based eigenvector computation for Latin-square bipartite min-max-plus systems, and the comparison with the power algorithm would be of interest to the discrete-event-systems community. The paper is clearly organized and the worked example is explicitly verified. However, the main theorem is not established: the proof relies on an invalid min-max interchange, and the efficiency claim rests on numerical experiments that are not reported. The manuscript gives no machine-checked proofs, reproducible code, or experimental data, so the significance is currently unsupported.","major_comments":[{"comment":"The proof passes the shifted map N through the componentwise maximum ⊕ without justification. For the w-block this requires Bλ⊗'(⊕_l u*(l)) = ⊕_l (Bλ⊗'u*(l)), that is, for each j, min_i max_l (b_{ji}+u_i(l)) = max_l min_i (b_{ji}+u_i(l)). Only the inequality max_l min_i ≤ min_i max_l holds in general; for the 2×2 array [[0,10],[9,1]], the left-hand side is 9 and the right-hand side is 1. Since no Latin-square-specific argument is given to restore equality on the orbits produced by Algorithm 2, Theorem 2 is not established for general bipartite min-max-plus systems. This is load-bearing: Algorithm 2 uses Theorem 2 to stop at step 5 without the restart loop, and the claimed efficiency advantage over Algorithm 3 depends on that step.","section":"Section 2, proof of Theorem 2"},{"comment":"Lemma 2 and Theorem 3 only prove N(v)≥v, not N(v)=v. The proof of Algorithm 1 then assumes that the restarted orbit is eventually periodic and concludes that a nondecreasing eventually periodic orbit must be constant. No proof of eventual periodicity of the restarted orbit is supplied; the periodicity of the original orbit x*(l) is not transferred to the orbit starting from v. Since min-max-plus maps can have bounded orbits that are not exactly eventually periodic, the termination of step 7 of Algorithm 2 is not justified.","section":"Section 2, 'Proof of Algorithm 1' and Lemma 2"},{"comment":"Remark 4 states that numerical experiments were made to compare Algorithms 2 and 3, but no experimental data, tables, or reproducibility details are included. The only evidence is the single 4×4 example in Example 3. The conclusion that the proposed method is 'more efficient' is therefore unsupported; at minimum a table of iteration counts or a complexity analysis for several Latin-square sizes is needed.","section":"Section 3, Remark 4"}],"minor_comments":[{"comment":"The abstract contains small writing errors: 'propose method' should be 'proposed method', and 'Islam abad' should be 'Islamabad'.","section":"Abstract"},{"comment":"The initial state vector is chosen without explanation; it would be useful to state whether the resulting eigenvector is independent of the starting vector, or to explain the choice.","section":"Section 3, Example 3"},{"comment":"The componentwise order '≥' used for vectors in R^{m+n} is not explicitly defined; the manuscript should state that inequalities are understood componentwise.","section":"Section 2, Theorem 3"},{"comment":"The symbol n is used both for the order of the Latin square and for the set {1,...,n}; the surrounding text would be clearer if the set were denoted by a different symbol such as [n] or n̲.","section":"Section 3, general notation"}],"recommendation":"reject","confidential_remarks":"The central theorem appears unsound as written, and the efficiency claim is not backed by reported experiments. The manuscript might be salvageable if the authors can prove a Latin-square-specific version of the min-max interchange or revise the algorithm to avoid relying on it; as it stands, the main contribution is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the central proof doesn't go through. Theorem 2 claims that if the shifted orbit x* is periodic from s to r, then the componentwise max v of one period is already an eigenvector. The proof needs N(v)=N(x*(s))⊕...⊕N(x*(r−1)). That holds for the A-block because max-plus matrix multiplication distributes over max, but for the B-block it requires min_i max_l(b_ji+u_i(l)) = max_l min_i(b_ji+u_i(l)). In general only one inequality holds. The 2×2 array [[0,10],[9,1]] separates them. The paper gives no Latin-square-specific reason equality holds on the orbit it generates, so Theorem 2 is unproved and Algorithm 2's step-5 early stop is unsupported. The proof of Algorithm 1 also slips: Lemma 2 is about the restarted orbit, not the original cycle, so the contradiction 'x(r)=x(s) implies no coordinate can increase' does not follow.