{"id":"9ba7a4d7-be65-48ea-acf9-08808d2b64c3","arxiv_id":"2607.07526","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The vanishing viscosity approximation for first-order MFGs with nonlocal coupling converges at rate O(ε^{1/2}) for the value function and O(ε^{1/8}) for the density in Wasserstein distance.","lead":"This paper proves quantitative convergence rates for the vanishing viscosity approximation of Mean Field Games with nonlocal coupling, improving the known rate for the value function and establishing a new rate for the density. A generalist might read it to understand how fast noisy game models converge to deterministic ones, which matters for numerical and analytical modeling.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The L^1 bound on ∆u^ε (Lemma 4.1) is used to control the adjoint term in Step 2, but the proof of (4.2) additionally requires ∥ρ^ε∥_{L∞} independent of ε, which is not established.","rationale":"The reader identified assumption (4.4) — the uniform W^{2,∞} bound on u^ε — as the weakest link, noting it limits the generality of the density convergence result. This is a fair observation about the scope of the Wasserstein rate (4.5), but it is not the most load-bearing concern for the correctness of the paper's central claims. The reader accepted the proofs of (4.1)–(4.3) at face value. My concern is that the proof of (4.2) — and by extension (4.3), which is used in the Wasserstein estimate — contains a gap: the uniform L∞ bound on the adjoint ρ^ε is asserted but not justified. This is a correctness issue, not merely a scope issue. If ∥ρ^ε∥_{L∞} is not uniformly bounded, the L^1 estimate (4.2) does not close, and since (4.3) depends on the L∞ or L^1 bound on u^ε - u, the gradient estimate (4.3) may also fail, which would undermine the Wasserstein rate (4.5) entirely — even under assumption (4.4). I note that the L∞ estimate (4.1) in Step 1 uses a different duality argument where ρ^ε appears inside an integral with |D²u^ε|², and the Bernstein method is used to control that term; the L∞ bound on ρ^ε is not needed there in the same way. So the concern is specifically about Step 2. It is possible that the one-sided bound on div(D_pH(x, Du^ε)) — which the authors note is controlled via semiconcavity and (H4) in the proof of Theorem 4.2 — does suffice for a uniform L∞ bound on ρ^ε via the De Giorgi or maximum principle machinery for Fokker-Planck equations. If so, the gap is fillable and the concern does not land. But this argument is not present in the paper, and the assertion is stated without justification or citation. The verdict moves from ACCEPT to CONDITIONAL: the result should hold conditional on filling this gap. The paper's contributions are substantial and the overall strategy is sound, but this step needs verification.","tokens_in":13948,"tokens_out":1359,"duration_ms":1131181,"concrete_test":"Verify whether ∥ρ^ε∥_{L∞(Q_T)} is uniformly bounded in ε for the adjoint equation (4.6). Specifically: (1) Write the explicit form of div(D_pH(x, Du^ε)) using the chain rule, which yields terms involving D²_{pp}H · D²u^ε and D²_{px}H. (2) Check whether the semiconcavity bound D²u^ε ≤ C (Lemma 4.1) combined with (H4) (uniform convexity, D²_{pp}H ≥ θI_d) provides a one-sided bound on div(D_pH) that is sufficient for a uniform L∞ bound on ρ^ε via the maximum principle for the Fokker-Planck equation. (3) If the one-sided bound suffices, write out the explicit maximum principle argument. If it does not, identify what additional regularity on u^ε is needed and whether (4.4) provides it.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Step 2 of the proof of Theorem 4.6 (the L^1 estimate (4.2)), the authors bound the term ∫|∂_ε u^ε| dx by duality against the adjoint ρ^ε solving (4.6). The estimate reads: ∫|∂_ε u^ε| dx ≤ ∥∆u^ε∥_{L^1} ∥ρ^ε∥_{L∞} + ∥∂_ε F[m^ε]∥_{L^1} ∥ρ^ε∥_{L∞} + ∥∂_ε G∥_{L^1}. The text then states 'Since ∥∆u^ε∥_{L^1}, ∥ρ^ε∥_{L∞} are bounded independent of ε, we get...' The uniform bound on ∥∆u^ε∥_{L^1} is justified by Lemma 4.1 (semiconcavity). However, the uniform L∞ bound on the adjoint ρ^ε is not established anywhere in the paper. The adjoint ρ^ε solves a Fokker-Planck equation with drift D_pH(x, Du^ε) and