{"id":"6fe61661-fbc7-486e-b63b-ad27e6501d57","arxiv_id":"2412.02762","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit counterexamples show the Hölder exponent 2s/(1-β) is optimal for semilinear fractional Laplacian equations in one dimension when 2s≤1-β, with 2s+β optimal otherwise.","lead":"The authors construct explicit one-dimensional solutions to a fractional Laplacian equation whose smoothness is exactly what a recent regularity bound predicts, showing that bound cannot be improved. The examples reveal a combined effect of nonlocality and nonlinearity: when the nonlinearity is not very smooth, solutions can be rougher than either local or linear theories suggest.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The periodic optimality result is conditional on an unproved existence claim for a smooth 8-periodic interpolant satisfying properties (a)-(f) of Proposition 3.1; no such function is constructed or shown to exist.","rationale":"The reader correctly identified the unproved periodic interpolant as the weakest link. The nonperiodic construction in Section 2 is explicit and self-contained: Lemma 2.3 computes (-Δ)^s of u in (2.1), and Lemma 2.4 produces f. The periodic version, however, needs a global 8-periodic u with six prescribed properties, and Proposition 3.1 simply assumes it. Lemma 3.2's proof is a case analysis whose A_k^1 estimate explicitly relies on property (e); without an existence proof one cannot certify that property (e) is compatible with smoothness, monotonicity, and quadratic behavior near x=2. I see no evidence that the claim is false; the gap is likely repairable by a standard bump-function interpolation, which is why CONDITIONAL rather than REJECT is appropriate. I would not change the reader's verdict. A related but secondary defect is that the nonperiodic u in (4.1) is not globally C^{2s+β} when 2s+β>1 because of corners at x=±1; this does not affect the local equation but should be corrected in any revision.","tokens_in":19004,"tokens_out":23504,"duration_ms":240964,"concrete_test":"Fill the gap by constructing the missing interpolant explicitly: on (1,2), set u(x)=1+∫_1^x φ(t)dt with φ∈C^∞([1,2]) satisfying 0≤φ≤r, φ(1)=r, φ'(1)=r(r-1), φ vanishing to all orders near 2, and ∫_1^2 φ = u(2)-1; set u(x)=x^r on (0,1) and extend by the symmetries (d) and periodicity. Then verify properties (a)-(f) and rerun the proof of Lemma 3.2 with this explicit u. If the A_k^1 bound (3.3) or the H-Lipschitz estimate fails for every admissible φ, the periodic theorem requires a different construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.1 postulates an 8-periodic u that is ±|x|^r on (-1,1), smooth on (-4,0)∪(0,4), increasing on (0,2), odd about 0, even about 2, flat on (1,2) in the sense sup_{x∈(1,2)}u'(x)=u'(1)=inf_{x∈(0,1)}u'(x), and quadratic near 2. No proof of existence of such an interpolant is supplied. Lemma 3.2 uses the flatness property essentially in the A_k^1 estimate (3.3): the claim |u(x^{1/r}+y)-u(z^{1/r}+y)|≤|x-z| depends on the derivative bound (3.4), and the later f∈C^β argument uses the quadratic behavior (f). Thus the periodic proof is conditional on an existence assertion that could hide an incompatibility among the six properties. The nonperiodic construction is explicit and independent, so the nonperiodic sharpness claims are not affected; but Theorem 1.2 as stated, with its periodic conclusion, is not fully proved. The same issue appears in the outline of Theorem 1.2(ii), where the periodic u is said to coincide with (4.1) on [-1,1] while being smooth on R\\4Z, although (4.1) has a corner at x=±1 when 2s+β>1; this needs clarification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Hölder regularity of solutions to the one-dimensional semilinear fractional Laplacian equation (-Δ)^s u = f(u) with f ∈ C^β. It constructs examples showing that, for 2s ≤ 1-β, the regularity exponent 2s/(1-β) from the upper bound in Theorem 1.1 cannot be improved, and for 2s > 1-β that the exponent 2s+β is sharp. In the nonperiodic case the construction