{"id":"4e63e16d-8e9b-4deb-b6f8-6d2dd6ff2175","arxiv_id":"2608.00438","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A perturbative homogenization theorem for viscous Hamilton-Jacobi equations with u/epsilon-dependent Hamiltonians, proved via periodic-parabolic correctors constructed by Fredholm theory and a fixed point argument.","lead":"This paper proves that a class of viscous Hamilton-Jacobi equations with Hamiltonians oscillating rapidly in both space and solution value converge to a simpler effective equation. The result extends homogenization theory for contact-type equations to the second-order viscous setting under a small perturbation assumption.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Perturbative construction depends on the under-proved uniform Schauder/Fredholm lemma on p-dependent skew tori (Lemma 3.1); Theorem 5 is not fully supported until that proof is completed.","rationale":"The reader's verdict accepts the paper while flagging (F2) as the weakest assumption and noting that the periodic-parabolic Schauder estimates on skew tori are only sketched. I focus on the latter because it is the point where the perturbative construction is most exposed. The proof of Lemma 3.1 is the foundation for Proposition 3.2, which provides the uniform invertibility needed for the contraction argument in Proposition 3.3. Without a fully verified uniform estimate, the smallness threshold η_{K0} in Theorem 5 has no rigorous basis. The concern is not that the lemma is false; the standard theory likely supports it. But the paper's sketch does not demonstrate the uniformity with respect to p, and the skew tori vary with p in a way that is not covered verbatim by the cited references. Therefore the appropriate verdict is conditional acceptance: the mathematical strategy is coherent and promising, but the central perturbative step should be supplied with a complete proof of Lemma 3.1 before the homogenization theorem can be regarded as fully established. This is a verifiability gap rather than a demonstrated error, so the reader's positive assessment is not overturned entirely.","tokens_in":33197,"tokens_out":41077,"duration_ms":362834,"concrete_test":"Complete the proof of Lemma 3.1 for the specific operator ~L_p by writing the skew torus as (θ,σ) ∈ T^{N+1} with y = θ, r = σ - p·θ, so that ~L_p becomes c0(p)(∂_σ - p·∂_θ) + b_p·∂_θ + d_p - Δ_θ. Then apply the standard periodic-parabolic Schauder theory after normalizing the time-derivative coefficient, tracking every constant's dependence on p and on the Hölder norms of b_p,d_p on T^{N+1}_p. In particular, check whether the skew first-order term -c0(p)p·∂_θ and the p-dependence of the coefficients are controlled uniformly by ‖χ^0_p - z‖_{C^2(T)} and by the compactness of K; if the Schauder constant or the number of cylinders in the covering of the fundamental domain depends on p in any other way, the uniform claim fails. Passing this check would confirm Proposition 3.2 and the fixed-point construction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim of Theorem 5 is established by constructing perturbed correctors via a contraction argument on a linearized periodic-parabolic operator. That construction is only as solid as Lemma 3.1, which asserts uniform periodic-parabolic Schauder estimates and a Fredholm alternative on the p-dependent skew tori T^{N+1}_p. The proof given is a sketch: it cites Lieberman/LSU and says that translations patch local estimates, and that the Fredholm property follows from compactness of the period map. It does not explicitly verify that the coefficients b_p = D_pF(χ^0_p, p a_p) and d_p = F_s(χ^0_p, p a_p) are uniformly Hölder in the parabolic distance on T^{N+1}_p, nor does it track how the covering of the fundamental domain and the Schauder constant depend on p. If the constant degenerates as p varies in the compact set K, then Proposition 3.2's uniform invertibility fails, the fixed-point ball in Proposition 3.3 has no uniform radius, and the correctors required for the homogenization theorem are not produced. The paper's own Remark 3.1 and the bootstrap in Lemma 3.2 also rely on this lemma. Since no complete proof is supplied, a referee cannot rule out a hidden p-dependence in the patching constants or a failure of the Fredholm alternative caused by the varying lattice. This is the most load-bearing gap in the argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies periodic homogenization of the viscous Hamilton--Jacobi