{"id":"9c3eb314-57fd-46a0-8564-3bfe64e0743e","arxiv_id":"2412.15111","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For closed arithmetic hyperbolic surfaces, the possible values of the first nonzero Laplace eigenvalue form a dense subset of [0, 1/4].","lead":"This paper proves that the possible spectral gaps of closed arithmetic hyperbolic surfaces are dense in the interval from 0 to 1/4, meaning these shapes can have first Laplace eigenvalues filling that whole range. It answers a question of Michael Magee and completes the compact case for the 'bass note' spectrum of locally uniform geometries.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.2, the key spectral-convergence step, is stated without proof and depends on unpublished work [Mag24b]; if it fails, the density construction in Theorem 6.4 collapses.","rationale":"The reader's verdict is well-founded. Theorem 6.2 is indeed the least secure load-bearing input: it is a strong-convergence-to-spectral-gap theorem for covers of a compact hyperbolic surface, stated for a general uniform lattice and used crucially in Theorem 6.4. The proof sketch is short and explicitly depends on unpublished work. Because the central construction requires λ1(Y_φ^(n)) and the selected 2-cover to be close to λ1(T_φ), any gap in Theorem 6.2 is fatal. The paper deserves credit for the computer-assisted verification of Proposition 3.2, which is reproducible via the attached notebook, and for clearly describing the probabilistic collar/systole input with references to multiple published sources. The abstract's 'equals [0,1/4]' overclaims the proven density statement, but that is a presentation issue that can be fixed without changing the main theorem; it does not by itself change the verdict. I recommend keeping CONDITIONAL: the proof is plausible but not self-contained at a key step.","tokens_in":15450,"tokens_out":13618,"duration_ms":115538,"concrete_test":"Independently derive Theorem 6.2 from the published arguments in [HM23, LM23, MT23] without citing [Mag24b]. In particular, verify that strong convergence of std_n ∘ ρ_n ∘ p implies strong convergence of the induced representations Ind_Γ^{PSL(2,R)}(std_n ∘ ρ_n ∘ p) to Ind_Γ^{PSL(2,R)}(ρ_reg ∘ p), and that the min-formula for λ1 follows. If this derivation cannot be completed, the proof of Theorem 6.4 is incomplete and the paper should either include a full proof or explicitly mark the result as conditional on [Mag24b].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing premise is Theorem 6.2 (Section 6.3). It asserts that, under strong convergence of std_n ∘ ρ_n to ρ_reg, the spectral gaps of the random covers Y_φ^(n) converge to min{λ1(K\\H^2), λ1(Γ\\H^2)}. This is used in Theorem 6.4 to guarantee that Y_φ^(n) and a suitable degree-two cover have λ1 > 1/4 − η. The paper provides only a proof sketch and states that the precise statement is not yet in the literature, citing the unpublished manuscript [Mag24b] for the key steps (Theorem 3.1 and Proposition 2.9). If this theorem fails or does not apply in the present setting, then the existence of covers with near-1/4 gap, and hence the η-density conclusion of Theorem 6.4, breaks. The abstract also overclaims that the set of limit points equals [0,1/4], while Theorem 1 only proves density in [0,1/4]; the reverse inclusion is not addressed. This is a presentation gap separate from the proof gap in Theorem 6.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Hide and Petri study the set of first nonzero eigenvalues (spectral gaps) of closed arithmetic hyperbolic surfaces. Their main theorem asserts that this set is dense in [0, 1/4]. The strategy starts from the Bolza surface X_B, a fixed handlebody attachment, and a meridian pants decomposition, then forms random covers Y_φ^(n) whose monodromy factors through a free group F2 via a homeomorphism φ. A tree cover T_φ is constructed with large systole, wide collars, and λ1 close to 1/4, using Mirzakhani's distribution of random pants decompositions and Cheeger's inequality. Strong convergence of permutation representations (Bordenave–Collins) together with a spectral-convergence theorem for covers (Theorem 6.2) is used to obtain random covers and degree-two covers with gap close to 1/4. Intermediate gaps are reached by switching between degree-two covers: Proposition 5.3 bounds the change in the Rayleigh quotient under a switch, and a Nica-type cycle-count statement is used to ensure that the switched curves have wide collars and long systole with positive probability. A