{"id":"49f7c994-8164-4e9f-938e-a722d5b3ba68","arxiv_id":"1908.08234","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The normalized Perron eigenvector of exp(kA) converges to a point in the tropical max-plus eigenspace, and two conjectures locate that point from the shape of the eigenspace.","lead":"This paper studies how the leading eigenvector of a matrix behaves in the very cold limit, using tropical mathematics. It reports two unproven conjectures that would describe the limit point from the shape of the tropical eigenspace.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conjecture 3.0.1 is contradicted by an explicit 3x3 matrix whose tropical eigenspace has the stated form but whose Perron limit is not the predicted endpoint.","rationale":"The reader rejected the paper because the central characterization is asserted without proof and the limit P_infinity is assumed from numerics. My review finds a stronger reason to reject: the central conjecture is false, as shown by an explicit 3x3 matrix satisfying the conjecture's hypothesis but violating its conclusion. The matrix A = [[0, -M, -M], [-M, -1, 0], [-M, 0, -1]] has a tropical eigenspace that is exactly the diagonal segment required by Conjecture 3.0.1, yet the normalized log-Perron limit is (0, M-1, M-1), not the endpoint (0, M, M). This is not a subtle missing proof or a numerical artifact; it is a direct algebraic counterexample. It also shows that the paper's 'complete characterization' is not merely unsupported but incorrect, so the abstract's overstatement is materially misleading. The reader's identified weakest assumption about the existence of the limit is less central, since the limit exists and is computable in this example. I therefore retain the REJECT verdict without further adjustment.","tokens_in":7044,"tokens_out":16527,"duration_ms":162557,"concrete_test":"Run the paper's numerical procedure on A = [[0, -2, -2], [-2, -1, 0], [-2, 0, -1]] for k up to 100 with high precision. The tropical eigenspace is {(0, t, t) : -2 <= t <= 2}, so Conjecture 3.0.1 predicts P_infinity(A) = (0, 2, 2). The Perron eigenvector of Ak = [[1, q, q], [q, r, 1], [q, 1, r]] with q = e^{-2k}, r = e^{-k} satisfies x2 = x3 and x1 / x2 = 2q / ([r + sqrt(r^2 + 8q^2)] / 2), giving (1/k) log(x2 / x1) -> 1 as k -> infinity. Thus the numerical limit should approach (0, 1, 1), not (0, 2, 2). If it does, Conjecture 3.0.1 is decisively refuted; if it does not, the discrepancy should be explained.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised characterization rests on Conjecture 3.0.1, but the conjecture is not merely unproved: it is false. Take M > 1 and A = [[0, -M, -M], [-M, -1, 0], [-M, 0, -1]]. The max-plus eigenvalue is 0, with critical graph consisting of the self-loop at 1 and the 2-cycle 2<->3, so the tropical eigenspace in TP^2 is exactly {(0, t, t) : -M <= t <= M}. Thus every eigenvector has the form (0, -M + alpha, -M + alpha) for 0 <= alpha <= 2M, matching the hypothesis of Conjecture 3.0.1 with beta = 2M; the conjecture predicts P_infinity(A) = (0, M, M). However, for Ak = exp(kA), write q = e^{-kM} and r = e^{-k}. The Perron equations for [[1, q, q], [q, r, 1], [q, 1, r]] give x2 = x3 and x1 / x2 = 2q / mu, where mu = [r + sqrt(r^2 + 8q^2)] / 2, so x1 / x2 ~ 2 e^{-k(M-1)}. Consequently (1/k) log(x2 / x1) -> M - 1, i.e. P_infinity(A) = (0, M-1, M-1). For M = 2 this is (0, 1, 1), strictly inside the segment [-2, 2] and not the endpoint (0, 2, 2) predicted by Conjecture 3.0.1. The limit exists in this example, so the reader's weakest-assumption concern about the limit is moot; the central conjecture itself fails.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotic behavior of the Perron eigenvector of the matrix exponential A_k = exp(kA) as k tends to infinity, aiming to characterize the limit P_∞(A) = lim_k (1/k) log L(A_k) in terms of the max-plus (tropical) eigenspace of A. It proves elementary results: Proposition 2.3.3 identifies the limit of the normalized Perron eigenvalue with the max-plus eigenvalue of A, and Proposition 2.4.1 claims that the normalized Perron eigenvector lies in the max-plus eigenspace. The paper then presents numerical experiments for 3x3 matrices with several critical classes, states Conjecture 3.0.1 (a characterization of P_∞(A) when the tropical eigenspace is a one-dimensional segment), states Conjecture 3.0.2 (P_∞(A) depends only on the tropical eigenspace), and discusses the failure of the Akian et al. algorithm in this setting. The main advertised result is the conjectured complete characterization.","tokens_in":7388,"tokens_out":9019,"duration_ms":83257,"significance":"The question addressed is natural and potentially useful for transfer-matrix asymptotics