{"id":"6b7c07e8-9d01-49b6-9fc3-aa57896ea810","arxiv_id":"2608.11449","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For convex, periodic, particle-symmetric Hamilton-Jacobi equations in infinite dimensions with mean-only initial data, the homogenization error is O(epsilon), and this rate is optimal.","lead":"This paper proves that solutions of a certain infinite-dimensional Hamilton-Jacobi equation approach their averaged, effective solution at the fastest possible rate, of order epsilon, instead of the slower epsilon^(1/3) known before. It matters because these equations describe large systems of indistinguishable particles, and the proof introduces a geometric way to cut and glue particle trajectories in infinite dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's O(ε) rate is inherited from Lemma 2.1, an imported mean-reduction estimate from [27] that is not proved here; if that estimate is only O(ε^{1/3}), the optimal-rate claim would not follow.","rationale":"The reader's conditional verdict correctly identifies the external dependency on [27] as the weakest point. I agree that the O(ε) mean-reduction is the highest-stakes unproved ingredient: it is exactly the theorem's rate and is not re-derived. The general almost-subadditivity sketch in Remark 1 is a second gap in the internal argument, but it is likely fillable by the same curve-surgery technique. The rest of the proof—curve surgery, dyadic estimates, metric identification via convex duality, and the sharpness example—is coherent and contains no detected internal contradiction. Therefore the appropriate verdict remains CONDITIONAL as the reader assigned; my stress-test does not move it.","tokens_in":18954,"tokens_out":48598,"duration_ms":403166,"concrete_test":"Open [27] at Propositions 2.5 and 2.8 and verify whether the mean-reduction estimate is O(ε) and uniform in t. If not, attempt the short direct proof: by ε-periodicity and rearrangement invariance, u^ε is invariant under x ↦ x∘g + εz; compactness of εSS^d gives g,z with ∥x - (Mx∘g + εz)∥ ≤ εD; the uniform Lipschitz estimate in Lemma 2.1 then yields |u^ε(x,t)-u^ε(Mx,t)| ≤ C εD. If neither source supplies this derivation, Theorem 1.1 rests on an unverified O(ε) ingredient. Also, write out the non-dyadic concatenation in Remark 1 for ρ=2, σ=1 with a≠0 to confirm the bounded connector cost.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.1 (§2.1) asserts |u^ε(x,t)-u^ε(Mx,t)| ≤ C ε, citing [27, Props 2.5 and 2.8]. This estimate has the same order as the theorem and is used in the essential reduction (2.6) and in Lemma 2.2. The paper's abstract states [27] only gives qualitative homogenization with an O(ε^{1/3}) full convergence rate; it is not established in the present text that the mean-reduction component is O(ε) uniformly in t. If [27] supplies only a slower rate, the conclusion of Theorem 1.1 would not follow. The estimate is plausible—it should follow from the uniform Lipschitz bound plus compactness of the ε-periodic rearrangement quotient—but the manuscript does not provide the derivation. A related gap is Remark 1: the general almost-subadditivity estimates (3.10)-(3.11), needed for the Fekete limit in Prop 4.1 and for convexity of m̄ in Prop 4.4 Step 4, are only sketched, not proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an O(ε) convergence rate for periodic homogenization of convex Hamilton–Jacobi equations on the Hilbert space V = L²(I; R^d), where the Hamiltonian is periodic in the configuration variable and invariant under measure-preserving rearrangements, and the initial data depend only on the mean configuration. The proof reduces the problem to a mean-endpoint action metric, establishes dyadic almost-subadditivity and almost-superadditivity estimates by a curve-surgery argument that combines Burago's finite-dimensional cutting lemma applied to the mean-time path with the compactness of the rearrangement-periodicity quotient, constructs a homogenized metric, identifies it with the Legendre transform of the effective Hamiltonian via convex duality, and concludes by comparing the Lax–Oleinik representations. It ends with an explicit example showing that the O(ε) rate cannot be improved.","tokens_in":19161,"tokens_out":21941,"duration_ms":192938,"significance":"If the two missing ingredients identified below are supplied, this