{"id":"b9dd7770-f2c7-48aa-9376-5cf8625e33c5","arxiv_id":"2411.16939","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For conservative horseshoes and for the classical Lagrange spectrum, the equality HD(k^{-1}(t)) = HD(k^{-1}(-∞,t]) holds precisely for t in a strictly increasing 'J' set, up to a countable exceptional set.","lead":"This paper studies when the Hausdorff dimension of a fiber of the dynamical Lagrange spectrum equals the dimension of the whole initial segment, and proves that this happens exactly on a natural strictly increasing set up to a countable exception. The result applies to smooth area-preserving maps with horseshoes and recovers the same structure for the classical Lagrange spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 is conditional on unproved black-box results from [13] and [15]; if either holds only under generic assumptions rather than for every f∈R^r_{φ,Λ}, the 'for any f' theorem does not follow.","rationale":"The reader's conditional verdict is well supported. The paper gives a detailed, plausibly correct construction of the homeomorphism Θ and the Hölder inverse in §3.3, and the internal chain of inequalities there appears coherent. The remaining obstacle is not a visible contradiction inside the text but the heavy dependence of the proof on external statements that are not verified here. The most important of these is Proposition 3.6 of [15], because the universal connection of subhorseshoes before max{t_{n+1},t_{m+1}} is the step that generates the nested sequence {Λ̃_n} and hence the equality D(η_-)=HD(k^{-1}(η_-)). Proposition 1 of [13] is equally load-bearing for the existence of the initial subhorseshoes with large dimension. My stress-test concern is specifically about quantifiers: Theorem 1.1 asserts a conclusion for every f∈R^r_{φ,Λ}, while the cited preprints are framed around generic or residual results, and the paper itself elsewhere distinguishes the two regimes. Thus the key check is to verify that the imported propositions are pointwise statements, not generic ones. The verdict should remain CONDITIONAL: acceptance is warranted only after the cited results are confirmed to apply in the stated generality. No change from the reader's verdict is needed, but the conditionality should be explicit and tied to this quantifier issue.","tokens_in":23261,"tokens_out":19217,"duration_ms":185662,"concrete_test":"Re-derive Proposition 3.6 of [15] in the exact hypotheses of Theorem 1.1, with no residual or generic assumptions, and check the quantifier of Proposition 1 of [13]: does it hold for every f∈R^r_{φ,Λ} or only for a residual subset? To make the check concrete, run the O-function/good-positions construction on the explicit classical model Λ(4) with f(x,y)=x+y from Section 4: enumerate the relevant finite alphabet words, verify the stated 98% good-position estimate, and test for two indices with O(m)=O(n) whether the connecting subhorseshoe before max{t_{n+1},t_{m+1}} can actually be built using only Proposition 2.9 and Corollary 2.10. If any equal-O pair fails to connect, or if either cited result requires a generic hypothesis, Theorem 1.1 must be weakened or its proof amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is proved in §3.2–3.4, but the two ingredients that make the construction work are imported verbatim. Proposition 1 of [13] is invoked at the start of §3.2 to produce, for arbitrary f∈P^r_{φ,Λ}, subhorseshoes Λ_n⊂Λ_{t_{n+1}-δ_n} with HD(Λ_n)>R_{φ,Λ,f}(t_n) and max f|Λ_n>t_n. Proposition 3.6 of [15], built on the O-function and the 'good positions' construction, is then the sole source of the fact that the Λ_n connect before max{t_{n+1},t_{m+1}}; this is what allows the nested sequence {Λ̃_n} and the Hölder map Θ in §3.3. Neither statement is proved or even fully stated in this paper, and both come from the author's own preprints whose stated scopes concern generic conservative diffeomorphisms and functions. Since Remark 1.2 and Theorem 2.5 carefully separate residual/generic conclusions from pointwise ones, the absence of a pointwise version of the two cited results is a live gap. In addition, the 