and tropical spectral theory. The paper contains a few correct elementary observations, and the availability of the code on GitHub is a plus. If the conjectures were true, they would indeed provide a robust characterization of the Perron-eigenvector limit for a wider class than the single-critical-class case. However, the central conjecture is not merely unproved: it is contradicted by an explicit 3x3 matrix given below. Since the abstract claims a 'complete characterization for a larger class of matrices', and the paper's own main tool is conjectural and false as stated, the significance of the paper in its present form is very limited.","major_comments":[{"comment":"The conjecture is false. Take A = [[0, -M, -M], [-M, -1, 0], [-M, 0, -1]] with M > 1. The max-plus eigenvalue of A is 0, with critical graph consisting of the self-loop at node 1 and the 2-cycle 2<->3. The max-plus eigenspace in TP^2 is exactly {(0, t, t): -M <= t <= M}, i.e. all eigenvectors have the form (0, -M+alpha, -M+alpha) for 0 <= alpha <= 2M. Thus the hypothesis of Conjecture 3.0.1 holds with beta = 2M, and the conjecture predicts P_∞(A) = (0, M, M). However, for A_k = exp(kA) = [[1, e^{-kM}, e^{-kM}], [e^{-kM}, e^{-k}, 1], [e^{-kM}, 1, e^{-k}]], symmetry gives x_2 = x_3, and the Perron equations yield x_1/x_2 = 2 e^{-kM}/s with s = (e^{-k} + sqrt(e^{-2k} + 8e^{-2kM}))/2 ~ e^{-k}. Therefore (1/k) log(x_2/x_1) -> M-1, so P_∞(A) = (0, M-1, M-1). For M = 2, this is (0, 1, 1), strictly inside the segment [-2, 2] and not the endpoint (0, 2, 2) predicted by the conjecture. This directly invalidates the claimed complete characterization.","section":"Section 3, Conjecture 3.0.1"},{"comment":"The existence of the limit P_∞(A) is asserted as an 'immediate observation from the plots' after the iteration is cut off around k = 30. No proof or error bound is supplied, and the heuristic derivation of Proposition 2.4.1 (that the normalized Perron vector lies in the eigenspace in the limit) rests on an informal 'as k→∞' step rather than a proved convergence statement. Since both conjectures are formulated in terms of this limit, a rigorous existence or convergence statement, or at least a conditional formulation, is needed before the conjectures can be meaningfully tested.","section":"Sections 2.4 and 3"},{"comment":"The assertion that σ_max+(A) = σ_max+(B) implies P_∞(A) = P_∞(B) is supported only by two numerical examples (Figures 8 and 9). No argument is given, and it is not a formal consequence of Conjecture 3.0.1. As this is a second load-bearing claim in the advertised complete characterization, it needs either a proof or an explicit statement that it is only numerical evidence. The current text does not supply either.","section":"Section 3, Conjecture 3.0.2"}],"minor_comments":[{"comment":"There is a typo: 'convengence' should be 'convergence'.","section":"Section 1"},{"comment":"The phrase 'Proofs of the conjectures in section 4' is incorrect: the conjectures appear in Section 3, not Section 4. Also, 'asymtotics' should be 'asymptotics'.","section":"Section 4"},{"comment":"The reference to 'Synchronization and linearity' lists 'B. Francois' as the first author and the year 2001; the correct authors are F. Baccelli, G. Cohen, G.J. Olsder, and J.-P. Quadrat, and the book was published by Wiley in 1992.","section":"References"},{"comment":"The displayed predicted vector from Theorem 6.1 of [Akian et al., 2006] is typeset ambiguously, mixing exponents and weights in a single column. It should be presented separately as a weight vector and an exponent vector so that the subsequent arrows to TP^2 are clear.","section":"Section 3, Akian counterexample"}],"recommendation":"reject","confidential_remarks":"The paper is an early exploratory note. The elementary eigenvalue-limit result (Proposition 2.3.3) is correct, but the paper's main advertised claim is a conjecture that is false as stated, as shown by the explicit counterexample in my report. I do not see a way to repair the central claim within the scope of the current manuscript; the appropriate venue would be a shorter note presenting the numerical experiments as conjectures without claiming a complete characterization."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: the main characterization claim, Conjecture 3.0.1, is false. The stress-test counterexample holds up. For M > 1, take A = [[0,-M,-M],[-M,-1,0],[-M,0,-1]]. Its max-plus eigenspace is exactly {(0,t,t) : -M ≤ t ≤ M}, so every eigenvector has the claimed form with beta = 2M. If the conjecture were true, the Perron limit would be (0,M,M). But for Ak = exp(kA), symmetry forces x2 = x3, and the Perron ratio solves q y^2 + r y - 2q = 0 with q = e^{-kM}, r = e^{-k}. The positive root gives (1/k) log(x2/x1) → M - 1, so P_infinity(A) = (0, M-1, M-1), strictly inside the segment, not the endpoint. The limit exists, so the existence assumption is not the issue; the conjecture itself fails.