\n\nWhat is good: the problem is real and the setup is clear. For Latin squares, using the known eigenvalue formula from [11] and then shifting by λ is sensible. The worked example is fully worked and numerically correct; they verify M(v)=λ⊗v. The algorithm is understandable. It is an honest, modest extension of Subiono's power algorithm.\n\nSoft spots: the load-bearing issue above is not minor; it is the difference between a one-pass algorithm and a restart. Also, Remark 4 mentions numerical experiments comparing algorithms 2 and 3, but no experiment is reported. The efficiency claim rests on one illustrative example. That is a separate weakness but easy to fix.\n\nWho it's for: specialists in max-plus/min-max-plus and discrete-event systems who care about Latin-square eigenproblems. It is not a field-changer.\n\nRecommendation: I'd send it to a referee who knows min-max-plus well, because the flaw is precise and possibly repairable—if the authors prove the missing inequality on this class, or show the algorithm still works after the restart, the note has value. But as it stands, I would not accept it; rejection with an encouraging report is the right outcome.","headline":"The paper's efficiency trick rests on a min-max interchange that is false in general, so the central theorem does not hold as stated; the worked example works, but the proof gap is real and the efficiency comparison is data-free.","tokens_in":11415,"tokens_out":4385,"would_cite":false,"duration_ms":41604,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A18","05B20","08A99"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Latin squares in a bipartite min-max-plus system, once the normalized iteration repeats a state, the componentwise maximum over that cycle is an eigenvector, so no restart loop is needed.","keywords":["bipartite min-max-plus systems","Latin squares","eigenvalue","eigenvector","max-plus algebra","min-plus algebra","power algorithm","periodic orbit"],"falsifier":"Choose a Latin-square pair $A,B$ of order $n$ in one of the four entry-type cases, run Algorithm 2, and after detecting $x^*(r)=x^*(s)$ compute both $M(v)$ and $\\lambda\\otimes v$; if the two vectors differ, Theorem 2 fails for that case.","tokens_in":10497,"feed_emoji":"🔁","tokens_out":9724,"duration_ms":81339,"temperature":0.7,"pith_summary":"The paper's target is the eigenproblem for Latin-square matrices inside a bipartite min-max-plus system, where the state update alternates a max-plus step and a min-plus step. It claims that once the normalized iteration $x^*(l+1)=N(x^*(l))$ revisits a state, the componentwise maximum $v=x^*(s)\\oplus\\dots\\oplus x^*(r-1)$ over the detected cycle is already an eigenvector, so $M(v)=\\lambda\\otimes v$ holds. If this is right, the computation shortens: the restart-and-iterate fallback used by earlier power algorithms is unnecessary, and the eigenvector comes from one cycle of the normalized map. The paper illustrates the construction on Latin squares of order 4 and argues by example that it is simpler than the existing power algorithm.","feed_headline":"For Latin squares, a repeated state already gives the eigenvector","feed_subtitle":"One pass through the normalized map settles the eigenproblem, skipping the restart loop.","key_machinery":"The carrying device is the normalized map $N$, defined by $N((u,w)^\\top)=(A_\\lambda\\otimes w,\\,B_\\lambda\\otimes' u)^\\top$ with $A_\\lambda=-\\lambda\\otimes A$ and $B_\\lambda=-\\lambda\\otimes B$. It turns the eigenproblem $M(v)=\\lambda\\otimes v$ into the search for a fixed point $N(v)=v$, and its iterates produce the sequence $x^*(0),x^*(1),\\dots$ in which a repeated state marks a cycle. The componentwise maximum $\\oplus$ over that cycle is the candidate eigenvector; the argument that $N$ distributes over this maximum is what makes the candidate exact. For Latin squares, the explicit eigenvalue formula supplies $\\lambda$ before any iteration begins.","core_discovery":"The central claim is Theorem 2: if the normalized orbit satisfies $x^*(r)=x^*(s)$ for integers $r>s\\ge 0$, then $M(v)=\\lambda\\otimes v$ for $v=x^*(s)\\oplus\\dots\\oplus x^*(r-1)$, meaning the pointwise maximum over one full cycle is an eigenvector. For Latin squares, the eigenvalue is known in advance, $\\lambda=(\\max(A)+\\min(B))/2$, so Algorithm 2 only has to form $A_\\lambda=-\\lambda\\otimes A$ and $B_\\lambda=-\\lambda\\otimes B$, iterate the normalized map until a state repeats, and take the componentwise maximum over the period. The paper claims that in the Latin-square cases it examines this yields the correct eigenvector directly, without the fallback of restarting from $v$ and iterating until a fixed point of $N$ is reached.","pith_inferences":["A natural next