diffusion ε. While the maximum principle gives ∥ρ^ε∥_{L^1} = 1, an L∞ bound requires controlling the divergence of the drift, i.e., div(D_pH(x, Du^ε)), which involves D²_{pp}H · D²u^ε. The semiconcavity bound D²u^ε ≤ C from Lemma 4.1 gives only an upper bound on ∆u^ε, not |∆u^ε|, and the negative part of the divergence [−div(D_pH(x, Du^ε))]⁻ is controlled (as noted in the proof of Theorem 4.2, citing [Eva10b]). But a uniform L∞ bound on ρ^ε requires controlling the full divergence or its negative part in a way that yields a parabolic Harnack-type bound independent of ε. The paper does not provide this argument. If ∥ρ^ε∥_{L∞} grows as ε → 0 (which is typical for adjoint equations with vanishing diffusion), the L^1 estimate (4.2) would not close, and the subsequent gradient estimate (4.3) — which feeds into the Wasserstein rate (4.5) — would also be affected, since (4.3) relies on (4.1) or (4.2) for the ∥u^ε - u∥_{L∞} factor.","agreement_with_reader":"disagree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript studies quantitative convergence rates for the vanishing viscosity approximation of first-order, time-dependent Mean Field Games (MFGs) with a nonlocal regularizing coupling. Under standard structural assumptions on the Hamiltonian (H1)-(H4) and regularizing properties of the coupling F, the authors improve the previously known convergence rate for the value function u^ε from O(ε^{1/4}) to O(ε^{1/2}) in both L^1 and L^∞. Furthermore, by exploiting duality methods and the semiconcavity of the solutions, they establish a quantitative convergence rate for the density m^ε in the 2-Wasserstein distance, W_2(m^ε, m) ≤ C ε^{1/8}, provided an additional uniform W^{2,∞} bound on u^ε holds. The proofs rely on integral methods, Bernstein-type estimates, and stability estimates for linear Fokker-Planck equations.","tokens_in":14469,"tokens_out":1590,"duration_ms":643880,"significance":"The paper makes a solid contribution to the analysis of MFGs by providing the first quantitative convergence rate for the Fokker-Planck component in the vanishing viscosity limit under these structural assumptions. The improvement of the rate for the Hamilton-Jacobi equation from O(ε^{1/4}) to O(ε^{1/2}) is a notable sharpening of the results in [TZ25]. The strategy of partially decoupling the system—first transferring the HJ convergence rate to the drift field, and then propagating it to the Fokker-Planck equation via stochastic characteristics—is elegant and exploits the regularizing structure of the coupling effectively. The authors clearly delineate the conditions under which the additional W^{2,∞} bound (Assumption 4.4) is satisfied, such as convex data or short time horizons, which provides concrete, falsifiable applicability criteria for the main results.","major_comments":[{"comment":"§4, Proof of Theorem 4.6, Step 2 (Eq. 4.2): The skeptic's concern regarding the uniform L^∞ bound on the adjoint state ρ^ε is valid and represents a load-bearing gap in the current proof. In the derivation of the L^1 estimate (4.2), the authors bound the term ∫|∂_ε u^ε| dx by duality against the adjoint ρ^ε solving (4.6). The text states: 'Since ∥∆u^ε∥_{L^1(Q_T)}, ∥ρ^ε∥_{L^∞(Q_T)} are bounded independent of ε, we get...'. While the uniform bound on ∥∆u^ε∥_{L^1} is justified by Lemma 4.1 (semiconcavity), the uniform L^∞ bound on ρ^ε is not established anywhere in the manuscript. The adjoint ρ^ε solves a Fokker-Planck equation with drift D_pH(x, Du^ε) and vanishing diffusion ε. A uniform L^∞ bound on ρ^ε requires controlling the divergence of the drift, which involves D^2_{pp}H · D^2u^ε. The semiconcavity bound from Lemma 4.1 only provides an upper bound on ∆u^ε, not |∆u^ε|. The authors do","section":null},{"comment":"not provide a parabolic Harnack-type argument or any other justification for why ∥ρ^ε∥_{L^∞} remains bounded independently of ε as the diffusion vanishes. If ∥ρ^ε∥_{L^∞} grows as ε → 0, the L^1 estimate (4.2) does not close. This issue must be addressed, either by providing the missing estimate for ρ^ε or by modifying the duality argument. Note that this concern does not affect the L^∞ estimate (4.1) in Step 1, where the initial datum for the adjoint is a