is explicit: for 2s < 1-β the authors define u(x) = sign(x)|x|^r, truncated outside [-1,1], with r = 2s/(1-β), and they compute (-Δ)^s u in closed form in Lemma 2.3, then define f by f(u(x)) = (-Δ)^s u(x) and prove f ∈ C^β. The case 2s = 1-β is handled separately in Lemma 2.5. For the periodic case, the paper announces an 8-periodic analogue in Proposition 3.1 by postulating a smooth interpolant satisfying six properties (a)-(f), and Lemma 3.2 is used to establish the needed Lipschitz property of H(t^{1/r}). For 2s > 1-β, the nonperiodic example is u(x) = sign(x)|x|^{2s+β} + x (truncated), and the periodic case is only outlined. The main theorem, Theorem 1.2, states both cases in the periodic setting.","tokens_in":19314,"tokens_out":4492,"duration_ms":47009,"significance":"If fully established, Theorem 1.2 would be a valuable contribution: it proves that the unusual exponent 2s/(1-β) discovered in Theorem 1.1 is not an artifact of the iteration argument but a genuine regularity threshold for semilinear nonlocal equations. The nonperiodic construction is explicit and transparent, and Lemma 2.3 gives a precise formula for the fractional Laplacian of the power-like function, which is a solid basis for the f ∈ C^β conclusion. The paper also includes a simple local example in Appendix A that motivates the nonlocal construction. The main weakness is that the periodic optimality claims depend on an unproved existence assertion for an interpolant satisfying properties (a)-(f) in Section 3, and the proof of Lemma 3.2 is sketchy at a load-bearing point. The nonperiodic results, however, are not affected by this gap.","major_comments":[{"comment":"The proof of Proposition 3.1 postulates the existence of an 8-periodic function u satisfying all six properties (a)-(f), including the flatness condition (e) and the quadratic behavior near x=2 in (f), but no construction or existence proof is given. Lemma 3.2 uses property (e) essentially in the A_k^1 estimate leading to (3.3), and the proof of Proposition 3.1 uses property (f) to prove that f is Lipschitz near u(2). Since every step of the periodic optimality argument depends on this interpolant, Theorem 1.2(i) is conditional on an unproved existence assertion that could hide an incompatibility among the six properties. Please supply an explicit interpolant or a rigorous existence proof.","section":"Section 3, Proposition 3.1(a)-(f)"},{"comment":"The proof that |u(x^{1/r}+y) - u(z^{1/r}+y)| ≤ |x-z| for y ∈ A_k^1 is the cornerstone of the Lipschitz estimate for H_+, but the argument is only sketched. In the first case the sentence 'using (3.4) ... one easily obtains (3.3)' omits the verification when the points x^{1/r}+y and z^{1/r}+y lie in different monotonicity regions, and in the second case the reflection argument is described geometrically without a complete algebraic check. Since Lemma 3.2 is needed for the periodic construction, please provide a fully detailed proof of (3.3).","section":"Lemma 3.2, A_k^1 estimate and Eq. (3.3)"},{"comment":"The periodic u for the case 2s > 1-β is said to coincide with (4.1) on [-1,1] and to be smooth on R \\ 4Z, but (4.1) has a corner at x = ±1 when 2s+β > 1: the derivative from the left at x=1 is 2s+β+1, while the derivative from the right is 0. This contradicts smoothness at x=1, which is an interior point of (0,4). The construction therefore needs modification near x=±1, and the required interpolation properties (with a proof of existence) should be stated explicitly, as in the previous comment.","section":"Section 4, outline of Theorem 1.2(ii)"}],"minor_comments":[{"comment":"There are several typographical issues in the TeX source, such as 'H¨older' and 'flatter', which should be corrected in the final version.","section":"Throughout"},{"comment":"The proof defines f only on [0,1/2] and then concludes the equation for all x ∈ [-1/2,1/2] using that u is even; this is correct because u([-1/2,0]) ⊂ [0,1/2], but the extension of f to the full real line by zero for negative arguments should be stated explicitly.","section":"Lemma 2.5"},{"comment":"The sentence 'using the smoothness of g and a Taylor expansion we get g'(x) = O(|x-2|)' should specify the neighborhood and the constant, since property (f) is used only locally near x=2.","section":"Proof of Proposition 3.1"},{"comment":"The claim that the function t ↦ 1 - t^β - ((y^{1+β}+1) - t(x^{1+β}+1))^β is negative 'due to convexity' would benefit from a one-line justification, as it is not immediate.