{"id":"9fabeb05-6835-4ccf-8a6e-adcffaec3226","arxiv_id":"2607.22001","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Constructs non-self-similar C∞ fractal systems with non-Rajchman stationary measures and flat-derivative diffeomorphisms that produce polynomial Fourier decay, with a sharp C∞ obstruction for Pisot parameters.","lead":"Fractal measures can be smooth-looking yet keep echoes at high frequencies, or be reshaped so that echoes decay polynomially even when the reshaping is flat on the fractal. This paper maps exactly where the known sufficient conditions for Fourier decay fail and succeed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main results hinge on unproved companion-paper theorems; no internal flaw found in the Fourier arguments.","rationale":"I read the paper carefully. The central claims are Theorem 1.1 (C∞ pseudo-affine IFS, not C^1-conjugate to self-similar, with non-Rajchman stationary measure) and Theorem 1.2 (flat-on-support diffeomorphism with polynomial Fourier decay). The proofs have two layers: (i) the pseudo-affine framework imported from the companion paper [ABRS26], and (ii) the Fourier-analytic arguments developed in this paper. I checked the internal layer: the matrix Riesz product in Section 3.2, the perturbation sum in Lemma 1, the two-sided product limit in Lemma 2, the Erdős-type non-decay at ξ=λ^{-k}, and the Salem-type random estimates in Section 4.2. All appear correct and internally coherent. The constant arithmetic checks out, including the choice of τ in Proposition 8. The non-conjugacy argument in Proposition 2 is sound assuming Theorem 2.3. The only serious vulnerability is that Theorems 2.1–2.3 from [ABRS26] are taken as black boxes, and the paper provides no way to verify them from the text. Since the reader's verdict was CONDITIONAL for exactly this reason, my stress-test does not change that verdict. I agree with the reader's weakest_assumption. There is no internal inconsistency I can find; the concern is about external verifiability and possible hidden gaps in the companion results.","tokens_in":28433,"tokens_out":25042,"duration_ms":210885,"concrete_test":"Verify Theorem 2.2 in the special case of the root-defect proportions (12): construct explicitly (or locate in [ABRS26]) a C∞ pseudo-affine IFS whose dynamical proportions are λ everywhere except λ_1(e)=κ. If such an IFS cannot exist, Theorem 1.1's existence claim collapses; if it can, the non-conjugacy and Fourier non-decay arguments in Sections 3.2–3.3 can then be checked directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Most load-bearing assumption: the entire construction of Φ in Theorem 1.1 and the conjugacy claims in Theorems 1.1–1.2 depend on Theorems 2.1–2.3 from the companion paper [ABRS26] (to appear), summarized in Section 2.2. In particular, Theorem 2.2 asserts that prescribed dynamical proportions are realizable by a C^{r,α} pseudo-affine IFS, and Theorem 2.3 gives a necessary and sufficient conjugacy criterion. These are not proved here. If either is false or not applicable (e.g., the C∞ condition in Theorem 2.2 has a hidden issue at the root defect), Theorem 1.1 fails. The paper's own Fourier analysis—matrix Riesz product, Lemma 2, Salem-type moment estimates—is coherent and internally checkable, so the risk is concentrated in the imported framework, not in the arithmetic of Sections 3.3/4.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies sharpness of sufficient conditions for polynomial Fourier decay of self-conformal measures on the line. Theorem 1.1 constructs a C^∞ pseudo-affine IFS Φ on [0,1] that is not C^1-conjugate to a self-similar IFS, yet its uniform Cantor–Lebesgue measure is not Rajchman; this shows that the 'linear' hypothesis in [AHW23] cannot be replaced by 'self-similar' at C^1 regularity. Theorem 1.2 shows that for every λ∈(0,1/2), r∈N, and α∈(0,1], there exists a C^{r+α} diffeomorphism g with g' constant on supp μ_λ such that g μ_λ has polynomial Fourier decay with an explicit exponent, even when μ_λ is non-Rajchman because λ^{-1} is Pisot. The paper also proves a matching Pisot obstruction (Prop. 9), a C^∞ stretched-exponential variant (Prop. 10), and a non-affine conjugacy preserving non-decay (Prop. 11). The proofs use a matrix Riesz-product representation for the Fourier transform, a one-scale cosine moment estimate, and a Borel–Cantelli net argument, all built on the pseudo-affine framework of [ABRS26].","tokens_in":28608,"tokens_out":23283,"duration_ms":221196,"significance":"If the companion results are valid, Theorem 1.1 is a substantial contribution: it provides the first example separating C^1-conjugacy to self-similar from the linearity criterion, and Proposition 9 gives a sharp