{"id":"3a9ec273-3210-4216-b23a-027119e56bed","arxiv_id":"2507.23731","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Power Fourier decay is proved for equilibrium states of nonlinear area-preserving Axiom A surface diffeomorphisms, giving positive lower Fourier dimension for certain C^{1+} self-conformal measures and for the Fibonacci Hamiltonian density of states.","lead":"A new proof shows that equilibrium states of nonlinear area-preserving Axiom A surface diffeomorphisms have power Fourier decay. This yields the first positive Fourier dimension results for low-regularity self-conformal measures and for the density of states of the Fibonacci Hamiltonian.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Fibonacci application rests on a single unverified expansion in §1.4; as printed, the leading Anosov-cocycle value is inconsistent with the preceding two terms, so the nonvanishing claim needs an independent symbolic check.","rationale":"The reader's weakest assumption is exactly the explicit Anosov cocycle computation in Section 1.4, and my read agrees: the application to the Fibonacci Hamiltonian and to the density of states measure is carried by the nonvanishing of that coefficient. My stress-test adds a concrete observation: the displayed intermediate sum in the computation does not equal the displayed final value, which makes an independent check mandatory. I do not regard this as evidence that the main Axiom A theorem is false; the general proof from (QNL) to Fourier decay is elaborate but internally plausible, and the rigidity/Appendix C portion is at least coherent modulo the usual reliance on imported technical lemmas. The verdict should therefore stay CONDITIONAL as the reader proposed: the core result is promising, but the headline application needs the arithmetic of Section 1.4 to be settled by an independent computation before the claim is treated as established.","tokens_in":79218,"tokens_out":16811,"duration_ms":164696,"concrete_test":"Use exact symbolic computation (SymPy, Mathematica, or Sage) to: (1) expand y_V(x,z), form f_V and the adapted-coordinate map f̃_V to order V^{-2}; (2) compute all four terms of the Anosov cocycle formula in Proposition 1.6 as rational functions in V; (3) compare the leading V^{-2} coefficient with -(140+76√5)/3 and check whether it is nonvanishing. This settles whether the trace-map application goes through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The gate from the general Axiom A theorem to Theorem 1.9 is the §1.4 assertion that the Anosov cocycle of T^2|S_V at p_V is -(140+76√5)/3 V^{-2}+O(V^{-1}), hence nonzero for small V. This computation is not independently verified or machine-checked, and as printed it contains an arithmetic inconsistency. From the displayed Taylor expansion of f̃_V, the two leading terms shown in the cocycle formula are A = (160/9)(5-3√5)/(7-3√5) V^{-2} and B = -(40/9)(2/(5+3√5)) V^{-2}. Their sum is -(380+252√5)/9 V^{-2}, not the displayed -(140+76√5)/3 V^{-2}; reversing the sign of B exactly yields the displayed value. A single sign slip alone would not change nonvanishing, but it shows the derivation of the Taylor coefficients in adapted coordinates is currently unchecked, and each quadratic and cubic coefficient enters the cocycle through products. If an independent symbolic computation returned cancellation at V^{-2}, the trace-map application would collapse and Theorem 1.9 would no longer follow from the methods of this paper. The general Theorem 1.4 does not depend on this arithmetic, but the paper's advertised Fibonacci-Hamiltonian corollary does.