{"id":"c5f31331-7460-43f2-ad1c-636961af2c28","arxiv_id":"2506.06608","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A computable vertical-displacement threshold implies unbounded diffusion and rotational horseshoes in annular maps, with computer-assisted proofs for standard and related families.","lead":"This paper gives simple numeric formulas: if one orbit of an annular map jumps farther vertically than a bound computed from the map's largest single-step jump, then the map has chaotic, unbounded motion. The authors verify such jumps rigorously with computer-assisted proofs for standard and dissipative standard map families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantitative theorems and CAPs inherit their entire force from unpublished criteria in [31]; until Corollary B, Proposition 4.9, and Theorem C of [31] are independently verified, the central claims remain conditional.","rationale":"The paper's central contribution is a quantitative extension of [31] into metric bounds and CAPs. All new theorems route through [31]'s topological criteria; the proof of Theorem 3.2 explicitly leans on Lemma 3.3 whose proof is 'essentially the same as that of [31]', and Theorem 3.6 uses Proposition 4.9 of [31] and Theorem C of [31] for the rotation-number behavior of circloids. Corollary B is imported directly for the dissipative CAP. Since [31] is listed as 'submitted' and no version is provided, the reader cannot check this foundation. This is exactly the weakest link: if [31] has a gap, the quantitative constants and the existence of rotational horseshoes are unsupported, even though each Krawczyk validation is individually strict. The 95.95%/98.47% discrepancy in Theorem 1.4 is a mild additional signal that the code output needs pinning to a hash, but it is not the main theoretical risk. A conditional verdict is appropriate: the argument is coherent and the CAP methodology is sound, but acceptance should require either an accessible, refereed version of [31] or a self-contained proof of the imported propositions. No change from the reader's verdict is needed.","tokens_in":26293,"tokens_out":12940,"duration_ms":126831,"concrete_test":"Obtain the full text of [31] and independently verify Proposition 4.9, Theorem C, and Corollary B from first principles, especially the step converting an N-d.p.n. with Birkhoff-related orbits into a rotational horseshoe; then re-derive Lemma 3.3 and Lemma 3.7 without invoking [31]. If any of these steps fails, Theorem 3.2 and Theorem 3.6 lose their quantitative bounds, and the CAPs built on them are unsubstantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 (Theorem 3.2), Theorem 2 (Theorem 3.6), and the dissipative CAP (Section 6) all rest on black-box results from [31]: Corollary B, Proposition 4.9, and Theorem C are invoked directly (Sections 2, 3.2, 6), and Lemmas 3.3 and 3.7 are declared 'essentially the same as [31]' or 'mostly done in [15]', with the crucial prime-end and circloid steps imported from [31]. If any of these [31] statements has a gap, the constants M = min{3N+2, N+4} and M1, M2, Mi collapse along with the rotational-horseshoe conclusions. This is a correctness risk, not a stylistic one: no accessible proof of [31] is supplied, and the manuscript itself flags the dependence. A secondary concrete inconsistency (Theorem 1.4 states 95.95%, Section 6.4 reports 98.47%) shows the numerical reporting needs checking, but the [31] dependency is the load-bearing issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives quantitative sufficient conditions for unbounded diffusion and rotational horseshoes in annular homeomorphisms, building on topological criteria from the companion paper [31] (Passeggi and Tal, 'Conditions implying annular chaos', submitted). Theorem 1 gives the bound M = min{3N(f)+2, N(f)+4} for non-wandering annulus homeomorphisms homotopic to a Dehn twist, and Theorem 2 gives M1 = 6N(f)+2, M2 = 4N(f)+2, and Mi = 2N(f)+2 for i ≥ 3 for maps with rotational difference ρ; Corollary 1 converts these into a computable bound on deviations from a rotation interval. Theorem 3 gives a parameter-uniform horseshoe statement for the Non-Twist Standard Family and a crossing criterion for total unbounded diffusion. The rest of the paper presents computer-assisted proofs for variations of the Standard Family, for the Non-Twist Standard Family on a 1000×1000 mesh, and for the Dissipative Standard Family, using parallel shooting and Krawczyk-verified interval arithmetic, with code at github.com/mcapinsk/cac-cap.","tokens_in":26503,"tokens_out":8916,"duration_ms":81673,"significance":"If the companion criteria in [31] are correct, this is a substantial practical advance: explicit and easily evaluated bounds depending only on N(f) and ρ, no dependence on derivatives, and