{"id":"152cd056-0166-48ce-80e7-375bd7d7defa","arxiv_id":"2608.00802","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"For every n≥3, the authors construct a nonzero real smooth solution u of -Δu+V u=0 with bounded real V and |u(x)|≤C exp(-c|x|^{4/3}), giving a counterexample to the Landis conjecture in that setting.","lead":"The paper constructs, in every dimension three and higher, a nonzero smooth solution of the stationary Schrödinger equation with a bounded real potential that decays like exp(-c|x|^{4/3}). This disproves the Landis conjecture for real-valued potentials in those dimensions, a long-open question in mathematical analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader accepted the paper with moderate confidence, identifying Lemma 11 as the weakest assumption. My pass agrees that the tangential-frequency separation is the pivotal point where n≥3 enters and where a defect would break the construction: without uniform separation, local superpositions of cosines could cancel identically, as in the n=2 example. However, I found no actual flaw. Lemma 11's greedy coloring is justified by the positive-dimensional tangent sphere and a uniform cap-measure bound, and Lemma 20 transfers the separation to the rescaled frequencies with a controlled error. The remaining chain from the sparse sublevel estimate through the Dirichlet bounds, harmonic replacement, and variable smoothing is internally coherent; all identities I re-derived match the text, and the parameter ordering avoids circularity. The main residual risk is the sheer length of the proof and the absence of machine verification, which reasonably justifies moderate rather than high confidence but does not amount to a specific mathematical objection. The verdict should remain unchanged.","tokens_in":22365,"tokens_out":41078,"duration_ms":349941,"concrete_test":"Independently verify the cap measure bound in Lemma 10 for m=n−2 (worst case n=3) and confirm that 2D_n C_{n−2} δ^{n−2}<1 admits a positive δ_n; if this check fails, Lemma 11 collapses and with it the uniform sublevel estimate of Proposition 21.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the full proof of Theorem 1, giving special scrutiny to the point the reader flags: Lemma 11 and the sparse sublevel estimate that depends on it. Lemma 11 is sound: for n≥3, T_j is an (n−2)-sphere, Lemma 10 bounds the forbidden caps with a constant independent of the vertex, and the greedy coloring succeeds because 2D_n C_{n−2} δ^{n−2}<1 for a positive δ_n. The scaled frequencies in Lemma 20 inherit separation via |r_j−1|≤C/L^2, and Lemma 17 excludes identical cancellation, so the uniform sublevel bound of Lemma 19 and Proposition 21 follow. I also re-checked the chain rule in Proposition 16, the weighted Poincaré inequality (Lemma 23), the pointwise Dirichlet bound (Lemma 27), the harmonic replacement (Proposition 28), the variable-kernel smoothing (Lemmas 34–36), and the gluing of V in Step 2 of Theorem 1. The parameter order is consistent: θ and ρ are fixed first, then τ and L0, then L, and only then the atlas, q, Ω, w, and F. No circular step, omitted identity, or post-hoc fitting appears. The construction is long and not machine-checked, but internally coherent; I find no load-bearing concern.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every n≥3, a nonzero smooth real-valued function u on R^n and a bounded smooth real potential V satisfying -Δu+Vu=0, with |u(x)|≤C_n exp(-c_n|x|^{4/3}). The construction builds an approximate solution f=Eq from a radial envelope E and a patched superposition of tangential plane waves whose frequencies are separated on overlapping patches; this separated-coloring step is where n≥3 enters. The set {|q|<τ} is shown to be sparse on the microscopic scale, a harmonic-repair step replaces f there, and a variable-radius radial smoothing yields a smooth F satisfying |ΔF|≤C|F| and |F|≤CE. The potential is then defined as V=ΔF/F, glued with V=0 on the region where F is harmonic. The proof is self-contained, the parameter ordering is explicit, and the main theorem gives a negative answer to the Landis conjecture for real-valued potentials in every dimension n≥3.","tokens_in":22515,"tokens_out":36637,"duration_ms":295186,"significance":"If correct, the theorem settles the long-standing real-valued Landis problem in all dimensions n≥3 and shows that the dimensional restriction is genuinely geometric, consistent with the known two-dimensional positive results. The paper is a substantial technical achievement: the proof is detailed and well structured, with the key mechanisms isolated as sparse sublevel estimates, harmonic repair, and variable-kernel smoothing. I found no circularity or post-hoc fitting; the 'safe quotient' gluing of V is legitimate. The