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims power Fourier decay for equilibrium states of smooth, area-preserving, Axiom A diffeomorphisms on surfaces, provided the stable or unstable distribution is not C^2. This is encoded as positivity of the lower Fourier dimension in all C^{1+α} charts and is applied to the density of states measure of the weakly coupled Fibonacci Hamiltonian via the trace map. The proof strategy is to reduce Fourier decay to a quantitative non-concentration estimate for a temporal distance function Δ, using the sum-product phenomenon, and then to prove that estimate by adapting the template and nonstationary normal coordinate machinery of Tsujii–Zhang to this Axiom A setting. The Fibonacci application rests on an explicit computation of the Anosov cocycle for T^2 restricted to the invariant surface S_V.","tokens_in":118,"tokens_out":12945,"duration_ms":378402,"significance":"If the result is correct, it is a significant advance: it would give the first positive Fourier dimension results for equilibrium states of low-regularity, nonlinear hyperbolic surface diffeomorphisms, and the first power Fourier decay for the density of states measure of the Fibonacci Hamiltonian, with implications for phase-averaged dispersive estimates in quasicrystals. The paper contains a substantial amount of original technical work: the sum-product reduction, the template construction, the local asymptotic analysis of Δ, and the appendices on regularity. However, the confidence in the advertised applications is conditional: one load-bearing displayed computation in §1.4 is arithmetically inconsistent as printed, and several central reductions are quoted from the author's own thesis and earlier papers rather than proved here. These issues are fixable but must be addressed before the claims can be accepted.","major_comments":[{"comment":"The displayed chain evaluating the Anosov cocycle is arithmetically inconsistent. The two leading terms shown are A = (160/9)(5−3√5)/(7−3√5) V^{-2} and B = −(40/9)(2/(5+3√5)) V^{-2}. Their sum is −(380+252√5)/9 V^{-2}, not the displayed −(140+76√5)/3 V^{-2}; reversing the sign of B gives exactly the displayed value. Since the only verification of the hypothesis E^s or E^u not C^2 for the Fibonacci application is the nonvanishing of this cocycle, the computation must be corrected and, ideally, independently checked by symbolic computation. The conclusion may survive the correction, but as printed the proof of Theorem 1.9 does not follow from the displayed estimates.","section":"§1.4, Anosov cocycle computation"},{"comment":"Proposition 4.1 is the bridge from the hypothesis E^s ∉ C^2 to the existence of p with r ↦ ∂_s Δ^+_p(r) not C^1; Theorem 4.3 then converts this into (QNL). The proof in Appendix C contains several steps that are only sketched: Lemma C.3 identifies ∂_sΔ^+ with ∂_uE^s using coordinates whose regularity is stated but not fully proved; §C.3 invokes a Cesàro averaging construction and says 'one can adapt the argument to show C^{2−} convergence' without writing the details; §C.4 concludes via Journé's lemma without checking its hypotheses in the Cantor-set setting. Because Theorem 1.4 inherits this step, the manuscript should provide complete arguments or precise references for those claims.","section":"§4 and Appendix C, Proposition 4.1"},{"comment":"Several load-bearing reductions are imported from the author's earlier work rather than proved. For example, Lemma 2.3, Lemma 2.4, and Lemma 2.6 are quoted from [Le24]/[Le23a]; the reduction of Proposition 3.11 is said to follow the arguments of [Le24]; Lemma 4.8's proof refers to '[Le24], section 5.9' for the key doubling estimate; and Corollary 3.2 says 'see [Le24] for details'. Since these sources are the author's own thesis and papers, the manuscript should either include the needed statements with proofs or clearly delimit the dependency. As written, a referee cannot verify the claimed generality for Axiom A surface diffeomorphisms without consulting those works.","section":"§3.1, §3.3, §4.1, and Lemma 4.8"}],"minor_comments":[{"comment":"There is a typo: after stating ∂_yF(0,0)=∂_xG(0,0)=0, the text writes ∂_xF(0,0)=μ; this should be ∂_yG(0,0)=μ.","section":"§1.3, Proposition 1.6"},{"comment":"The statement says the conclusion holds for any α>0, while the proof fixes a small α and then works with C^{1+α} regularity. The inclusion C^{1+β}⊂C^{1+α} for β≥α makes the passage harmless, but it would be helpful to state this explicitly.","section":"Theorem 1.4 and Definition 1.2"},{"comment":"There are typos such as 'Fibonnaci' and 'density of sate'; these should be corrected.","section":"Abstract and §1.1"},{"comment":"The reference [Mc34] appears in the bibliography but does not seem to be cited in the text; please check whether it is needed.