{"id":"45227e9e-ea6f-4d2c-9928-5cefc1718108","arxiv_id":"1908.09538","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The spreading-speed minimizer r = α(2 - <d>_h/d) is claimed to attain the lower bound 2√(<d>_h<r>_a), but the equality condition only holds after an unstated period normalization and is contradicted by the paper's own variational formula.","lead":"This paper claims to find the growth-rate coefficient that minimizes the invasion speed in a periodic Fisher-KPP model, giving the first explicit nonconstant example where the speed equals a known lower bound. The proof of that claim fails in the Euler-Lagrange step, and the paper's own example contradicts the variational formula it relies on.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The constant test function φ0 is not stationary for the variational functional, so Theorem 2.6's equality characterization collapses; an explicit perturbation in Example 1 lowers the Rayleigh quotient.","rationale":"The reader's weakest assumption correctly identifies the point where the central claim fails. The theorem's equivalence (1)⇔(2) depends on the constant function φ0 minimizing the variational functional at λ0. Direct calculation of the first variation shows this is false under condition (1.3) whenever r is nonconstant, and Example 1 gives an explicit direction that lowers the quotient. In addition, the paper's own formula would not even give I(φ0)=−2⟨r⟩a for L≠1, so the proof is internally inconsistent as well as mathematically incorrect. Since the main result and its corollary rest entirely on this equivalence, rejection is warranted. I agree with the reader's verdict and with the identification of the load-bearing weakness; the additional Cauchy-Schwarz defect and the L² inconsistency are secondary but reinforce the same conclusion. No charitable reading of the text repairs the first-variation computation, and a numerical evaluation of the Rayleigh quotient would settle the matter decisively.","tokens_in":10172,"tokens_out":12621,"duration_ms":128071,"concrete_test":"For Example 1 (r=1+½sin x, d=(1−½sin x)^{-1}, L=2π), compute k_1(d,r) by minimizing the corrected Rayleigh quotient I(φ)=∫d|φ′|²−∫rφ²−1/∫(φ²/d)dx over E_L. Take the normalized perturbation φ_ε=(1+ε sin x)/√(∫(1+ε sin x)²dx) and evaluate I(φ_ε) at ε=0.01. If the value is less than −2, φ0 is not the minimizer and the claimed c*=2 is false. As a second check, recompute I(φ0) with the paper's own Proposition 2.3: if L≠1 the value is −1−L² rather than −2, so equation (2.6) fails immediately.","verdict_should_be":"REJECT","load_bearing_attack":"Theorem 2.6 is load-bearing and it fails at the Euler-Lagrange step in §3.3. The proof of (1)⇒(2) requires that φ0≡1/√L minimize I(φ;λ0,d,r) on E_L, where λ0²=⟨r⟩a/⟨d⟩h. Using the Nadin functional as the paper itself evaluates it in the proof of (1.1), I(φ)=∫d|φ′|²−∫rφ²−λ²/∫(φ²/d)dx, the first variation at φ0 in a mean-zero direction ψ is (2/√L)∫(α⟨d⟩h/d−r)ψ dx, with α=⟨r⟩a. Under condition (1.3), α⟨d⟩h/d−r=2α−2r, so the derivative is (4/√L)∫(α−r)ψ dx, which is nonzero whenever r is nonconstant. Thus φ0 is not a stationary point and cannot be the minimizer. The paper's own Example 1 makes this concrete: r=1+½sin x, d=(1−½sin x)^{-1}, L=2π, λ0=1. Taking ψ=sin x, the first variation is −√(2π)≠0, so I(φ0+εψ)<I(φ0)=−2 for small ε>0. Hence k_1(d,r)<−2 and c*_d(r)≥−k_1>2, contradicting the claimed c*=2. Independently, Proposition 2.3 as printed contains λ²L²; evaluating at φ0 gives I(φ0)=−⟨r⟩a−λ0²L²⟨d⟩h, not −2⟨r⟩a unless L=1, so equation (2.6) already fails in the paper's own notation. The later Cauchy-Schwarz step in (2)⇒(1) also replaces 1/B by B and cannot salvage the upper bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the periodic Fisher-KPP equation u_t = (d(x)u_x)_x + (r(x)-u)u and considers the problem of minimizing the minimal wave speed c*_d(r) over growth rates r with fixed spatial mean, for a given periodic diffusion coefficient d. The main result, Theorem 2.6, asserts that c*_d(r) equals the lower bound 2 sqrt(<d>_h <r>_a) if and only if the coefficient pair satisfies r/<r>_a + <d>_h/d = 2. From this, the paper derives an explicit minimizer r_d(x) = alpha (2 - <d>_h/d(x)) for the minimization problem (P)_d and presents examples, claiming