potential weak point identified by the reader, Lemma 11 (the tangential-frequency coloring), withstands scrutiny: the forbidden-cap estimate and greedy coloring are sound, and the separation is inherited by the frozen local frequencies in Lemma 20. The paper is appropriate for a leading analysis journal.","major_comments":[],"minor_comments":[{"comment":"In the outline, the phrase 'Φ(r)∼3/4r^{4/3} for larger' appears to be a typo for 'for large r'; please correct it.","section":"Outline"},{"comment":"In the proof of Lemma 27, just after the display 'Lemma 26 yields the bounds', the constant is written 'Cloc_n' in the first integral with the subscript 'loc' dropped; the notation should be made consistent.","section":"Lemma 27"},{"comment":"The phrase 'a real-analytic function that is not identically zero on any connected open set' is imprecise; the zero-set theorem should be quoted for a function that is not identically zero on a nonempty open set, or on a connected component of its domain.","section":"Lemma 19"},{"comment":"In the proof of Lemma 26, the subharmonicity of H in B(x,R) is used without explicitly noting that the term g1_{D\\setminus B(x,R)} vanishes on B(x,R); a one-sentence justification would improve readability, although the argument is correct.","section":"Lemma 26"},{"comment":"Given the number of parameters (L, ρ, θ, τ, σ) and the order in which they are fixed, a short summary table of dependencies would help the reader; the current ordering is consistent, but it is hard to track.","section":"Sections 2-10"}],"recommendation":"accept","confidential_remarks":"No confidential concerns. I found no citation or novelty issues, and the manuscript is squarely within the scope of a mathematical analysis/PDE journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the one-line: this is a real result, not a routine extension. It constructs, for every n≥3, a nonzero smooth real u and bounded real V satisfying -Δu+Vu=0 with decay exp(-c|x|^{4/3}). That falsifies the real-valued Landis conjecture in all dimensions n≥3, a gap that has been open since Meshkov's complex example. The proof is long but the architecture is clear: a radial envelope E=exp(-Φ), an oscillating factor q built from cosines on a variable-scale atlas, a harmonic repair on the set where q is small, and a final smoothing. I went through the main steps: Proposition 16 gives the approximate equation with the crucial tangential cancellation; Proposition 21 gives the sparse small-value set; Propositions 28, 31 and 37 handle the repair, the harmonic/nonvanishing alternatives, and the global envelope; the gluing of V in Theorem 1 works. I found no circular step: u is constructed first, V=Δu/u afterwards. The debt to Meshkov and to Filonov–Krymskii is explicit and honest.\n\nThe genuinely new ingredient is Lemma 11, the tangential-frequency coloring. It uses n≥3 to have a positive-dimensional tangent sphere, so a greedy coloring can separate frequencies on overlapping patches. This is also where n=2 fails—only one frequency pair modulo sign exists, so local terms can cancel identically, which matches the known positive 2D result. The dimensional threshold looks correct, not accidental.\n\nSoft spots? The paper is long and not machine-checked; with this many estimates there is room for a subtle constant or index error. I checked the usual trouble spots—the uniform sublevel estimate, the weighted Poincaré inequality, the pointwise Dirichlet bound, the kernel derivative bounds—and they hold together. The appendix's scale-balance derivation of 4/3 is heuristic, but the proof does not rely on it; the parameters are fixed before the construction, not fit to the target. So the risk is a local typo, not a load-bearing flaw. The AI-tools acknowledgment is transparent, and the citation pattern is appropriate.\n\nWho should read it: anyone in elliptic PDE or unique continuation. It deserves a serious referee. My recommendation: send it to peer review. If I were refereeing, I would ask for a careful check of Sections 4–7, but I would not desk-reject.","headline":"A long, careful proof that the real-valued Landis conjecture fails in every dimension n≥3, with the n≥3 threshold genuinely explained by the tangent-sphere coloring.","tokens_in":23146,"tokens_out":2146,"would_cite":true,"duration_ms":20493,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","35B40","35B60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Landis conjecture fails for real-valued potentials in every dimension n≥3, by constructing nonzero solutions with decay at the borderline rate exp(-c|x|^{4/3}).","keywords":["Landis conjecture","unique continuation at infinity","Schrödinger equation","real-valued potential","exponential decay","harmonic replacement","tangential