computer-assisted proofs over large parameter ranges in conservative and dissipative settings. The CAP methodology is a genuine strength: interval arithmetic with Krawczyk-type verification is used, the code is provided, and the theoretical constants are explicit and falsifiable through the stated numerical results. The main caveat is that Theorems 1, 2, Theorem 3(2), and the dissipative CAP inherit their force from the unpublished source [31], whose Corollary B, Proposition 4.9, and Theorem C are imported as black boxes; consequently the significance is conditional on independent verification of that companion work.","major_comments":[{"comment":"The central theoretical results depend on unpublished black-box statements from [31]. Specifically, Corollary B of [31] is restated in §2 and applied directly in §6; Lemma 3.3 in §3.1 is disposed of by saying the proof is 'essentially the same as [31]'; the proof of Theorem 3.6 invokes Proposition 4.9 of [31] and Theorem C of [31]; and Lemma 3.11 invokes both Theorem C and Proposition 4.9 of [31]. Since [31] is a submitted manuscript that is not publicly available and is co-authored by two of the present authors, the proofs of Theorems 1, 2, Theorem 3(2), and the dissipative CAP are not independently verifiable as written. If any of these imported statements has a gap, the constants M, M1, M2, Mi and the rotational-horseshoe conclusions collapse. I ask that the authors either prove the needed statements within this paper, or make [31] publicly available and, ideally, have the specific imported results independently checked; otherwise the theorems should be explicitly labeled as conditional on [31].","section":"§2, §3.1, §3.2, §3.3, §6"},{"comment":"Theorem 1.4 states that chaos is obtained 'in at least 95.95% of the area' of the parameter domain, while Section 6.4 reports that the validated area 'exceeds 98.47% of the domain'. Since the quantified area is precisely the content of Theorem 1.4, this is not a cosmetic discrepancy. The authors must determine which number is correct and ensure that the theorem statement, the reported computation, and the code output agree.","section":"§1.4 vs §6.4"},{"comment":"The proof of Theorem 3.8(1) begins by assuming that the map 'has no topological entropy' and then asserts 'from the considerations of the previous subsection' that the map has bounded diffusion and that every orbit stays in a horizontal strip of uniformly bounded width. The previous subsection establishes only a sufficient condition for unbounded diffusion (a crossing from below -Ma,b to above Ma,b), not an equivalence, so the implication 'no entropy ⇒ bounded diffusion' needs a proof or a precise citation. The argument then invokes Theorem A of [25], Theorem 1.4 of [20], and Ushiki's theorem as cited in [16]; please state the exact hypotheses and verify explicitly that they apply for all a ∈ (0,∞) and b ≠ 0.","section":"§3.3, proof of point (1) of Theorem 3.8"}],"minor_comments":[{"comment":"The statement of Lemma 3.3 contains a redundancy: after the case p = 2, the clause 'if p ≥ 3 then we can show the same for σ ∪ f^2(σ)' should presumably read 'σ ∪ f(σ)' or should be corrected; in the proof, 'the cases where p = 1 or p + 2 are similar' appears to be a typo for 'p = 1 or p = 2'.","section":"§3.1, Lemma 3.3"},{"comment":"The proof of Lemma 3.10 writes fa,b(1/4, y) = (x', y + |b|), but for b > 0 the second coordinate is y - b = y - |b|; the argument works after choosing x = 3/4 instead for b > 0. Please make the sign choice explicit.","section":"§3.3, Lemma 3.10"},{"comment":"The sentence claiming that the parametrization of the initial and final points 'ensures that there can be only one solution for (8)' overstates the matter; uniqueness is guaranteed by the Krawczyk verification on the chosen interval box, not by the parametrization alone. I suggest rephrasing.","section":"§4.1, paragraph after Eq. (8)"},{"comment":"The manuscript contains numerous typos and infelicities that should be cleaned up in revision, for example 'copared' in the caption of Figure 1, 'supremmun' in §3, 'employees' in the proof of Theorem 3.2, 'estasblish' in §4.1, and 'the taks' in the footnote of §5.2.","section":"Throughout"},{"comment":"For reproducibility of the computer-assisted proofs, the repository should be pinned to a specific commit or release, and the computational environment (compiler/interpreter and interval arithmetic library versions) should be stated.","section":"Code availability [1]"},{"comment":"Reference [31] should indicate its current status and availability; the citation to Ushiki's theorem through [16, p.289] would be easier to verify if the original source or a more precise statement were given.