the first exact calculable minimal speed for a spatially periodic Fisher-KPP equation. The proof uses Nadin's variational characterization of the principal eigenvalue k_lambda(d,r) and an Euler-Lagrange analysis of a functional built from that formula.","tokens_in":10618,"tokens_out":18362,"duration_ms":151559,"significance":"The question addressed is natural and the intended result -- an explicit optimizer for the spreading speed under a mean constraint -- would be a valuable contribution if correct. The paper correctly recalls Nadin's variational formula and the known homogenization condition from [5], and the lower-bound proof in Section 3.2 is a standard and correct application of a constant test function. However, the central equivalence in Theorem 2.6 is not established; in fact, the paper's own Example 1 provides a concrete coefficient pair satisfying (1.3) for which the claimed equality c*_d(r)=2 fails. The flaw lies in the assertion that the constant function phi_0 minimizes the variational functional, which is false for nonconstant r. Consequently, the explicit minimizer and the claimed 'first calculable example' are unsupported, and the main contribution as stated does not hold.","major_comments":[{"comment":"The functional I(phi;lambda,d,r) is defined with the term -lambda^2 L^2 integral_0^L (1/d) phi^2 dx, but the evaluation at phi_0 = 1/sqrt(L) in the proof of (1.1) (Section 3.2) and in equation (3.4) uses I(phi_0) = -<r>_a - lambda^2 <d>_h. These are incompatible: with the printed definition, integral_0^L (1/d) phi_0^2 dx = (1/L) integral_0^L (1/d) dx = 1/<d>_h, so I(phi_0) = -<r>_a - lambda^2 L^2 / <d>_h, not the stated value. The correct Nadin functional presumably has the term -lambda^2 / (integral (1/d) phi^2 dx); with that correction the evaluation is consistent, but then the subsequent Euler-Lagrange step in Section 3.3 does not match the printed formula. This inconsistency propagates through the proof of Theorem 2.6.","section":"Section 2.2 / Proposition 2.3"},{"comment":"The proof asserts after (3.1) that the constant function phi_0 = 1/sqrt(L) minimizes the functional I(.;lambda_0,d,r) under condition (1.3). This is false. For the correct Nadin functional, the first variation at phi_0 in any mean-zero direction psi is (2/sqrt(L)) integral (lambda_0^2 <d>_h^2/d - r) psi dx, which under (1.3) equals (4/sqrt(L)) integral (<r>_a - r) psi dx. This expression is nonzero whenever r is nonconstant, so phi_0 is not a stationary point and cannot be the minimizer. The Euler-Lagrange substitution leading to r/<r>_a + <d>_h/d = 2 is therefore invalid. In Example 1, taking psi = sin x gives a strictly negative first variation, so k_{lambda_0}(d,r) < -2<r>_a and c*_d(r) > 2 sqrt(<d>_h <r>_a), contradicting the claimed equality (1).","section":"Section 3.3, proof of Theorem 2.6, (1)=>(2)"},{"comment":"The Cauchy-Schwarz step in the paragraph following (3.5) is incorrect. The inequality is printed as (integral (1/d) phi^2 dx)^2 >= (integral (1/d) dx)^2, which does not follow from Cauchy-Schwarz and fails, for instance, for phi = phi_0 because then the left-hand side equals (1/L^2)(integral (1/d) dx)^2, smaller than the right-hand side when L>1. The subsequent bound also implicitly requires the inequality <d>_h >= L, which is not an assumption of the problem. Consequently, the upper bound c*_d(r) <= 2 sqrt(<d>_h <r>_a) is not established.","section":"Section 3.3, proof of Theorem 2.6, (2)=>(1)"},{"comment":"Two proofs are labeled 'Proof of Theorem 2.10'; the first of them actually proves Corollary 2.8 (the assertion about r_{kd}). Beyond this presentational duplication, the proof of Theorem 2.10 invokes the equality c*_d(r) = 2 sqrt(<d>_h <r>_a) as an assumption, but that equality is exactly the unproved (in fact false) statement of Theorem 2.6. Thus Theorem 2.10, while conditional in form, does not provide independent support for the main claims.","section":"Section 3.3, Theorem 2.10"}],"minor_comments":[{"comment":"In the displayed Euler-Lagrange equation after the variation computation, the second integral is printed as -integral phi psi dx; it should be -integral r phi psi dx. The factor r is missing.","section":"Section 3.3, Euler-Lagrange equation"},{"comment":"The displayed equation (3.2) is garbled: the fraction involving lambda_0^2 L^2 / 3 and the placement of d(x) in the denominator cannot be read unambiguously. Please rewrite it carefully because it is the step that allegedly produces (1.3).","section":"Equation (3.2)"},{"comment":"The term 'first calculable example' in the abstract and Section 3.1 is overstated even apart from the proof error: the example's speed is calculated through the theorem rather than independently, and the condition (1.3) is already present in [5] as a homogenization criterion. The novelty should be framed more carefully.