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is heavily dependent on the author's own thesis and earlier papers, which is a legitimate but potentially delicate dependency; the editor may wish to confirm the journal's policy on citing unpublished work for load-bearing steps. The arithmetic inconsistency in §1.4 should be resolved before a second round, since the Fibonacci application is the paper's most visible advertised consequence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know three things about this one. The main theorem is genuine: Theorem 1.4 proves positive lower Fourier dimension for equilibrium states of area-preserving surface Axiom A diffeomorphisms under a nonlinearity condition (Es or Eu not C^2), and the machinery is a serious extension of Tsujii–Zhang and Bourgain–Dyatlov to this setting. That is the first positive result in the C^{1+} self-conformal / quasicrystal direction, so the paper matters. The proof is long but the global structure is clear: reduce Fourier decay to a non-concentration estimate on the temporal distance function ∆, then prove that via the sum-product phenomenon and a detailed blowup/rigidity argument for the functions X_bx. Credit is due for the nonstationary normal coordinates and templates, which are genuinely new in this low-regularity surface setting.\n\nWhere the soft spots are. The paper leans heavily on the author's own prior work — [Le24], [Le23a], [Le23b] — for several technical lemmas, including the cohomology reductions and the transfer-operator machinery. That is a self-citation burden, not a flaw per se, but a referee will need to verify those quoted results actually say what is needed. The bigger issue is the Fibonacci application. In §1.4, the Anosov cocycle computation that gates Theorem 1.9 is not independently checked, and as printed it contains an arithmetic inconsistency: the two displayed leading terms sum to −(380+252√5)/9 V^{−2}, not the claimed −(140+76√5)/3 V^{−2}. Flipping the sign of the second term exactly produces the displayed value, so there is almost certainly a sign slip in the Taylor expansion of f̃_V in adapted coordinates. The nonvanishing conclusion survives either way, so this particular slip is minor, but it shows the computation is unchecked. An independent symbolic verification of that expansion should be requested before Theorem 1.9 is taken as established. The general Theorem 1.4 does not depend on this arithmetic, and I see no comparable problem in the main argument, though its completeness rests on the quoted lemmas.\n\nBottom line: the central theorem is a significant advance and is worth a serious referee. The Fibonacci corollary is separable and currently rests on an unverified computation. Send it to peer review, with a referee instruction to check §1.4 and to audit the transfer from the author's earlier papers.","headline":"Strong paper with a genuinely new main theorem; the Fibonacci corollary depends on an unchecked §1.4 cocycle computation that has a concrete sign inconsistency.","tokens_in":80005,"tokens_out":5525,"would_cite":true,"duration_ms":52307,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D20","37D35","42B10","37C45","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlinear surface hyperbolicity forces power Fourier decay of equilibrium states, and the weakly coupled Fibonacci Hamiltonian inherits it.","keywords":["Fourier decay","Axiom A diffeomorphism","Equilibrium state","Fibonacci Hamiltonian","Density of states","Anosov cocycle","Sum-product phenomenon","Lower Fourier dimension"],"falsifier":"Compute, at a sequence of small couplings $V\\to 0$, the trace map $T^2|_{S_V}$ in adapted coordinates at the fixed point $p_V$ and evaluate the Anosov cocycle $\\partial^3_{xyy}G/\\mu - \\partial^2_{xy}F\\partial^2_{xy}G/(\\lambda-1) - \\partial^2_{xx}G\\partial^2_{yy}F/(1-\\mu^3) - \\partial^2_{xy}G\\partial^2_{yy}G/\\mu^2$; matching the leading term $-(140+76\\sqrt{5})V^{-2}/3$ within controlled error supports the paper's conclusion, while a zero or sign change at some $V\\in(0,V_0)$ would refute