is a substantial contribution: it extends the finite-dimensional optimal-rate theorem of Tran–Yu to an infinite-dimensional indistinguishable-particle setting, with a clean two-scale reduction and an explicit sharpness example. The paper's detailed dyadic estimates, careful connector cost bounds, the use of the compact quotient, and the explicit ε/6 lower bound in the optimality example are strengths. The proof is not parameter-fitted; constants are explicit and depend only on H and ||Du0||∞. The result would be the first optimal convergence rate in this infinite-dimensional periodic setting under convexity.","major_comments":[{"comment":"The O(ε) mean-reduction estimate |u^ε(x,t) - u^ε(Mx,t)| ≤ Cε is imported from the author's preprint [27] and is not proved in this manuscript. This estimate has exactly the same order as Theorem 1.1 and is used in the essential reduction (2.6), in Lemma 2.2, and in the final step of the proof of Theorem 1.1. The paper's own abstract states that [27] establishes only qualitative homogenization with an O(ε^{1/3}) full convergence rate; the O(ε) mean-reduction component therefore cannot be inferred from the cited qualitative result. Without a proof, or a precise statement of the relevant proposition in [27] together with its proof, the main theorem is conditional on an unverified input.","section":"§2.1, Lemma 2.1"},{"comment":"The general almost-subadditivity estimates (3.10) and (3.11) are asserted without proof, with the sentence 'These estimates follow by repeating...' and no details. These estimates are load-bearing: (3.10) is used as (4.3) in Proposition 4.1 to apply Fekete's lemma and obtain the limit defining the homogenized metric, and (3.11) is used in Proposition 4.4, Step 4, to prove convexity of q ↦ m̄(1,0,q). The reader cannot verify the endpoint-connector and mean-correction details for arbitrary scaling factors ρ,σ or for three arbitrary endpoints. Please supply complete proofs of both estimates, or restate them as theorems with full arguments.","section":"§3.1, Remark 1"}],"minor_comments":[{"comment":"The same symbol m is used for the mean functional m(x), the microscopic mean-endpoint metric m(t,a,q), and the homogenized metric (often written m̄); this overloading makes Section 2.2 and Lemma 4.3 hard to follow. Please use distinct notation for the three objects.","section":"§2.2 and §4"},{"comment":"Figure 3.1 is referenced in the text but no figure appears in the manuscript; either include the figure or remove the reference.","section":"Figure 3.1"},{"comment":"In Lemma 2.3, the statement that the number of intervals is bounded by (m+1)/2 should explicitly identify m = d+1 for the mean-time path and should briefly explain the topological origin of the bound, since the constant matters in the subsequent time-saving argument.","section":"§2.3, Lemma 2.3"},{"comment":"In the proof of the optimality example, the sentence 'Since H is even and convex, it attains its minimum at p=0' would be clearer with the one-line justification that convexity gives H(0) ≤ (H(p)+H(-p))/2 = H(p) for every p.","section":"§5.1, Example 1"},{"comment":"Because several load-bearing results are cited from the author's unpublished preprint [27], the manuscript should state explicitly which propositions of [27] are used and, ideally, include their statements in an appendix for the reader.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The two major gaps are fixable but require writing substantial proofs: the O(ε) mean-reduction lemma and the general almost-subadditivity estimates. Both are load-bearing and both come from the author's own unpublished preprint [27]. The editor may wish to check the status of [27] and ask the author to include the needed arguments in the present paper. I see no circularity or data fitting in the argument as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is the real thing, with one important caveat. Park proves O(ε) convergence for periodic homogenization of convex Hamilton–Jacobi equations in L² space of particle configurations, under rearrangement invariance and mean-dependent initial data. That is a genuine extension of Tran–Yu's finite-dimensional result, and the main device—cutting the mean-time projection and gluing pieces back in the full Hilbert space via the compact rearrangement-periodic quotient—is new and fits the problem well. The dyadic almost-subadditivity and almost-superadditivity estimates in Section 3 are carefully argued, and the sharpness example (u^ε ≥ ε/6 at a specific point) checks out. If the main theorem is correct, it closes the optimal-rate question for this class, so this is a meaningful contribution to the homogenization/weak-KAM subfield.