'without loss of generality' after the finite-range O-function in §3.2 is an unstated subsequence reduction; it is probably repairable, but it needs to be written. If either imported proposition requires a residual hypothesis, an extra transversality condition, or a 'good positions' estimate that is not available for every f∈R^r_{φ,Λ}, then Theorem 1.1 as stated is not established. A third imported criterion, Corollary 3.9 of [14], is used in Proposition 3.10 to obtain strict monotonicity of D|J; if that criterion fails, the strict increase part of Theorem 1.1 also fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the dynamical Lagrange spectrum of a smooth conservative diffeomorphism with a mixing horseshoe. It defines, for each point in the unstable Cantor set, a quantity k_{φ,Λ,f} generalizing the best constant of Diophantine approximation, and analyzes the dimension function D_{φ,Λ,f}(t)=HD(k_{φ,Λ,f}^{-1}(-∞,t]). The main theorem claims that for every r≥2 and every f in R^r_{φ,Λ}, D is continuous, its image is [0,HD(Λ)/2], and the positive-dimension part of the Lagrange spectrum decomposes as J∪F∪J̃, where D|J is strictly increasing, D(t)=HD(k^{-1}(t)) on J, F is countable, and equality fails on J̃. Under an additional generic hypothesis with HD(Λ)<1, the paper also claims agreement of local and global dimension functions on J and strict disagreement on J̃. The corresponding statements are then transferred to the classical Lagrange spectrum. The proofs rely substantially on results imported from the author's preprints, most notably Proposition 1 of [13] and Proposition 3.6 of [15], which are neither stated in full nor proved here.","tokens_in":23623,"tokens_out":6130,"duration_ms":53147,"significance":"If the imported results hold in the pointwise form used, the theorem is a substantial contribution: it gives a complete description of the set of spectral points where the level-set dimension equals the half-line dimension and shows that this set is a strictly increasing arc. The Hölder homeomorphism construction in §3.3 is self-contained and carefully written, and the author is explicit about the generic assumptions in Theorem 1.3 and Remark 1.2. The main weakness is that the central theorem is conditional on unproved statements from three preprints by the same author and collaborators. The paper would be much stronger if the author either stated and proved those imported propositions or clearly restricted the main theorem to the hypotheses under which they are known to hold. As it stands, the result is plausible but not independently verifiable from the manuscript alone.","major_comments":[{"comment":"The proof of Proposition 3.7 begins by invoking 'proposition 1 of [13]' to produce, for arbitrary f in R^r_{φ,Λ}, subhorseshoes Λ_n ⊂ Λ_{t_{n+1}-δ_n} with HD(Λ_n) > R_{φ,Λ,f}(t_n) and max f|Λ_n > t_n. The precise hypotheses of that proposition are not stated, and [13] is a preprint by the same author. In this paper, Theorem 2.5 and Remark 1.2 are carefully phrased for residual sets, whereas Theorem 1.1 is claimed for every f in R^r_{φ,Λ}. The pointwise validity of Proposition 1 of [13] is therefore a live assumption. If it holds only under generic or residual hypotheses, or requires extra transversality conditions, then the existence of the sequence {Λ_n} and hence Theorem 1.1 is not established.","section":"§3.2"},{"comment":"Immediately after Proposition 3.6, the text says that since the function O takes only finitely many values, 'without loss of generality, we can suppose that Λ_n connects with Λ_m before max{t_{n+1},t_{m+1}} for any n,m'. A finite range alone does not imply that all pairs connect; this is an unstated subsequence reduction. The reduction needs to be written explicitly and checked against the requirements t_n → η_-, t_0 = η_- - ǫ, and R(t_n) < HD(Λ̃_n) ≤ R(t_{n+1}). The same issue affects the construction of the nested sequence {Λ̃_n} in §3.2 and hence the Hölder homeomorphism Θ in §3.3. As written, this is a gap in the proof.","section":"§3.2"},{"comment":"Proposition 3.10 asserts that if Λ̃_1 ⊈ Λ̃_2 are subhorseshoes, then HD(Λ̃_1) < HD(Λ̃_2), and