it.","tokens_in":79006,"feed_emoji":"🌀","tokens_out":7170,"duration_ms":74460,"temperature":0.7,"pith_summary":"The paper tries to prove that genuine nonlinearity in surface dynamics forces Fourier transforms of equilibrium measures to decay at a power rate. The main theorem says that if $f$ is a $C^\\infty$ area-preserving Axiom A diffeomorphism on a surface and, on a basic set $\\Omega$, either the stable or unstable distribution is not $C^2$, then the measure of maximal entropy (more generally, every equilibrium state) has positive lower Fourier dimension. The proof converts Fourier decay into a non-concentration estimate for a temporal distance function of a suspension flow, and then shows that non-concentration follows from the failure of the foliations to be $C^2$. Via the Fibonacci trace map, this yields power Fourier decay for the density of states measure of the weakly coupled Fibonacci Hamiltonian, giving the first phase-averaged decay statement for a quasicrystal without an extra time average.","feed_headline":"Fibonacci spectrum gains power Fourier decay","feed_subtitle":"The density of states of the weakly coupled Fibonacci Hamiltonian now provably has positive Fourier dimension.","key_machinery":"The load-bearing object is the temporal distance function $\\Delta(p,q)=\\sum_{n\\in\\mathbb{Z}} \\tau_f(f^n p)-\\tau_f(f^n[p,q])-\\tau_f(f^n[q,p])+\\tau_f(f^n q)$ for the suspension flow over $\\Omega$ with roof function $\\tau_f=\\ln\\|df|_{E^u}\\|$. It measures the joint non-integrability of the stable and unstable foliations, and its failure to concentrate near $0$ is the quantitative nonlinearity condition (QNL). The proof has two halves: via the sum-product phenomenon, QNL implies power Fourier decay of equilibrium states; via nonstationary normal coordinates and template vector fields adapted from three-dimensional Anosov flows, the assumption that $E^s$ or $E^u$ is not $C^2$ produces a Weierstrass-type autosimilarity that oscillates at every scale modulo polynomials, hence QNL holds. The Anosov cocycle at a fixed point provides a checkable criterion for this non-$C^2$ nonlinearity.","core_discovery":"The central discovery is that the condition $E^s \\notin C^2$ or $E^u \\notin C^2$ for a basic set of an area-preserving Axiom A surface diffeomorphism implies $\\dim_{F,C^{1+\\alpha}}(\\mu)>0$ for every $\\alpha>0$, where $\\mu$ is the measure of maximal entropy and also for equilibrium states of H\\\"older potentials. In the trace-map setting, this becomes a statement about the Fibonacci Hamiltonian: for all sufficiently small coupling $V>0$, the density of states measure $N_V$ has positive lower Fourier dimension, and the phase-averaged correlation $\\int_\\omega \\langle\\delta_0,e^{-itH_{V,\\omega}}\\delta_0\\rangle d\\omega$ decays like a power of $t$. The nonlinearity of the hyperbolic dynamics, rather than merely the fractal dimension of the spectrum, is what creates the oscillatory randomness behind the Fourier decay.","pith_inferences":["The template argument is phrased for a suspension flow, so a natural extension is to prove exponential mixing for genuine three-dimensional Axiom A flows whose strong distributions are not $C^2$; the paper only hints at this possibility.","The Fourier decay of the density of states does not immediately give the pointwise dispersive bound $\\|\\int_\\omega e^{-itH_{V,\\omega}}\\delta_0\\,d\\omega\\|_{\\ell^\\infty}\\le C|t|^{-\\rho}$; promoting the phase-averaged statement to this pointwise form would settle an open problem mentioned in the paper.","The Anosov-cocycle computation is a leading-order asymptotic in $V$, so for intermediate couplings the same criterion could be checked numerically, potentially extending positive Fourier dimension beyond the weakly coupled regime.","The rigidity result suggests a testable dichotomy: if a surface Axiom A map has $C^2$ stable and unstable distributions and is area preserving, its equilibrium states may lack power Fourier decay, so measuring Fourier dimension could serve as a numerical indicator of foliation regularity."],"forward_implications":["Power Fourier decay holds for every equilibrium state of a nonlinear