computer-assisted linear programming verification (Proposition 3.2) shows all degree-two covers of X_B have λ1 > 1/4, which is used to identify the limiting gap in the strong-convergence step. The paper also claims, in the abstract and introduction, that the set of limit points of all such spectral gaps equals [0, 1/4].","tokens_in":15754,"tokens_out":11540,"duration_ms":101969,"significance":"If the proof is completed, this is a substantial contribution: it answers Magee's question for closed arithmetic surfaces and provides a new proof of the Louder–Magee theorem on covers with near-optimal spectral gaps. The paper has clear strengths: the order of constants is explicit in Section 6.1; the computer verification is shipped as a Jupyter notebook using interval arithmetic; the geometric inputs are stated as precise propositions; and the overall architecture is coherent and well motivated. However, the central result is currently conditional on an unpublished theorem and on an unproven extension of Nica's cycle-count result, and the abstract overstates what is proved. These issues need to be resolved before the paper can be accepted.","major_comments":[{"comment":"Theorem 6.2 is the spectral-convergence engine: it is used in Theorem 6.4 to guarantee that λ1(Y_φ^(n)) and the chosen degree-two cover have λ1 > 1/4 − η. The proof is only a sketch and explicitly states that the precise statement is not yet available in the literature, citing the unpublished manuscript [Mag24b] for its two key steps (Theorem 3.1 and Proposition 2.9). If [Mag24b] is not available, or if its hypotheses do not match the present setting, the construction of good near-1/4 covers collapses. The paper must either include a complete proof of Theorem 6.2 or be revised to state the main theorem as conditional on a precisely cited published result.","section":"Section 6.3 (Theorem 6.2)"},{"comment":"The positive-probability control of short geodesics and collar widths relies on a vector-independence and Poisson-limit statement for cycle counts of random permutations under word maps. The text says this statement \"is not literally contained in Nica's paper, but it can be proven using similar methods\", referring to [LP10, BP23, PZ24] without giving a precise statement. This is load-bearing for the switching argument, specifically for avoiding a geodesic γ satisfying (6.1). Please provide a self-contained proof or a precise citation of a theorem whose hypotheses match the use.","section":"Section 6.6 (Nica-type cycle statistics)"},{"comment":"The abstract claims that the set of limit points of spectral gaps of closed arithmetic hyperbolic surfaces equals [0, 1/4]. What Theorem 1 actually proves is that the set of spectral gaps is dense in [0, 1/4]. Denseness implies that every point of [0, 1/4] is a limit point, but it does not rule out limit points larger than 1/4; indeed the Bolza surface itself has λ1 ≈ 3.8389. No argument is given excluding accumulation outside [0, 1/4]. Please either prove the reverse inclusion or rephrase the abstract and the introductory \"can be stated as\" sentence as a density result.","section":"Abstract and Section 1.2"},{"comment":"The existence of a degree-two cover of Y_φ^(n) with λ1 > 1/4 − η is obtained by asserting that θ ⊗ ρ_n strongly converges to ρ_reg \"using the same argument as in [Mag24a, Section 4]\". This is a second, unstated strong-convergence input with a quantitative conclusion that is essential to the construction. It should be stated as a lemma with either a proof or an exact reference rather than an appeal to an analogous argument elsewhere.","section":"Section 6.4 (proof of Theorem 6.4)"}],"minor_comments":[{"comment":"In the switch bound, the displayed inequality says ⟨∆f′, f′⟩/⟨f′, f′⟩ ≤ λ1( bY2) + η; from context it should be λ1( bY1) + η, since f is an eigenfunction on bY1 and f′ is constructed there. Please correct the label.","section":"Section 6.6"},{"comment":"The phrase \"asymptotically strictly positive\" for the probability of the good event is imprecise; the proof gives a uniform lower bound a > 0 for the cycle-count event and a probability tending to 1 for the spectral event, which together imply a positive lower bound. Consider stating this directly.","section":"Section 6.6"},{"comment":"There are several typos: \"satsify\" in Section 3, \"homomoprhism\" in Section 3, \"desciption\" in Section 6.5, and \"Mirakhani\" instead of \"Mirzakhani\" in the proof of Proposition 6.1.","section":"Global"},{"comment":"The caption \"A tree