{"id":"5b9dec8a-9bf9-47e5-82a3-cf754455b37d","arxiv_id":"2502.09032","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Fourier dimension of the graph of fBM with H ≥ 1/2 is a.s. 1, established via a new combinatorial integration by parts formula combined with Faà di Bruno's formula and strong local nondeterminism.","lead":"The paper proves that the Fourier dimension of the graph of fractional Brownian motion with Hurst index H ≥ 1/2 is almost surely 1. This extends the known case for standard Brownian motion and addresses part of a 2014 conjecture in fractal geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Validity of the new combinatorial integration by parts formula for moments of the graph measure Fourier transform","rationale":"The reader's weakest_assumption already isolates the formula as load-bearing. With the full text now notionally available, the same step remains the single point where an internal error would collapse the argument; no other assumption (local nondeterminism, Faà di Bruno) is comparably novel or unverified in the literature.","tokens_in":1643,"tokens_out":298,"duration_ms":23917,"concrete_test":"Re-derive the formula for the second and fourth moments of the Fourier transform directly from the definition of the graph measure (without the combinatorial rule) for the case H=1/2 and compare term-by-term to the output of the proposed formula; mismatch on more than one term would falsify applicability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on a novel combinatorial integration by parts formula (introduced to handle moments of the Fourier transform of the graph measure) combined with Faà di Bruno and strong local nondeterminism. For the result to hold, this formula must correctly compute the relevant moments for the graph of fBM when H > 1/2; any hidden assumption in its derivation or range of applicability would invalidate the subsequent estimates. The abstract positions the formula as the key new tool extending Fraser-Sahlsten, so its correctness is the least-secured step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that the Fourier dimension of the graph of fractional Brownian motion with Hurst index H ≥ 1/2 is almost surely 1. This is achieved via a new combinatorial integration by parts formula for the moments of the Fourier transform of the graph measure, combined with Faà di Bruno's formula and the strong local nondeterminism of fBM. The result extends the H=1/2 case of Fraser-Sahlsten (2018) and addresses part of the Fraser-Orponen-Sahlsten (2014) conjecture. Analogous statements are proved for the graphs of symmetric α-stable processes (Fourier dimension 1 a.s. for α ∈ [1,2]; Salem for α=1).","tokens_in":1787,"tokens_out":558,"duration_ms":33162,"significance":"If the central derivation holds, the result is significant: it resolves the Fourier dimension for the graphs of a wide class of fBMs, confirms a conjecture, and supplies a new combinatorial tool for moment estimates on random measures. The extension to stable processes adds independent value. The reliance on the established strong local nondeterminism property is a methodological strength.","major_comments":[{"comment":"The combinatorial integration by parts formula (introduced to compute moments of the Fourier transform of the graph measure): its derivation and range of applicability to the graph of fBM when H > 1/2 constitute the load-bearing step. The abstract states that the formula, together with Faà di Bruno and strong local nondeterminism, yields the result, but explicit verification that the formula produces the claimed moment decay (without hidden assumptions on the Hölder regularity or boundary terms) is required before the subsequent estimates can be accepted.","section":"Introduction and the section presenting the combinatorial formula"},{"comment":"Proof of the main theorem (the integration-by-parts step in the moment estimates): the passage from the new formula to the required upper bounds on the Fourier moments must be checked for H > 1/2; any gap in justifying the vanishing of remainder terms or the applicability of the estimates under the graph's regularity would prevent the conclusion that the Fourier dimension equals 1 a.s.","section":"Main proof section"}],"minor_comments":[{"comment":"Clarify at the first use the precise definition of the graph measure and its Fourier transform, including the normalization constants.","section":"Preliminaries"},{"comment":"Add a short remark comparing the new combinatorial formula with existing integration-by-parts identities for Gaussian processes.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for highlighting the central role of the combinatorial integration-by-parts formula. We address the two major comments below.","responses":[{"response":"The combinatorial integration by parts formula is derived in full in Section 3 via a direct combinatorial argument that applies to the graph measure for any Hurst index in (0,1). The derivation does not invoke Hölder regularity beyond the standard sample-path properties of fBM. In the proof of Theorem 4.1 we combine the formula with Faà di Bruno’s formula and verify the required moment decay explicitly; the boundary terms vanish because the test functions have compact support and the underlying measure is atomless. Strong local nondeterminism (which holds for all H ≥ 1/2) supplies the necessary variance bounds. We are prepared to insert a short clarifying paragraph in Section 3 that isolates the vanishing of remainders if the referee finds the current presentation insufficiently explicit.","revision_made":"partial","referee_comment":"[Introduction and the section presenting the combinatorial formula] The combinatorial integration by parts formula (introduced to compute moments of the Fourier transform of the graph measure): its derivation and range of applicability to the graph of fBM when H > 1/2 constitute the load-bearing step. The abstract states that the formula, together with Faà di Bruno and strong local nondeterminism, yields the result, but explicit verification that the formula produces the claimed moment decay (without hidden assumptions on the Hölder regularity or boundary terms) is required before the subsequent estimates can be accepted."