\n\nWhat the paper does well: it cleanly restates the known result that the normalized Perron eigenvector limit lies in the tropical eigenspace, and it documents a real counterexample to the Akian-Bapat-Gaubert algorithm. That counterexample is a genuine, reproducible observation and is the most useful thing here.\n\nThe soft spots are severe. The abstract overclaims a 'complete characterization' when the main results are conjectures, and the three-by-three numerics are suggestive but not proof. Worse, the conjecture is not merely unproved; it is refuted by an explicit matrix. The paper would need major revision: either retract the conjecture and offer a weaker correct statement, or cut the paper down to a short note about the counterexample.\n\nThe proofs that are present (Lemma 2.3.2, Prop 2.4.1) are elementary and sound. The citation pattern is fine; reliance on Gaubert-Plus and Akian et al. is standard.\n\nWho is this for? Someone in tropical spectral theory might want the counterexample on record, but the paper in its current form is not a reliable reference because its central claim is wrong. I would not send this to a serious referee as is. If the author reworks it into a note about the Akian et al. counterexample, that would be worth a quick look.\n\nRegards.","headline":"The paper's central conjecture is false; the useful fragment is a counterexample to an existing algorithm.","tokens_in":7960,"tokens_out":7383,"would_cite":false,"duration_ms":55816,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A80","15A18","15A48"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the asymptotic Perron eigenvector of exp(kA) is governed by the tropical eigenspace of A, with the limit selecting a distinguished point—the far endpoint when the eigenspace is a segment—and depending only on that…","keywords":["tropical analysis","Perron-Frobenius eigenvector","transfer matrix","max-plus algebra","asymptotic eigenvector","tropical eigenspace","critical eigenvectors","Hadamard power"],"falsifier":"Compute P_infinity for a 3 by 3 matrix with a two-dimensional tropical eigenspace and check, using high-precision arithmetic past k = 30, whether the normalized eigenvector sequence oscillates between two accumulation points; two distinct subsequential limits would falsify the claimed well-definedness. Alternatively, find two matrices A and B with identical sigma_{max+} but different P_infinity, which would directly falsify Conjecture 3.0.2.","tokens_in":6796,"feed_emoji":"🌡️","tokens_out":7018,"duration_ms":65293,"temperature":0.7,"pith_summary":"This paper asks whether the zero-temperature limit of the Perron eigenvector of the transfer matrix exp(kA) can be read off from tropical data of A. It establishes that the normalized limit P_infinity(A) always lies in the tropical max-plus eigenspace, but that it need not be a critical eigenvector; the earlier single-critical-eigenvector characterization does not generalize. The paper's central conjectures are: when the tropical eigenspace is a line segment of the form (0, v1+alpha, ..., vn+alpha), the limit is the far endpoint (0, v1+beta, ..., vn+beta), and more generally P_infinity depends only on the tropical eigenspace, not on the other entries of A. A complete proof of these conjectures would let physicists and applied mathematicians compute asymptotic eigenvectors directly from a few extremal entries of the matrix.","feed_headline":"Perron eigenvector limit tracks the tropical eigenspace","feed_subtitle":"Two conjectures map the zero-temperature transfer-matrix eigenvector onto tropical geometry, making it computable from a few entries.","key_machinery":"The load-bearing object is the limit map P_infinity(A) = lim_{k->infty} (1/k) log(L(exp(kA))), a point in tropical projective space $TP^{{n-1}}$, together with the tropical max-plus eigenspace sigma_{max+}(A), the set of tropical eigenvectors of A. The argument connects the two through Hadamard powers: L(exp(kA)) = L($M^{{(k)}}$) with M = exp(A), and the eigenvectors of $M^{{(k)}}$ are the k-th Hadamard powers of tropical eigenvectors of M, so (1/k) log of the Perron eigenvector lies in sigma_{max+}(A). The conjectures identify which point of the eigenspace the limit selects: the far endpoint when the eigenspace is one-dimensional, and in general a function of the eigenspace alone.","core_discovery":"The central discovery is that the asymptotic normalized Perron eigenvector P_infinity(A) = lim_{k->infty} (1/k) log(L(exp(kA))) is controlled by the tropical eigenspace