{"id":"2132464c-f208-48ff-81ec-fe4d0732dbd5","arxiv_id":"1908.00472","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every Anosov map of the torus has a bi-infinite geodesic axis in the Farey graph, making its stable translation length a computable positive integer.","lead":"This paper proves that every Anosov map of the torus has a straight invariant path in the Farey graph of curves, so its stable translation length is always an integer. The proof is constructive and yields an algorithm to compute this integer, with applications to minimal words, length ratios, and the distribution of translation lengths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem depends on Proposition 13's under-derived concatenation lemma; the reluctant-move case analysis does not clearly establish that an extendable efficient geodesic exists, so the integer translation-length conclusion is not fully secured.","rationale":"The paper's central claim requires a constructive geodesic axis in the Farey graph for every Anosov map, and the integer stable translation length follows from that axis. The load-bearing step is indeed Proposition 13: without a proof that some efficient geodesic in a finite concatenation of prime subladders extends to a bi-infinite efficient geodesic, Theorem 18 does not follow. The proof as written is a high-level case analysis with figures standing in for several key assertions, and Lemma 14's argument is compressed enough that I cannot certify the concatenation step from the text. I read the paper in good faith: the ladder setup, the PSL(2,Z) action, the periodicity theorem, and the worked examples are concrete and plausible, and I did not find a clear counterexample. The concern is therefore about completeness and rigor of a central combinatorial lemma, not about internal inconsistency or an obviously false conclusion. The reader's conditional verdict is appropriate: the main idea is reasonable, but the proof of Proposition 13 should be expanded or verified before the paper is accepted as fully rigorous. A brute-force check over small periods would settle whether the specific combinatorial configurations asserted in the proof actually occur, and it would either produce a counterexample or give strong evidence that the concatenation step is sound. I also noticed a minor issue in Theorem 23, where the word tau_a^{-m} tau_b^n is claimed to be formed from a set S containing only positive powers, but that issue is secondary to the axis theorem and does not change the verdict.","tokens_in":20668,"tokens_out":6268,"duration_ms":72692,"concrete_test":"Write a small exhaustive checker for Proposition 13: for every minimal period (a1,...,an) with n <= 8 and coefficients in {1,2,3,4}, generate the prime subladder, enumerate all paths with endpoints on the same side that satisfy the efficient-moving condition locally, and test whether at least one such path remains efficient when repeated bi-infinitely across the period boundary. If any period fails, that period gives an explicit counterexample to Proposition 13 and Theorem 18; if all pass, repeat the check for random longer periods to test the claimed reduction to the reluctant-move obstruction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5's Theorem 18 is derived entirely from Proposition 13 and Theorem 17: after f stabilizes a periodic ladder L, one must find an efficient geodesic in a finite concatenation L' of prime subladders whose bi-infinite repetition remains efficient. The proof of Proposition 13 in Section 3.3 is the only place this is argued. It asserts that exactly two maximal efficient geodesics in L' have endpoints on the same side, that the only possible obstruction is a 'reluctant move' at the semi-final-to-final transition, and that if both candidate geodesics have problematic reluctant moves then the first and last coefficients of L' are 1. None of these assertions is derived from the definitions; Figure 9 is offered in place of a complete argument. Lemma 14 then claims that unless all coefficients are 1 the two geodesics intersect and thereafter agree, but its proof only says they cannot both avoid t-moves; it does not fully handle the case where the two geodesics meet at a pivot after different earlier choices, nor does it justify the subsequent claim that the all-1 case contradicts the presence of reluctant moves, since an efficient geodesic in an all-1 ladder still contains the final t-move. Because the concatenated path must be efficient in every finite subpath of the bi-infinite ladder, a single unchecked configuration at the period boundary invalidates the construction of the geodesic axis and hence the integer translation-length conclusion. The text itself signals that this is a sketchy case analysis, and no machine-checked or exhaustive verification is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stable translation length of Anosov (hyperbolic) elements of PSL(2,Z) acting on the Farey graph of the torus. The main result (Theorem 18) asserts that every Anosov map has a bi-infinite geodesic axis in the Farey graph, so its stable translation length is a positive integer; the proof is constructive and the authors give an algorithm for computing it from the continued-fraction expansion of a fixed point. The framework is a combinatorial object called a ladder, together with 'efficient' geodesics satisfying a local move rule. Three applications are then developed: identifying a minimal word among products of two Dehn twists, bounding the ratio of Teichmuller to curve-graph translation lengths, and showing the length spectrum is evenly spread.","tokens_in":20898,"tokens_out":18766,"duration_ms":188880,"significance":"If the central theorem is established, this is a clean sporadic-case strengthening of Bowditch's rationality result: on the Farey graph the stable translation length is not merely rational but an integer. The constructive ladder-and-efficient-geodesic method is a useful feature, and the two worked examples give concrete integer lengths. The paper is self-contained against standard facts on cutting sequences and continued fractions, and the main construction has no fitted parameters. However, the central combinatorial concatenation lemma (Proposition 13) is not yet proved rigorously, and since Theorem 18 depends directly on it, the paper currently does not fully secure the advertised result.","major_comments":[{"comment":"Proposition 13 is load-bearing for Theorem 18, but its proof is incomplete. The