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main issue for the editor is the heavy reliance on [31], an unpublished submission co-authored by two of the present authors. I would not recommend acceptance until [31] is available or the needed results are proved in this paper, since the core theorems are otherwise conditional on an unverifiable source. The discrepancy in the reported percentage of Theorem 1.4 must also be resolved, and the code repository should be pinned to a commit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nBottom line: this is a useful, honest paper with a load-bearing dependency. The new content is real: explicit thresholds M = min{3N+2, N+4} and M1=6N+2, M2=4N+2, Mi=2N+2 give computable criteria for unbounded diffusion and rotational chaos, and Corollary 1 turns that into a concrete deviation bound for rotation intervals. The CAP sections are broad: twist and non-twist variations of the standard family, NTSF on a 10^6 mesh, and the dissipative standard family over [3,10]×[0.1,0.8]. Code is on GitHub. The parallel-shooting/Krawczyk setup is standard interval-arithmetic practice, applied cleanly, and the authors are right to emphasize that the method needs no derivative information. Theorem 3's analytic part is a solid piece of work.\n\nThe soft spot is exactly the one the reader flagged. Theorems 1, 2, and the dissipative CAP import central results from [31]—Corollary B, Proposition 4.9, and Theorem C are invoked directly, and Lemmas 3.3 and 3.7 are declared \"essentially the same\" as [31]. Since [31] is by two of the present authors and not publicly available, this is a genuine circularity burden. It is not stylistic: the constants are only as good as the unverified topological criteria. The CAP validations are rigorous in themselves, but the conclusions drawn from them—rotational horseshoes, diffusion—inherit the [31] assumptions. If [31] has a gap, the quantitative theorems and the CAP conclusions collapse. My own read of Section 3 does not show a visible hole in the imported results, but I cannot verify them from this manuscript. The fix is straightforward: make [31] available to referees, or include its key statements and proofs in an appendix, and referee the two papers in parallel.\n\nMinor but real: Theorem 1.4 states 95.95% while Section 6.4 reports 98.47%. In a CAP paper, that numerical inconsistency matters. Also, the code is not pinned to a commit; for interval-arithmetic proofs, a commit hash is the norm. Both are trivial to fix.\n\nWho this is for: people in torus/annulus dynamics and validated computing. It deserves a serious referee, conditionally—I would not desk-reject. I would not, however, cite the thresholds as established until [31] is accessible and the 95.95/98.47 discrepancy is resolved.","headline":"Useful quantitative thresholds and broad CAPs, but the whole theorem stack rests on the authors' unpublished [31] and a numerical inconsistency needs fixing.","tokens_in":27030,"tokens_out":2243,"would_cite":false,"duration_ms":23511,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37E30","37B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a single orbit whose vertical displacement exceeds a threshold depending only on the map's maximal step forces unbounded diffusion; for area-preserving maps with zero vertical drift it also forces rotational chaos…","keywords":["annular homeomorphisms","rotational horseshoe","unbounded diffusion","Dehn twist","rotation set","computer-assisted proof","interval arithmetic","Non-Twist Standard Family"],"falsifier":"Exhibit a non-wandering Dehn-twist homeomorphism of the annulus with $N(f)\\le1$ and an orbit satisfying $|pr_2(f^n(z)-z)|\\ge5$, while every orbit of the map remains in a bounded vertical strip. That would refute the diffusion part of Theorem 1.","tokens_in":26109,"feed_emoji":"🌀","tokens_out":14430,"duration_ms":137058,"temperature":0.7,"pith_summary":"This paper converts the qualitative theory of annular chaos into a finite numerical test. For a non-wandering homeomorphism of the annulus whose lift behaves like a Dehn twist $(x,y)\\mapsto(x+ky,y)$, it proves that a single orbit with vertical displacement at least $M=\\min\\{3N(f)+2,N(f)+4\\}$, where $N(f)$ is the largest one-step vertical displacement, forces unbounded diffusion; if the map preserves an absolutely continuous measure with zero vertical drift, the same displacement forces a rotational horseshoe and total unbounded diffusion. For non-wandering maps homotopic to the identity that have fixed points with rotational difference $\\rho\\ge1$, the thresholds are $M_1=6N(f)+2$, $M_2=4N(f)+2$, and $M_i=2N(f)+2$ for $i\\ge3$, and crossing $M_\\rho$ again gives unbounded diffusion and a non-empty rotation-set interior. These bounds give computable control on deviations from a horizontal rotation interval, and the paper demonstrates the method