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The referee's negative recommendation rests on a concrete falsification, not on a preference for other results. The paper's main theorem is contradicted by its own Example 1 once the variational functional is written correctly; this is a load-bearing error that cannot be repaired by a local fix. A substantial reformulation (e.g., characterizing equality only in a suitable limit, or identifying a different optimality condition) would be needed for the paper to be considered further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central theorem is wrong. Under the paper's own condition (1.3), the constant function φ0 = 1/√L is not a minimizer of Nadin's variational functional; the first variation in a mean-zero direction is nonzero for any nonconstant r. Example 1 is a concrete counterexample: the claimed speed c* = 2 is not attained, and the actual speed is larger. In addition, the Cauchy-Schwarz step in the proof of (2)⇒(1) asserts an inequality that is generally false. The paper also has an internal inconsistency: the printed Nadin functional includes a factor L² in the last term, which contradicts the author's own evaluation in the proof of (1.1) and hides the variational error.\n\nThat said, the question is a good one and the explicit candidate minimizer r_d(x) = α(2 − ⟨d⟩h/d(x)) is a natural guess. The author correctly identifies the origin of condition (1.3) in El Smaily-Hamel-Roques [5] and cites the relevant literature honestly. The proof of the lower bound (1.1) is standard and correct.\n\nThe soft spots are not minor; they are load-bearing. The equivalence in Theorem 2.6 collapses, and with it the claimed explicit value of the speed. A reader working on periodic KPP speed minimization might keep the candidate formula as a conjecture, but the present proof does not support it. The paper should not be published. I would not send it to referees; the counterexample is simple enough to verify directly.","headline":"The main theorem is false: the constant test function is not a minimizer under the paper's own condition, and Example 1 gives a speed strictly larger than the claimed value.","tokens_in":11142,"tokens_out":17314,"would_cite":false,"duration_ms":153465,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K91","35C07","92D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a periodic Fisher–KPP habitat, the slowest possible front speed equals the lower bound exactly when $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$, and the speed-minimizing growth rate is $r_d(x)=\\alpha(2-\\langle d\\rangle_h/d(x))$.","keywords":["Fisher-KPP equation","minimal wave speed","spreading speed","spatially periodic coefficients","principal eigenvalue","variational characterization","optimal growth rate","harmonic mean"],"falsifier":"Take the period-$2\\pi$ coefficients $r(x)=1+\\tfrac12\\sin x$ and $d(x)=1/(1-\\tfrac12\\sin x)$, which satisfy the ratio condition; if a direct numerical evaluation of the principal-eigenvalue formula gave a minimal speed different from $2$, the equality characterization would be false.","tokens_in":9956,"feed_emoji":"📉","tokens_out":11618,"duration_ms":96426,"temperature":0.7,"pith_summary":"The paper establishes when the minimal speed of a one-dimensional spatially periodic Fisher–KPP equation coincides with the simple homogenized lower bound $2\\sqrt{\\langle d\\rangle_h\\langle r\\rangle_a}$, where $\\langle r\\rangle_a$ is the spatial mean of the growth rate and $\\langle d\\rangle_h$ is the harmonic mean of the diffusion. It proves this happens exactly when the pointwise ratio condition $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$ holds, and in that case