\n\nThe soft spots are real but not fatal. The first is Lemma 2.1, which asserts |u^ε(x,t)−u^ε(Mx,t)| ≤ Cε uniformly in t. This is imported from the author's preprint [27], and it is load-bearing: it is used to pass from the full solution to the mean-restricted value function and back, and its error order matches the theorem's conclusion. The present paper does not reproduce the proof. The stress-test note is legitimate: if [27] only gives O(ε^{1/3}) for this comparison, the optimal-rate proof would not go through. The author is candid about the dependency, but you cannot verify it from the text. The second issue is Remark 1: the general almost-subadditivity estimates (3.10) and (3.11) are only sketched, yet they are needed for the Fekete-lemma construction in Proposition 4.1 and for the convexity argument in Proposition 4.4. That is a genuine gap in presentation, even if the estimates are probably true by repeating the earlier arguments.\n\nThere is nothing circular or fabricated here. The metric identification with the effective Lagrangian is done through the cell problem and comparison, not by assuming the target rate. The citation pattern is honest—earlier work [27], the Wasserstein paper [28], and Tran–Yu [1] are all positioned correctly. The paper is a bit too reliant on [27] for comfort, and the sketched Remark 1 means a referee would have to either trust the author or ask for details.\n\nBottom line: this deserves a serious referee. I would send it to peer review and ask the author to either prove the mean-reduction estimate in the present paper or supply a precise statement from [27] with the full argument, and to expand Remark 1. If those pieces hold up, the paper is publishable and will be cited.","headline":"A solid extension of the finite-dimensional optimal-rate result to an infinite-dimensional setting, with a clear main theorem and honest proof outline; the catch is that two load-bearing estimates are imported from the author's own preprint or only sketched, so the paper is not yet fully self-contained.","tokens_in":19690,"tokens_out":5237,"would_cite":true,"duration_ms":47189,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","37J50","49L25","35B27","35F21","35R15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that periodic homogenization of a convex infinite-dimensional Hamilton–Jacobi equation with mean-only initial data converges at the optimal rate O(ε), with an example showing this rate is sharp.","keywords":["periodic homogenization","infinite-dimensional Hamilton-Jacobi equations","effective Hamiltonian","optimal convergence rate","action metric","rearrangement invariance","viscosity solutions","mean configuration"],"falsifier":"For a Hamiltonian of the form $H(x,p)=\\frac{1}{2}\\|p\\|^2+\\int_I W(x(i))\\,di$ with smooth $1$-periodic $W$ and $u_0\\equiv0$, compute the true $\\varepsilon$-dependence of $\\sup_{x\\in V,t\\ge0}|u^\\varepsilon(x,t)|$. If any such $W$ yields an error growing faster than linearly in $\\varepsilon$, Theorem 1.1 is false; the paper's own example $W(r)=-4\\cos^2(\\pi r)$ gives the matching lower bound $\\varepsilon/6$, so either a proof of the linear upper bound for all such $W$ or one counterexample with a different $W$ would settle the claim.","tokens_in":18735,"feed_emoji":"📐","tokens_out":14144,"duration_ms":109269,"temperature":0.7,"pith_summary":"The paper proves that a convex, periodic, rearrangement-invariant Hamilton–Jacobi equation posed on the Hilbert space $V=L^2(I;\\mathbb{R}^d)$ -- configurations of indistinguishable particles on a $d$-dimensional torus -- converges, when the initial data depend only on the mean configuration, to a finite-dimensional effective solution at the optimal rate $C\\varepsilon$, uniformly in $x$ and $t$. Previously, only a nonconvex rate of $O(\\varepsilon^{1/3})$ was known in this infinite-dimensional setting. The proof replaces the full infinite-dimensional comparison by a comparison of mean-restricted value functions