equation (1.1) with Hamiltonian H(y,s,p)=F(s,p)+\\eta W(y,s,p), where the unknown enters through the fast scale u^\\epsilon/\\epsilon. In the unperturbed case the author constructs one-dimensional monotone correctors, proves uniqueness and continuity of the cell speed, and establishes local uniform convergence to u_t-c(Du)=0. In the perturbed case, for sufficiently small \\eta depending on the Lipschitz constant of the initial datum, periodic-parabolic correctors on p-dependent skew tori are constructed by linearizing around the one-dimensional profile, applying a Fredholm alternative, and performing a contraction argument; this yields the homogenization theorem with effective Hamiltonian c^\\eta(p). The paper also proves a large-time averaging result for a general class of cell evolutions and identifies the averaged constant with the corrector speed in the perturbative class, followed by three examples.","tokens_in":33413,"tokens_out":34299,"duration_ms":343218,"significance":"If fully substantiated, the paper is a genuine contribution to the homogenization of second-order contact-type Hamilton--Jacobi equations, since it combines fast x/\\epsilon and u^\\epsilon/\\epsilon oscillations with a viscous term and gradient-dependent spatial perturbations. The unperturbed corrector construction is detailed and convincing, and the perturbative strategy based on periodic-parabolic Fredholm theory is natural and potentially reusable. The explicit examples in Section 5 are useful. However, the central perturbative claim, Theorem 5, currently rests on a uniform Schauder/Fredholm lemma and a bootstrap regularity step that are only sketched, and the continuity proof for the effective speed has a nontrivial passage-to-the-limit gap. These points are local and likely repairable, but they are load-bearing.","major_comments":[{"comment":"Lemma 3.1 is the foundation of Propositions 3.2 and 3.3 and therefore of Theorem 5, but its proof is a sketch. The manuscript does not explicitly verify that the coefficients b_p(y,r)=D_pF(\\chi^0_p(p\\cdot y+r), p a_p(p\\cdot y+r)) and d_p(y,r)=F_s(\\chi^0_p(p\\cdot y+r), p a_p(p\\cdot y+r)) are uniformly H\\\"older continuous in the parabolic distance on T^{N+1}_p, nor does it track how the covering of the fundamental domain and the Schauder constant depend on p. The Fredholm alternative on the varying skew lattice is also asserted rather than proved. Please supply either a complete proof of the uniform estimates and Fredholm property or a precise statement of an existing theorem with all hypotheses verified on the skew tori.","section":"Section 3.1, Lemma 3.1"},{"comment":"The bootstrap regularity result is sketched but is load-bearing: Definition 3.1 requires w^\\eta_p\\in C^2(T^{N+1}), and the viscosity test in Theorem 5 uses (\\chi^\\eta_p)_{zz} and D^2_{yz}\\chi^\\eta_p. The differentiated equation for U=\\Xi_r is written down, but the Schauder step is not justified in detail: one needs the coefficients F_s(\\Xi,D_y\\Xi)+\\eta W_s(\\Xi,D_y\\Xi) and D_pF(\\Xi,D_y\\Xi)+\\eta D_pW(\\Xi,D_y\\Xi) to be uniformly H\\\"older in the parabolic distance on T^{N+1}_p, uniformly in p\\in K, and the argument that the resulting bounds pass back to C^2 in the original variables is only summarized.","section":"Section 3.1, Lemma 3.2 and Remark 3.1"},{"comment":"The proof of continuity of p\\mapsto c^\\eta(p) passes to a subsequence and then identifies the limit using the uniqueness of the small solution from Proposition 3.3. As written, it does not show that the limiting pair (\\psi^*,\\kappa^*) belongs to the small ball B^\\eta_p in X_p, nor does it explain how the X_{p_n}-norms behave under the varying skew lattice when passing to the limit. Without such control the uniqueness step is not justified. Please add a compactness argument that keeps the limit inside the contraction class or otherwise identify the limit by a stability argument for the fixed-point equation.","section":"Section 3.1, Proposition 3.5"},{"comment":"The smallness threshold \\eta_K in Proposition 3.3 is compact-dependent, so for a fixed \\eta\\le\\eta_{K_0} the constructed cell speed c^\\eta(p) is a priori defined only on K_0=B_{Lip(u_0)}(0). Theorem 5 nevertheless states convergence to the viscosity solution of u_t-c^\\eta(Du)=0 on R^N, which requires a Hamiltonian defined on all of R^N. Either prove that c^\\eta admits a continuous extension to R^N, or formulate and justify a restricted viscosity solution whose test- function gradients lie