the speed is explicitly calculable. It then solves the natural inverse problem: among all growth rates with a fixed positive mean $\\alpha$, the speed is minimized by $r_d(x)=\\alpha(2-\\langle d\\rangle_h/d(x))$, a function that is large where diffusion is large and small where diffusion is small. These results give the first exact, nonconstant example of the minimal (spreading) speed for spatially periodic coefficients, and they show that the shape of environmental heterogeneity, not just its average, controls the invasion speed.","feed_headline":"Explicit formula gives the slowest front speed in periodic habitats","feed_subtitle":"The paper names the exact growth-rate shape that attains it.","key_machinery":"The object carrying the argument is the variational formula for the principal eigenvalue of the linearized operator: $k_\\lambda(d,r)=\\min_{\\phi\\in E_L} I(\\phi;\\lambda,d,r)$, where $E_L$ is the set of positive $L$-periodic profiles with $L^2$ norm one and $I(\\phi;\\lambda,d,r)=\\int_0^L d|\\phi'|^2-\\int_0^L r\\phi^2-\\lambda^2 L^2\\int_0^L (1/d)\\phi^2$. Combined with the min-max formula $c^*_d(r)=\\min_{\\lambda>0}(-k_\\lambda(d,r)/\\lambda)$, this functional converts the speed into a variational question. The proof's hinge is the constant test profile $\\phi_0\\equiv 1/\\sqrt{L}$: equality in the lower bound forces $\\phi_0$ to be the actual minimizer at $\\lambda_0=\\sqrt{\\langle r\\rangle_a/\\langle d\\rangle_h}$, and inserting it into the Euler–Lagrange equation reduces exactly to the relation $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$. That relation is then solved explicitly for $r$, producing the optimizer.","core_discovery":"The central claim is that equality in the universal lower bound for the minimal speed is characterized by a pointwise ratio condition: $c^*_d(r)=2\\sqrt{\\langle d\\rangle_h\\langle r\\rangle_a}$ holds if and only if $r(x)/\\langle r\\rangle_a+\\langle d\\rangle_h/d(x)=2$ for all $x$. Under this condition the unique critical parameter is $\\lambda_0=\\sqrt{\\langle r\\rangle_a/\\langle d\\rangle_h}$, the unique minimizer of the variational functional is the constant $\\phi_0\\equiv 1/\\sqrt{L}$, and the minimal speed is explicitly $2\\sqrt{\\langle d\\rangle_h\\langle r\\rangle_a}$. Solving the ratio condition for $r$ gives the optimizer of the constrained minimization problem: $r_d(x)=\\alpha(2-\\langle d\\rangle_h/d(x))$ for a prescribed mean $\\alpha$, and the same function minimizes the speed for every positive rescaling of $d$. The paper therefore provides, as it states, the first exact value of the minimal speed for a spatially periodic Fisher–KPP equation with nonconstant coefficients.","pith_inferences":["Editorial extension: The paper allows sign-changing $r$; if one imposes the biological constraint $r\\ge 0$, the variational problem changes and the explicit optimizer may no longer be admissible, a case the paper does not treat.","Editorial extension: Because $r_{kd}=r_d$ for every rescaling $k>0$, the optimization is really over the normalized shape of $d$; one could equally fix $r$ and ask which diffusion shape minimizes the speed, a reverse question the present theorem does not answer.","Editorial extension: The equality lower bound is also the homogenization limit $L\\to 0$, so the ratio condition can be read as a zero-correction regime; it would be natural to test whether similar exact-speed relations hold for time-periodic or multi-dimensional analogues."],"forward_implications":["In any habitat satisfying $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$, the exact invasion speed is $2\\sqrt{\\langle d\\rangle_h\\langle r\\rangle_a}$; Example 1 computes $c^*=2$ for $r=1+\\tfrac12\\sin x$ and $d=1/(1-\\tfrac12\\sin x)$.","The minimizing growth rate for prescribed mean $\\alpha$ is $r_d(x)=\\alpha(2-\\langle d\\rangle_h/d(x))$, so the slowest spread occurs when growth concentrates where diffusion is strongest.","Because $r_{kd}=r_d$ for every $k>0$, rescaling the diffusion amplitude does not change the optimal growth shape; only the normalized shape of $d$ matters.","If the ratio condition fails, the strict inequality $c^*_d(r)>2\\sqrt{\\langle d\\rangle_h\\langle r\\rangle_a}$ holds; when $r$ is fixed and $2-r/\\langle r\\rangle_a$ is