cover of a surface of genus two\" does not explain the relationship to the handlebody attachment or the meridian pants decomposition described in Sections 2.2–2.3; a more informative caption would help.","section":"Figure 1"},{"comment":"The identification of H^(n) with (Z/2Z)^k depends on a choice of spanning tree in W^(n), making the Hamming distance and the switch operation generating-set dependent. It may be worth noting explicitly that any two elements of H_φ^(n) are joined by switches in this generating set, which is what the connectivity argument needs.","section":"Section 6.5"}],"recommendation":"major_revision","confidential_remarks":"The dependence on [Mag24b] is the main concern: the central theorem's proof is not self-contained, and the abstract claims more than Theorem 1 establishes. If the authors can supply the missing proof of Theorem 6.2 and the Nica-type statement, or cite published versions of them, the paper would be a strong contribution to the spectral geometry of arithmetic surfaces. I would not recommend rejection solely for reliance on a preprint, but the manuscript as submitted is not yet suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper answers Magee's question: the spectral gaps of closed arithmetic hyperbolic surfaces are dense in [0,1/4]. That is the headline. The proof is a genuine new construction, not a cosmetic variant of the non-compact case. They build random covers of the Bolza surface via maps to F2, use Mirzakhani's pants-decomposition asymptotics to get a tree cover with long systole, wide collars, and λ1 close to 1/4, then interpolate to lower gaps by switching among degree-two covers. The computer check that every degree-two cover of Bolza has λ1 > 1/4 is shipped as a notebook, and the interval arithmetic looks reproducible. Credit where due: this is a serious piece of work.\n\nThe soft spots are the usual ones for this area. Theorem 6.2, the strong-convergence step that identifies the limit of λ1 of the random covers, is stated with a proof sketch and depends on Magee's unpublished [Mag24b]. The text says the precise statement is not yet in the literature. That is load-bearing: if [Mag24b] fails to appear or doesn't cover this setting, Theorem 6.4 doesn't go through. The reliance is at least explicit and the authors cite closely related published work (Hide–Magee, Louder–Magee, Magee–Thomas) that plausibly contains the needed ingredients. The Nica-type independence claim in Section 6.6 is also stated as not literally in the literature, but they list four references with versions of it; that's a minor gap, not a red flag. On the presentation side, the abstract says 'limit points equals [0,1/4]' while the body proves density. Density plus a standard upper bound λ1 ≤ 1/4 + O(1/g) gives the reverse inclusion, but the upper bound isn't stated, so the abstract does a bit more work than the text. That's a fixable presentation gap.\n\nMy bottom line: the central argument is coherent and the result is significant. I would send this to a serious referee. The referee should be asked to verify Theorem 6.2's provenance and whether the proof sketch can be made complete without [Mag24b]. If the authors can either prove Theorem 6.2 in the paper or point to a public version of Magee's manuscript, I'd be comfortable accepting after minor revisions. If [Mag24b] remains unavailable, this should be conditional pending that paper.\n\nI'd bring this to our reading group and I'd cite it.","headline":"Answers Magee's question with a genuine new construction; the proof is coherent and the computation is reproducible, but the key convergence theorem rests on unpublished work.","tokens_in":16230,"tokens_out":5488,"would_cite":true,"duration_ms":49240,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"The set of limit points of spectral gaps of closed arithmetic hyperbolic surfaces is exactly the full interval $[0,\\frac14]$.","keywords":["spectral gap","arithmetic hyperbolic surfaces","limit points","Laplacian eigenvalues","random covers","Bolza surface","strong convergence","degree-two covers"],"falsifier":"Find a uniform lattice $\\Gamma$, an epimorphism $\\Gamma\\to\\Lambda$, and a sequence of permutation representations of $\\Lambda$ whose standard representations converge strongly to the regular representation, yet for which the smallest positive eigenvalues of the resulting covers $\\Gamma_n\\backslash\\mathbb{H}^2$ do not converge to $\\min\\{\\lambda_1(K\\backslash\\mathbb{H}^2),\\lambda_1(\\Gamma\\backslash\\mathbb{H}^2)\\}$; such