it is attributed to 'corollary 3.9 of [14]'. This strict monotonicity is used to prove that D|J is strictly increasing, a central bullet of Theorem 1.1. The statement is not proved in the manuscript, and it is not a general property of arbitrary hyperbolic sets; it depends on the specific conservative horseshoe setting and on the dimension formula HD(Λ) = 2 HD(K^u(Λ)). If the cited corollary fails or has additional hypotheses, the strict-increase conclusion is unsupported. The author should either include a proof or provide the precise statement of the quoted corollary.","section":"§3.4"},{"comment":"The proof of Theorem 4.2 for t > t_1 uses the assertion from [14] that for any j in J̃(t,ǫ) one has HD(Λ̃_j) < 0.99, together with the estimate HD(M\\L) < 1 from [24]. These are imported results. They are less central to Theorem 1.1 than the issues above, but they are load-bearing for the local-dimension statement in the classical case. The author should state the precise form of the needed bound and its hypotheses, or prove it.","section":"§4"}],"minor_comments":[{"comment":"The section heading 'Preliminares' should be 'Preliminaries'.","section":"Table of contents"},{"comment":"The phrase 'modulus some countable subset' should be 'modulo some countable subset'.","section":"§3.4"},{"comment":"In equation (3.2), the expression 'log|Cu(Λ_{η_-}, r_0)|/r_0 - c_1 < 1.001 Du' is ambiguous; parentheses should make clear that c_1 is subtracted before forming the quotient or after, as intended.","section":"§3.2"},{"comment":"The phrase 'By the spectral theorem' should likely be 'By the spectral decomposition theorem', since the reference is to the decomposition of hyperbolic sets into basic pieces.","section":"§3.3"},{"comment":"The definition of F_{φ,Λ,f} as the set of points of J^0_{φ,Λ,f} isolated on the left is called 'enumerable' without an explicit argument; the countability is plausible because such points are separated by intervals with no points of J^0, but a short justification would improve clarity.","section":"§3.4"}],"recommendation":"major_revision","confidential_remarks":"The paper depends very heavily on three preprints by the same author and coauthors (arXiv:2305.07819, arXiv:2403.18940, and arXiv:2309.14646), and the crucial statements are not reproduced or proved. The editor may wish to request that the author provide the exact statements of Proposition 1 of [13], Proposition 3.6 of [15], and Corollary 3.9 of [14], together with a clear indication of their hypotheses, or else restrict Theorem 1.1 to the settings in which those statements are known to hold. The unstated subsequence reduction in §3.2 also needs to be written out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take.\n\nThe paper proves a real, natural extension of [14]: for a conservative horseshoe and a smooth function f that is locally monotone along stable and unstable directions, the set of points in the Lagrange spectrum with positive dimension splits as J ∪ F ∪ J̃, where on J the fiber dimension equals the half-line dimension and D|J is strictly increasing, F is countable, and on J̃ the two dimensions differ. The same dichotomy is recovered for the classical Lagrange spectrum in §4. The construction of the Hölder embedding Θ in §3.3 is new and is written out in detail; the chain of estimates checking the Hölder inverse is the kind of thing one wants to see, and it is here.\n\nThe main soft spot is the paper's dependence on three external results, all from the author's own preprints. Proposition 1 of [13] produces the subhorseshoes with Hausdorff dimension arbitrarily close to R(t_{n+1}); Proposition 3.6 of [15], built on the O-function and 'good positions', gives the connection property that lets the nested sequence {Λ̃_n} be built; and Corollary 3.9 of [14] supplies strict monotonicity of dimension under strict inclusion of subhorseshoes. None of these is stated fully or proved in this text. If any of them carries a residual or generic hypothesis rather than holding for every f ∈ R^r_{φ,Λ}, Theorem 1.1 as stated does not follow. The introduction describes [13] as a generic result, which makes the pointwise status of its Proposition 1 genuinely unclear to me from this paper alone. This is a load-bearing conditionality, not a cosmetic one.