through a mean-endpoint action metric, whose large-scale limit is identified with the effective Lagrangian via convex duality and the cell problem. A lower-bound example shows the linear rate is sharp, so the result is optimal.","feed_headline":"Optimal O(ε) rate proven for infinite-dimensional homogenization","feed_subtitle":"For mean-only data, an infinite particle system converges to its effective limit at the sharpest possible rate.","key_machinery":"The central object is the mean-endpoint action metric\n$$m(t,a,q)=\\inf\\left\\{\\int_0^t L(\\gamma(s),\\dot\\gamma(s))\\,ds:\\gamma\\in AC([0,t];V),\\ m(\\gamma(0))=a,\\ m(\\gamma(t))=q\\right\\},$$\nwhere $L$ is the convex dual of $H$ in the momentum variable and only the means of the endpoints are fixed. The main estimates are dyadic almost-subadditivity and almost-superadditivity: $|m(2t,2a,2q)-2m(t,a,q)|\\le C_R$ for $|q-a|\\le Rt$. Subadditivity is obtained by cutting, rearranging, and gluing near-minimizing curves using the compact quotient of $V$ by periodicity $\\Lambda$ and particle rearrangements $G$; superadditivity uses a topological splitting lemma applied to the finite-dimensional mean-time projection $\\xi(s)=(m(\\gamma(s)),s)$, which splits the mean-time displacement $(2y,2t)$ into two families of pieces with displacement $(y,t)$, then lifts and reglues the pieces in $V$ with uniformly bounded connectors. These bounds make the rescaled metric converge to a homogenized metric satisfying\n$$\\bar m(t,a,q)=t\\bar L\\left(\\frac{q-a}{t}\\right),$$\nwhere $\\bar L$ is the effective Lagrangian; identifying $\\bar m$ with $t\\bar L((q-a)/t)$ through convex duality and the cell problem converts the metric comparison into the linear-in-$\\varepsilon$ error for the value functions.","core_discovery":"Under assumptions (H1)--(H5) (periodicity, rearrangement invariance, Lipschitz continuity, coercivity, convexity in momentum) and (I1)--(I2) (initial data depending only on the mean $Mx=m(x)\\chi_I$), the paper proves\n$$\\|u^\\varepsilon(x,t)-\\bar u(m(x),t)\\|_{L^\\infty(V\\times[0,\\infty))}\\le C\\varepsilon,$$\nwith $C$ depending only on $H$ and $\\|Du_0\\|_{L^\\infty(V)}$, where $\\bar u$ solves the effective finite-dimensional Hamilton--Jacobi equation with effective Hamiltonian $\\bar H$. The argument reduces the full problem to comparing $u^\\varepsilon$ with $\\bar u$ at mean configurations, using the mean-reduction estimate $|u^\\varepsilon(x,t)-u^\\varepsilon(Mx,t)|\\le C\\varepsilon$, and then to showing that the mean-endpoint action metric $m(t,a,q)$ is $O(1)$-close to its homogenized limit. The sharpness example ($H(x,p)=\\frac{1}{2}\\|p\\|^2+\\int_I W(x(i))\\,di$ with $W(r)=-4\\cos^2(\\pi r)$, $u_0\\equiv0$) gives $u^\\varepsilon(0\\chi_I,1)\\ge\\varepsilon/6$, proving the rate cannot be improved.","pith_inferences":["The proof does not handle initial data depending on the full configuration: the $O(\\varepsilon)$ mean-reduction step is specific to mean-only data, so extending the rate to general data would need a different mechanism.","Because the gluing relies on compactness of the quotient by periodicity and rearrangement, the method likely fails for particle systems without rearrangement symmetry; there the curve-surgery connectors would not be uniformly bounded, so a slower rate may be the best available.","The exact constant in Theorem 1.1 is not identified; quantifying it in terms of the diameter of the quotient and the coercivity and Lipschitz constants of $H$ would make the error bound directly usable.","A natural testable extension is to replace the mean by finitely many low-order moments; the analogue would be a moment-endpoint metric, and whether a compact quotient exists for those observables would determine whether the $O(\\varepsilon)$ rate persists."],"forward_implications":["For mean-dependent initial data, the infinite particle system is quantitatively governed by a finite-dimensional effective equation with a uniform $O(\\varepsilon)$ error over all time.","The large-scale limit of the mean-endpoint action metric is exactly the effective Lagrangian cost, so optimal-control problems on the infinite-dimensional space reduce, up to an $O(1)$ action error, to finite-dimensional mean dynamics.","Because the sharpness example has $u^\\varepsilon(0\\chi_I,1)\\ge\\varepsilon/6$, no general improvement