{"id":"ff2cb021-c5b0-4535-a070-38925381ebba","arxiv_id":"2607.04917","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For minimal immersions ϕ(y,z,t)=(f(t)y,f₂(t)z,f₁(t)), the functions ω(t)y_i z_j with ω=f^{-k}f₂^{-ℓ} are eigenfunctions of the stability operator for eigenvalue −n, implying index ≥ kℓ+3k+3ℓ+8.","lead":"The paper finds new explicit eigenfunctions of the stability operator for a family of minimal hypersurfaces in spheres, all sharing the eigenvalue −n. This raises the known lower bound on their stability index and generalizes earlier results from the equal-dimension case.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the only non-trivial hypothesis (the minimality ODE) and notes that the algebra is elementary once that ODE is inserted. Because the paper writes every intermediate identity explicitly, the verification is self-contained and does not rely on unstated estimates or asymptotic regimes. The comparison argument that converts the unexpected eigenvalue of S_{22} into the numerical lower bound kℓ+3k+3ℓ+8 is the standard one already used for the Clifford and Carlotto–Schulz cases; it introduces no new risk. Consequently the ACCEPT verdict with high confidence stands.","tokens_in":8003,"tokens_out":507,"duration_ms":4284,"concrete_test":"Independently recompute the final expanded expression for Q (the polynomial that appears just before the last display of the proof of Theorem 2.2) by symbolic substitution of ξ_{1}=f_{1} cos \theta+f_{2} sin \theta, ξ_{2}=f_{2} cos \theta-f_{1} sin \theta and h^{2}=1-ξ_{2}^{2} into a computer-algebra system; confirm that every monomial cancels identically.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 2.2) is an elementary but lengthy algebraic identity: after substituting the arc-length TreadmillSled ODEs that encode minimality, the quantity Q that appears in the Jacobi equation for ζ_ij reduces identically to zero. The cancellations are fully expanded in the text (terms involving n cancel, the coefficient of θ' simplifies to f^{2}ξ_{2}/h^{4}, and the remaining trigonometric polynomial in f_{1},f_{2},\theta vanishes after substituting the definitions of ξ_{1},ξ_{2}). No hidden analytic assumption or circular step is present once the Laplacian formula (Lemma 2.1) and the first-order system for (f_{1},f_{2},\theta) are granted; both are standard for this family and already used in the equal-dimension case. The subsequent index lower bound (Corollary 2.3) follows from the standard Rayleigh comparison of the four reduced operators S_{ij} and is likewise free of gaps.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the stability operator J of compact minimal immersions φ: S^k \times S^ℓ \times S^1 \to S^{k+ℓ+2} of the form (1), φ(y,z,t)=(f(t)y,f_{2}(t)z,f_{1}(t)). Theorem 2.2 asserts that the functions ζ_{ij}=ω(t) y_i z_j with ω=f^{-k} f_{2}^{-ℓ} are eigenfunctions for the eigenvalue -n (n=k+ℓ+1). The proof reduces J(ζ_{ij})=-n ζ_{ij} to the vanishing of an explicit scalar quantity Q, which is verified by substituting the arc-length TreadmillSled ODEs that encode minimality and performing a fully expanded algebraic cancellation. Corollary 2.3 then obtains the index lower bound ind(M)≥ kℓ+3k+3ℓ+8 by Rayleigh comparison of the four reduced operators S_{11}, S_{21}, S_{12}, S_{22} on the profile curve, using that -n is an unexpected eigenvalue of S_{22} and at least the second (resp. third) eigenvalue of the remaining operators.","tokens_in":8183,"tokens_out":845,"duration_ms":6378,"significance":"The result supplies the first explicit family of eigenfunctions for -n beyond the Gauss-map coordinates on this class of generalized rotational minimal hypersurfaces, and it extends the author’s earlier multiplicity count from the equal-dimension case k=ℓ to arbitrary positive integers k,ℓ. The resulting index lower bound is sharp enough to be useful for comparison with known numerical values (e.g., for k=ℓ=1) and with the author’s conjectured exact index for the Carlotto–Schulz family. The derivation is elementary, fully explicit, and free of circularity once the standard Laplacian formula (Lemma 2.1) and the first-order minimality system are granted; those inputs are already established in the literature for this family. The paper therefore constitutes a clean, self-contained advance on the spectral geometry of these examples.","major_comments":[],"minor_comments":[{"comment":"The title page and running header contain several typographical slips (“HYPERSURF ACES”, “UNEXPECTED MULTIPLICITY STABILITY OPERATOR”). These should be corrected for the published version.","section":null},{"comment":"In the introduction the author notes a typo in the formula for a_{1} in the earlier paper [8] and supplies the corrected expression a_{1}=f^{2}/h^{2}. A one-sentence reminder of the geometric meaning of a_{1} (or a brief re-derivation) would make the correction self-contained.","section":null},{"comment":"The final trigonometric identity that shows the remaining polynomial vanishes is expanded in full; a short remark that the same cancellation can be read as the identity | (f_{1},f_{2}) |^{2} sin^{2}\theta + \tau =0 after substituting the definitions of ξ_{1},ξ_{2} would improve readability without lengthening the argument.","section":null},{"comment":"References [3] and [6] appear with arXiv identifiers dated 2026; if these are still preprints, the citation style should be made uniform with the other arXiv entries.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, technically clean, and fits squarely in the spectral geometry of minimal hypersurfaces in spheres. The algebraic verification is lengthy but fully written out; I see no load-bearing gap. Acceptance is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new content is exactly what the title claims: for the product immersions φ(y,z,t)=(f(t)y,f_{2}(t)z,f_{1}(t)), the functions ζ_{ij}=ω(t)y_i z_j with ω=f^{-k}f_{2}^{-ℓ} are eigenfunctions of the Jacobi operator for eigenvalue -n, even when k\neqℓ. That immediately gives the index lower bound kℓ+3k+3ℓ+8. The equal-dimension case was already done by the same author; this removes the restriction and writes the cancellation out in full.\n\nThe algebra is long but elementary and appears correct. After the TreadmillSled ODEs that encode minimality are substituted, every intermediate expression for Q is expanded, the n-terms cancel, the coefficient of \theta' simplifies cleanly, and the remaining trigonometric polynomial vanishes by the definitions of ξ_{1},ξ_{2}. No hidden analytic assumption or circular step shows up once Lemma 2.1 and the first-order system are granted. The subsequent Rayleigh comparison of the four reduced operators S_{ij} is standard and free of gaps. Self-citation is heavy but legitimate: the Laplacian formula and the k=ℓ case are used as black boxes whose statements do not depend on the new result.\n\nSoft spots are minor. The comparison argument that turns the unexpected eigenvalue into the index count is a bit sketchy (it relies on “at least the second/third eigenvalue” without writing the variational characterization in detail), and the paper inherits the coordinate formulae from earlier work without re-deriving them. Neither issue threatens the main identity. The result does not touch the index=n+3 conjecture for n>2, so broader impact is limited to people already working on these generalized rotational examples.\n\nThis is for specialists who track Morse index of minimal hypersurfaces in spheres, especially anyone computing or bounding indices for Carlotto–Schulz-type constructions. The math is solid enough that a serious editor should send it to referees; I would cite the bound if I needed a uniform lower estimate for this family.","headline":"Clean extension of Perdomo's own equal-dimension eigenfunctions to k \neq ℓ, with a solid algebraic verification and a usable index lower bound.","tokens_in":8815,"tokens_out":573,"would_cite":true,"duration_ms":4636,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","53A10"],"pacs":[],"model":"grok-4.5","headline":"Minimal products of three spheres admit new explicit eigenfunctions for the stability operator at eigenvalue -n, forcing the index to be at least kℓ+3k+3ℓ+8.","keywords":["minimal hypersurfaces","stability operator","eigenfunctions","stability index","generalized rotational hypersurfaces","Gauss map","spheres"],"falsifier":"Directly compute the second variation of a known numerical example (for instance a Carlotto–Schulz surface with small k and ℓ) and check whether the stated product functions really give eigenvalue -n and whether the counted multiplicity matches the claimed lower bound.","tokens_in":8863,"feed_emoji":"📐","tokens_out":925,"duration_ms":6552,"temperature":0.7,"pith_summary":"On any compact minimal hypersurface in a sphere the coordinate functions of the Gauss map are eigenfunctions of the stability operator for the eigenvalue -n. This paper constructs