probability measure and the estimate relies on the L^1 conservation of ρ^ε rather than its L^∞ norm.","section":null}],"minor_comments":[{"comment":"§1.3, Main contributions: The text mentions 'new quantitative convergence of gradients in L^∞_t(L^2_x)'. It would be helpful to explicitly state the rate O(ε^{1/4}) here, matching the statement in Theorem 4.6 (4.3), for consistency and immediate impact.","section":null},{"comment":"§4, Theorem 4.2: The proof involves a double parameter η ≤ ε and then lets η → 0. The notation and the logic flow here are slightly terse; expanding the explanation of how the limit η → 0 recovers the estimates with m (rather than m^η) would improve readability.","section":null},{"comment":"§4, Proof of Theorem 4.6, Step 1: The claim that '2ε ∫∫ |D^2u^ε|^2 ρ^ε dxdt ≤ C(...)' is justified via the Bernstein method on ω^ε = |Du^ε|^2. The duality argument transferring the bound from the ω^ε equation to the integral against ρ^ε is sketched very briefly. A few additional lines detailing this transfer would strengthen the rigor of this key step.","section":null},{"comment":"Remark 4.5: The reference to [TZ25, Remark 7.1] for the verification of (F1)-(F2) is noted. It might be beneficial to briefly sketch the core idea of why the convolution structure F[m] = f(x, k ⋆ m) ⋆ k satisfies these assumptions, to make the manuscript more self-contained.","section":null},{"comment":"Typo in §4, Proof of Theorem 4.6, Step 2: 'Since ∥∆u^ε∥_{L^1(Q_T)}, ∥ρ^ε∥_{L^∞(Q_T)} are bounded independent of ε, we get, integrating in ε, the following bound ∫ |(u^ε - u)(τ, x)| dx ≤ C_1 ε + 2C_2 √ε.' The term C_1 ε appears to be a typo, as the preceding estimate yields a bound of C_1 + C_2/√ε for ∂_ε u^ε, which upon integration in ε should give C_1 ε + 2C_2 √ε. This is consistent, but the intermediate display equation showing the bound on ∂_ε u^ε (currently '≤ C_1 + C_2/√ε') should be clearly stated before the integration step.","section":null}],"recommendation":"major_revision","confidential_remarks":"The central claims of the paper are interesting and the overall strategy is sound. However, the gap in the proof of the L^1 estimate (4.2) identified by the skeptic is a substantive issue that needs to be resolved before the paper can be accepted. The authors need to either provide a reference or a proof for the uniform L^∞ bound on the adjoint ρ^ε, or adjust their duality argument to circumvent this requirement. If they can resolve this, the paper should be acceptable for publication. I recommend major revision to give them the opportunity to address this specific technical gap."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the constructive assessment. The referee raises one major concern regarding the uniform L^∞ bound on the adjoint state ρ^ε in Step 2 of the proof of Theorem 4.6, which we address below.","responses":[{"response":"The referee is correct that the uniform L^∞ bound on ρ^ε is not currently justified in the manuscript, and we acknowledge this gap in Step 2. We have carefully re-examined the argument and agree that the semiconcavity estimate from Lemma 4.1 provides only an upper bound on ∆u^ε, which is insufficient to control div(D_pH(x,Du^ε)) and hence insufficient to establish a uniform L^∞ bound on ρ^ε via standard maximum principle or Harnack-type arguments for the adjoint Fokker-Planck equation with vanishing diffusion. We will revise the proof of (4.2) to eliminate the dependence on ∥ρ^ε∥_{L^∞}. The key observation is that the duality argument in Step 2 can be restructured to use an L^1-adjoint rather than an L^∞-adjoint. Specifically, instead of taking ρ(τ) = sgn(∂_ε u^ε(τ)) ∈ L^∞ and bounding ∫|∂_ε u^ε| dx by duality against ρ^ε with the L^∞ norm of ρ^ε appearing, one can work directly with the L^1 norm of the adjoint state (which is conserved and equals 1) combined with the L^1 bound on ∆u^ε from Lemma 4.1 and the L^1 bound on ∂_ε F[m^ε] from Corollary 4.4. Concretely, the term ∫∫ ∆u^ε ρ^ε dxdt is bounded by ∥∆u^ε∥_{L^1} · ∥ρ^ε∥_{L^∞} in the current draft, but can alternatively be controlled by ∥∆u^ε∥_{L^1} · ∥ρ^ε∥_{L^1} = ∥∆u^ε∥_{L^1} when the adjoint is initialized as a probability measure (as in Step 1), using the mass conservation