an example would invalidate Theorem 6.2 and break the construction of covers with gaps near $\\frac14$.","tokens_in":15259,"feed_emoji":"🎵","tokens_out":14946,"duration_ms":122047,"temperature":0.7,"pith_summary":"This paper answers a question left open by the recent non-compact case: for closed arithmetic hyperbolic surfaces, the possible spectral gaps are dense in the interval $[0,\\frac14]$. The main theorem states that the values $\\lambda_1(X)$ attained by closed arithmetic surfaces, intersected with $[0,\\frac14]$, form a dense subset of $[0,\\frac14]$. Since there are only finitely many arithmetic surfaces of any fixed genus, this is the same as saying that the set of limit points of all such spectral gaps is exactly $[0,\\frac14]$. The proof builds, inside one fixed arithmetic family (covers of the Bolza surface), finite covers whose spectral gaps approximate every number in the interval. This matters because it turns a countable, seemingly sparse set into a continuum, showing that arithmeticity alone imposes no further restriction on possible gaps in this range.","feed_headline":"Spectral gaps of arithmetic surfaces fill [0, 1/4]","feed_subtitle":"Every number between 0 and 0.25 appears as a limit of possible spectral gaps.","key_machinery":"The proof is carried by a random-model construction. Fix a handlebody attachment on the Bolza surface (the genus-two surface obtained from a regular octagon with opposite sides identified) and a pants decomposition by meridians; a homeomorphism $\\phi$ produces an epimorphism from the surface group $\\Gamma_B$ to a free group $F_2$, whose kernel gives an infinite planar 'tree cover' $T_\\phi$. Choosing a uniformly random permutation representation $\\rho\\in\\mathrm{Hom}(F_2,S_n)$ yields a random finite cover $Y_\\phi^{(n)}\\to X_B$, and homomorphisms of $\\pi_1(Y_\\phi^{(n)})$ to $\\mathbb{Z}/2\\mathbb{Z}$ give the degree-two covers whose spectral gaps are studied. Three ingredients make the argument work: random pants-decomposition asymptotics select $\\phi$ so that $T_\\phi$ has long systole, wide standard collars, and spectral gap close to $\\frac14$; a strong-convergence theorem for permutation representations transfers this gap to the random covers; and a 'switching' move changes one degree-two cover into another along a preferred pants curve without moving the spectral gap by more than a prescribed $\\eta$, using eigenfunction delocalization and flattening near collars. A separate computer-assisted linear-programming check based on the Selberg trace formula certifies that every connected degree-two cover of the Bolza surface has $\\lambda_1>\\frac14$, which is what makes the near-$\\frac14$ double covers possible.","core_discovery":"The central claim is Theorem 1: the set $\\{\\lambda_1(X): X \\text{ a closed arithmetic surface}\\}\\cap[0,\\frac14]$ is dense in $[0,\\frac14]$. Equivalently, as stated in the abstract, the limit points of the spectral gaps of closed arithmetic hyperbolic surfaces are precisely the whole interval $[0,\\frac14]$. The surfaces that realize this density are built as degree-two covers of a certain sequence of random finite covers of the Bolza surface, so the density is achieved using only finite-index torsion-free subgroups of a single uniform arithmetic group, the $(2,3,8)$-triangle group (generated by rotations of orders 2, 3 and 8 fixing a hyperbolic triangle). The same method proves an analogous statement for any closed orientable hyperbolic surface $X=\\Gamma\\backslash\\mathbb{H}^2$: the spectral gaps of its finite-index subgroups are dense in $[0,\\Lambda]$, where $\\Lambda$ is the minimum spectral gap among double covers of $X$. It also yields a new proof that some sequence of finite-sheeted covers of any such $X$ has genus tending to infinity and spectral gap tending to $\\frac14$.","pith_inferences":["The authors do not spell this out, but the switching construction suggests a testable generalization: any base surface whose double covers all have spectral gap above $\\frac14$ and which admits meridian pants decompositions with arbitrarily long curves should yield the same density statement for its finite covers.","If the unpublished strong-convergence ingredient is published in the stronger form sketched here, the same random model would likely give convergence not only of the first eigenvalue but of the full spectral measure of the random covers to that of the tree cover, making the density proof part of a finer spectral equidistribution statement.","The constants in