regularity threshold for the phenomenon in Theorem 1.2. The Fourier-analytic arguments in Sections 3.3 and 4.2 are coherent and largely self-contained; the matrix Riesz product and the one-scale cosine moment bound are elegant, and the constants in Lemma 1 and Proposition 8 are carefully tracked. The main vulnerability is the paper's heavy dependence on the to-appear companion [ABRS26]: all of the dynamical constructions and the conjugacy conclusions rest on Theorems 2.2 and 2.3 stated there. The paper also frankly acknowledges the overlap with Ekström [Eks16] for the existence part of Theorem 1.2, while adding an explicit self-conformal structure and a C^∞ variant.","major_comments":[{"comment":"The central results are conditional on theorems that are stated but not proved. Theorem 2.2 is used to construct the IFS Φ in Proposition 2 (Theorem 1.1) and the random IFS Φ_ω in Proposition 7 (Theorem 1.2); Theorem 2.3 is used to conclude non-conjugacy in Theorem 1.1 and to derive the derivative identity h'_ω ≡ 1/L_0(ω) in Proposition 7. These are load-bearing: if any of these statements is false or has a hidden hypothesis, the main theorems fail. Since [ABRS26] is to appear, the submitted manuscript is not independently verifiable. The authors should either include full proofs of the needed parts of Theorems 2.1–2.3, or make the companion manuscript available to referees, or state precisely which parts of the present results are unconditional without it.","section":"Section 2.2 (Theorems 2.1–2.3)"},{"comment":"The reduction from an arbitrary C^1 conjugacy to the canonical address-preserving map is only implicit. In the proof of Proposition 2, after normalizing by an affine map, the authors apply Theorem 2.3 to show that Φ is not C^1-conjugate to Φ_λ. This is valid because any conjugacy on the attractor sends cylinders to cylinders and therefore agrees with the canonical map on the attractor; however, this fact is not stated explicitly in the proof. Since the 'if and only if' in Theorem 2.3 is for the canonical map, the authors should include a sentence explaining that every conjugacy on the attractor is the canonical map (up to the affine normalization), so the criterion applies. This is a presentation point, but it is load-bearing for the non-conjugacy conclusion.","section":"Section 3.1 (proof of Proposition 2)"}],"minor_comments":[{"comment":"Typo: '0=f_0(1)<f_0(1)<...' should presumably be '0=f_0(0)<f_0(1)<...'.","section":"Section 2.1"},{"comment":"The notation M_j for negative j uses λ^j = q^j with j<0; this is implicit and should be stated explicitly, since Lemma 1 relies on it.","section":"Equation (26)"},{"comment":"The overlap with Ekström [Eks16] is handled honestly, but the abstract and introduction state Theorem 1.2 as if the existence part were new. Consider moving the acknowledgement of the overlap to the introduction or making the incremental nature explicit in the theorem statement.","section":"Section 1.2 / Remark 1"},{"comment":"The induction argument for h^{(j)}(0)=0 when r≥2 is compressed. It would be clearer to spell out that h' is constant on a perfect set, so the derivative of h' at each point of the set vanishes, and then iterate.","section":"Proposition 9"}],"recommendation":"major_revision","confidential_remarks":"The key issue is the dependence on [ABRS26]. If that companion paper is not yet accepted, the editor should secure it before publication. The internal Fourier analysis appears sound, and the authors' acknowledgment of the Ekström overlap is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper does a good thing: it shows that the natural-looking replacement of \"linear\" by \"self-similar\" in the AHW23 Fourier-decay criterion fails at C^1 regularity (Theorem 1.1), and it pushes the flat-phase pushforward results of Ekström into an explicit pseudo-affine, self-conformal framework with a Riesz-product representation (Theorem 1.2). The Pisot obstruction (Prop 9) and the stretched-exponential C∞ variant (Prop 10) are genuinely new and sharpen the picture. The authors are honest in Remark 1 about what Ekström already did; that is credit where it is due.\n\nThe internal math I checked — the one-scale cosine moment bound, the Borel–Cantelli net argument, the matrix Riesz product and its convergence proof, the choice of ε in (23) giving β = λ + λ^N — is coherent. I did not find a private flaw in the Fourier arguments. The explicit exponents are plausible and the constant arithmetic checks out. The paper is carefully written; the proofs are detailed.