area-preserving Axiom A surface diffeomorphism whose basic set has a non-$C^2$ stable or unstable distribution, not only for the measure of maximal entropy.","The density of states measure of the weakly coupled Fibonacci Hamiltonian has positive lower Fourier dimension, so any $C^{1+}$ image of the spectrum has positive Fourier dimension.","Phase-averaged quantum correlations for the Fibonacci Hamiltonian decay as a power of time, without the additional time average used in earlier results.","Circle extensions over hyperbolic surface maps satisfying the same nonlinearity condition obtain a spectral gap and exponential mixing.","Self-conformal measures generated by $C^{1+}$ iterated function systems that are factors of hyperbolic diffeomorphisms have positive lower Fourier dimension, a first result in this low-regularity setting."],"supporting_citations":[{"why":"Supplies the exponential-mixing and template machinery for three-dimensional Anosov flows that the paper generalizes to control oscillations of the temporal distance function.","marker":"[TZ20]"},{"why":"Introduces the sum-product reduction from Fourier decay to non-concentration of temporal distance functions, the starting point of Section 3.","marker":"[BD17]"},{"why":"Provides the discretized sum-product theorem used to convert non-concentration of $\\Delta$ into power Fourier decay of equilibrium states.","marker":"[SS20]"},{"why":"Identifies the density of states measure of the Fibonacci Hamiltonian with the pushforward of the maximal entropy measure of the trace map, the bridge used in Theorem 1.8 and Theorem 1.9.","marker":"[DG09]"},{"why":"Characterizes the Fibonacci spectrum through bounded orbits of the trace map, the foundational input for the trace-map application.","marker":"[Su87]"},{"why":"States the Anosov cocycle criterion used to certify that $E^s$ or $E^u$ is not $C^2$ for the trace map restricted to $S_V$.","marker":"[An67]"},{"why":"Contains the precise reduction of power Fourier decay for equilibrium states to the quantitative nonlinearity condition, including the lower Fourier dimension formulation used here.","marker":"[Le24]"},{"why":"Gives the local product structure for equilibrium states that supports the localization argument for checking the non-concentration estimates.","marker":"[Cl20]"}],"fun_headline_variants":["Fibonacci spectrum now provably has power Fourier decay","Power Fourier decay proven for Fibonacci quasicrystal","First quasicrystal power Fourier decay result","Fibonacci density of states shows Fourier decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Fibonacci application rests on the explicit asymptotic computation that the Anosov cocycle of $T^2|_{S_V}$ equals $-(140+76\\sqrt{5})V^{-2}/3+O(V^{-1})$, nonvanishing for all sufficiently small $V>0$; if that computation is wrong, or if the identification of the density of states with the pushforward of the maximal entropy measure fails, Theorem 1.9 would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Fibonacci spectrum now provably has power Fourier decay","Power Fourier decay proven for Fibonacci quasicrystal","First quasicrystal power Fourier decay result","Fibonacci density of states shows Fourier decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001198,"raw_usage":{"total_tokens":4946,"prompt_tokens":958,"completion_tokens":3988,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":3932}},"tokens_in":574,"tokens_out":3988,"duration_ms":26336,"temperature":1.0,"reasoning_tokens":3932,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:26:15.838750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, at a sequence of small couplings $V\\to 0$, the trace map $T^2|_{S_V}$ in adapted coordinates at the fixed point $p_V$ and evaluate the Anosov cocycle $\\partial^3_{xyy}G/\\mu - \\partial^2_{xy}F\\partial^2_{xy}G/(\\lambda-1) - \\partial^2_{xx}G\\partial^2_{yy}F/(1-\\mu^3) - \\partial^2_{xy}G\\partial^2_{yy}G/\\mu^2$; matching the leading term $-(140+76\\sqrt{5})V^{-2}/3$ within controlled error supports the paper's conclusion, while a zero or sign change at some $V\\in(0,V_0)$ would refute it.","supporting_citations":[],"review_version":1}