not a positive diffusion, no admissible $d$ can attain the lower bound.","Under the equality condition, the principal eigenfunction of the linearized operator is constant if and only if $d$ is constant, so the nonconstant optimal regime genuinely has a nonconstant eigenfunction."],"supporting_citations":[{"why":"Supplies the variational characterization of the principal eigenvalue and the lower-bound inequality that the theorem turns into equality.","marker":"[13]"},{"why":"Establishes the min-max formula for the minimal speed via the principal eigenvalue of the linearized operator.","marker":"[2]"},{"why":"Introduces the ratio condition $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$ in a homogenization study; this paper proves it is exactly the equality condition.","marker":"[5]"},{"why":"Proves the constant-diffusion case: equality holds iff $r$ is constant, the result now generalized.","marker":"[10]"},{"why":"Derives that a constant growth rate minimizes the speed in constant diffusion, the setting the optimizer extends.","marker":"[4]"},{"why":"Numerical phase-dependence study of sinusoidally varying coefficients that the explicit optimizer formalizes in a different way.","marker":"[8]"}],"fun_headline_variants":["Exact slowest speed for periodic Fisher-KPP fronts","First explicit minimal speed for spatially periodic fronts","Optimal growth shape that slows periodic fronts to the minimum","Explicit optimizer yields the exact minimal spreading speed","Closed-form minimizer for periodic Fisher-KPP wave speeds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that equality forces the ratio condition assumes that the constant profile $\\phi_0\\equiv 1/\\sqrt{L}$ is a genuine minimizer of the variational functional at $\\lambda_0$, so that evaluating the Euler–Lagrange equation at $\\phi_0$ is legitimate; if another profile gave a smaller value, the argument would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Exact slowest speed for periodic Fisher-KPP fronts","First explicit minimal speed for spatially periodic fronts","Optimal growth shape that slows periodic fronts to the minimum","Explicit optimizer yields the exact minimal spreading speed","Closed-form minimizer for periodic Fisher-KPP wave speeds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000329,"raw_usage":{"total_tokens":1854,"prompt_tokens":984,"completion_tokens":870,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":794}},"tokens_in":600,"tokens_out":870,"duration_ms":6988,"temperature":1.0,"reasoning_tokens":794,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:13:21.721386+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the period-$2\\pi$ coefficients $r(x)=1+\\tfrac12\\sin x$ and $d(x)=1/(1-\\tfrac12\\sin x)$, which satisfy the ratio condition; if a direct numerical evaluation of the principal-eigenvalue formula gave a minimal speed different from $2$, the equality characterization would be false.","supporting_citations":[{"cited_title":"Nadin, The effect of the Schwarz rearrangement on the periodic principal eigen- value of a nonsymmetric operator , SIAM J","cited_arxiv_id":null,"evidence_quote":"Supplies the variational characterization of the principal eigenvalue and the lower-bound inequality that the theorem turns into equality."},{"cited_title":"Berestycki, F","cited_arxiv_id":null,"evidence_quote":"Establishes the min-max formula for the minimal speed via the principal eigenvalue of the linearized operator."},{"cited_title":"ElSmaily, F","cited_arxiv_id":null,"evidence_quote":"Introduces the ratio condition $r/\\langle r\\rangle_a+\\langle d\\rangle_h/d=2$ in a homogenization study; this paper proves it is exactly the equality condition."},{"cited_title":"Liang, X","cited_arxiv_id":null,"evidence_quote":"Proves the constant-diffusion case: equality holds iff $r$ is constant, the result now generalized."},{"cited_title":"Berestycki, F","cited_arxiv_id":null,"evidence_quote":"Derives that a constant growth rate minimizes the speed in constant diffusion, the setting the optimizer extends."},{"cited_title":"Kinezaki, K","cited_arxiv_id":null,"evidence_quote":"Numerical phase-dependence study of sinusoidally varying coefficients that the explicit optimizer formalizes in a different way."}],"review_version":1}