property of the Fokker-Planck equation. This mirrors the structure already used successfully in Step 1, where the referee correctly notes the argument closes because it relies on L^1 conservation rather than L^∞ bounds. We will rewrite Step 2 to make this explicit, removing the unjustified appeal to ∥ρ^ε∥_{L^∞}. We note that this modification does not affect the final rate O(√ε) for (4.2), nor does it impact the subsequent estimates (4.3) and (4","revision_made":"yes","referee_comment":"The referee identifies a load-bearing gap in Step 2 (Eq. 4.2) of the proof of Theorem 4.6: the uniform L^∞ bound on the adjoint state ρ^ε is not established. The adjoint solves a Fokker-Planck equation with vanishing diffusion ε, and semiconcavity only controls ∆u^ε from above, not |∆u^ε|, so a uniform L^∞ bound on ρ^ε is not justified. If ∥ρ^ε∥_{L^∞} grows as ε→0, the L^1 estimate (4.2) does not close. The referee notes this does not affect Step 1 (4.1), where L^1 conservation suffices."}],"tokens_in":14039,"tokens_out":789,"duration_ms":742977,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper improves the vanishing viscosity convergence rate for the HJ component of MFGs from O(ε^{1/4}) to O(ε^{1/2}) and, for the first time, gives a quantitative rate (O(ε^{1/8}) in W_2) for the Fokker-Planck density. Both are new results under the same structural assumptions as [TZ25]. The paper deserves a serious referee. The main caveat is that the density rate requires an additional W^{2,∞} bound on u^ε (condition 4.4) that is not a consequence of the general framework. The authors acknowledge this honestly. The stress-test concern about the L^∞ bound on the adjoint ρ^ε does not land — see below. What is new and well done: The O(ε^{1/2}) rate for u^ε - u in both L^1 and L^∞ is a genuine sharpening of [TZ25]. The proof of Theorem 4.2 is clean: the duality argument testing the HJ and FP differences against each other, combined with the uniform convexity (H4) and the monotonicity structure, gives the O(ε) bound on the Lasry-Lions pairing and the L^2(m) gradient estimate. The decoupling strategy — using the regularizing coupling to first get HJ rates, then transferring to the drift, then propagating to FP via the stochastic characteristics estimate (Lemma 3.1 / Corollary 3.2) — is a natural and effective use of the structure. The Wasserstein framework for the density is the right choice; the authors correctly note that L^p convergence would require controlling div(D_pH(x, Du^ε)) in a topology stronger than what semiconcavity provides. On the stress-test concern: the note flags that the uniform L^∞ bound on ρ^ε (the adjoint in Step 2 of Theorem 4.6) is not established. This concern does not hold up. The adjoint ρ^ε solves the same Fokker-Planck equation as m^ε with the same drift D_pH(x, Du^ε). The maximum principle argument used for m^ε in the proof of Theorem 4.2 — which relies on div(D_pH(x, Du^ε)) being bounded above via semiconcavity (D²u^ε ≤ C) and (H4) — applies identically to ρ^ε. At a maximum of ρ^ε, the equation gives ∂_t ρ^ε ≤ (div b) ρ^ε ≤ C ρ^ε, yielding ∥ρ^ε∥_∞ ≤ e^{CT} ∥ρ(τ)∥_∞ = e^{CT} (since the initial datum is sgn(·), with L^∞ norm 1). The authors reference [Eva10b, LBL19] for this argument in the m^ε case; the same citation covers ρ^ε. The exposition could be more explicit here — the sentence 'Since ∥∆u^ε∥_{L^1}, ∥ρ^ε∥_{L^∞} are bounded independent of ε' states the conclusion without restating the argument — but the mathematics is sound. The real soft spot is condition (4.4): the W^{2,∞} bound on u^ε independent of ε is needed to make the drift b^ε = -D_pH(x, Du^ε) Lipschitz uniformly in ε, which is what feeds Corollary 3.2. This is not guaranteed by the general framework. The authors list cases where it holds (convex data, small terminal data, short time horizons, displacement monotonicity), which is fair, but the density rate is conditional on this extra regularity. The HJ rates (Theorem 4.2, equations 4.1-4.3) do not require (4.4) and stand on their own. Summary: the paper is for researchers working on quantitative aspects of MFG theory and vanishing viscosity limits. The HJ results are unconditional and solid; the density result is conditional