with validated computer-assisted proofs in the Standard Family, its non-twist and dissipative variations, and the Non-Twist Standard Family.","feed_headline":"One orbit's drift can prove chaos on the annulus","feed_subtitle":"Chaos detection becomes a single displacement check, validated on standard, non-twist, and dissipative families.","key_machinery":"The load-bearing mechanism is the invariant circloid: a minimal essential continuum that separates the annulus into two ends. The proof shows that bounded diffusion produces such a circloid $C$, and then bounds its vertical diameter $VD(C)$ in terms of $N(f)$ and the rotational difference $\\rho$. This is done with prime-end rotation numbers: arcs joining $C$ to its vertical translate are stretched by $f^2$ or $f^3$, and Lemmas 3.3 and 3.7 turn the stretching into inequalities such as $VD(C)<3N(f)+1$ or $VD(C)<2nN(f)+1$. For the computer-assisted proofs, the indispensable tool is parallel shooting: a candidate orbit from $y=-B$ to $y>B$ is encoded as a zero of a finite-dimensional map $F$, whose existence is proved by the Krawczyk interval-Newton method; the method needs only interval enclosures of iterates, not derivatives. In the dissipative case, the imported $N$-disjoint-pair criterion of [31] is verified by building small segments near fixed points and using validated trajectories from them across the map.","core_discovery":"On its own terms, the central claim is that unbounded diffusion is forced by a single, quantitatively bounded displacement. For a non-wandering Dehn-twist map of the annulus, Theorem 1 states that an orbit with $|pr_2(f^n(z)-z)|\\ge \\min\\{3N(f)+2,N(f)+4\\}$ has unbounded diffusion, and under zero vertical drift it has rotational chaos and total unbounded diffusion. Theorem 2 gives the analogous constants $M_\\rho$ for non-wandering maps homotopic to the identity with a rotational difference $\\rho\\ge1$, and adds that the projected torus map has a rotation set with non-empty interior. Corollary 1 extracts from Theorem 2 a computable bound on how far orbits can deviate from a horizontal rotation interval containing two rational points. The paper also proves for the Non-Twist Standard Family a parameter-dependent bound $M_{a,b}$ such that an orbit crossing from below $-M_{a,b}$ to above $M_{a,b}$ yields total unbounded diffusion, and that every $a>0$, $b\\ne0$ parameter pair has a rotational horseshoe.","pith_inferences":["The constants are likely not sharp: the proof fixes the number of iterates used in the prime-end argument, so a finer prime-end count could push the thresholds down without changing the topology.","The CAP routine is a natural template for an automated chaos screen: on any parameter mesh it needs only a crossing orbit candidate plus interval arithmetic, and a failure means “no proof found yet” rather than “no chaos.”","Zero vertical drift enters only through the final horseshoe step, so the paper proves unbounded diffusion without it but a rotational horseshoe only with it; whether the drift condition is actually necessary for the horseshoe conclusion is left open."],"forward_implications":["If Theorem 1 is right, chaos detection for Dehn-twist-type maps reduces to computing $N(f)$ and locating one orbit with vertical displacement at least $\\min\\{3N(f)+2,N(f)+4\\}$; no invariant curves or derivative estimates are needed.","If Theorem 2 is right, then any non-wandering area-preserving map whose rotation set is a horizontal interval containing two rational points has all its vertical deviations inside the explicit strip $[-M_\\rho,M_\\rho]$; leaving that strip certifies a non-empty rotation-set interior.","The Non-Twist Standard Family result implies that every parameter pair $a>0$, $b\\ne0$ in that family carries a rotational horseshoe, so the twist-breaking region is not an obstacle to symbolic chaos.","The CAP validations establish diffusion and chaos in reproducible instances: for all listed twist and non-twist variations of the Standard Family, for at least 240,359 parameter pairs in the Non-Twist Standard Family mesh, and for at least 95.95% of the area of the dissipative parameter domain considered.","Because the CAP validation uses only $C^0$ information, the same algorithm is directly portable to maps whose derivatives are unavailable, such as Poincaré return maps, a direction the paper explicitly says it plans to pursue."],"supporting_citations":[{"why":"Supplies the companion topological criteria: a pair of fixed points with rotational difference and an N-disjoint pair of neighborhoods produces a rotational horseshoe; imported as the foundation of Sections 2, 3, and 6.","marker":"[31]"},{"why":"Gives the topological-horseshoe theorem used to conclude rotational chaos from two Birkhoff-related ends in Proposition 3.1 and