the proof are uniform, so a quantitative version could in principle be extracted: for a given tolerance $\\eta$, one could bound the cover degree needed to place a spectral gap within $\\eta$ of any prescribed value in $[0,\\frac14]$."],"forward_implications":["Because the density is achieved by covers of the Bolza surface, the interval $[0,\\frac14]$ is realized inside a single arithmetic commensurability class, not by combining many different classes.","For any closed orientable hyperbolic surface $X$, the spectral gaps of its finite-index covers are dense in $[0,\\Lambda]$, where $\\Lambda$ is the smallest spectral gap among its double covers.","There exists a sequence of finite-sheeted covers of any such $X$ with genera tending to infinity and spectral gaps tending to $\\frac14$, recovering a previously known result by a different argument.","No value strictly above $\\frac14$ can be a limit point of spectral gaps of closed arithmetic surfaces; values above the interval can occur only as isolated points, such as those coming from low-genus exceptional surfaces like the Bolza surface and the Klein quartic."],"supporting_citations":[{"why":"Supplies the non-compact analogue and poses the question this paper answers; its switching strategy is adapted here.","marker":"[Mag24a]"},{"why":"Unpublished source of Theorem 6.2, the strong-convergence step that transfers spectral gaps from infinite covers to finite random covers.","marker":"[Mag24b]"},{"why":"Proves strong convergence of uniformly random permutation representations of free groups, used to make the random covers inherit the tree cover's gap.","marker":"[BC19]"},{"why":"Gives the random pants-decomposition asymptotics used to find homeomorphisms with long systole and wide standard collars.","marker":"[Mir16]"},{"why":"Supplies the linear-programming and Selberg-trace-formula certification that every degree-two cover of the Bolza surface has spectral gap above $\\frac14$.","marker":"[FBP23, FBGMPP23]"},{"why":"Provides the delocalization bound for eigenfunctions used in the switch estimate.","marker":"[GLMST21]"},{"why":"Provides the Poisson statistics of cycles in random word maps used to avoid short geodesics with positive probability.","marker":"[Nic94]"},{"why":"Provides the earlier strong-convergence framework and the appendix on spectral gaps of covers from which the model is adapted.","marker":"[LM23]"}],"fun_headline_variants":["Limit points of spectral gaps on arithmetic surfaces are exactly [0, 1/4]","Spectral gaps of arithmetic surfaces hit every value between 0 and 0.25","Arithmetic surface gaps: limit set is the whole interval [0, 1/4]","Every spectral gap limit on arithmetic hyperbolic surfaces is in [0, 1/4]","Spectral gaps of arithmetic surfaces: limit points fill entire [0, 1/4]"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the unpublished strong-convergence theorem cited as [Mag24b] is true as stated, with only a proof sketch supplied; a secondary gap is that the cycle-counting statement used in Section 6.6 is asserted to follow by similar methods but is not contained in any cited reference.","fun_headline_variants_meta":{"raw":{"variants":["Limit points of spectral gaps on arithmetic surfaces are exactly [0, 1/4]","Spectral gaps of arithmetic surfaces hit every value between 0 and 0.25","Arithmetic surface gaps: limit set is the whole interval [0, 1/4]","Every spectral gap limit on arithmetic hyperbolic surfaces is in [0, 1/4]","Spectral gaps of arithmetic surfaces: limit points fill entire [0, 1/4]"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000696,"raw_usage":{"total_tokens":3067,"prompt_tokens":788,"completion_tokens":2279,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":2162}},"tokens_in":404,"tokens_out":2279,"duration_ms":12986,"temperature":1.0,"reasoning_tokens":2162,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:37:21.395818+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a uniform lattice $\\Gamma$, an epimorphism $\\Gamma\\to\\Lambda$, and a sequence of permutation representations of $\\Lambda$ whose standard representations converge strongly to the regular representation, yet for which the smallest positive eigenvalues of the resulting covers $\\Gamma_n\\backslash\\mathbb{H}^2$ do not converge to $\\min\\{\\lambda_1(K\\backslash\\mathbb{H}^2),\\lambda_1(\\Gamma\\backslash\\mathbb{H}^2)\\}$; such an example would invalidate Theorem 6.2 and break the construction of covers with gaps near $\\frac14$.","supporting_citations":[],"review_version":1}