\n\nThere is also a small, repairable gap: the 'without loss of generality' that reduces to the case where all Λ_n connect before max{t_{n+1},t_{m+1}} is really a subsequence argument (O takes finitely many values), and it should be written as such.\n\nIf the external results are correct, Theorems 1.1, 1.3, 4.1, and 4.2 appear to follow. The classical section is a clean application once the dynamical theorem is granted. I would send this to a serious referee who is willing to read the three preprints side by side with the manuscript; the paper deserves that. It is not ready to be accepted on its own, but it is ready to be reviewed.","headline":"A well-structured proof of a genuine extension of [14], but the main theorem is conditional on three black-box results from the author's own preprints.","tokens_in":24197,"tokens_out":4489,"would_cite":true,"duration_ms":39737,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D20","28A80","11J06","37C45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For mixing horseshoes of conservative surface diffeomorphisms, the Hausdorff dimension of the Lagrange spectrum concentrates along a strictly increasing set J, with only countably many exceptional points.","keywords":["Hausdorff dimension","Lagrange spectrum","Markov spectrum","mixing horseshoe","surface diffeomorphism","Diophantine approximation","Hölder homeomorphism"],"falsifier":"Take the horseshoe $\\Lambda(4)$ that models $L\\cap(-\\infty,\\sqrt{32}]$, and for a dense sample of $\\eta\\in(0,1)$ compute the left endpoint $\\eta_-=\\min\\{t:D(t)=\\eta\\}$ together with $\\operatorname{HD}(k^{-1}(\\eta_-))$ via continued fractions with coefficients bounded by 4; the theorem predicts $D(\\eta_-)=\\operatorname{HD}(k^{-1}(\\eta_-))$ at every sampled $\\eta_-$, so a single discrepancy at a left endpoint would refute Theorem 4.1 and, by the transfer argument, Theorem 1.1.","tokens_in":103,"feed_emoji":"📈","tokens_out":17128,"duration_ms":265059,"temperature":0.7,"pith_summary":"The paper establishes a precise dichotomy for where Hausdorff dimension accumulates in Lagrange spectra, the sets of Diophantine approximation constants attached to irrational numbers. In the dynamical setting, where the classical spectrum is replaced by one built from a mixing horseshoe (a hyperbolic invariant Cantor-like set) and a smooth function, the cumulative dimension function $D_{\\varphi,\\Lambda,f}(t)=\\operatorname{HD}(k^{-1}_{\\varphi,\\Lambda,f}(-\\infty,t])$ is continuous with image $[0,\\operatorname{HD}(\\Lambda)/2]$, and the dimension-carrying part of the spectrum splits into a strictly increasing spine $J$, a countable set $F$, and a set $\\tilde J$ where the dimension of the half-line differs from that of the level set. The classical Lagrange spectrum obeys the same law, $L\\cap(3,\\infty)=J\\cup F\\cup\\tilde J$ with $D(J)=(0,1)$. This matters because it localizes the dimension-carrying levels to a one-parameter family of left endpoints, and generically in the low-dimensional case it forces local and cumulative dimensions to agree exactly on that spine.","feed_headline":"Lagrange-spectrum dimension concentrates on a strict increasing spine","feed_subtitle":"Same spine governs both dynamical and classical Lagrange spectra.","key_machinery":"The carrying object is the cumulative dimension function $D_{\\varphi,\\Lambda,f}(t)=\\operatorname{HD}(k^{-1}_{\\varphi,\\Lambda,f}(-\\infty,t])$, tied to the horseshoe's unstable dimension by $D=\\frac12 R_{\\varphi,\\Lambda,f}$. The proof's engine is a nested family of subhorseshoes $\\tilde\\Lambda_n$ chosen inside $\\Lambda_{t_{n+1}-\\delta_n}$ with $\\operatorname{HD}(\\tilde\\Lambda_n)$ squeezed between $R(t_n)$ and $R(t_{n+1})$; these come from a subhorseshoe approximation result and an $O$-function/good-positions connection criterion, and are inserted into the level set $k^{-1}(\\eta_-)$ by a Hölder homeomorphism $\\Theta$ whose inverse is Hölder with exponent arbitrarily close to $1$. That