in K_0.","section":"Section 3.2, Theorem 5"},{"comment":"The maximum-principle proof of the oscillation estimate uses a penalized functional \\Phi over the unbounded domain R^{2N}, but it does not establish the growth control needed for the maximum to be attained, nor does it justify the claim that the terms \\alpha|\\zeta_\\alpha| and \\alpha|\\eta_\\alpha| are O_\\alpha(1) when passing to the limit \\alpha\\to 0. A standard barrier comparison using the linear growth estimate |H(\\zeta,s,q)|\\le\\chi(|q|) and w(\\zeta,0)=0 should close this gap, but it is not included. Since Theorem 6 depends on this proposition, the argument should be completed.","section":"Section 4.2, Proposition 4.1"}],"minor_comments":[{"comment":"The symbol c_0 is used both for the positivity constant in assumption (F1) and for the unperturbed cell speed c_0(p) in Section 3; this is confusing and should be resolved by a change of notation.","section":"Section 1.2 and Section 2"},{"comment":"In the sentence before the displayed identity, 'there exists \\epsilon and maximum points' should read 'there exist s_\\epsilon and maximum points'; the parameter \\epsilon is already fixed.","section":"Appendix C, Lemma C.1"},{"comment":"In the contradiction argument it would help to state explicitly that the compact embedding is applied on a fixed fundamental domain after pulling back by (y,r)=(\\theta,\\sigma-p_n\\cdot\\theta), since the skew tori vary with p_n.","section":"Section 3.1, Proposition 3.2"},{"comment":"The notation (4.3) is referenced in the proof but does not appear to be labeled in the text; please add the number or refer to the displayed comparison inequality directly.","section":"Section 4.3, Theorem 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its main ideas, but the perturbative construction is not yet fully rigorous at the points identified in the report. The author should be asked to complete the proof of Lemma 3.1 or to state a precise reference theorem for skew tori, to detail the bootstrap in Lemma 3.2, and to repair the compactness/uniqueness passage in Proposition 3.5. The global definition of the effective Hamiltonian also needs clarification. These are repair tasks within the manuscript's scope, not indications of a fundamentally wrong approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves homogenization for viscous Hamilton-Jacobi equations of contact type H(y,s,p)=F(s,p)+ηW(y,s,p), when F is strictly monotone along rays and the perturbation is small. The main novelty is real: the linearization at the one-dimensional ODE profile gives a periodic-parabolic operator on a p-dependent skew torus, and correctors are constructed by a Fredholm alternative plus contraction. That technique is new, and the unperturbed theorem (with H=F depending on s and gradient) is a useful extension of both the semilinear heat equation setting and the first-order contact theory.\n\nThe unperturbed cell problem is solved completely, with explicit ODE construction, uniqueness, and careful continuity of the cell speed near p=0. The perturbative construction is coherent: the linearized operator has kernel spanned by a_p, the adjoint kernel yields the speed shift, and the fixed point argument produces small correctors. The large-time averaging result for a general class and its identification with the cell speed are natural and well integrated. The examples are honest and illustrate the assumptions.\n\nThe load-bearing point is Lemma 3.1, the uniform periodic-parabolic Schauder estimates and Fredholm alternative on skew tori T_p^{N+1}. The proof as written is a sketch: it cites Lieberman and LSU, then says translations patch the local estimates. A referee will want the dependence on p tracked explicitly—verifying that the Hölder norms of the coefficients b_p, d_p are uniform in the parabolic metric on T_p^{N+1}, and that the covering number and Schauder constants do not degenerate as p varies. I do not think the statement is false; the lattice generators are bounded on compact K and the fundamental domain has fixed volume, so the uniformity should be true. But as written, Section 3 rests on an under-proved lemma. This is a technical gap, not a conceptual one.\n\nMinor caveats: the bootstrap regularity (Lemma 3.2 and Remark 3.1) also relies on the same lemma and is only sketched, and the smallness threshold for η is existential and depends on the initial datum's Lipschitz constant. These are stated honestly but limit the scope: this is a perturbative result, not a resolution of the general cell problem (1.2).