frequencies","dimension threshold"],"falsifier":"Exhibit a nonzero real $C^\\infty$ solution in $\\mathbb{R}^2$ of $-\\Delta u+Vu=0$ with bounded real $V$ and decay $\\exp(-c|x|^{4/3})$; this would contradict the two-dimensional positive Landis-type theorem. Short of that, compute the local model for two overlapping patches in $n=2$ with equal weights and opposite tangent directions: $P(z)=\\frac12\\cos(\\xi\\cdot z)+\\frac12\\cos(\\xi\\cdot z+\\pi)=0$ identically, which verifies that the separation condition (37) and the sublevel estimate of Proposition 21 fail exactly where the paper's dimensional restriction enters.","tokens_in":22075,"feed_emoji":"📉","tokens_out":11909,"duration_ms":103144,"temperature":0.7,"pith_summary":"The paper proves that the Landis conjecture is false for real-valued potentials in every dimension $n\\ge 3$. It constructs a nonzero real smooth function $u$ on $\\mathbb{R}^n$ and a bounded real potential $V$ such that $-\\Delta u+Vu=0$ and $|u(x)|\\le C_n\\exp(-c_n|x|^{4/3})$. A sympathetic reader would care because this closes a long-standing gap: earlier counterexamples with this decay were complex-valued in two dimensions (and by products, in even dimensions), while a real-valued analogue was known only on a cylinder. The proof's architecture—a radial envelope modulated by tangent plane waves, with harmonic repair in the sparse region where the wave sum is small—locates the obstruction to dimension two in a geometric frequency-separation lemma.","feed_headline":"Landis conjecture fails for real potentials in every dimension ≥3","feed_subtitle":"Separated tangential frequencies on overlapping patches make this possible only for dimensions n≥3.","key_machinery":"The load-bearing object is the approximate solution $f=Eq$, where $E=\\exp(-\\Phi)$ with $\\Phi(r)\\simeq \\frac{3}{4}r^{4/3}$ and $q(x)=\\sum_j \\chi_j(x)\\cos(A_j t_j\\cdot(x-x_j))$ is a patched superposition of real plane waves. The frequencies $A_j t_j$ are frozen on macroscopic balls of radius $\\kappa_2 A_j$; each $t_j$ is tangent to the sphere through $x_j$, and Lemma 11 assigns these tangent directions to overlapping patches with uniform separation modulo sign, a task possible only when the tangent sphere $S^{n-2}$ has positive dimension, i.e. $n\\ge 3$. Two cancellations make the approximate equation work: $\\Delta q+A^2q$ is uniformly bounded (radial cancellation) and $(x/A^2)\\cdot\\nabla q$ is uniformly bounded (tangential cancellation). Balancing the errors produced by the partition of unity and by freezing a tangent vector across a curved patch forces $a(r)\\simeq r^{1/3}$, hence decay power $4/3$. The repair mechanism is the safe-quotient step: where $|q|<\\tau$, the paper solves $-\\Delta w=\\Delta f$ in that sparse set and obtains $|w|\\le (\\tau/20)E L^2/A^2$; a radial mollification then makes $F$ smooth while preserving harmonicity in the small-value region.","core_discovery":"The central claim, Theorem 1, is that for every $n\\ge 3$ there exists a nonzero real function $u\\in C^\\infty(\\mathbb{R}^n)$, a real function $V\\in C^\\infty(\\mathbb{R}^n)\\cap L^\\infty(\\mathbb{R}^n)$, and positive constants $C_n,c_n$ such that $-\\Delta u+Vu=0$ and $|u(x)|\\le C_n\\exp(-c_n|x|^{4/3})$. The solution is built explicitly: the approximate solution is $f=Eq$, with radial envelope $E=e^{-\\Phi}$ and an oscillating factor $q$ formed by patching real plane waves $\\cos(A_j t_j\\cdot(x-x_j))$ on a covering whose ball radii are proportional to $A(x)\\simeq |x|^{1/3}$. The paper proves $|\\Delta f|\\le C E(1+L^6 A^{-4}|q|)$, then solves a Dirichlet problem on the sparse set $\\{|q|<\\tau\\}$ to replace $f$ by a function $f^*$ that is harmonic there, and finally smooths on the microscopic scale $A^{-1}$ to obtain $F$ with $|\\Delta F|\\le C|F|$ and $|F|\\le C E$. Setting $V=\\Delta F/F$ where $F\\neq 0$ and $V=0$ where $F$ is harmonic gives the bounded real potential. The key alternative is that $F$ is exactly harmonic near every small value of $q$ and bounded below by $c\\tau E$ away from that region, so the ratio $\\Delta F/F$ is uniformly bounded.","pith_inferences":["The appendix's scale balance suggests the same envelope-plus-tangential-wave architecture should realize every subcritical decay exponent $p<4/3$, not just the endpoint; the paper proves only $p=4/3$.","The harmonic-repair step is modular: any elliptic equation admitting an approximate solution with a sparse small-value set and a pointwise correction estimate could use the same Dirichlet-replacement-then-mollify recipe; the Schrödinger operator is one instance.","One quantitative question left implicit is how the constants $C_n,c_n$ and the sup-norm of $V$ grow with $n$; the geometric separation lemma works for all $n\\ge 3$ but gives no