{"id":"7e1833c0-3341-4369-8453-65c4a9379d63","arxiv_id":"2607.21380","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For Leray's structure theorem, the eventual-regularity time is shown to be at most of order R log R for large Reynolds numbers R, improving Leray's high-power bound.","lead":"A new estimate is proposed for the time after which a weak solution of the 3D Navier–Stokes equations in a bounded domain becomes smooth, improving Leray's classical bound for large initial data. The new bound grows only logarithmically with the Reynolds number instead of as a high power.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4's contradiction requires unproved strict positivity of kinetic energy at the right endpoint; without it the uniform small-gradient time t_m and hence θ0 bound are not established.","rationale":"The paper's proof strategy is coherent and follows a known Leray-type construction, but the decisive step in Lemma 4 is not rigorous as written. The reader's weakest assumption identifies exactly the missing strict positivity of kinetic energy at the right endpoint of the contradiction interval. I agree with that assessment. The gap is not a stylistic quibble: the uniform small-gradient time t_m is used to feed the arctangent differential inequality (2.23), which yields the uniform gradient bound (2.24), which in turn produces the regularity on (θ0,∞) via the limit argument in Section 3. If t_m cannot be guaranteed, the claimed improvement of Leray's endpoint estimate is not established. The central qualitative claim may still be true—there is no obvious counterexample to the existence of such a time—so the appropriate verdict remains CONDITIONAL rather than REJECT. The secondary numerical inconsistencies (different formulas for θ0 and s across Theorem 1.2, Lemma 4, and Section 3) reinforce the need for revision but are not the main obstruction. The paper would need either a proof that ||v^m(t)||₂ stays strictly positive for all finite t, or a different argument producing t_m, before the theorem can be accepted.","tokens_in":10943,"tokens_out":15373,"duration_ms":157710,"concrete_test":"Test Lemma 4's contradiction by deriving a uniform-in-m lower bound for the kinetic energy of the Galerkin system (2.13)-(2.14): at truncation p, finite-dimensionality gives y_p(t) ≥ ||v_p^m(t0)||₂² exp(−2λ_p(t−t0)/R) > 0. Check whether this bound survives the limits p→∞ and m-fixed mollification with a constant independent of p and m. If no such uniform positive lower bound can be shown, the energy identity (2.11) does not rule out ||v^m(t_end)||₂² = 0, so the t_m claim in Lemma 4 is unsupported. If a uniform bound exists (e.g., via backward uniqueness of the Stokes semigroup), the contradiction can be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.2 rests on Lemma 4: for every approximating solution v^m one must find t_m in (t0, t0 + R||v(t0)||₂²/[2(1−η)]) with ||∇v^m(t_m)||₂² < 1−η. The proof is by contradiction. Assuming ||∇v^m||₂² ≥ 1−η on that interval, the energy identity (2.11) gives ||v^m(t_end)||₂² + (2/R)∫_{t0}^{t_end}||∇v^m||₂² = ||v^m(t0)||₂², while the assumption forces (2/R)∫ ≥ ||v(t0)||₂². This is not a contradiction: it only forces ||v^m(t_end)||₂² = 0. The text never proves strict positivity of the kinetic energy at the right endpoint. Estimate (2.12) is an upper bound, not a lower one, and the energy inequality permits downward jumps, so finite-time complete dissipation is not excluded. In Section 3 the displayed equality '||v^m(s)||₂² = (2/R)∫...' explicitly omits ||v^m(t_end)||₂², which is exactly the missing positive term. Without a valid t_m, the uniform arctangent bound (2.24) and the regularity interval (θ0,∞) collapse. The separate mismatches between Theorem 1.2 and Lemma 4 in the constants (missing R; 4/π vs 2/π in s) are secondary; the positivity gap is the load-bearing defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims an improvement of Leray's structure theorem for weak solutions of the 3D Navier–Stokes equations in a bounded domain. In dimensionless form, Theorem 1.2 asserts the existence of a final regularity interval (θ0, ∞) and gives the estimate θ0 ∈ (s, s + π/[4(1−η)(α+β)]) with s = 0 for small data and s = (R/2γ²) log[4/π (α+β)||v0||₂²] for large data. The proof uses the Leray approximating sequence, uniform energy estimates, and a differential inequality for the gradient to show that after a controlled waiting time the gradient is uniformly bounded; then a standard 'tower of intervals' argument establishes the structure theorem on (0, θ0). The claimed contribution is the logarithmic dependence