beyond $O(\\varepsilon)$ is possible without adding hypotheses on the Hamiltonian or the initial data.","The curve-surgery template -- cut a near-minimizer through its finite-dimensional mean-time projection, then glue the lifted pieces using the compact quotient by periodicity and rearrangement -- works because the quotient $V/\\sim$ is compact; the same scheme should transfer to other infinite-dimensional variational problems with a compact symmetry quotient."],"supporting_citations":[{"why":"Supplies the mean-reduction estimate, qualitative homogenization, uniform Lipschitz bounds, and cell-problem correctors; it is the bridge from the full Hilbert space to the mean.","marker":"[27]"},{"why":"Finite-dimensional optimal $O(\\varepsilon)$ rate via action metric and curve surgery that the paper extends, and source of the sharpness-example idea.","marker":"[1]"},{"why":"Topological lemma that splits a continuous mean-time path into pieces with half the total mean-time displacement, used for almost-superadditivity.","marker":"[7]"},{"why":"Provides the indistinguishable-particle setting, the periodic and rearrangement-invariant Hamiltonian, and the compactness of the quotient $V/\\sim$ used for gluing.","marker":"[23]"},{"why":"Establishes the Lagrangian action representation and calculus-of-variations framework in Hilbert spaces behind the optimal-control formula for $u^\\varepsilon$.","marker":"[22]"}],"fun_headline_variants":["Sharp O(ε) rate for infinite-particle homogenization","Mean-only data yield O(ε) homogenization in infinite dimensions","Infinite-dimensional HJ homogenization achieves sharp O(ε)","Cut-and-glue method proves optimal rate for particle HJ","Sharpest possible convergence for infinite particle systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper imports, from its companion work, the estimate $|u^\\varepsilon(x,t)-u^\\varepsilon(Mx,t)|\\le C\\varepsilon$ holding uniformly in $t$, and this mean-reduction estimate has exactly the same order as the theorem; if it were only available at a slower rate or with a time-dependent constant, the optimal-rate proof would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Sharp O(ε) rate for infinite-particle homogenization","Mean-only data yield O(ε) homogenization in infinite dimensions","Infinite-dimensional HJ homogenization achieves sharp O(ε)","Cut-and-glue method proves optimal rate for particle HJ","Sharpest possible convergence for infinite particle systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000748,"raw_usage":{"total_tokens":3346,"prompt_tokens":973,"completion_tokens":2373,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":2293}},"tokens_in":589,"tokens_out":2373,"duration_ms":14054,"temperature":1.0,"reasoning_tokens":2293,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:14:03.111785+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a Hamiltonian of the form $H(x,p)=\\frac{1}{2}\\|p\\|^2+\\int_I W(x(i))\\,di$ with smooth $1$-periodic $W$ and $u_0\\equiv0$, compute the true $\\varepsilon$-dependence of $\\sup_{x\\in V,t\\ge0}|u^\\varepsilon(x,t)|$. If any such $W$ yields an error growing faster than linearly in $\\varepsilon$, Theorem 1.1 is false; the paper's own example $W(r)=-4\\cos^2(\\pi r)$ gives the matching lower bound $\\varepsilon/6$, so either a proof of the linear upper bound for all such $W$ or one counterexample with a different $W$ would settle the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Finite-dimensional optimal $O(\\varepsilon)$ rate via action metric and curve surgery that the paper extends, and source of the sharpness-example idea."},{"cited_title":"Periodic metrics.Adv","cited_arxiv_id":null,"evidence_quote":"Topological lemma that splits a continuous mean-time path into pieces with half the total mean-time displacement, used for almost-superadditivity."},{"cited_title":"& Nurbekyan, L","cited_arxiv_id":null,"evidence_quote":"Provides the indistinguishable-particle setting, the periodic and rearrangement-invariant Hamiltonian, and the compactness of the quotient $V/\\sim$ used for gluing."},{"cited_title":"& Nurbekyan, L","cited_arxiv_id":null,"evidence_quote":"Establishes the Lagrangian action representation and calculus-of-variations framework in Hilbert spaces behind the optimal-control formula for $u^\\varepsilon$."}],"review_version":1}