proof asserts, without derivation, that the only possible obstruction to efficiency of the bi-infinite concatenation is a 'reluctant move' at the semi-final-to-final transition; that there are exactly two maximal efficient geodesics in a prime subladder with endpoints on the same side; and that if both candidates are problematic then the first and last coefficients of L' are 1. Figure 9 is used in place of a complete case analysis for the last assertion. Lemma 14, which is supposed to resolve the remaining case, only treats the situation in which the two geodesics do not intersect; it concludes that both are p...p and all coefficients are 1. It does not analyze intersections that occur after different earlier choices, and it does not establish the final claim in the proof of Proposition 13 that an all-1 ladder contradicts the presence of reluctant moves, since the proof does not rule out the possibility that an efficient path in an all-1 finite ladder contains a final t-move. A complete proof of this lemma is required before Theorem 18 can be accepted.","section":"§3.3, Proposition 13 and Lemma 14"},{"comment":"The proof of Theorem 17 asserts that L' is a finite concatenation of copies of the prime subladder L''. This is needed to apply the 'more generally' clause of Proposition 13 in Theorem 18. The common-divisor argument states that minimality of the prime subladder forces the common divisor d to be odd, and then infers from the even length of a prime subladder that p and f(p) lie on different sides of L. Neither the minimality assertion nor the side-switching inference is proved. Please replace this passage with a formal argument, or prove directly that the f-translate of a rung in the periodic part is separated from the rung by an integral number of prime subladders.","section":"§4, Theorem 17"},{"comment":"The correctness of the computation algorithm is not established. After Proposition 21, the paper asserts that if the subladder ~L has even length then p is a pivot point and ~L is a finite concatenation of a prime subladder, and that if it has odd length then calibration (Algorithm 2) converts it into an even-length subladder whose efficient geodesic computes the translation length. No proof is supplied that the calibrated ladder is of the required form or that the length of its efficient geodesic equals l_C(f) rather than a multiple of it. Since the advertised contribution includes an algorithm that computes the exact translation length, this missing justification should be supplied.","section":"§5, algorithm after Proposition 21"}],"minor_comments":[{"comment":"The abstract says f acts 'transitively' on the invariant geodesic; the precise statement, as in Theorem 18, is that f acts by translation. Please correct this wording.","section":"Abstract and Introduction"},{"comment":"The displayed matrix in the text is (277 60; 337 73), but the figure caption and the stated fixed points (77±sqrt(26149))/337 correspond to the matrix (227 60; 337 73). Please correct this inconsistency.","section":"Example 1"},{"comment":"The proof of Proposition 11 uses the undefined term 'locus' and dismisses endpoint cases as 'obvious'. Please define the term and expand the exceptional cases, since the local-to-global geodesic argument is not fully formal.","section":"§3.3, Proposition 11"},{"comment":"Lemma 3 is cited with the one-line proof 'Refer to [Ser15]'. Since the lemma is used in the proof of Theorem 17, a precise reference to the statement in [Ser15] or a short self-contained explanation would be helpful.","section":"§2, Lemma 3"}],"recommendation":"major_revision","confidential_remarks":"The ladder framework is promising and the main theorem is plausible, but the gap in Proposition 13 is real and currently blocks the central claim. If the authors can supply a complete proof of the concatenation lemma and tighten the surrounding arguments, the paper would be publishable after revision. I would not accept the current version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that every Anosov map on the torus curve graph has a bi-infinite geodesic axis, hence integral stable translation length, with a constructive algorithm. That is a clean result, and the worked examples check out. The ladder/cutting-sequence setup is clearly explained, and the applications (minimal word, ratio bound, length spectrum density) are reasonable. I think the main theorem is likely true.\n\nWhat is actually new is the geodesic-axis statement in the Farey graph and the exact-integer consequence, plus the algorithmic computation. The machinery overlaps heavily with Series's cutting sequences and the Beardon–Hockman–Short ancestor paths, which the paper cites honestly. That does not make the result trivial, but the novelty is moderate and incremental.\n\nWhere the paper gets shaky is Proposition 13, the concatenation lemma for efficient geodesics in periodic ladders. Theorem 18 depends on it, and the proof is a quick case analysis with Figure 9 doing a lot of work. The stress-test note is right: Lemma 14 does not fully pin down every configuration where the two candidate geodesics meet, and the 'all coefficients are 1' contradiction is asserted rather than shown. This is the single load-bearing step, and it is the least rigorous part of the paper. I do not see a counterexample, and the examples work, so I would call it a proof gap rather than a false result. A referee should demand a complete proof.\n\nSmaller issues: a few notational slips (the 277/227 matrix in Example 1), and Theorem 23's proof is only a few sentences and handwaves the key comparison. Those are fixable. The trace-count argument in Theorem 29 also glosses over the PSL vs SL conjugacy class count, though the conclusion is likely right.\n\nThis paper is for anyone working on curve graphs, Farey graph geodesics, or translation-length rigidity. It deserves a serious referee; with the Proposition 13 gap filled and the typos cleaned up, it would be a solid contribution. I would send it out rather than desk reject.","headline":"The integer-translation-length theorem for Anosov maps on the Farey graph is plausible and likely true, but the paper's main construction rests on a genuine proof gap in Proposition 13 that must be filled before the theorem is fully secured.","tokens_in":21477,"tokens_out":5636,"would_cite":true,"duration_ms":58030,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:55:43.119648+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}