{"id":"9504af26-62b5-4f94-9376-fc2b38b6b39a","arxiv_id":"2504.17986","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Explicit Teichmüller geodesic rays exist that are sublinearly Morse yet have minimal non-uniquely ergodic vertical foliations, giving the first such examples.","lead":"The authors build the first explicit Teichmüller geodesic rays that are sublinearly Morse, the geometric signature of a generic random direction, while their vertical foliations are minimal and non-uniquely ergodic. The result shows that the 'generic directions' captured by random walks on the mapping class group can be atypical at the boundary in a way that no unique invariant measure can describe.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof leans on the unpublished 'strong passing-up' theorem [Dur23, Prop 4.7]; if that theorem or its application to the slit subsurfaces fails, Proposition 3 and the log-bounded projections estimate collapse.","rationale":"The reader's CONDITIONAL verdict already identifies the same core fragility: the paper's central combinatorial step imports an unpublished theorem from a co-author's preprint and gives no independent verification. My reading agrees that this is the single most load-bearing concern. I considered other possible objections. Proposition 2 proves filling only for slit curves whose indices differ by six, while Claim 1 invokes pairwise filling for an arbitrary four-tuple; this is likely repairable by passing to a subsequence with index gaps that are multiples of six, but it is not written down. Lemma 3's overlap control relies on taking L large and is consistent with standard hyperbolicity of the curve graph. The σ=1/100 choice in Claim 1 conflicts with both the hypothesis σ≥10E of Proposition 5 and the desired separation L_1/100; this looks like a typo, with σ intended to scale with L_1, but it underscores that the proof of Claim 1 is not fully transparent. None of these issues, by itself, demonstrates that the theorem is false: the construction is explicit, the non-uniquely ergodic vertical foliation follows from a published theorem of Chaika–Masur–Wolf, and the sublinear Morseness criterion from Durham–Zalloum is a cited prior result. The decisive question remains whether [Dur23, Prop 4.7] is true and applies to this V. Since the paper does not establish that, CONDITIONAL is the appropriate verdict; I would not move to ACCEPT, and I would not reject on the basis of a suspected typo or an unpublished citation alone.","tokens_in":13567,"tokens_out":9746,"duration_ms":105300,"concrete_test":"Check [Dur23, Prop 4.7] directly in the case needed here: S of genus 2, the Teichmüller geodesic with α=[1,4,9,...], and V the collection of subsurfaces supplied by Proposition 4. Write out the hypotheses of Proposition 4.7, verify them for this V, and prove the effective constants: for each K_2≥K_1≥50E and σ≥10E, exhibit P_2 and a container W with the required four boundary curves in order and separated by L_1/100. In particular, rerun Claim 1 with σ=L_1/100 instead of σ=1/100 and check that the inductive counting using N_1^6 consecutive domains still works. If Proposition 4.7 cannot be verified in this setting, or if its effective constants force σ to stay bounded independently of L_1, Proposition 3 fails and Theorem A is unsupported. If it verifies cleanly, the CONDITIONAL verdict stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3 is the combinatorial heart of the paper: it converts short slit curves into linear progress in the curve graph, and Theorem 2's log-bounded projections estimate feeds directly on it. The proof of Proposition 3 is built entirely on Proposition 5, a statement quoted verbatim from the unpublished preprint [Dur23, Prop 4.7]. The present paper neither proves Proposition 5 nor verifies that the specific family V={V_k} produced by Proposition 4 satisfies the hypotheses of that theorem, such as the nesting relations V<W, K_1-relevance with the required cardinality, and the effectiveness of the constants P_1 and P_2. If Proposition 5 is false or inapplicable here, the ordered four-tuple of slit curves required by Claim 1 need not exist, and the claimed linear divergence of the slit curves collapses. There is also an internal warning in Claim 1: it says the subdivision constant is chosen as σ=1/100, but Proposition 5 only applies for σ≥10E, and a pairwise separation of L_1/100 cannot be certified by intervals of length 1/100 unless L_1 is bounded. This is likely a typo, with σ needing to grow with L_1, but as written it shows that the passage from the counting conclusion of Proposition 5 to the separated, ordered four-tuple is not automatic. Both points are fixable in principle, but they are load-bearing: no part of the construction substitutes for the missing verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an explicit Teichmüller geodesic ray γ in the genus-two surface, built from the Chaika–Masur–Wolf slit-torus construction with the continued fraction α=[1,4,9,16,...] and subsequence n_k=2k+1, whose vertical foliation is minimal and non-uniquely ergodic. The main theorem (Theorem A) asserts that this ray is sublinearly Morse, and the authors prove this by verifying a log-bounded projections criterion from [DZ22] via a combinatorial analysis of the slit curves in the curve graph. The proof has two main parts: Proposition 3 shows that the slit curves make linearly growing distance in C(S) at logarithmically spaced times, and Theorem 2 converts this into log-bounded projections and hence sublinear Morseness. The construction also yields Corollary B about a pair of sublinearly divergent rays with the same underlying vertical foliation.","tokens_in":13873,"tokens_out":13062,"duration_ms":126361,"significance":"If the proof is sound, the paper provides the first explicit examples of sublinearly Morse Teichmüller geodesic rays with minimal non-uniquely ergodic vertical foliations, sharpening the known contrast between generic directions and non-generic ones. The use of a concrete continued fraction and the reduction of sublinear Morseness to an explicit combinatorial statement in the curve graph are valuable and potentially exportable. The paper also carefully separates the roles of the curve graph boundary and the PMF limit set, giving a new illustration of why the injection of Cordes for Morse rays does not extend to sublinearly Morse rays.","major_comments":[{"comment":"The proof applies Proposition 5 with the subdivision constant σ=1/100, but Proposition 5 is stated only for σ≥10E, where E=E(S) is a fixed constant. Since 1/100 is certainly smaller than 10E for the relevant E, the application as written is outside the theorem's hypotheses. This is load-bearing because Claim 1 is the mechanism that converts the counting conclusion of Proposition 5 into the ordered, separated four-tuple of slit curves, and Proposition 3 depends on it. The argument can likely be repaired by choosing σ to be a function of L1 (e.g., σ = L1/100) and letting P2 depend on L1, but as written the proof is invalid at this step.","section":"Section 4, Claim 1 (proof of Proposition 3)"},{"comment":"The central combinatorial step of the paper rests on Proposition 5, which is quoted verbatim from the unpublished preprint [Dur23, Proposition 4.7] by one of the authors. Since Proposition 5 is neither proved nor independently established in this manuscript, the paper's main result is contingent on an unreviewed external statement. At minimum, the authors should either include a proof of Proposition 5 in an appendix or provide a reference to a peer-reviewed publication containing it. Additionally, the paper should explicitly verify that the collection V={V_k} produced by Proposition 4 satisfies the quantitative hypotheses of Proposition 5, including K1-relevance with K1≥50E and the cardinality requirements.","section":"Section 4, Proposition 5 and its role"},{"comment":"Proposition 4 is a quantitative version of Rafi's short-curve/big-projection theorem, stated as 'following from Rafi's original proof in [Raf05], though it is not commonly stated this way in the literature.' Since the subsequent construction of the subsurfaces V_k relies directly on the exact quantitative constants in Proposition 4, the authors should supply a proof or a precise reference that establishes this version. Without this, the verification that the V_k are sufficiently relevant is not self-contained.","section":"Section 4, Proposition 4"},{"comment":"The proof of Claim 2 refers to 'item (4) of Claim 1', but Claim 1 has only items (1), (2), and (3). The intended argument appears to be that applying item (3) to two distinct blocks produces two distinct slit curves contained in the same subsurface W, contradicting the fact that the slit curves are filling. This is a fixable error, but it must be corrected because Claim 2 is used to ensure that the container subsurfaces W_i are distinct, which is needed for the complexity induction.","section":"Section 4, Claim 2"},{"comment":"The active-interval argument establishing that some slit curve ζ_j is contained in the container subsurface W is only sketched. In particular, the assertions that the active intervals I_{V_i}, I_{V_j}, I_{V_k} can be arranged to be contained in I_W and to appear in the same order as the projections to C(W), and that simultaneous shortness of ζ_j and ∂W implies ζ_j⊂W by the Collar Lemma, require a more detailed and precise justification. This step is essential for carrying the induction that produces the final pair of domains with large distance in C(S).","section":"Section 4, proof of Claim 1, item (3)"}],"minor_comments":[{"comment":"There is a repeated phrase: 'we can we can construct two distinct rays' should read 'we can construct two distinct rays'.","section":"Corollary B, first paragraph"},{"comment":"The text says the subsurface V_k is 'provided by item (2) of Proposition 4', but the relevant implication is item (1), which produces a large-projection subsurface from a short curve. The reference should be corrected.","section":"Proof of Proposition 3, first paragraph after defining V_k"},{"comment":"The proof of Proposition 3 concludes that the slit curves for the first and last domains in a block project about L0/100 apart, but it does not explicitly state how the index gap between those domains is controlled. Adding a sentence explaining that the domains are chosen from consecutive blocks of size N_1^6 would improve clarity.","section":"Section 4, statement of Proposition 3 and proof"},{"comment":"Hyperbolic and extremal length notation is used without a formal definition; in particular the comparison H(ζ_k)/π ≤ E(ζ_k) in the proof of Proposition 1 would benefit from a citation to Maskit's original paper, which is already provided, but the notation could be defined explicitly.","section":"Several places"}],"recommendation":"major_revision","confidential_remarks":"The paper's main result is interesting and plausibly correct, but the refereeing is made difficult by the reliance on the unpublished preprint [Dur23] for the key combinatorial proposition. In addition to the σ issue and the other proof gaps, I would encourage the editor to ask the authors to make the dependence on [Dur23] fully verifiable, either by proving the needed statement or by obtaining and citing a peer-reviewed version. If the authors can fix the local but load-bearing issues in Section 4, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it builds explicit flat surfaces from the slit-torus machinery of Chaika–Masur–Wolf, using the continued fraction α = [1,4,9,16,...] and subsequence n_k = 2k+1, and then verifies the Durham–Zalloum bounded-projections criterion for sublinear Morseness. The genuinely new step is Section 4, where the authors prove that the slit curves make linear progress in the curve graph. That is what converts CMW's non-uniquely ergodic rays into sublinearly Morse rays. If the argument holds, it answers a natural question and gives the first such examples. The exposition is mostly clear and the overall strategy is sound.\n\nThe soft spots are real but not obviously fatal. The main proof rests on Proposition 5, the 'strong passing-up' theorem, imported verbatim from Durham's unpublished preprint [Dur23]. The paper neither proves that statement nor verifies in detail that the specific family of slit subsurfaces V_k satisfies its hypotheses—the nesting, the relevance threshold, the cardinality, and the effectiveness constants. This is not circular, since the proposition is independent of the target result, but it is load-bearing: if Proposition 5 is false or does not apply, Claim 1 fails and the linear divergence estimate collapses. The stress-test note is on target here.\n\nThere is also a concrete internal typo that should be fixed: Claim 1 chooses subdivision constant σ = 1/100, while Proposition 5 only applies for σ ≥ 10E. As written, that passage is not automatic; likely σ needs to grow with L_1. This is fixable, but it needs correcting before the paper is final. Some geometric estimates, like the 'grid of rectangles' in Proposition 2, are sketched rather than fully quantified; I think they are probably right, but they deserve a careful pass. Corollary B also has a duplicated phrase, a minor editing issue.\n\nWho is this for? People working on sublinear Morse boundaries, random walks on mapping class groups, and Teichmüller geodesics. It deserves a serious referee: the question is natural, the construction is explicit, and the main idea is credible. I would send it out, but with an instruction to the authors to either prove or precisely cite and verify Proposition 5, and to sort out the σ inconsistency. With those addressed, I expect the paper to be accepted.","headline":"First examples of sublinearly Morse Teichmüller geodesic rays with minimal non-uniquely ergodic vertical foliations; the construction is plausible and the paper deserves refereeing, but the proof has a load-bearing dependency on an unproved proposition from an unpublished preprint.","tokens_in":14436,"tokens_out":2225,"would_cite":true,"duration_ms":23629,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32G15","30F60","37E35","37D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"There exist Teichmüller geodesic rays that are sublinearly Morse yet have minimal non-uniquely ergodic vertical foliations.","keywords":["Teichmüller space","Teichmüller