},{"response":"Section 4 carries out the passage from the combinatorial formula to the moment upper bounds. For H > 1/2 the graph is Hölder continuous of order H; this regularity is used only to control the size of increments when applying the nondeterminism estimates. Each remainder term arising after Faà di Bruno expansion is bounded by a direct Gaussian tail argument that is uniform in the range H ≥ 1/2. The same estimates that work for H = 1/2 extend immediately once the nondeterminism constant is adjusted; no additional vanishing arguments are required. We therefore see no gap that would invalidate the conclusion that the Fourier dimension is 1 almost surely.","revision_made":"no","referee_comment":"[Main proof section] Proof of the main theorem (the integration-by-parts step in the moment estimates): the passage from the new formula to the required upper bounds on the Fourier moments must be checked for H > 1/2; any gap in justifying the vanishing of remainder terms or the applicability of the estimates under the graph's regularity would prevent the conclusion that the Fourier dimension equals 1 a.s."}],"tokens_in":1410,"tokens_out":584,"duration_ms":37710,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that this paper shows the Fourier dimension of the graph of fractional Brownian motion is almost surely 1 when the Hurst index is at least 1/2. It also gets the same conclusion for symmetric alpha-stable processes with alpha in [1,2]. This extends the Fraser-Sahlsten result for ordinary Brownian motion and settles part of the 2014 conjecture by Fraser, Orponen and Sahlsten.","headline":"The paper proves the Fourier dimension of fBM graphs is a.s. 1 for H ≥ 1/2 by introducing a combinatorial integration by parts formula.","tokens_in":2308,"tokens_out":166,"would_cite":false,"duration_ms":56295,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean (J-uniqueness, Aczél classification)","rs_theorem":null,"paper_passage":"We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure... together with Faà di Bruno's formula and strong local nondeterminism"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/ArithmeticFromLogic.lean (LogicNat, orbit embedding)","rs_theorem":null,"paper_passage":"Proposition 3.4 (combinatorial IBP) and Proposition 6.9 (Faà di Bruno on exp(π g_a))"}],"headline":"Fourier dimension of fBM graphs via combinatorial IBP and Faà di Bruno; no RS structures","alignment":"orthogonal","rationale":"Paper's core is a new combinatorial integration-by-parts formula (Prop. 3.4) plus Faà di Bruno on exp(g) for moments of graph-measure Fourier transforms (Prop. 6.9, Thm. 5.2), combined with strong LND of fBM. This machinery lives entirely in harmonic analysis/probability and has zero overlap with RS cost functions, ratio symmetry, φ-ladders, 8-tick periodicity or J-cost forcing. No contradiction with any RS theorem is possible; the domains are disjoint.","tokens_in":68547,"confidence":"high","tokens_out":339,"duration_ms":12202,"cache_read_input_tokens":32896,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1.","keywords":["Fourier dimension","fractional Brownian motion","graph","almost surely","integration by parts","stable processes","Salem set"],"falsifier":"A single sample path of fractional Brownian motion with Hurst index strictly greater than 1/2 whose graph has Fourier dimension strictly less than 1 would falsify the claim.","tokens_in":2534,"feed_emoji":"","tokens_out":523,"duration_ms":37903,"temperature":0.7,"pith_summary":"The paper proves that the graph of fractional Brownian motion with Hurst index H greater than 1/2 has Fourier dimension equal to 1 almost surely. This extends the corresponding result for standard Brownian motion and confirms part of a conjecture from 2014. The argument relies on a new combinatorial integration by parts formula that evaluates moments of the Fourier transform of the graph measure, together with Faà di Bruno's formula and the strong local nondeterminism of the process. The same approach also yields Fourier dimension 1 for the graphs of symmetric alpha-stable processes when alpha lies in [1,2], with the additional conclusion that the graph is a Salem set when alpha equals 1.","feed_headline":"Fourier dimension of fBM graph equals 1 almost surely","feed_subtitle":"New combinatorial integration by parts formula extends the Brownian motion case to Hurst index above 1/2.","key_machinery":"The combinatorial integration by parts formula that computes the moments of the Fourier transform of the graph measure.","core_discovery":"We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Faà di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric alpha-stable process has Fourier dimension 1 almost surely when α ∈","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["fBM graph Fourier dim 1 almost surely","fBM graph dim 1 almost surely for H above 1/2","Fourier dim 1 almost surely for fBM graphs","Symmetric alpha stable graphs Fourier dim 1 almost surely"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The combinatorial integration by parts formula correctly computes the moments of the Fourier transform of the graph measure.","fun_headline_variants_meta":{"raw":{"variants":["fBM graph Fourier dim 1 almost surely","fBM graph dim 1 almost surely for H above 1/2","Fourier dim 1 almost surely for fBM graphs","Symmetric alpha stable graphs Fourier dim 1 almost surely"]},"model":"grok-4.3","cost_usd":0.009877,"raw_usage":{"total_tokens":4308,"prompt_tokens":661,"num_sources_used":0,"completion_tokens":66,"cost_in_usd_ticks":98765500,"prompt_tokens_details":{"text_tokens":661,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3581,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":661,"tokens_out":66,"duration_ms":80382,"temperature":1.0,"reasoning_tokens":3581,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T03:49:16.533347+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single sample path of fractional Brownian motion with Hurst index strictly greater than 1/2 whose graph has Fourier dimension strictly less than 1 would falsify the claim.","supporting_citations":[],"review_version":1}