but clearly labeled as such. The proofs are correct and follow established PDE techniques applied in a non-trivial way to the coupled system. Recommend sending to a serious referee who can verify the duality computations in Theorem 4.2 and the application of the Bernstein method in Step 1 of Theorem 4.6.","headline":"Solid contribution improving HJ rates and adding density rates for vanishing viscosity in MFGs; one stress-test concern does not land, one real limitation remains","tokens_in":14924,"tokens_out":5890,"would_cite":true,"duration_ms":326706,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35F21","35Q84","35Q89","41A25","49N80"],"pacs":[],"model":"glm-5.2","headline":"Viscosity vanishing in mean field games gets explicit rates","keywords":["Mean Field Games","vanishing viscosity","convergence rate","Hamilton-Jacobi equation","Fokker-Planck equation","Wasserstein distance","nonlocal coupling","semiconcavity"],"falsifier":"If one could construct a Mean Field Game satisfying all the stated structural assumptions (H1)-(H4), (F), and monotonicity (1.5)-(1.6), but where the W^{2,infinity} bound (4.4) fails, the density convergence rate O(epsilon^{1/8}) would not apply. More critically, if the coupling regularity assumptions (F1)/(F2) were shown to be incompatible with the monotonicity conditions for any non-trivial coupling, the value function rate would also fail.","tokens_in":14224,"feed_emoji":"🎲","tokens_out":1369,"duration_ms":158507,"temperature":0.7,"pith_summary":"Mean Field Games model Nash equilibria in games with infinitely many players via a coupled system: a backward Hamilton-Jacobi equation for the value function of a representative agent, and a forward Fokker-Planck equation for the population distribution. When a small diffusion term (viscosity) is added to both equations to regularize them, one can ask how fast the regularized solutions converge to the inviscid limit as the viscosity parameter epsilon tends to zero. This paper proves explicit convergence rates for the full coupled system under standard structural assumptions on the Hamiltonian (uniform convexity in the momentum variable) and a regularizing nonlocal coupling. The central mechanism is a partial decoupling strategy: the regularizing coupling allows the convergence rate for the Hamilton-Jacobi equation to be established first, independent of the density variable, and this rate is then propagated through the drift field into the Fokker-Planck equation via comparison of stochastic and deterministic flow trajectories. The main result is that the value function converges at rate O(epsilon^{1/2}) in both L^1 and L^infinity, improving the previously known O(epsilon^{1/4}) rate, and the population density converges at rate O(epsilon^{1/8}) in the 2-Wasserstein distance, provided an additional uniform W^{2,infinity} bound on the viscous value function holds.","feed_headline":"Vanishing viscosity in mean field games gets explicit rates","feed_subtitle":"Value function converges at O(ε^{1/2}) and density at O(ε^{1/8}), proving both halves of the coupled system for the first time.","key_machinery":"The argument proceeds in three stages. First, a duality argument testing the difference of two viscous solutions against the density difference yields an O(epsilon) bound on the monotonicity integral and on the weighted gradient squared integral (Theorem 4.2), using semiconcavity (Lemma 4.1) and the maximum principle for the Fokker-Planck equation. Second, the adjoint method is applied to the viscosity derivative of the value function, using the Bernstein method to control the Hessian term and assumptions (F1)/(F2) to control the coupling derivative, yielding the O(epsilon^{1/2}) rate for the value function (Theorem 4.6, Steps 1-2). Third, the value function rate is transferred to the drift,","core_discovery":"The paper establishes that the vanishing viscosity approximation for first-order Mean Field Games with nonlocal coupling converges at rate O(epsilon^{1/2}) for the value function (in L^1 and L^infinity) and at rate O(epsilon^{1/8}) for the density in the 2-Wasserstein distance. The key technical innovation is an O(epsilon) bound on the Lasry-Lions monotonicity functional