geodesic","sublinearly Morse","non-uniquely ergodic foliation","minimal foliation","curve graph","translation surface","mapping class group"],"falsifier":"Compute the curve graph distances $d_{\\mathcal C(S)}(\\zeta_{k_n},\\zeta_{k_n+1})$ for the explicit surface $X$ at times $t_k=\\log q_{2k+1}$, with $\\alpha=[1,4,9,16,\\ldots]$ and $n_k=2k+1$; Proposition 3 predicts that for any fixed $L>0$ these distances eventually exceed $L$ on a subsequence while the time gaps are $O(\\log k)$. A subsequence with bounded or sublinear curve-graph distances would refute the claim.","tokens_in":13340,"feed_emoji":"📐","tokens_out":14494,"duration_ms":125568,"temperature":0.7,"pith_summary":"This paper proves that a geodesic ray in Teichmüller space can satisfy the weak hyperbolicity condition called sublinear Morseness while its vertical foliation is minimal but not uniquely ergodic. That existence separates sublinear Morseness from the unique-ergodicity property of the vertical foliation, so the class of sublinearly Morse rays is strictly larger than the class of rays with uniquely ergodic vertical foliations. The construction is explicit: a flat genus-two surface built from two skewed tori glued along a slit, using the continued fraction $\\alpha=[1,4,9,16,\\ldots]$ with the subsequence $n_k=2k+1$. The paper verifies sublinear Morseness by showing the geodesic has log-bounded subsurface projections, using quantitative estimates on how the associated slit curves spread out in the curve graph. If correct, the result marks a concrete limit on how much of the random-walk genericity picture can be recovered from sublinear Morseness alone.","feed_headline":"Sublinearly Morse rays need not be uniquely ergodic","feed_subtitle":"Explicit flat-surface examples separate sublinear Morseness from unique ergodicity.","key_machinery":"The central object is the explicit flat surface $X$: two identically oriented copies of the skewed torus $Y=\\begin{pmatrix}1&-\\alpha\\\\0&1\\end{pmatrix}T$ glued along a slit with holonomy $(b,0)$, where $\\alpha=[1,4,9,16,\\ldots]$ and $b=2\\sum_{k=1}^{\\infty}(q_{2k+1}\\alpha-p_{2k+1})$, with the subsequence $n_k=2k+1$. The Teichmüller geodesic $\\gamma(t)=g_tX$ has first-return rotation by $\\alpha$; at times $t_k=\\log q_{2k+1}$ a sequence of slit curves $\\zeta_k$ becomes extremely short. The workhorse of the proof is the combination of the paper's quantitative short-curves-to-large-projections statement (Proposition 4) with the strong passing-up proposition (Proposition 5), which together force boundary curves of many relevant subsurfaces to appear in order along a curve-graph geodesic, and therefore force the slit curves to spread out linearly. Lemma 3 packages this spread as intervals with bounded overlap, which yields the log-bounded projections estimate needed to conclude $\\log^{2p}$-Morseness through the criterion quoted from [DZ22].","core_discovery":"On its own terms, the central claim is Theorem A: there exist Teichmüller geodesic rays which are sublinearly Morse but have minimal non-uniquely ergodic vertical foliations. The rays come from the flat surface $X$ given by gluing two copies of a skewed torus along a slit, with $\\alpha=[1,4,9,16,\\ldots]$ and $n_k=2k+1$; Lemma 2 verifies non-unique ergodicity, and Theorem 2 verifies that the ray is $\\log^{2p}$-Morse for some $p=p(S)>0$. The proof of Theorem 2 checks the log-bounded projections criterion from [DZ22]: at times $t_k=\\log q_{2k+1}$ the slit curves $\\zeta_k$ become hyperbolically short, and the combinatorial argument shows these curves spread out linearly in the curve graph while the time gaps grow only logarithmically. A corollary records that two such rays with the same underlying vertical foliation can diverge at a sublinear rate, and the construction can be adjusted so the limit set in the space of projectivized measured foliations is an interval rather than a point.","pith_inferences":["This suggests that any attempt to characterize random-walk genericity purely by sublinear Morseness must add a second condition, such as recurrence or unique ergodicity, to exclude a measure-zero population of atypical generic rays.","The explicit family invites a testable spectrum: choosing other subsequences of the partial quotients of $\\alpha=[1,4,9,16,\\ldots]$ in place of $n_k=2k+1$ should produce rays with different curve-graph divergence rates and possibly different ergodicity properties, if the estimates from the slit-torus construction continue to apply.","One could try to prove the linear spread of the slit curves directly from the flat geometry, without the imported strong passing-up proposition; a direct proof would make the construction self-contained and might extend to a broader class of translation surfaces."],"forward_implications":["Sublinear Morseness