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The nonperiodic results appear sound and are explicitly proved. The periodic results in Theorem 1.2 are the main advertised contribution, and they currently rest on an unproved existence claim for a smooth interpolant with six specific properties. This is a load-bearing gap that can likely be fixed by a direct construction, so I recommend major revision rather than rejection. There is no indication of circularity or inappropriate citation practice; the use of [2] as context is standard."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: the paper proves that the Hölder exponent 2s/(1-β) from the authors' companion paper [2] cannot be improved, by explicit examples. The nonperiodic construction is rigorous and does the job. The periodic construction in Theorem 1.2, as written, is conditional on an existence claim that is not proved.\n\nWhat is actually new: the sharpness examples. For 2s < 1-β, they take u(x) = |x|^r near the origin with r = 2s/(1-β), extend by constants outside [-1,1], and show (-Δ)^s u = f(u) with f ∈ C^β. The core work is Lemma 2.3, which computes the fractional Laplacian of such a function and decomposes it into a C^1 part and a tail H whose composition with x^{1/r} is Lipschitz. That lemma is the substantive contribution. The case 2s = 1-β is handled by an explicit computation, and the regime 2s > 1-β with exponent 2s+β is also done explicitly in the nonperiodic setting. These parts are solid.\n\nThe soft spot: Section 3. Proposition 3.1 postulates an 8-periodic function u satisfying six properties (a)-(f) — matching |x|^r on (-1,1), smooth away from integers, increasing on (0,2), odd/even symmetric, flat on (1,2) in the sense that the maximum slope equals u'(1), and quadratic near 2. No such function is constructed or shown to exist. The proof of Lemma 3.2, which is the periodic analogue of the Lipschitz tail estimate, uses all of these properties, in particular the flatness condition to control the A_k^1 estimate. So the periodic optimality claim holds only if such an interpolant exists. The same issue appears in the outline for Theorem 1.2(ii), where an 8-periodic function is said to coincide with (4.1) on [-1,1] and be smooth on R\\4Z; since (4.1) has a corner at x=1 when 2s+β>1, the gluing must be specified. None of this is provided.\n\nI don't think the gap is fatal to the main idea. Constructing such an interpolant is likely a routine bump-function exercise, and the authors could fill it in with a paragraph or an appendix. But as it stands, Theorem 1.2 is not fully proved; only the nonperiodic part is.\n\nThe citation practice is fine: the upper bound from [2] is cited for context, not used to build the examples. No circularity.\n\nWho gets value: anyone working on regularity for nonlocal semilinear equations. The nonperiodic examples are a useful reference even if the periodic theorem is patched. This deserves a serious referee, with the request to supply the missing periodic construction.