{"id":"be5e5893-9e12-42c1-9172-7a0802ca8801","arxiv_id":"2608.12131","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every non-negative diagonal planar minimizer is unstable under one-dimensional perturbations throughout -3/2 < L < 0, with L=0 as the sharp endpoint.","lead":"This paper proves that planar isotropic-nematic interfaces in the Landau-de Gennes model are unstable for every negative elastic anisotropy L above -3/2, removing a technical condition from earlier work. It also shows L=0 is the exact endpoint of this instability and corrects a normalization error in a known interface profile.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof's internal estimates are coherent; the only load-bearing external input is the imported existence/regularity theorem [22, Thm 3.2], which is standard and not circular.","rationale":"The reader's ACCEPT verdict is appropriate. I followed the main proof line in detail. Proposition 3.3 is the only delicate step: it uses minimality to get J[u+,v+]≥0, derives uv≥0, then rules out sign changes via a W=0/uniqueness argument; the logic is valid, including the final strict positivity step, because a simultaneous zero of S' and T' forces (S,T) to be one of the two wells by the first integral, and then Cauchy uniqueness contradicts the boundary conditions. Lemma 3.4's bound T-S<1/6 is algebraically correct, and it is exactly what makes F0>0 in Lemma 5.1. Lemma 5.2's identity and sign are correct, and the cutoff argument for the non-L2 mode S+T is justified by exponential decay. Proposition 6.1's factorization q0[p]=∫s^2((p/s)')^2 dz is correct and gives Λ(0)=0. The only premise not proved in this paper is the existence/regularity of a non-negative diagonal planar minimizer from [22, Theorem 3.2]. That is an independent imported result, not circular, and no evidence here contradicts it. Therefore no load-bearing internal concern was found.","tokens_in":14608,"tokens_out":22448,"duration_ms":183825,"concrete_test":"Verify Wu [22, Theorem 3.2] independently for every L∈(-3/2,0): re-prove existence of a non-negative minimizer of E_L over A_p and the exponential convergence in (3.1) plus the bounds in (3.2). A concrete sub-check is to confirm that the degenerate-metric construction in [22, §6] yields a minimizer not just for α≥1 but for all α∈(0,1), and that the positivity argument in Proposition 3.1 does not require an extra assumption such as (1.13). If any of these fails, the range of Theorem 1.1 would have to be restricted accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the chain from Proposition 3.3 through Lemma 5.1 and Lemma 5.2. The derivative inequalities S'>0, T'>0, and 3T'-S'>0 are obtained by sound maximum-principle and second-variation arguments; the sign identity q_L[S+T]=(L/6)∫(S'+T')(3T'-S') dz is algebraically correct, and the cutoff localization is justified by exponential decay. Proposition 6.1's factorization and zero-resonance conclusion are also correct. I found no internal inconsistency. The only genuinely load-bearing premise is the imported existence/regularity result [22, Theorem 3.2], which supplies a non-negative minimizer with exponential convergence (3.1) and bounds (3.2). If that theorem failed for part of the interval (-3/2,0), or if its minimizer lacked the required decay or bounds, Theorem 1.1 would not follow. But this is a standard external dependence, not a circular use of the instability conclusion, and nothing in the present text suggests it is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies one-dimensional planar isotropic–nematic interfaces in the Landau–de Gennes model with elastic anisotropy constant L. The authors prove that for every L in the interval (-3/2,0), every non-negative diagonal planar minimizer (S,T) of the reduced energy is unstable under one-dimensional off-diagonal perturbations, thereby removing an extra structural condition (1.13) required in earlier work. The proof combines new pointwise derivative bounds S'>0, T'>0, and 3T'-S'>0 with the exact identity q_L[S+T] = (L/6)∫(S'+T')(3T'-S') dz < 0 and a cutoff localization argument that converts the non-L^2 trial mode S+T into a compactly supported negative direction. At L=0, the paper proves the factorization q_0[p] = ∫ s^2((p/s)')^2 dz ≥ 0, shows that the stability index Λ(0) equals 0, and identifies s as a zero-energy threshold resonance. It also defines the optimal stability index Λ(L) = inf σ(L_L) and corrects a factor-of-two normalization inconsistency in a previously stated interface profile.","tokens_in":14779,"tokens_out":29150,"duration_ms":256416,"significance":"The result is significant because it settles the sharp negative-L range for instability in the reduced planar problem: the condition L > -3/2 is exactly the coercivity range of the reduced energy, and the paper establishes instability