explicit dimensional dependence.","A direct test of the dimensional threshold is to attempt the construction in $n=2$: the model superposition $\\frac12\\cos(\\xi\\cdot z)+\\frac12\\cos(\\xi\\cdot z+\\pi)$ vanishes identically, so the sparse-small-value proposition cannot hold."],"forward_implications":["In every dimension $n\\ge 3$, real-valued bounded potentials admit nonzero solutions with decay $\\exp(-c|x|^{4/3})$, so the Landis conjecture in its real form is false there.","The exponent $4/3$ is the optimal borderline: a previously known theorem still forces $u=0$ if the solution decays like $\\exp(-\\tau |x|^{4/3})$ for arbitrarily large $\\tau$, so the new example achieves the borderline rate rather than exceeding it.","The two-dimensional positive result (decay $\\exp(-c|x|(\\ln|x|)^{1/2})$ forces zero) is consistent: the new construction is blocked in $n=2$ by the absence of separated tangent directions on the zero-dimensional tangent sphere.","Because the constructed $u$ is real and nonzero, it is also a complex-valued solution; this settles the Landis question in odd dimensions $n\\ge 3$ as well as even dimensions."],"supporting_citations":[{"why":"Supplies the complex-valued two-dimensional counterexample with the same $4/3$ decay, the optimality theorem for that exponent, and the structural template (safe quotient, annular scale balance) that the present construction parallels.","marker":"[11]"},{"why":"Introduces the $f\\to f^*\\to F$ harmonic-repair scheme on a cylinder that the present proof adapts to genuinely radial decay in $\\mathbb{R}^n$.","marker":"[5]"},{"why":"Gives the two-dimensional positive Landis-type result that the new construction does not contradict, marking the dimensional threshold.","marker":"[10]"},{"why":"Records Landis's Problems 5 and 6, the conjecture this paper disproves in dimensions three and higher.","marker":"[12]"}],"fun_headline_variants":["Landis conjecture shattered in 3D and higher with explicit counterexamples","Landis conjecture disproved in every dimension ≥3 by explicit solutions","Explicit decaying solutions kill Landis conjecture for all n≥3","Landis conjecture fails for all dimensions ≥3 with bounded real potentials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on Lemma 11, which colors the overlap graph with tangent directions separated modulo sign; if such a coloring did not exist, local superpositions of cosines could cancel identically—as they do in dimension two—and the uniform sublevel estimate that produces the sparse small-value region would fail.","fun_headline_variants_meta":{"raw":{"variants":["Landis conjecture shattered in 3D and higher with explicit counterexamples","Landis conjecture disproved in every dimension ≥3 by explicit solutions","Explicit decaying solutions kill Landis conjecture for all n≥3","Landis conjecture fails for all dimensions ≥3 with bounded real potentials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000662,"raw_usage":{"total_tokens":3009,"prompt_tokens":913,"completion_tokens":2096,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":2020}},"tokens_in":529,"tokens_out":2096,"duration_ms":12973,"temperature":1.0,"reasoning_tokens":2020,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:16:37.812598+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a nonzero real $C^\\infty$ solution in $\\mathbb{R}^2$ of $-\\Delta u+Vu=0$ with bounded real $V$ and decay $\\exp(-c|x|^{4/3})$; this would contradict the two-dimensional positive Landis-type theorem. Short of that, compute the local model for two overlapping patches in $n=2$ with equal weights and opposite tangent directions: $P(z)=\\frac12\\cos(\\xi\\cdot z)+\\frac12\\cos(\\xi\\cdot z+\\pi)=0$ identically, which verifies that the separation condition (37) and the sublevel estimate of Proposition 21 fail exactly where the paper's dimensional restriction enters.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the complex-valued two-dimensional counterexample with the same $4/3$ decay, the optimality theorem for that exponent, and the structural template (safe quotient, annular scale balance) that the present construction parallels."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the $f\\to f^*\\to F$ harmonic-repair scheme on a cylinder that the present proof adapts to genuinely radial decay in $\\mathbb{R}^n$."},{"cited_title":"Logunov, E","cited_arxiv_id":null,"evidence_quote":"Gives the two-dimensional positive Landis-type result that the new construction does not contradict, marking the dimensional threshold."},{"cited_title":"Differential Equations and their Applications","cited_arxiv_id":null,"evidence_quote":"Records Landis's Problems 5 and 6, the conjecture this paper disproves in dimensions three and higher."}],"review_version":2}