\n\nI would send this to peer review. The novel technique and the clean unperturbed part justify referee time. The referees should ask for a complete proof of Lemma 3.1 and a fuller write-up of the bootstrap. My own verdict would be accept after revision, provided those details are supplied. I would cite it if I work on HJ homogenization.","headline":"Genuinely new perturbative construction of periodic-parabolic correctors for viscous contact-type HJ homogenization; the main theorem is credible, but the uniform Schauder/Fredholm lemma on skew tori is under-proved and will need a fuller statement before Theorem 5 is sealed.","tokens_in":33968,"tokens_out":6879,"would_cite":true,"duration_ms":65155,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B27","35F21","35D40","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that viscous Hamilton–Jacobi equations with fast dependence on the phase $u^\\epsilon/\\epsilon$ and on $x/\\epsilon$ homogenize to a first-order effective equation $u_t-c^\\eta(Du)=0$ whenever the spatially oscillatory part…","keywords":["viscous Hamilton-Jacobi equations","homogenization","u/epsilon-periodic Hamiltonians","contact-type Hamilton-Jacobi equations","viscosity solutions","periodic-parabolic correctors","cell problem","large-time averaging"],"falsifier":"Take a smooth positive 1-periodic family $F_\\delta(s,p)=\\delta+\\sqrt{\\delta^2+|p|^2}$ and plot $T_p(\\mu)=\\int_0^1 v_\\mu(s)^{-1}\\,ds$ for the periodic ODE $|p|^2vv'=-\\mu v+F_\\delta(s,pv)$ as $\\delta$ varies. If for some $\\delta>0$ the map $\\mu\\mapsto T_p(\\mu)$ is not strictly increasing, or two different $\\mu$ satisfy $T_p(\\mu)=1$, then the uniqueness of the cell speed in Theorem 3 fails; a single such numerical example would settle the claim.","tokens_in":32938,"feed_emoji":"🌀","tokens_out":13257,"duration_ms":111409,"temperature":0.7,"pith_summary":"This paper proves a homogenization result for a viscous Hamilton-Jacobi equation whose Hamiltonian depends on the unknown through the fast phase $u^\\epsilon/\\epsilon$, in addition to the fast space variable and the gradient. For Hamiltonians of the form $F(s,p)+\\eta W(y,s,p)$ with $|\\eta|$ small enough, the allowed size being set by the Lipschitz constant of the initial data, the solutions converge locally uniformly to the viscosity solution of the first-order effective equation $u_t-c^\\eta(Du)=0$. In the unperturbed case $F(s,p)$, the effective speed comes from a one-dimensional monotone cell ODE, while in the perturbed case it comes from a periodic-parabolic corrector built by linearizing that one-dimensional profile. The paper also proves that the long-time average of the associated cell evolution equals the same cell speed. A reader should care because this is the viscous counterpart of contact-type $u/\\epsilon$-periodic homogenization, combining first-order Hamilton-Jacobi structure with semilinear heat-equation structure.","feed_headline":"Viscous Hamilton–Jacobi: small spatial oscillations still homogenize","feed_subtitle":"Even with the fast u/ε phase, a tiny spatial perturbation still yields an effective first-order equation.","key_machinery":"The central object is the periodic-parabolic corrector $\\chi^\\eta_p(y,z)=z+w^\\eta_p(y,z)$ together with its cell speed $c^\\eta(p)$, solving equation (3.2) on the skew torus $\\mathbb{T}^{N+1}_p$, the torus obtained from the identifications $(y,r)\\sim(y+k,r-p\\cdot k)$ and $(y,r)\\sim(y,r+1)$. In the unperturbed case the corrector reduces to a monotone one-dimensional profile $\\chi_p$, whose slope $v(s)=\\chi'_p$ solves the first-order periodic ODE $|p|^2vv'=cv+F(s,pv)$, with the speed fixed by the normalization $\\int_0^1 v^{-1}\\,ds=1$. For the perturbation, the paper linearizes at $\\eta=0$ and obtains the periodic-parabolic operator $\\tilde L_p\\psi=c^0(p)\\psi_r+F_s(\\chi^0_p,pa_p)\\psi+D_pF(\\chi^0_p,pa_p)\\cdot D_y\\psi-\\Delta_y\\psi$; the strict positivity of $a_p=(\\chi^0_p)'$ ensures the kernel is exactly the span of $a_p$, so the Fredholm alternative makes $\\tilde A_p(\\psi,\\kappa)=\\tilde L_p\\psi+\\kappa a_p$ an isomorphism, and a contraction mapping on a small ball gives the corrector branch with $\\|\\tilde\\psi^\\eta_p\\|_{C^{2+\\alpha,1+\\alpha/2}}+|\\kappa^\\eta_p|\\le C_K|\\eta|$.","core_discovery":"The central claim is that, under positivity, strict subhomogeneity along rays, recession-at-infinity, and regularity assumptions on $F$, and a natural Lipschitz/periodicity condition on $W$, the equation $u^\\epsilon_t+F(u^\\epsilon/\\epsilon,Du^\\epsilon)+\\eta W(x/\\epsilon,u^\\epsilon/\\epsilon,Du^\\epsilon)=\\epsilon\\Delta u^\\epsilon$ homogenizes to $u_t-c^\\eta(Du)=0$ for $|\\eta|$ below a threshold determined by the compact gradient set $K_0=B_{\\mathrm{Lip}(u_0)}(0)$ (Theorem 5). The speed $c^\\eta(p)$ is defined through a corrector $\\chi^\\eta_p(y,z)=z+w^\\eta_p(y,z)$ solving the periodic-parabolic cell equation (3.2), with $\\chi^\\eta_p(y,z+1)=\\chi^\\eta_p(y,z)+1$ and $(\\chi^\\eta_p)_z>0$. The construction linearizes the cell operator at the unperturbed one-dimensional profile; the linearized operator is a periodic-parabolic operator on a skew torus whose kernel is spanned by the strictly positive slope $a_p=(\\chi^0_p)'$, and the Fredholm alternative turns this into an isomorphism, after which a contraction argument yields a unique small corrector branch. For general periodic Hamiltonians satisfying only (H1)-(H2), the paper shows $w(\\zeta,\\tau)/\\tau\\to\\lambda(p)$ for the cell evolution $w_\\tau+H(\\zeta,p\\cdot\\zeta+w,p+Dw)=\\Delta w$, and in the perturbative class this averaged constant coincides with the corrector speed $c^\\eta(p)$.","pith_inferences":["Beyond the paper, Example 5.3 reveals a gauge-type recipe: any smooth periodic $\\varphi(y)$ produces an admissible perturbation $W_\\eta$ by the quotient $[\\Delta_y\\varphi+F(s-\\eta\\varphi,q-\\eta D_y\\varphi)-F(s,q)]/\\eta$, so the perturbative class contains many Hamiltonians with exact correctors $\\chi^0_p(z)+\\eta\\varphi(y)$ and unchanged speed $c^0(p)$.","Beyond the paper, the contraction argument suggests that the smallness threshold $\\eta_K$ can be quantified in terms of the Holder norms of $F$ and $W$ on an enlarged phase-gradient region, which would turn the qualitative homogenization theorem into a rate-bearing construction.","Beyond the paper, the large-time averaging theorem holds under only (H1)-(H2), before any corrector is available; this opens a numerical route to approximate $\\lambda(p)$ by simulating the cell evolution and comparing its long-time average to the corrector speed in the perturbative regime."],"forward_implications":["The unperturbed Hamiltonian $F(s,p)$ alone yields homogenization to $u_t-c(Du)=0$, so a viscous equation with fast phase dependence and no spatial oscillation has a first-order effective equation whose speed is selected by a one-dimensional travelling-wave profile.","When $|\\eta|$ is below the threshold controlled by $\\mathrm{Lip}(u_0)$, the effective speed $p\\mapsto c^\\eta(p)$ is continuous on compact gradient sets, and the corrector admits uniform $C^2$ bounds, which is exactly what Evans' perturbed test function argument needs.","The long-time average of the cell evolution $w(\\zeta,\\tau)/\\tau$ converges to the same speed as the corrector, so in the perturbative class the homogenized Hamiltonian can be read off either from the static cell problem or from the evolutionary cell problem.","The spatial Lipschitz estimate for the half-relaxed limits keeps every gradient appearing in the viscosity test inside the compact set $K_0$ on which the corrector was built, making the smallness condition on $\\eta$ quantitative and datum-dependent.","The explicit example with $\\eta_0=1/10$ shows the perturbative class is non-empty even with a nontrivial spatial oscillation, and in that example the corrector and speed are known in closed form."],"supporting_citations":[{"why":"Establishes the foundational periodic cell-problem framework for Hamilton-Jacobi homogenization that the present cell ODE generalizes.","marker":"[16]"},{"why":"Supplies the phase-shift contact lemma and the embedded $u/\\epsilon$-periodic cell evolution used in Sections 3 and 4.","marker":"[13]"},{"why":"Provides the space-time periodic solution and long-time behavior methods underlying the large-time averaging theorem.","marker":"[3]"},{"why":"Gives the perturbed test function method that converts corrector bounds into viscosity inequalities for the half-relaxed limits.","marker":"[9]"},{"why":"Supplies the semilinear heat equation model whose travelling-wave/cell ODE is recovered when $F$ has no gradient dependence.","marker":"[6]"},{"why":"Gives the periodic-parabolic Schauder estimates used on skew tori in Lemma 3.1.","marker":"[15]"},{"why":"Provides the parabolic regularity theory used to patch local Schauder estimates and to differentiate the corrector