does not imply unique ergodicity of the vertical foliation, so the class of sublinearly Morse Teichmüller geodesic rays is strictly larger than the class of rays with uniquely ergodic vertical foliations.","The example rays are non-recurrent, because minimal non-uniquely ergodic vertical foliations fail a standard recurrence criterion; they are therefore atypical among the directions that random walks track almost surely, despite being sublinearly Morse.","Two distinct rays with the same underlying topological vertical foliation can diverge at a sublinear rate, so the sublinearly Morse boundary does not separate such rays at a linear scale.","Varying the weights of the two ergodic measures can make the limit set of one of these rays in projectivized measured foliation space an interval, while the ray still determines a unique point in the Gromov boundary of the curve graph.","The genus-two examples lift to higher-genus surfaces by covering constructions, so the phenomenon is not an accident of genus two."],"supporting_citations":[{"why":"Supplies the slit-torus construction, the assumptions on the continued fraction, and Theorem 2.3/2.7 establishing minimal non-uniquely ergodic vertical foliations and the limit-set behavior.","marker":"[CMW19]"},{"why":"Provides the criterion (Theorem K, part 2, and Theorem A) used to conclude sublinear Morseness from log-bounded subsurface projections and to compare rays through the curve graph boundary.","marker":"[DZ22]"},{"why":"Contributes the strong passing-up proposition (Proposition 4.7) that turns a large collection of relevant subsurfaces into ordered boundary curves along a curve-graph geodesic, the key combinatorial step of Proposition 3.","marker":"[Dur23]"},{"why":"Gives the quantitative theorem that short curves along a Teichmüller geodesic force large subsurface projections, used to attach a large-projection subsurface to each short slit curve.","marker":"[Raf05]"},{"why":"Provides the active-interval description and the fact that Teichmüller geodesics project to uniform unparameterized quasigeodesics, used to order projections and control overlap of the intervals.","marker":"[Raf14]"},{"why":"Supplies the distance formula relating Teichmüller distance to sums of subsurface projection distances, used in the log-bounded projections estimate.","marker":"[Raf07]"},{"why":"Gives the augmented marking complex and the distance formula version needed to pass from subsurface projections to Teichmüller distance in the estimate.","marker":"[Dur16]"},{"why":"Establishes that random-walk generic directions track Teichmüller geodesic rays with uniquely ergodic vertical foliations, the property that the new sublinearly Morse rays fail.","marker":"[KM96]"},{"why":"Originally proves the non-unique ergodicity criterion for the associated interval exchange that the paper inherits through [CMW19].","marker":"[Vee69]"}],"fun_headline_variants":["First flat-surface rays: sublinear Morse but not uniquely ergodic","Sublinear Morse rays with minimal non-unique ergodic foliations","Atypical rays: sublinear Morse, not uniquely ergodic","First examples separate sublinear Morseness from unique ergodicity","Flat-surface rays: sublinear Morse, minimal foliation not uniquely ergodic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the strong passing-up proposition imported from a separate unpublished manuscript applies to the family of relevant subsurfaces produced by the short slit curves; if it fails there, the proof that these curves spread linearly in the curve graph collapses.","fun_headline_variants_meta":{"raw":{"variants":["First flat-surface rays: sublinear Morse but not uniquely ergodic","Sublinear Morse rays with minimal non-unique ergodic foliations","Atypical rays: sublinear Morse, not uniquely ergodic","First examples separate sublinear Morseness from unique ergodicity","Flat-surface rays: sublinear Morse, minimal foliation not uniquely ergodic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000706,"raw_usage":{"total_tokens":3127,"prompt_tokens":833,"completion_tokens":2294,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":2202}},"tokens_in":449,"tokens_out":2294,"duration_ms":12861,"temperature":1.0,"reasoning_tokens":2202,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:29:32.033916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the curve graph distances $d_{\\mathcal C(S)}(\\zeta_{k_n},\\zeta_{k_n+1})$ for the explicit surface $X$ at times $t_k=\\log q_{2k+1}$, with $\\alpha=[1,4,9,16,\\ldots]$ and $n_k=2k+1$; Proposition 3 predicts that for any fixed $L>0$ these distances eventually exceed $L$ on a subsequence while the time gaps are $O(\\log k)$. A subsequence with bounded or sublinear curve-graph distances would refute the claim.","supporting_citations":[],"review_version":1}