in the Non-Twist Standard Family proof.","marker":"[25]"},{"why":"Provides the invariant-annulus and circloid argument (Theorem 5.1) used in Lemma 3.7 for maps homotopic to the identity with bounded diffusion.","marker":"[15]"},{"why":"Establishes the vertical rotation number and Dehn-twist fixed-point behavior used in Proposition 3.1 for the Dehn-twist class.","marker":"[4]"},{"why":"Supplies the fixed-point existence for Dehn-twist-type torus maps needed in Proposition 3.1.","marker":"[5]"},{"why":"Gives the Krawczyk interval-Newton inclusion theorem underlying all computer-assisted validations in Sections 4–6.","marker":"[6]"},{"why":"Birkhoff's curve theorem is used in Lemma 3.12 to force invariant circloids in twist regions to be graphs.","marker":"[8]"},{"why":"Theorem A establishes a single rotation number on invariant circloids of non-wandering maps, used in the proofs of Theorems 3.2 and 3.6.","marker":"[19]"},{"why":"The solution of Boyland's conjecture, used in Proposition 3.1 to place the Lebesgue rotation vector in the interior of the rotation set under zero drift.","marker":"[26]"}],"fun_headline_variants":["One displacement check guarantees annular chaos","Quantitative bound turns drift into diffusion proof","Chaos in annulus maps from a single orbit shift","Computer-assisted proof: small drift, big chaos","A single orbit's motion certifies diffusion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction assumes that the topological criterion from [31] is correct: whenever a map has two fixed points differing by a whole-number vertical rotation and a pair of small disjoint neighborhoods whose orbits interlink, a chaotic rotational horseshoe necessarily exists.","fun_headline_variants_meta":{"raw":{"variants":["One displacement check guarantees annular chaos","Quantitative bound turns drift into diffusion proof","Chaos in annulus maps from a single orbit shift","Computer-assisted proof: small drift, big chaos","A single orbit's motion certifies diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1338,"prompt_tokens":840,"completion_tokens":498,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":430}},"tokens_in":456,"tokens_out":498,"duration_ms":5039,"temperature":1.0,"reasoning_tokens":430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:54:43.324824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a non-wandering Dehn-twist homeomorphism of the annulus with $N(f)\\le1$ and an orbit satisfying $|pr_2(f^n(z)-z)|\\ge5$, while every orbit of the map remains in a bounded vertical strip. That would refute the diffusion part of Theorem 1.","supporting_citations":[{"cited_title":"Passeggi and F","cited_arxiv_id":null,"evidence_quote":"Supplies the companion topological criteria: a pair of fixed points with rotational difference and an N-disjoint pair of neighborhoods produces a rotational horseshoe; imported as the foundation of Sections 2, 3, and 6."},{"cited_title":"Le Calvez and F","cited_arxiv_id":null,"evidence_quote":"Gives the topological-horseshoe theorem used to conclude rotational chaos from two Birkhoff-related ends in Proposition 3.1 and in the Non-Twist Standard Family proof."},{"cited_title":"Guelman, A","cited_arxiv_id":null,"evidence_quote":"Provides the invariant-annulus and circloid argument (Theorem 5.1) used in Lemma 3.7 for maps homotopic to the identity with bounded diffusion."},{"cited_title":"Addas-Zanata, F","cited_arxiv_id":null,"evidence_quote":"Establishes the vertical rotation number and Dehn-twist fixed-point behavior used in Proposition 3.1 for the Dehn-twist class."},{"cited_title":"Some extensions of the poincar´ e–birkhoff theorem to the cylinder and a remark on mappings of the torus homotopic to dehn twists","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-point existence for Dehn-twist-type torus maps needed in Proposition 3.1."},{"cited_title":"Inclusion methods for systems of nonlinear equations – the interval Newton method and modi- fications, In: Topics in validated computations, Stud","cited_arxiv_id":null,"evidence_quote":"Gives the Krawczyk interval-Newton inclusion theorem underlying all computer-assisted validations in Sections 4–6."},{"cited_title":"Birkhoff","cited_arxiv_id":null,"evidence_quote":"Birkhoff's curve theorem is used in Lemma 3.12 to force invariant circloids in twist regions to be graphs."},{"cited_title":"Koropecki","cited_arxiv_id":null,"evidence_quote":"Theorem A establishes a single rotation number on invariant circloids of non-wandering maps, used in the proofs of Theorems 3.2 and 3.6."},{"cited_title":"Le Calvez and F","cited_arxiv_id":null,"evidence_quote":"The solution of Boyland's conjecture, used in Proposition 3.1 to place the Lebesgue rotation vector in the interior of the rotation set under zero drift."}],"review_version":1}