\n\nThe soft spot is not in the Fourier analysis; it is upstream. The entire construction of Φ in Theorem 1.1 and the conjugacy claims in Theorems 1.1–1.2 rely on Theorems 2.1–2.3 from the authors' companion [ABRS26], which is to appear. Theorem 2.2 prescribes dynamical proportions realizable by a C^{r,α} pseudo-affine IFS; Theorem 2.3 states a necessary-and-sufficient conjugacy criterion. These are not proved here, and they are load-bearing. If either has a hidden issue (say, a problem at the root of the C∞ case, or a missing hypothesis in the C^1 conjugacy criterion), the main theorems fail regardless of the good behavior of the Fourier estimates. The reader's conditional verdict is accurate. This is a structural dependency, not a circularity: the decay arguments are not fitted to a desired conclusion, but the framework that produces the examples is external.\n\nA minor note: the same dependency infects Proposition 11, which is otherwise a neat complement.\n\nWho this is for: researchers working on Fourier decay of self-conformal measures and smooth conjugacy rigidity. It deserves a serious referee, but the referee needs access to [ABRS26]. I would not cite the paper as a standalone source until the companion is available and the two can be checked together.","headline":"Careful sharpness results at the AHW23 boundary; the Fourier analysis is solid, but the construction's foundations live in an unpublished companion paper.","tokens_in":29234,"tokens_out":2604,"would_cite":false,"duration_ms":24793,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","37C45","42A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs a C∞ self-conformal iterated function system whose derivative is constant on its attractor but which is not C^1-conjugate to any self-similar system, and shows its uniform stationary measure is not Rajchman; separately,","keywords":["pseudo-affine IFS","self-conformal measures","Rajchman property","Fourier decay","dynamical proportions","Bernoulli convolutions","Pisot numbers","matrix Riesz products"],"falsifier":"Compute the cylinder-length ratios of the constructed IFS from Proposition 2 at all finite words: the ratio Ψ(w)/λ^{|w|} must equal 1 except at the root word, where it equals κ/λ. If the ratios deviate at any non-root word, the cocycle-criterion argument for non-conjugacy is void. Separately, numerically evaluate the Fourier transform of the stationary measure at the frequencies λ^{-k} for large k; the proof predicts a fixed positive lower bound, so observing decay to zero would indicate an error in the matrix-product estimate.","tokens_in":28208,"feed_emoji":"📉","tokens_out":5561,"duration_ms":56687,"temperature":0.7,"pith_summary":"The paper studies how sharp recent sufficient conditions for polynomial Fourier decay of self-conformal measures actually are. It proves two things. First, there exists a C∞ iterated function system that is linear in the sense that all derivatives on the attractor equal one constant, yet is not C^1-conjugate to any self-similar system, and its uniform (1/2,1/2) stationary measure has a Fourier transform that does not vanish at infinity. This shows that a known C^2 Fourier-decay criterion, which requires failure of conjugacy to a linear system, cannot be relaxed to failure of conjugacy to a self-similar system at this regularity. Second, for every λ in (0,1/2), it constructs random C^{r,α} diffeomorphisms whose derivatives are constant on the support of the Bernoulli convolution μ_λ, but whose push-forwards μ_λ nonetheless enjoy polynomial Fourier decay, even when μ_λ itself is not Rajchman. The proofs rely on pseudo-affine IFS and replace the classical infinite-convolution product with a matrix-valued Riesz product.","feed_headline":"Smooth non-self-similar IFS blocks Fourier-decay criterion","feed_subtitle":"A C∞ system with constant-on-attractor derivatives carries a non-Rajchman measure, showing the conjugacy test must stay linear.","key_machinery":"The central object is the pseudo-affine IFS: an IFS in which every map's derivative equals the same λ at every point of the attractor. Its geometry is encoded in the dynamical proportions, the ratios between a gap and its two descendant gaps; regularity of the system is characterized by how fast these proportions converge to λ, and conjugacy between two pseudo-affine systems is characterized by the pointwise convergence of the ratio of their cocycles. For the Fourier arguments, the central identities are the matrix Riesz product for the two-state renewal recursion (used in Theorem 1.1) and the level-homogeneous independent cosine product for the random construction (used in Theorem 1.2). The","core_discovery":"The central claim is that two phenomena coexist. On the negative side, a C∞ pseudo-affine IFS with slope λ exists whose dynamical proportions equal λ except at the root word, where a single defect κ=λ+ε is inserted. The