throughout its entire negative half, with L=0 as the endpoint. The proof is essentially self-contained and rigorous: the derivative inequalities, the exact second-variation computation, the spectral analysis via Kato–Rellich and Weyl's theorem, and the cutoff localization are all written in detail. The paper has no fitted parameters, and its central instability criterion is a concrete falsifiable spectral statement. The main external input is the existence and regularity theorem for non-negative diagonal planar minimizers quoted from [22, Theorem 3.2]; this is a standard, non-circular dependency rather than an internal weakness. Overall, the claims are well supported and the paper makes a solid contribution to the stability theory of Landau–de Gennes interfaces.","major_comments":[],"minor_comments":[{"comment":"The displayed definition of a_L in (1.10) is typeset as a_L = 2 + L/2, which is inconsistent with the subsequent text and with Lemma 5.2, where a_L = 1 + L/2 = (2+L)/2; the displayed formula should be corrected.","section":"§1, Eq. (1.10)"},{"comment":"The normalized second variation (1.9) is defined only at profiles where the first variation of F_0 vanishes, but the manuscript does not explicitly verify that the diagonal planar minimizer Q_p is critical for the off-diagonal perturbation (4.1); this follows from the orthogonality relations Q_p:P_p = 0 and tr(Q_p^2 P_p) = 0, and a one-sentence verification would remove any doubt.","section":"§4, Eq. (4.2)"},{"comment":"The passage from J[ζ_R u_+, ζ_R v_+] to J[u_+, v_+] is terse: the quadratic form J is introduced only for C_0^∞ test pairs, and its use for H^1 pairs such as (u_+, v_+) should be justified by density and by the vanishing of the boundary terms in the subsequent integration by parts.","section":"§3, Proposition 3.3"},{"comment":"The notation q_L[p] is initially defined for p ∈ H^1(R) in (1.11), but Lemma 5.2 applies it to p = S+T ∉ L^2(R); although the lemma explains that the integral is meant as an improper integral, a remark at the first use of the notation would avoid confusion.","section":"§5, Lemma 5.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript relies on the authors' own earlier theorem [22, Theorem 3.2] for existence and fine properties of the non-negative diagonal planar minimizer. This is a legitimate external input, but the main theorem of the present paper inherits any gap in that result, so the dependence should be stated clearly in the introduction. No other concerns about scope or novelty arise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper settles the planar instability question that the second author left open: for every L in (-3/2,0), any non-negative diagonal planar minimizer is unstable under one-dimensional perturbations, and no extra condition like (1.13) is needed. The key is the new pointwise inequality 3T' - S' > 0, proved here by maximum-principle arguments together with the auxiliary bound T - S < 1/6. Once you have that, the exact identity q_L[S+T] = (L/6) ∫ (S'+T')(3T'-S') dz is sign-definite and gives the negative mode. The mode is not in L^2, but the cutoff localization with explicit error estimates is handled correctly. The L=0 factorization q_0[p] = ∫ s^2((p/s)')^2 dz ≥ 0 is clean and shows the endpoint is a zero-energy resonance, not an eigenvalue.\n\nWhat is genuinely new: the derivative inequality, the removal of (1.13), the optimal stability index Λ(L), and a small factor-of-two normalization correction to an earlier explicit profile. The proofs are written in full; the spectral part is standard Kato-Rellich/Weyl but done carefully. There are no fitted parameters and no circular step: the negative direction is an exact computation from the Euler-Lagrange equations. The one genuinely imported premise is the existence/regularity theorem [22, Thm 3.2], which supplies a smooth non-negative minimizer with exponential decay and the bounds 0 < S,T ≤ 1/2. That is an independent existence result, not the instability conclusion, so calling it circular would be wrong. Still, the sharp 'any minimizer' statement inherits whatever caveats that theorem carries. If the referee checks that the hypotheses line up, the main theorem stands.\n\nI would send this to a serious PDE or liquid-crystal journal. It deserves a full referee report, and I expect it to be accepted after a routine check of the imported theorem. For a reading group, it is a nice example of how a maximum-principle inequality plus an exact spectral identity can replace a clumsy trial-function condition.