on R for large data, improving Leray's polynomial estimate.","tokens_in":11340,"tokens_out":19961,"duration_ms":189574,"significance":"If correct, the result would be a genuine improvement: for large Reynolds numbers the endpoint s would grow only logarithmically in R, while the length of the waiting interval shrinks like 1/(α+β). The proof strategy is transparent, uses standard energy identities and differential inequalities, and the paper carefully writes out the approximating-sequence constructions. However, the central lemma has a load-bearing gap (the contradiction argument only forces zero kinetic energy at the endpoint, not a contradiction), and the endpoint formulas in Theorem 1.2, Lemma 4, and Section 3 are mutually inconsistent. These issues are serious but potentially repairable, so the paper is not acceptable in its present form.","major_comments":[{"comment":"The contradiction step is invalid. Assuming ||∇v^m(t)||² ≥ 1−η on (t0, t0+L), L = R||v(t0)||₂²/[2(1−η)], and using (2.11) with s=t0 gives ||v^m(t0+L)||₂² = ||v(t0)||₂² − (2/R)∫ ||∇v^m||² ≤ 0, hence equality and ||v^m(t0+L)||₂² = 0. This does not contradict (2.11); it merely forces complete dissipation at the right endpoint. The proof never establishes strict positivity of the kinetic energy for finite times — (2.12) is an upper bound, not a lower bound. Without a strict lower bound, the existence of t_m with ||∇v^m(t_m)||² < 1−η is not proved, and (2.24) and the regularity interval (θ0,∞) are not established. The same gap occurs in Section 3.","section":"Lemma 4, Eqs. (2.21)–(2.24)"},{"comment":"The stated endpoint formulas are mutually inconsistent and do not follow from the proof. Solving R(α+β)||v0||₂² e^{−2γ²s/R} = π/2, as in (3.27), gives s = (R/2γ²) log(2R(α+β)||v0||₂²/π); the R inside the logarithm is missing from (3.27) and from Theorem 1.2, and the factor is 4/π rather than 2/π. Also, Lemma 4 states θ0 ∈ (t0, t0 + π/[4R(1−η)(α+β)]), whereas the preceding inequality R||v(t0)||₂²/[2(1−η)] ≤ π/[4(1−η)(α+β)] has no R in the denominator. Theorem 1.2 uses the version without R, and Section 3's small-data case uses π/8. The correct numerical bound must be derived and stated coherently; as it stands, the claimed endpoint of Theorem 1.2 is not proved.","section":"Theorem 1.2, Lemma 4, Section 3 (3.27)"}],"minor_comments":[{"comment":"The interval θ0 ∈ (0, π/[8R(1−η)(α+β)]) should presumably be π/[4(1−η)(α+β)] to match Lemma 4 and the algebra; the R in the denominator also needs removal.","section":"Section 3, small-data case"},{"comment":"The formula for s is missing the factor R inside the logarithm; the factor 4/π in Theorem 1.2 is inconsistent with the 2/π in (3.27) and with the derivation from (2.12).","section":"Equation (3.27) and Theorem 1.2"},{"comment":"Typographical: the set should be R_+ (time), not R³_+; also 'R^3_+' appears to be a typo.","section":"Theorem 1.2"},{"comment":"The Hölder-exponent notation is garbled: exponents such as 'p/2' and 'pq/(2q−p)' appear without correct placement, making the estimate hard to check.","section":"Proof of Lemma 2, Eq. (2.7)"},{"comment":"The notation oscillates between v^m and v^m_p in the Galerkin proof; this should be cleaned up.","section":"Theorem 2.1, Eqs. (2.15)–(2.17)"}],"recommendation":"major_revision","confidential_remarks":"The structural part of the proof is largely taken from the author's prior work [6] (and Lemma 3 from [2]); the novelty rests on the endpoint estimate. The editor may wish to verify the extent of overlap with arXiv:2512.05598 and [2], and whether the new estimate is sufficiently distinguished to justify publication. The two major issues above are repairable in principle, but the positivity gap in Lemma 4 may require a nontrivial argument to close."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is on the right track and the quantitative claim is genuinely new for large data: it replaces Leray's polynomial endpoint with a logarithmic one, using exponential energy decay in a bounded domain. The comparison with [1] and Leray is honest, and the dimensionless setup keeps the scaling transparent. The differential-inequality machinery is standard and the arctangent bound is plausible if the main lemma holds.\n\nThe main soft spot is Lemma 4. The contradiction argument does not produce a contradiction. Assuming the gradient norm stays at least 1−η on the stated interval, the energy identity (2.11) gives ||v^m(t_end)||² + (2/R)∫||∇v^m||² = ||v(t0)||². The lower bound on the integral forces (2/R)∫ ≥ ||v(t0)||², hence ||v^m(t_end)||² ≤ 0, not an impossibility. You need either strict positivity of the kinetic energy at the right endpoint or a continuity argument to integrate slightly past that point and violate the energy identity strictly. The stress-test note is right. As written, the existence of t_m and therefore θ0 is not established. This is central, not cosmetic.