equation.","marker":"[14]"},{"why":"Supplies the periodic-parabolic Fredholm alternative and positivity theory that make the linearized operator an isomorphism.","marker":"[12]"},{"why":"Underpins the viscosity comparison principle proved in Appendix A and used twice in the homogenization argument.","marker":"[8]"}],"fun_headline_variants":["Small spatial wiggle doesn't break homogenization in viscous HJ","Tiny spatial oscillations still homogenize with u/ε dependence","Viscous HJ: small spatial perturbations don't stop homogenization","Sublinear u/ε HJ homogenizes even with tiny spatial oscillations","Small η spatial term doesn't alter homogenized law in viscous HJ"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The strict decrease of the ratio $F(s,\\rho p)/\\rho$ as $\\rho$ grows is the load-bearing premise; without it, the one-dimensional cell problem may have several speeds or a non-positive corrector slope, and the later perturbative step would lose the unique linearized solution it needs.","fun_headline_variants_meta":{"raw":{"variants":["Small spatial wiggle doesn't break homogenization in viscous HJ","Tiny spatial oscillations still homogenize with u/ε dependence","Viscous HJ: small spatial perturbations don't stop homogenization","Sublinear u/ε HJ homogenizes even with tiny spatial oscillations","Small η spatial term doesn't alter homogenized law in viscous HJ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000516,"raw_usage":{"total_tokens":2535,"prompt_tokens":1005,"completion_tokens":1530,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":1441}},"tokens_in":621,"tokens_out":1530,"duration_ms":11457,"temperature":1.0,"reasoning_tokens":1441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:21:59.785853+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth positive 1-periodic family $F_\\delta(s,p)=\\delta+\\sqrt{\\delta^2+|p|^2}$ and plot $T_p(\\mu)=\\int_0^1 v_\\mu(s)^{-1}\\,ds$ for the periodic ODE $|p|^2vv'=-\\mu v+F_\\delta(s,pv)$ as $\\delta$ varies. If for some $\\delta>0$ the map $\\mu\\mapsto T_p(\\mu)$ is not strictly increasing, or two different $\\mu$ satisfy $T_p(\\mu)=1$, then the uniqueness of the cell speed in Theorem 3 fails; a single such numerical example would settle the claim.","supporting_citations":[{"cited_title":"Homogenization of hamilton–jacobi equations","cited_arxiv_id":null,"evidence_quote":"Establishes the foundational periodic cell-problem framework for Hamilton-Jacobi homogenization that the present cell ODE generalizes."},{"cited_title":"Homogenization of first- order equations with (u/ǫ)-periodic Hamiltonians","cited_arxiv_id":null,"evidence_quote":"Supplies the phase-shift contact lemma and the embedded $u/\\epsilon$-periodic cell evolution used in Sections 3 and 4."},{"cited_title":"Space-time periodi c solutions and long-time behavior of solutions to quasi-linear parabolic equations","cited_arxiv_id":null,"evidence_quote":"Provides the space-time periodic solution and long-time behavior methods underlying the large-time averaging theorem."},{"cited_title":"The perturbed test function method fo r viscosity solutions of nonlinear pde","cited_arxiv_id":null,"evidence_quote":"Gives the perturbed test function method that converts corrector bounds into viscosity inequalities for the half-relaxed limits."},{"cited_title":"Homogenizatio n of a semilinear heat equation","cited_arxiv_id":null,"evidence_quote":"Supplies the semilinear heat equation model whose travelling-wave/cell ODE is recovered when $F$ has no gradient dependence."},{"cited_title":"Second Order Parabolic Differe ntial Equations","cited_arxiv_id":null,"evidence_quote":"Gives the periodic-parabolic Schauder estimates used on skew tori in Lemma 3.1."},{"cited_title":"Linear and Quasilinear Equations of Parabolic Type","cited_arxiv_id":null,"evidence_quote":"Provides the parabolic regularity theory used to patch local Schauder estimates and to differentiate the corrector equation."},{"cited_title":"Periodic-Parabolic Boundary V alue Problems and Positivity","cited_arxiv_id":null,"evidence_quote":"Supplies the periodic-parabolic Fredholm alternative and positivity theory that make the linearized operator an isomorphism."},{"cited_title":"User’s gui de to viscosity solutions of second order partial differ- ential equations","cited_arxiv_id":null,"evidence_quote":"Underpins the viscosity comparison principle proved in Appendix A and used twice in the homogenization argument."}],"review_version":2}