\n\nBest.","headline":"Sharp negative-L planar instability in Landau-de Gennes, with full proofs and only a standard existence theorem imported from the authors' earlier work.","tokens_in":15347,"tokens_out":2585,"would_cite":true,"duration_ms":22140,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B35","35Q35","82D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem 1.1: every non-negative diagonal planar minimizer is unstable for all L in (-3/2,0).","keywords":["Landau–de Gennes model","isotropic–nematic interface","planar anchoring","diagonal planar minimizer","second variation","spectral instability","zero-energy resonance","heteroclinic connection"],"falsifier":"For some $L\\in(-\\frac32,0)$, say $L=-0.5$, compute numerically the minimizer of $E_L$ in $A_p$ by solving the ODE system (1.8) with boundary conditions (1.6). If any such non-negative minimizer has a point with $3T'-S'\\le 0$, or if the computed value of $q_L[S+T]=\\frac{L}{6}\\int(S'+T')(3T'-S')\\,dz$ is non-negative, then Lemma 5.1 or identity (5.6) fails and Theorem 1.1 collapses; conversely, verifying the strict inequality $3T'-S'>0$ and $q_L[S+T]<0$ on a fine grid would support the theorem.","tokens_in":14343,"feed_emoji":"🌀","tokens_out":6678,"duration_ms":61052,"temperature":0.7,"pith_summary":"This paper settles a stability question for the interface between the isotropic and nematic phases of a liquid crystal, described by the Landau–de Gennes order-parameter tensor. The authors prove that for every anisotropy parameter $L$ in $(-3/2,0)$, any non-negative minimizer of the reduced energy within the diagonal planar class is unstable under one-dimensional perturbations. Earlier work only established this instability under an extra integral condition, which the paper removes. The proof exposes a rotation mode: infinitesimally rotating the planar director toward the interface normal lowers the energy. It also shows that $L=0$ is the sharp endpoint of the instability range from the negative side, because the critical spectral operator is non-negative at $L=0$ and has a zero-energy resonance there.","feed_headline":"Planar nematic interfaces unstable across full negative-L range","feed_subtitle":"New rotation-mode identity kills the extra condition and pins L=0 as the sharp endpoint.","key_machinery":"The load-bearing object is the scalar quadratic form $q_L[p]=\\int_{\\mathbb{R}}(a_L(p')^2+V_L p^2)\\,dz$ with $a_L=(2+L)/2$ and $V_L=1+3(S-3T)+2(S^2+3T^2)$, together with the associated self-adjoint operator $L_L=-a_L\\frac{d^2}{dz^2}+V_L$ on $L^2(\\mathbb{R})$. Substituting the off-diagonal perturbation (4.1) into the tensorial second variation reduces it to $\\frac{1}{3}q_L[p]$. The decisive identity is $L_L(S+T)=\\frac{L}{6}(S''-3T'')$, which, after integration by parts, yields $q_L[S+T]=\\frac{L}{6}\\int_{\\mathbb{R}}(S'+T')(3T'-S')\\,dz$. The new derivative inequalities $S'>0$, $T'>0$, and $3T'-S'>0$ make the integrand strictly positive, so the sign is fixed by $L<0$. The optimal stability index $\\Lambda(L)=\\inf\\sigma(L_L)$ is then strictly negative for $-\\frac{3}{2}<L<0$ and, by the factorization at $L=0$, equal to zero at the endpoint.","core_discovery":"The central claim is Theorem 1.1: let $-\\frac{3}{2}<L<0$ and let $(S,T)\\in A_p$ be a non-negative diagonal planar minimizer, with $Q_p=\\frac{1}{3}\\operatorname{diag}(S+3T,\\,S-3T,\\,-2S)$. Then there exists a smooth compactly supported $S_0$-valued perturbation $P$ such that the normalized second variation satisfies $E_{Q_p}(P,P)<0$, so $Q_p$ is unstable under one-dimensional perturbations. The instability is detected by the off-diagonal mode $p=S+T$, which is exactly the infinitesimal rotation of the planar director toward the interface normal. The proof establishes new pointwise derivative inequalities $S'>0$, $T'>0$, and $3T'-S'>0$, and then evaluates the relevant scalar quadratic form exactly as $q_L[S+T]=\\frac{L}{6}\\int_{\\mathbb{R}}(S'+T')(3T'-S')\\,dz<0$. A slow cutoff in the nematic tail converts the non-$L^2$ test function $S+T$ into a compactly supported negative direction. At $L=0$, the explicit profile $S=T=s(z)=\\frac14(1+\\tanh(z/2))$ gives the exact factorization $q_0[p]=\\int_{\\mathbb{R}} s^2\\,((p/s)')^2\\,dz\\ge 0$, so the optimal stability index vanishes and $s$ is a zero-energy threshold resonance rather than an $L^2$ eigenfunction.","pith_inferences":["The proof needs only the pointwise signs $S'>0$, $T'>0$, $3T'-S'>0$ and the algebraic identity for $q_L[S+T]$; the same mechanism should persist for any sufficiently regular planar heteroclinic profile with those signs, not exclusively for minimizers, a claim the paper does not make but that a numerical check could test directly.","The zero-energy resonance at $L=0$ suggests that perturbation theory near $L=0$ is a resonance problem, not a gap-opening one; a natural extension is to compute the leading-order behaviour of $\\Lambda(L)$ as $L\\uparrow 0$ and ask whether any positive-$L$ stability appears away from the endpoint.","The corrected $L=0$ profile $s(z)=\\frac14(1+\\tanh(z/2))$ fixes the interface length scale in the adopted