transfer forces $\\operatorname{HD}(K^u(\\tilde\\Lambda_0))\\le\\operatorname{HD}(k^{-1}(\\eta_-))$ and yields the equality at every left endpoint. In the classical case the machinery is transplanted through the realization of initial segments $L\\cap(-\\infty,\\sqrt{N^2+4N})$ as the dynamical spectra of horseshoes $\\Lambda(N)$.","core_discovery":"The central claim is Theorem 1.1: for every $C^2$ conservative diffeomorphism of a compact surface with a mixing horseshoe $\\Lambda$, every $r\\ge2$, and every $f$ in the open set of $C^r$ functions whose gradient is not perpendicular to the stable or unstable directions, $D_{\\varphi,\\Lambda,f}$ is continuous from $\\mathbb{R}$ onto $[0,\\operatorname{HD}(\\Lambda)/2]$. Writing $\\eta_-=\\min\\{t:D(t)=\\eta\\}$ and $J=\\{\\eta_-:\\eta\\in(0,\\operatorname{HD}(\\Lambda)/2]\\}$, one has $D(t)=\\operatorname{HD}(k^{-1}(t))$ for every $t\\in J$, $D|_J$ is strictly increasing, and the dimension-carrying part of the spectrum is $J\\cup F\\cup\\tilde J$ with $F$ countable and $D(t)\\ne\\operatorname{HD}(k^{-1}(t))$ on $\\tilde J$. Section 4 transfers the theorem to the classical Lagrange spectrum, where $L\\cap(3,\\infty)=J\\cup F\\cup\\tilde J$, $J=\\{\\eta_-:\\eta\\in(0,1)\\}$, $D(J)=(0,1)$, and generically $L_{\\mathrm{loc}}=L$ on $J$ while $L_{\\mathrm{loc}}<L$ on $\\tilde J$.","pith_inferences":["The paper notes that Theorem 4.1 does not settle injectivity of $D$ on the interior of the classical spectrum; a natural next question is whether the spine $J$ actually runs through every plateau of $D$, or whether some plateaus carry their dimension elsewhere in $\\tilde J$.","Because initial segments $L\\cap(-\\infty,\\sqrt{N^2+4N})$ are modeled by the horseshoes $\\Lambda(N)$, the identity $D(\\eta_-)=\\operatorname{HD}(k^{-1}(\\eta_-))$ can be checked numerically for small $N$ using continued fractions with bounded coefficients, giving a direct test of the transfer from dynamics to number theory.","The same connection-and-embedding mechanism is likely to work for any locally maximal hyperbolic set with local product structure and a real function transverse to the stable and unstable directions, not only for mixing horseshoes; if so, the three-part decomposition would be a general phenomenon for dimension functions over hyperbolic spectra."],"forward_implications":["For every $\\eta\\in(0,\\operatorname{HD}(\\Lambda)/2]$, the left endpoint $\\eta_-$ satisfies $D(\\eta_-)=\\operatorname{HD}(k^{-1}(\\eta_-))=\\eta$, so the dimension-carrying levels are exactly these left endpoints.","On the spine $J$, the function $D$ is strictly increasing and covers the interval $(0,\\operatorname{HD}(\\Lambda)/2]$, so the set where cumulative and level-set dimensions agree contains an uncountable interval-like family, with at most countably many extra coincidence points.","For the classical Lagrange spectrum, $L\\cap(3,\\infty)$ splits as $J\\cup F\\cup\\tilde J$, and at every point of $\\tilde J$ the cumulative dimension $D(t)$ is strictly larger than the dimension of the level set $k^{-1}(t)$.","Generically, when $\\operatorname{HD}(\\Lambda)<1$, local and cumulative dimensions of the dynamical spectrum agree on $J$ and disagree on $\\tilde J$, making $J$ the part of the spectrum where local dimension is as large as possible.","If the local-dimension function is non-decreasing on the derived set of $L$, then the exceptional set $\\tilde J$ contains no accumulation points of the spectrum (Corollary 4.3)."],"supporting_citations":[{"why":"Supplies the subhorseshoe approximation result (Proposition 1) used to find $\\Lambda_n$ inside $\\Lambda_{t_{n+1}-\\delta_n}$ with Hausdorff dimension arbitrarily close to $R(t_{n+1})$; without this the nested family $\\tilde\\Lambda_n$ cannot be built.","marker":"[13]"},{"why":"Supplies the $O$-function and good-positions construction that yields Proposition 3.6, the connection criterion letting any two subhorseshoes connect before the larger threshold; this is what makes the nested family and the embedding $\\Theta$ possible.","marker":"[15]"},{"why":"Provides the dynamical realization of initial segments of