sigma_{max+}(A), not by individual matrix entries. After proving P_infinity(A) lies in sigma_{max+}(A), the paper gives numerical evidence that it converges to a point in the interior or on the boundary of the eigenspace, and states two conjectures. Conjecture 3.0.1 says that if every tropical eigenvector has the form (0, v1+alpha, ..., vn+alpha) with 0 <= alpha <= beta, then P_infinity(A) = (0, v1+beta, ..., vn+beta), i.e. the limit is the endpoint of the eigenspace segment. Conjecture 3.0.2 says that equality of tropical eigenspaces, sigma_{max+}(A) = sigma_{max+}(B), implies equality of limits, P_infinity(A) = P_infinity(B). The paper also reports that the Schur-complement algorithm from cited perturbation theory often predicts a non-Perron eigenvector or no vector at all, and gives a concrete counterexample where a predicted vector lies in the eigenspace but is not the limit.","pith_inferences":["A testable extension of the paper's outlook is to compute P_infinity for matrices with identical tropical eigenspaces but different critical-graph cycle lengths; if the limits differ, Conjecture 3.0.2 would need refinement.","For higher-dimensional tropical eigenspaces, the selected limit point may be a weighted combination of critical eigenvectors, with weights possibly tied to cycle mean costs; the line-segment case would then be the special case where the weights collapse to an endpoint.","If the eigenspace-only dependence is correct, analogous statements should hold in the min-plus and max-times tropical algebras via exponentiation and logarithms, potentially yielding a tropical notion of eigenvector for tensors, as the paper suggests."],"forward_implications":["For matrices whose tropical eigenspace is a segment, the zero-temperature Perron eigenvector limit is determined by the eigenspace's endpoint, so perturbing entries that do not change that eigenspace leaves the limit unchanged.","The invariance conjectured in 3.0.2 would make P_infinity computable from the small set of critical entries that determine the tropical eigenspace, rather than from all n^2 entries.","The counterexample to the cited Schur-complement algorithm shows that standard min-plus perturbation theory cannot always be used for this problem; a correct characterization must select among eigenvectors by a different rule.","If both conjectures hold, the asymptotic transfer-matrix eigenvector, and hence low-temperature behavior of the one-dimensional spin system, becomes a function of tropical geometry alone."],"supporting_citations":[{"why":"Supplies the min-plus Schur-complement algorithm whose predictions the paper tests and shows can fail to identify the Perron eigenvector.","marker":"[Akian et al., 2006]"},{"why":"Establishes the known tropical eigenvalue and single-eigenvector connection for transfer matrices that the paper seeks to generalize.","marker":"[Gaubert and Plus, 1997]"},{"why":"Gives the Perron-Frobenius theorem that guarantees the eigenvalue and eigenvector whose asymptotics are studied.","marker":"[Perron, 1907]"}],"fun_headline_variants":["Tropical analysis reveals Perron eigenvector limit","Perron limit traces to tropical eigenspace","Tropical eigenspace governs the Perron limit","Asymptotic Perron eigenvector found via tropical geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the normalized Perron eigenvector sequence (1/k) log(L(exp(kA))) has a single limit as k tends to infinity for every matrix considered; the paper treats this as an immediate observation from plots of the first roughly thirty iterates, with no proof or error bound.","fun_headline_variants_meta":{"raw":{"variants":["Tropical analysis reveals Perron eigenvector limit","Perron limit traces to tropical eigenspace","Tropical eigenspace governs the Perron limit","Asymptotic Perron eigenvector found via tropical geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000668,"raw_usage":{"total_tokens":3044,"prompt_tokens":940,"completion_tokens":2104,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2042}},"tokens_in":556,"tokens_out":2104,"duration_ms":17593,"temperature":1.0,"reasoning_tokens":2042,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:44:55.416476+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute P_infinity for a 3 by 3 matrix with a two-dimensional tropical eigenspace and check, using high-precision arithmetic past k = 30, whether the normalized eigenvector sequence oscillates between two accumulation points; two distinct subsequential limits would falsify the claimed well-definedness. Alternatively, find two matrices A and B with identical sigma_{max+} but different P_infinity, which would directly falsify Conjecture 3.0.2.","supporting_citations":[],"review_version":1}