test is to run Algorithm 2 on random Latin-square pairs in all four entry-type cases; if the fallback step ever fires, the efficiency gain is not universal.","A cheap safeguard would be to check $N(v)=v$ before accepting the cycle-max vector, since the proof's interchange step is the only non-mechanical part of the argument.","The cycle-max rule can be read as a tropical analogue of averaging over a periodic orbit; in non-Latin systems one may need several cycles before such an aggregate stabilizes into an eigenvector."],"forward_implications":["For any Latin-square system in the paper's class, the eigenvector can be computed from one detected cycle of the normalized map, with no restart loop.","Because the eigenvalue is known from $\\lambda=(\\max(A)+\\min(B))/2$, the algorithm fully separates eigenvalue computation from eigenvector computation.","The same cycle-max rule is claimed to produce both trivial and nontrivial eigenvectors, depending on the starting vector.","Stopping at an exact state repetition $x^*(r)=x^*(s)$ is simpler to detect than the power algorithm's scalar-shift condition $x(r)=c\\otimes x(s)$.","The construction also suggests how an analogous algorithm might be derived for separated min-max-plus systems, as the paper's conclusion proposes."],"supporting_citations":[{"why":"Supplies the explicit eigenvalue formula for Latin squares in bipartite min-max-plus systems that Algorithm 2 uses as its first step.","marker":"[11]"},{"why":"The power algorithm for bipartite min-max-plus systems that serves as the baseline and whose restart fallback the new algorithm tries to avoid.","marker":"[12]"},{"why":"The authors' earlier efficient algorithm for nontrivial eigenvectors in max-plus algebra, extended here to the bipartite Latin-square setting.","marker":"[13]"},{"why":"The classical power algorithm in max-plus algebra, the background method for detecting cycles and building eigenvectors from them.","marker":"[2]"}],"fun_headline_variants":["Repeated state in Latin squares yields eigenvector directly","Latin squares: one cycle gives eigenvector, no restart needed","Efficient eigen solving for Latin squares via repeated states","New algorithm: repeated state gives eigenvector for Latin squares","One pass through the map settles eigenproblem for Latin squares"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes an interchange of order: combining several states by taking coordinate-wise maxima and then applying the evolution rule must give the same result as evolving each state first and then combining, which is not automatic in systems that mix max and min operations.","fun_headline_variants_meta":{"raw":{"variants":["Repeated state in Latin squares yields eigenvector directly","Latin squares: one cycle gives eigenvector, no restart needed","Efficient eigen solving for Latin squares via repeated states","New algorithm: repeated state gives eigenvector for Latin squares","One pass through the map settles eigenproblem for Latin squares"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000746,"raw_usage":{"total_tokens":3247,"prompt_tokens":791,"completion_tokens":2456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":2376}},"tokens_in":407,"tokens_out":2456,"duration_ms":15006,"temperature":1.0,"reasoning_tokens":2376,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:43:41.367674+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a Latin-square pair $A,B$ of order $n$ in one of the four entry-type cases, run Algorithm 2, and after detecting $x^*(r)=x^*(s)$ compute both $M(v)$ and $\\lambda\\otimes v$; if the two vectors differ, Theorem 2 fails for that case.","supporting_citations":[{"cited_title":"S.; Adzkiya, D","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit eigenvalue formula for Latin squares in bipartite min-max-plus systems that Algorithm 2 uses as its first step."},{"cited_title":"Power algorithms for (max,+)- and b ipartite (min,max,+)-systems","cited_arxiv_id":null,"evidence_quote":"The power algorithm for bipartite min-max-plus systems that serves as the baseline and whose restart fallback the new algorithm tries to avoid."},{"cited_title":"An Efficient Algorithm for Nontrivia l Eigenvectors in Max-Plus Algebra","cited_arxiv_id":null,"evidence_quote":"The authors' earlier efficient algorithm for nontrivial eigenvectors in max-plus algebra, extended here to the bipartite Latin-square setting."},{"cited_title":"The power algorithm in max algebra","cited_arxiv_id":null,"evidence_quote":"The classical power algorithm in max-plus algebra, the background method for detecting cycles and building eigenvectors from them."}],"review_version":1}