\n\nRecommendation: send to peer review, expect a minor-to-moderate revision.","headline":"Sharpness of the Hölder exponent 2s/(1-β) is proved with explicit nonperiodic examples, but the periodic version rests on an unproved interpolation.","tokens_in":19858,"tokens_out":7156,"would_cite":true,"duration_ms":65704,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J61","35B65","35S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Explicit examples show that for $(-\\Delta)^s u=f(u)$ with $f\\in C^\\beta$, the Hölder exponent $2s/(1-\\beta)$ is optimal when $2s\\le 1-\\beta$, and $2s+\\beta$ is optimal otherwise.","keywords":["fractional Laplacian","Hölder regularity","semilinear equations","optimal regularity","periodic solutions","one-dimensional","nonlocal operators","C^beta nonlinearity"],"falsifier":"Check whether the six properties (a)-(f) in Section 3 are consistent: if one can prove that no smooth 8-periodic function can simultaneously satisfy all of them, the periodic construction fails; conversely, producing any such interpolant and verifying the Lipschitz estimate (3.1) numerically would test the proof directly.","tokens_in":18759,"feed_emoji":"","tokens_out":11230,"duration_ms":99997,"temperature":0.7,"pith_summary":"This paper constructs explicit one-dimensional solutions of the fractional semilinear equation $(-\\Delta)^s u=f(u)$, with $f$ Hölder continuous of order $\\beta$, that have exactly the worst possible Hölder regularity allowed by the upper-bound theorem from the authors' earlier work. In the regime $2s\\le 1-\\beta$ the constructed $u$ lies in $C^{2s/(1-\\beta)}$ but in no better Hölder class on any interval; in the complementary regime $2s>1-\\beta$ the same non-improvement holds with exponent $2s+\\beta$. These examples show that the exponent $2s/(1-\\beta)$ is a real combined effect of nonlocality and semilinearity, not an artifact of the iteration argument that produced the upper bound. A reader should care because the result pins down the exact boundary of what regularity theory can guarantee for this class of equations and shows that the classical linear expectation $2s+\\beta$ fails precisely when $2s\\le 1-\\beta$.","feed_headline":"New Hölder exponent for fractional equations is optimal","feed_subtitle":"When 2s≤1−β, solutions can be no smoother than C^{2s/(1−β)}; the classical bound 2s+β is optimal otherwise.","key_machinery":"The engine is a corner-shaped model function. In the nonperiodic case one takes $u(x)=\\operatorname{sign}(x)\\,|x|^r$ on $[-1,1]$ with $r=2s/(1-\\beta)$, extended to $\\pm1$ outside. Lemma 2.3 computes $(-\\Delta)^s u$ for such functions as $c_1x^{r-2s}+c_2x^r+c_sG(x)-c_sH(x)$, proves $G\\in C^1$, and shows that $x\\mapsto H(x^{1/r})$ is Lipschitz; this is exactly what is needed to define $f(t)$ from $t=x^r$ and verify $f\\in C^\\beta$ with $\\beta=(r-2s)/r$. For the periodic version the same lemma is applied to an 8-periodic interpolant satisfying six structural properties (symmetry, monotonicity, flatness, and quadratic behaviour near the top). In the complementary regime $2s>1-\\beta$ the model is $u(x)=\\operatorname{sign}(x)|x|^{2s+\\beta}+x$, whose inverse is Lipschitz, which makes the composition $f\\circ u$ amenable to the same argument.","core_discovery":"The central result, Theorem 1.2, is a pair of optimality examples. For every $0<s<1$, $0<\\beta<1$, and every period $2L$, there is a $2L$-periodic function $u$ and a function $f\\in C^\\beta(\\mathbb{R})$ such that $(-\\Delta)^s u=f(u)$ holds on all of $\\mathbb{R}$. In the regime $2s\\le 1-\\beta$, $u\\in C^{2s/(1-\\beta)}(\\mathbb{R})$ yet $u\\notin C^{2s/(1-\\beta)+\\epsilon}([-\\rho,\\rho])$ for any $\\epsilon,\\rho>0$; in the complementary regime $2s>1-\\beta$, the same non-improvement holds with the classical exponent $2s+\\beta$. Together with the upper bounds of Theorem 1.1, this shows the new exponent $2s/(1-\\beta)$ is sharp and that the dichotomy between the two regimes is not a proof artefact. The boundary case $2s=1-\\beta$ is included and gives optimal Lipschitz ($C^1$) regularity.","pith_inferences":["The same corner construction likely extends to radially symmetric solutions in higher dimensions, where the fractional Laplacian of a radial profile has a similar one-dimensional kernel near the origin; the exponent $2s/(1-\\beta)$ would then control the radial Hölder regularity at the origin.","A natural stress test is $\\beta=0$, i.e. continuous nonlinearities: the formula predicts $C^{2s}$ as the limiting regularity, but the present proof requires $\\beta>0$ for the composition law, so whether the phenomenon survives for merely continuous $f$ is left open.","The explicit nonperiodic example could serve as a numerical benchmark: discretizing $(-\\Delta)^s$ on the corner function with fixed $s,\\beta$ should reproduce the predicted lack of $C^{r+\\epsilon}$ regularity at zero, giving a cheap computational check of the theory."],"forward_implications":["The regularity estimate $C^{2s/(1-\\beta)-\\epsilon}$ from Theorem 1.1(i) is optimal: no general theorem can replace the exponent by anything larger when $2s\\le 1-\\beta$.","In the complementary case $2s>1-\\beta$, the classical exponent $2s+\\beta$ is also optimal, so the full dichotomy in Theorem 1.1 is sharp.","At the boundary $2s=1-\\beta$, the sharp regularity is $C^1$ (Lipschitz), the same value the linear theory would predict.","Because the examples are periodic, they show optimality for the whole-line, periodic, and Dirichlet-type settings covered by the earlier theorem."],"supporting_citations":[{"why":"Provides the sharp $C^{2s+\\beta}$ regularity in the linear fractional case, the baseline against which the new exponent is compared.","marker":"[1]"},{"why":"States the upper-bound theorem (Theorem 1.1) whose exponents the present examples prove sharp, and introduced the exponent $2s/(1-\\beta)$.","marker":"[2]"},{"why":"Records the open question on $C^{2+\\beta}$ regularity for local equations with continuous nonlinearity, motivating the study of optimal regularity.","marker":"[3]"}],"fun_headline_variants":["Optimal Hölder exponent for fractional semilinear equations","Semilinear fractional Laplacian: smoothness bound is sharp","Fractional PDEs: new regularity ceiling found","Nonlocal semilinear problem: optimal C^α exponent","Sharper regularity limit for fractional Laplacian models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the unproved existence, in the periodic construction, of a smooth 8-periodic interpolant that is $|x|^r$ on $(-1,1)$, odd about 0, even about 2, increasing on (0,2), flatter on (1,2) than on (0,1), and quadratic near $x=2$; if no such function exists, the periodic optimality example in Theorem 1.2(i) is not established, although the nonperiodic example is explicit and does not rely on it.","fun_headline_variants_meta":{"raw":{"variants":["Optimal Hölder exponent for fractional semilinear equations","Semilinear fractional Laplacian: smoothness bound is sharp","Fractional PDEs: new regularity ceiling found","Nonlocal semilinear problem: optimal C^α exponent","Sharper regularity limit for fractional Laplacian models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1577,"prompt_tokens":921,"completion_tokens":656,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":578}},"tokens_in":537,"tokens_out":656,"duration_ms":7210,"temperature":1.0,"reasoning_tokens":578,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:08:10.624838+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the six properties (a)-(f) in Section 3 are consistent: if one can prove that no smooth 8-periodic function can simultaneously satisfy all of them, the periodic construction fails; conversely, producing any such interpolant and verifying the Lipschitz estimate (3.1) numerically would test the proof directly.","supporting_citations":[{"cited_title":"Bass, Regularity results for stable-like operators , J","cited_arxiv_id":null,"evidence_quote":"Provides the sharp $C^{2s+\\beta}$ regularity in the linear fractional case, the baseline against which the new exponent is compared."},{"cited_title":"Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity","cited_arxiv_id":"2404.06462","evidence_quote":"States the upper-bound theorem (Theorem 1.1) whose exponents the present examples prove sharp, and introduced the exponent $2s/(1-\\beta)$."},{"cited_title":"Shahgholian, Regularity issues for semilinear PDE-s (a narrative approa ch), Algebra i Analiz 27 (2015), no","cited_arxiv_id":null,"evidence_quote":"Records the open question on $C^{2+\\beta}$ regularity for local equations with continuous nonlinearity, motivating the study of optimal regularity."}],"review_version":1}