{"id":"051f1abb-d0ae-4e2c-9dcc-33b4c9e0b980","arxiv_id":"2505.14168","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove the existence and local uniqueness of multi-peak solutions with prescribed L2 mass for the critical Brézis-Nirenberg problem in dimensions at least six.","lead":"This paper proves that a nonlinear wave equation with a fixed total mass can have many peak-shaped solutions, and that these peaks are locally unique. It matters because normalized solutions describe laser beams in hollow-core fibers and Bose-Einstein condensates, where the total number of particles is fixed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's uniqueness proof assumes every solution with (1.5) has the spike expansion (1.6)-(1.7) with a common limit μ_k; this classification is neither proved nor cited, and without it the ♯Q_{a_k} count is unsupported.","rationale":"The reader's flagged concern (Lemma 3.2's omitted proof and the unverified o(1/μbar^2) rate) is real and should be supplied before the uniqueness proof is accepted. However, I see a more structural gap in the statement and proof of Theorem 1.2: the proof begins by assuming that any two solutions satisfying (1.5) have the standard spike expansion with a common limiting μ_k, and it never proves the classification needed for the counting claim. Theorem 1.1 is existential, not a structure theorem, so invoking it to represent arbitrary solutions is not legitimate. A fixed-frequency structure theorem is available in [12] for solutions of (1.8) satisfying (1.10), but the paper neither states it nor proves the required λ_ρ→0 and transfer to the normalized setting. Without this, the 'Moreover' sentence is unsupported, and the uniqueness argument only applies to solutions already known to share the same limiting rates. This reinforces the CONDITIONAL verdict rather than changing it: the main ideas are plausible, but the paper must either add the classification proposition or explicitly restrict the counting claim. I therefore recommend no change to the reader's verdict, with the classification gap recorded as the primary missing block.","tokens_in":28072,"tokens_out":29818,"duration_ms":272459,"concrete_test":"To settle this, supply the missing classification proposition: for N≥6, let u_ρ solve (1.1) with ||u_ρ||^2 = ρ and |∇u_ρ|^2 ⇀ S^{N/2} Σ δ_{a_i}; prove, or explicitly import from [12] after establishing λ_ρ→0, that up to a subsequence u_ρ = Σ_j PU_{x_{j,ρ},μ_{j,ρ}} + w_ρ with x_{j,ρ}→a_j, ρ^{1/2} μ_{j,ρ}→c_j>0, and (c_1,...,c_k) satisfying ∇_μ Ψ_k(a_k,c)=0 and ∇_x Ψ_k(a_k,c)=0. If this classification cannot be established, the 'Moreover' sentence of Theorem 1.2 should be removed or made conditional on such a proposition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 opens by taking two solutions u_ρ^(1), u_ρ^(2) of (1.1) satisfying only (1.5), then writes each in the form u_ρ^(l)=Σ_j PU_{x_j^(l),μ_j^(l)} + w^(l) with a common limiting scale μ_j (no superscript l). This is not a consequence of Theorem 1.1, which constructs a solution from a prescribed nondegenerate critical point (a_k,μ_k) but does not classify all solutions whose gradient energy concentrates at a_1,...,a_k. The proof also needs λ_ρ→0 and a normalized analogue of the structure theorem of [12] to attach to each such solution a limit (a_k, μ_k) ∈ Q_{a_k}; otherwise two solutions with different points of Q_{a_k} would have limiting scales differing by a nonzero factor and equation (3.9) would fail. The final counting assertion (the number of solutions satisfying (1.5) equals ♯Q_{a_k}) is therefore unproved: the text proves at most uniqueness within the class of solutions sharing the same (a_k, μ_k) via (1.6)-(1.7), and never shows that every solution satisfying (1.5) belongs to such a class. If the uniqueness argument were read literally for all solutions satisfying (1.5), it would contradict the 'Moreover' statement when ♯Q_{a_k}>1.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:38:57.981730+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}