the classical Markov and Lagrange spectra as spectra of horseshoes $\\Lambda(N)$, which transfers Theorems 1.1 and 1.3 to the classical setting.","marker":"[2]"},{"why":"Establishes the classical facts about $D$, $\\operatorname{HD}(k^{-1})$, and the relation $L(t)=\\min\\{1,2D(t)\\}$ that Theorem 4.1 refines and builds on.","marker":"[16]"},{"why":"Supplies the spectral decomposition, the strict dimension-growth corollary for nested subhorseshoes used to show $D|_J$ is strictly increasing, and the Hölder embedding pattern adapted in Section 3.3.","marker":"[14]"},{"why":"Provides the theorem that the image of a product of non-essentially affine regular Cantor sets under a non-degenerate $C^1$ map has dimension the minimum of 1 and the product dimension; this forces the local dimension to be 1 above $t_1$ in the classical proof.","marker":"[21]"},{"why":"Supplies the fact $\\operatorname{HD}(M\\setminus L)<1$, used in Theorem 4.2 to conclude the local dimension of $L$ equals 1 on $J$ above $t_1$.","marker":"[24]"}],"fun_headline_variants":["Dimension concentrates on a strict increasing spine in Lagrange spectra","Dimension of Lagrange spectra concentrates on a strictly increasing spine","Strictly increasing spine carries the dimension of Lagrange spectra","Dimension-spine in Lagrange spectra is strictly increasing","Concentration of dimension: Lagrange spectra share a strict increasing spine"],"cache_read_input_tokens":26112,"weakest_assumption_plain":"The load-bearing premise is that two results from the author's own earlier preprints are correct: near each threshold there exist subhorseshoes whose Hausdorff dimension is arbitrarily close to the dimension of the half-line set, and any two such subhorseshoes connect inside the larger of their thresholds; if either premise fails, the nested family and the Hölder embedding on which Theorem 1.1 rests cannot be constructed.","fun_headline_variants_meta":{"raw":{"variants":["Dimension concentrates on a strict increasing spine in Lagrange spectra","Dimension of Lagrange spectra concentrates on a strictly increasing spine","Strictly increasing spine carries the dimension of Lagrange spectra","Dimension-spine in Lagrange spectra is strictly increasing","Concentration of dimension: Lagrange spectra share a strict increasing spine"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000427,"raw_usage":{"total_tokens":2211,"prompt_tokens":998,"completion_tokens":1213,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":1135}},"tokens_in":614,"tokens_out":1213,"duration_ms":8915,"temperature":1.0,"reasoning_tokens":1135,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:45:11.622969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the horseshoe $\\Lambda(4)$ that models $L\\cap(-\\infty,\\sqrt{32}]$, and for a dense sample of $\\eta\\in(0,1)$ compute the left endpoint $\\eta_-=\\min\\{t:D(t)=\\eta\\}$ together with $\\operatorname{HD}(k^{-1}(\\eta_-))$ via continued fractions with coefficients bounded by 4; the theorem predicts $D(\\eta_-)=\\operatorname{HD}(k^{-1}(\\eta_-))$ at every sampled $\\eta_-$, so a single discrepancy at a left endpoint would refute Theorem 4.1 and, by the transfer argument, Theorem 1.1.","supporting_citations":[{"cited_title":"Lima and C","cited_arxiv_id":null,"evidence_quote":"Provides the dynamical realization of initial segments of the classical Markov and Lagrange spectra as spectra of horseshoes $\\Lambda(N)$, which transfers Theorems 1.1 and 1.3 to the classical setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the classical facts about $D$, $\\operatorname{HD}(k^{-1})$, and the relation $L(t)=\\min\\{1,2D(t)\\}$ that Theorem 4.1 refines and builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the spectral decomposition, the strict dimension-growth corollary for nested subhorseshoes used to show $D|_J$ is strictly increasing, and the Hölder embedding pattern adapted in Section 3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fact $\\operatorname{HD}(M\\setminus L)<1$, used in Theorem 4.2 to conclude the local dimension of $L$ equals 1 on $J$ above $t_1$."}],"review_version":1}