\n\nThere are also secondary inconsistencies: Theorem 1.2, Lemma 4, and Section 3 do not agree on the constants. The interval endpoint has an extra R in Lemma 4, s is written with 2/π in (3.27) and 4/π in Theorem 1.2, and the small-data case appears with π/8 vs π/4. These are easy to fix, but they make the paper hard to read.\n\nWhat the paper does well: the setup is careful, the energy decay estimate (2.12) is correct, and the use of the approximating sequence and strong convergence is standard. The import from [6] and [2] is not circular; the new endpoint follows from decay, not from the conclusion.\n\nWho this is for: people working on quantitative partial regularity for Navier–Stokes. It does not touch the open regularity problem, but it improves a classical bound in a way that looks true once Lemma 4 is repaired.\n\nRecommendation: send it to referees. The central idea is probably right and the gap is likely fixable, but the current version should not be accepted until Lemma 4 is reworked and the constants are unified.","headline":"A plausible and useful logarithmic improvement of Leray's endpoint bound for large data, but the written proof has a real gap in Lemma 4 and the constants are inconsistent; still worth refereeing.","tokens_in":11770,"tokens_out":6879,"would_cite":false,"duration_ms":68574,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35B65","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that for weak solutions of the Navier–Stokes equations in a bounded domain, the left endpoint of the final interval of regularity can be placed at a time that, at large Reynolds numbers, grows only logarithmically with the","keywords":["Navier-Stokes equations","weak solutions","Leray structure theorem","partial regularity","regularity interval","Reynolds number","energy decay","bounded domain"],"falsifier":"Take a concrete bounded domain (for example a cube with periodic boundary conditions), compute the constants c, C_S, C_P that define α, β, γ, choose a smooth initial datum with large L² norm, and simulate the Leray approximating sequence—for instance a Galerkin truncation with a Friedrichs mollifier. Measure the first time t_m at which ||∇v^m(t)||² drops below 1−η; if for some m this time falls outside (s, s + π/4[(1−η)(α+β)]^{-1}), or if the subsequent bound ||∇v^m(t)||² ≤ tan(π/2−σ) is violated on (θ0,∞), then Lemma 4 and Theorem 1.2 are refuted.","tokens_in":10855,"feed_emoji":"🌀","tokens_out":10333,"duration_ms":90780,"temperature":0.7,"pith_summary":"The paper revisits Leray's structure theorem, which guarantees that every weak solution becomes smooth on an unbounded time interval after some waiting time bounded by a high power of the Reynolds number. It proposes a sharper bound on the left endpoint of that interval: for large initial data, the waiting time grows like R log R instead of a high power of R, and for small data it can be taken arbitrarily close to the initial instant. Since Leray's structure theorem is the only general long-time regularity statement for the three-dimensional Navier–Stokes equations, any improvement of the waiting time directly sharpens what is known about the behavior of weak solutions after a finite time. The proof is built for bounded domains and carries over to space-periodic domains without change.","feed_headline":"New bound cuts Navier-Stokes regularity wait to R log R at high R","feed_subtitle":"Leray's classical bound diverges like a power of the Reynolds number; the new estimate shrinks the waiting time before the final smooth inte","key_machinery":"The central mechanism is the pairing of the exponential kinetic-energy decay ||v^m(t)||² ≤ ||v0||² exp(−2γ²t/R), where γ is the reciprocal of the Poincaré constant, with a uniform small-gradient time found by a contradiction argument from the energy identity: some t_m near s must satisfy ||∇v^m(t_m)||² < 1−η. Inserting that small gradient into the differential inequality for the gradient and rewriting it through the arctangent identity yields a uniform bound ||∇v^m(t)||² ≤ tan(π/2−σ) on (θ0,∞). The arctangent transformation is the clock that converts one small-gradient instant into permanent boundedness of the gradient.","core_discovery":"Theorem 1.2 asserts that for any divergence-free initial datum in L² there exists a weak solution whose final regularity interval begins