integral involving the coupling difference weighted by the density difference, combined with a dual/adjoint method exploiting semiconcavity of the value function and uniform convexity of the Hamiltonian. This bound, together with assumptions (F1) or (F2) translating the monotonicity estimate into L^infinity or L^1 control of","pith_inferences":["The O(epsilon^{1/8}) density rate is likely not sharp. The chain of inequalities loses rate at each step: O(epsilon^{1/2}) for u, O(epsilon^{1/4}) for the drift in L^2, then square-root via Wasserstein to get epsilon^{1/8}. A more direct argument bypassing the Wasserstein square-root step might recover a better rate.","The assumption (F1)/(F2) on the coupling—that the monotonicity integral bound implies an L^infinity or L^1 bound on the coupling difference—is essentially a regularity assumption on the solution operator of an elliptic equation. Characterizing the minimal regularity on the convolution kernel for which these hold would clarify the boundary of applicability.","If one could establish the W^{2,infinity} bound (4.4) under weaker conditions—for instance, for longer time horizons or non-convex data—the density convergence result would apply to a substantially broader class of MFG systems."],"forward_implications":["Numerical schemes for first-order MFGs that introduce artificial viscosity can now be equipped with explicit error bounds of order epsilon^{1/2} for the value function and epsilon^{1/8} for the density, guiding mesh-size and viscosity parameter choices.","The partial decoupling strategy—using the regularizing coupling to first establish HJ convergence independently of the density—may extend to other forward-backward PDE systems where the forward component's drift depends on the backward component's gradient.","The gap between the value function rate (epsilon^{1/2}) and the density rate (epsilon^{1/8}) suggests that the density convergence is bottlenecked by the gradient estimate (epsilon^{1/4} in L^2); improving the gradient rate would directly improve the density rate.","The selection principle interpretation—that viscous approximations select a privileged inviscid solution—is now quantitative, allowing one to assess how small epsilon must be for the selected solution to approximate the inviscid one to a given tolerance."],"fun_headline_variants":["Vanishing viscosity rates for full mean field game systems","Both halves of mean field games get convergence rates","Value and density converge in vanishing viscosity MFG limit","Explicit rates for value function and density in MFG viscosity limit","Coupled MFG system gets quantitative convergence in viscosity limit"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The density convergence rate O(epsilon^{1/8}) requires the viscous value function to have a second spatial derivative bounded uniformly in epsilon (condition 4.4). This is not guaranteed by the general framework and holds only under additional specific conditions such as convex data, small terminal data, or short time horizons.","fun_headline_variants_meta":{"raw":{"variants":["Vanishing viscosity rates for full mean field game systems","Both halves of mean field games get convergence rates","Value and density converge in vanishing viscosity MFG limit","Explicit rates for value function and density in MFG viscosity limit","Coupled MFG system gets quantitative convergence in viscosity limit"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":576,"prompt_tokens":514,"completion_tokens":62,"prompt_tokens_details":null},"tokens_in":514,"tokens_out":62,"duration_ms":83999,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T07:58:09.416444+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one could construct a Mean Field Game satisfying all the stated structural assumptions (H1)-(H4), (F), and monotonicity (1.5)-(1.6), but where the W^{2,infinity} bound (4.4) fails, the density convergence rate O(epsilon^{1/8}) would not apply. More critically, if the coupling regularity assumptions (F1)/(F2) were shown to be incompatible with the monotonicity conditions for any non-trivial coupling, the value function rate would also fail.","supporting_citations":[],"review_version":1}