a larger family of eigenfunctions, of the product form ω(t) y_i z_j, for the special minimal immersions that factor through three spheres. The weight ω is an explicit power of the two radial profile functions that define the immersion. Because these new eigenfunctions are linearly independent of the Gauss-map coordinates, the multiplicity of -n jumps, and a comparison of four reduced one-dimensional operators then yields a concrete lower bound on the stability index. The bound improves earlier estimates that treated only the equal-dimension case and applies whether or not the immersion is embedded.","feed_headline":"New eigenfunctions raise the stability index of product minimal hypersurfaces","feed_subtitle":"Explicit product functions force index at least kℓ+3k+3ℓ+8 for every such immersion","key_machinery":"The reduced second-order operator S_{22} obtained by restricting the stability operator to product functions of the form η(t)y_i z_j; Theorem 2.2 shows that -n is an unexpected eigenvalue of S_{22}, which, by Rayleigh comparison with the three companion operators S_{11}, S_{21} and S_{12}, forces three further negative eigenvalues and produces the index lower bound.","core_discovery":"For every minimal immersion ϕ(y,z,t)=(f(t)y,f₂(t)z,f₁(t)) of S^k×S^ℓ×S¹ into the sphere, the functions ζ_ij=ω(t)y_i z_j with ω=f^{-k}f₂^{-ℓ} satisfy J(ζ_ij)=-n ζ_ij. Consequently the multiplicity of the eigenvalue -n is at least (k+ℓ+3)+(k+1)(ℓ+1) and the stability index is at least kℓ+3k+3ℓ+8.","pith_inferences":["The same product construction may produce unexpected eigenfunctions for other eigenvalues once the profile ODE is solved, potentially tightening the index bound further.","If the long-standing conjecture that index n+3 characterises Clifford hypersurfaces is true, these examples lie strictly outside that class for every k,ℓ≥1.","The explicit weight ω suggests a recursive pattern that could generate still higher-multiplicity eigenfunctions when more spherical factors are present."],"forward_implications":["Every minimal immersion of the three-sphere product type has stability index at least kℓ+3k+3ℓ+8, independent of embedding.","The multiplicity of -n is at least (k+ℓ+3)+(k+1)(ℓ+1), strictly larger than the classical n+2 contribution coming from the Gauss map alone.","When k=ℓ the new bound recovers and extends the earlier estimate k²+6k+8.","Any further eigenfunctions or numerical spectrum computations for these immersions must account for this enlarged eigenspace before claiming completeness."],"fun_headline_variants":["Explicit product eigenfunctions raise stability index of sphere immersions","New -n eigenfunctions force index at least kℓ+3k+3ℓ+8 on products","Product functions ζ_ij give extra stability eigenfunctions for minimal immersions","Stability index of S^k×S^ℓ×S¹ immersions bounded below by new eigenfunctions","Explicit eigenfunctions lift multiplicity of -n on product minimal hypersurfaces"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The profile curve must satisfy the first-order ODE that encodes minimality; if that ODE fails, the algebraic cancellations that prove the new functions are eigenfunctions collapse.","fun_headline_variants_meta":{"raw":{"variants":["Explicit product eigenfunctions raise stability index of sphere immersions","New -n eigenfunctions force index at least kℓ+3k+3ℓ+8 on products","Product functions ζ_ij give extra stability eigenfunctions for minimal immersions","Stability index of S^k×S^ℓ×S¹ immersions bounded below by new eigenfunctions","Explicit eigenfunctions lift multiplicity of -n on product minimal hypersurfaces"]},"model":"grok-4.5","effort":"low","cost_usd":0.006434,"raw_usage":{"total_tokens":1571,"prompt_tokens":748,"num_sources_used":0,"completion_tokens":109,"cost_in_usd_ticks":64340000,"prompt_tokens_details":{"text_tokens":748,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":714,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":748,"tokens_out":109,"duration_ms":5166,"temperature":1.0,"reasoning_tokens":714,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T11:28:17.116351+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Directly compute the second variation of a known numerical example (for instance a Carlotto–Schulz surface with small k and ℓ) and check whether the stated product functions really give eigenvalue -n and whether the counted multiplicity matches the claimed lower bound.","supporting_citations":[],"review_version":1}