normalization; comparisons with earlier numerically computed interface widths should use this profile rather than the factor-of-two inconsistent version mentioned in the abstract."],"forward_implications":["The instability holds on the whole coercive negative range $(-\\frac{3}{2},0)$ of the planar reduction, with no extra integral condition required; in particular it covers $[-1,0)$, the full negative range where the three-dimensional elastic energy is non-negative.","$L=0$ is the sharp endpoint from below: $\\Lambda(L)<0$ for every $L\\in(-\\frac{3}{2},0)$ while $\\Lambda(0)=0$, and at $L=0$ the critical mode is a zero-energy resonance, so the threshold is not an ordinary isolated-eigenvalue crossing.","For every $L\\in(-\\frac{3}{2},0)$, $\\Lambda(L)$ is a simple eigenvalue of $L_L$ with a strictly positive eigenfunction, giving a distinguished optimal negative direction in the one-dimensional perturbation class.","Because the negative perturbation is one-dimensional, any larger perturbation class that contains one-dimensional perturbations also detects the instability of $Q_p$.","The geometric source of the instability is a rotation of the planar director toward the interface normal, which is consistent with the energetic preference for homeotropic or tilted anchoring when $L<0$."],"supporting_citations":[{"why":"Supplies the existence, smoothness, exponential decay, and bounds $0<S,T\\le \\frac12$ of the non-negative diagonal planar minimizer used throughout the proof.","marker":"[22, Theorem 3.2]"},{"why":"Provides the $L=0$ explicit interface profile and the homeotropic equilibrium used for comparison and for fixing the normalization.","marker":"[17]"},{"why":"Gives the degenerate-metric method that constructs the diagonal planar minimizer in the admissible class.","marker":"[23, Theorem 3.1]"},{"why":"Provides the spectral inequality and bulk-potential factorization that identify the wells and the zero set of the reduced potential $W$.","marker":"[13]"},{"why":"Supplies the spectral-analytic tools (Kato–Rellich, Weyl, max–min, ground-state theorem) behind the stability index $\\Lambda(L)$ and the simple eigenfunction conclusion.","marker":"[20]"},{"why":"States the $L=0$ rigidity result for the full energy that frames the threshold analysis at the endpoint.","marker":"[4]"}],"fun_headline_variants":["Planar interfaces unstable for all L in (-3/2,0)","Sharp endpoint L=0: planar interfaces unstable for every L<0","Removing extra condition: full negative-L instability of planar interfaces","L=0 marks sharp end of planar isotropic-nematic instability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes from earlier work that a smooth non-negative diagonal planar minimizer exists, converges exponentially to $(0,0)$ and $(\\frac12,\\frac12)$, and satisfies $0<S,T\\le\\frac12$; if such a minimizer does not exist or lacks these bounds, the derivative-sign and cutoff-localization arguments do not go through.","fun_headline_variants_meta":{"raw":{"variants":["Planar interfaces unstable for all L in (-3/2,0)","Sharp endpoint L=0: planar interfaces unstable for every L<0","Removing extra condition: full negative-L instability of planar interfaces","L=0 marks sharp end of planar isotropic-nematic instability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000349,"raw_usage":{"total_tokens":1940,"prompt_tokens":1012,"completion_tokens":928,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":853}},"tokens_in":628,"tokens_out":928,"duration_ms":8806,"temperature":1.0,"reasoning_tokens":853,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:15:33.442377+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For some $L\\in(-\\frac32,0)$, say $L=-0.5$, compute numerically the minimizer of $E_L$ in $A_p$ by solving the ODE system (1.8) with boundary conditions (1.6). If any such non-negative minimizer has a point with $3T'-S'\\le 0$, or if the computed value of $q_L[S+T]=\\frac{L}{6}\\int(S'+T')(3T'-S')\\,dz$ is non-negative, then Lemma 5.1 or identity (5.6) fails and Theorem 1.1 collapses; conversely, verifying the strict inequality $3T'-S'>0$ and $q_L[S+T]<0$ on a fine grid would support the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $L=0$ explicit interface profile and the homeotropic equilibrium used for comparison and for fixing the normalization."},{"cited_title":"Teschl.Mathematical Methods in Quantum Mechanics: With Applications to Schr¨ odinger Operators, volume 157 ofGraduate Studies in Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral-analytic tools (Kato–Rellich, Weyl, max–min, ground-state theorem) behind the stability index $\\Lambda(L)$ and the simple eigenfunction conclusion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the $L=0$ rigidity result for the full energy that frames the threshold analysis at the endpoint."}],"review_version":1}