at θ0, with θ0 in (s, s + π/4[(1−η)(α+β)]^{-1}); here s = 0 when R(α+β)||v0||² ≤ π/2, and s = R/(2γ²) log[4/π (α+β)||v0||²] otherwise. Because α grows like R³ and β like R, the interval's length shrinks like R^{-3} while s grows like R log R, so at large Reynolds numbers the waiting time is dramatically shorter than Leray's bound, which diverges like a high power of R. The estimates are established for the Leray approximating sequence and transferred to the weak solution via the uniqueness of the weak limit.","pith_inferences":["The argument suggests that the waiting time for regularity is governed by the rate of kinetic-energy decay (through the Poincaré constant) and by the size of the constants α, β; if that is the mechanism, then any improvement of the decay estimate for weak solutions—for instance in the form of stronger decay in higher norms—would directly translate into a shorter waiting time.","The arctangent differential-inequality construction is a parameter-free clock that may transfer to other dissipative parabolic systems with an a priori energy decay and a Poincaré-type inequality; the paper's own pointer to fluid–rigid-body interaction indicates one such direction.","A natural testable extension is to optimize the free parameter η (and the choice of 1−η) to minimise the endpoint θ0; the paper does not perform this optimisation, and a numerical evaluation of the constants could show which terms dominate the bound.","If combined with the patched local regularity intervals on (0, θ0), the uniform gradient bound on (θ0,∞) could sharpen control of the singular set near the endpoint, for instance by improving the Hausdorff-measure estimate in the vicinity of the transition time."],"forward_implications":["If Theorem 1.2 is correct, the final interval of regularity for Leray–Hopf weak solutions in bounded domains starts at a waiting time that grows only logarithmically in the Reynolds number, making eventual smoothness a much earlier event than Leray's polynomial bound suggested.","In the small-data regime, the waiting time can be taken arbitrarily close to the initial time, with the length of the gap controlled by the explicit constant [(1−η)(α+β)]^{-1}.","The same endpoint estimate holds for space-periodic weak solutions on the three-torus, since the proof needs no modification in that setting.","In physical units the waiting time scales like L²/(νγ²) log R for large R, a diffusive time scale, rather than a high-power Reynolds factor; this exposes the energy-decay rate as the controlling quantity."],"fun_headline_variants":["Navier-Stokes: Leray wait shrinks from power law to R log R","New estimate shortens final regularity wait to R log R","Leray's bound tightened: no more power-law divergence","Quicker smoothness: Navier-Stokes regularity at R log R","Improved Leray theorem: wait drops to R log R at high R"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on the existence, uniformly in the approximation index, of a time t_m in a short interval where the gradient is smaller than 1−η; the contradiction argument that establishes this requires the remaining kinetic energy at the right endpoint of that interval to be strictly positive, a fact the paper does not state.","fun_headline_variants_meta":{"raw":{"variants":["Navier-Stokes: Leray wait shrinks from power law to R log R","New estimate shortens final regularity wait to R log R","Leray's bound tightened: no more power-law divergence","Quicker smoothness: Navier-Stokes regularity at R log R","Improved Leray theorem: wait drops to R log R at high R"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000773,"raw_usage":{"total_tokens":3196,"prompt_tokens":623,"completion_tokens":2573,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":367,"completion_tokens_details":{"reasoning_tokens":2479}},"tokens_in":367,"tokens_out":2573,"duration_ms":19020,"temperature":1.0,"reasoning_tokens":2479,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:36:40.749694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete bounded domain (for example a cube with periodic boundary conditions), compute the constants c, C_S, C_P that define α, β, γ, choose a smooth initial datum with large L² norm, and simulate the Leray approximating sequence—for instance a Galerkin truncation with a Friedrichs mollifier. Measure the first time t_m at which ||∇v^m(t)||² drops below 1−η; if for some m this time falls outside (s, s + π/4[(1−η)(α+β)]^{-1}), or if the subsequent bound ||∇v^m(t)||² ≤ tan(π/2−σ) is violated on (θ0,∞), then Lemma 4 and Theorem 1.2 are refuted.","supporting_citations":[],"review_version":1}