defect vanishes at small scales, yielding C∞ smoothness, but it persists in the dynamical proportions, so the cocycle-ratio criterion from the companion theory shows the IFS is not C^1-conjugate to the self-similar system Φ_λ. The uniform stationary measure is not an infinite convolution; its finite approximations satisfy a two-state renewal recursion, and passing to Fourier transforms gives a matrix product whose factors are 2×2 matrices with phases concentrated near integer","pith_inferences":["If the companion realizability theorems are sound, a deterministic version of the random construction should be obtainable by replacing the Borel–Cantelli step with an equidistribution or Diophantine approximation argument on the perturbation variables, since the level-homogeneous structure makes the induced geometry particularly rigid.","The matrix Riesz product for two-state recursions likely extends to IFS with more than two maps, yielding block-matrix products whose decay is governed by the joint spectral radius of the associated matrices; the Pisot-type argument here suggests a general arithmetic obstruction to the Rajchman property when all phases are exponentially close to integers.","Proposition 9 places a ceiling on the decay rate for C∞ flat derivatives in the Pisot case; whether the polynomial-to-stretched transition at C∞ is sharp in the non-Pisot case is a natural next question.","These examples indicate that 'lack of conjugacy to self-similar' is not the right invariant for Fourier decay at finite regularity; the more relevant quantity may be the rate at which the dynamical proportions converge to their limiting value, rather than whether that limit system is affine."],"forward_implications":["The known C^2 criterion for polynomial Fourier decay of self-conformal measures is sharp in its regularity setting: at C^1 conjugacy level, absence of conjugacy to a self-similar system does not imply the Rajchman property.","Pseudo-affine IFS provide an exactly solvable class where regularity, conjugacy, and Fourier behaviour can be read off from a single sequence of gap-length ratios.","For Bernoulli convolutions with contraction ratio below 1/2, the standard phenomenon that smooth nonlinear images have polynomial Fourier decay persists even when the image map's derivative is flat on the support, so L^2-flattening is not necessary in that regime.","If λ^{-1} is Pisot, no C∞ diffeomorphism whose derivative is constant on the support of μ_λ can yield polynomial Fourier decay; the best possible in this class is stretched-exponential decay, which the paper realizes.","The construction yields an explicit, computable lower bound on the Fourier-decay exponent in terms of λ and the smoothness exponent s."],"fun_headline_variants":["Smooth IFS hides defect, breaks Fourier-decay test","C∞ map dodges conjugacy, still non-Rajchman","Pseudo-affine IFS: smooth yet evades Fourier decay","Defect at root word defeats smoothness criterion","Non-self-similar IFS retains stubborn Fourier modes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire construction relies on three theorems from the authors' companion paper: that prescribed dynamical proportions are realizable by a C^{r,α} pseudo-affine IFS with matching regularity, and that the convergence of the cocycle-ratio function is both necessary and sufficient for conjugacy to exist; if any of these has a flaw, both Theorem 1.1 and the conjugacy and self-conformal-structure parts of Theorem 1.2 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Smooth IFS hides defect, breaks Fourier-decay test","C∞ map dodges conjugacy, still non-Rajchman","Pseudo-affine IFS: smooth yet evades Fourier decay","Defect at root word defeats smoothness criterion","Non-self-similar IFS retains stubborn Fourier modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1152,"prompt_tokens":662,"completion_tokens":490,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":406}},"tokens_in":406,"tokens_out":490,"duration_ms":5686,"temperature":1.0,"reasoning_tokens":406,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:06:51.454275+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the cylinder-length ratios of the constructed IFS from Proposition 2 at all finite words: the ratio Ψ(w)/λ^{|w|} must equal 1 except at the root word, where it equals κ/λ. If the ratios deviate at any non-root word, the cocycle-criterion argument for non-conjugacy is void. Separately, numerically evaluate the Fourier transform of the stationary measure at the frequencies λ^{-k} for large k; the proof predicts a fixed positive lower bound, so observing decay to zero would indicate an error in the matrix-product estimate.","supporting_citations":[],"review_version":1}