{"id":"ab9b6b49-b3bf-440f-b863-5acb9f4efa0c","arxiv_id":"2504.12524","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A continuously parametrized family of kinetically constrained exclusion processes is shown to converge to the porous medium equation with non-integer exponent under general conditions.","lead":"This paper constructs a family of particle-jump models on a line whose large-scale density follows a generalized porous media diffusion equation with any real exponent, and proves a hydrodynamic limit for a broad class of constrained exclusion processes. The result gives a robust framework for deriving nonlinear diffusion equations from microscopic stochastic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Regime-I verification for the interpolating process is asserted, not proved: the mobile-cluster lower bound r⋆ is never derived from the coefficients in (2.3), and Lemma 4.6's auxiliary set (4.11) is garbled, so Theorem 2.13 is not fully established as written.","rationale":"The reader's conditional verdict is supported by this stress-test. The strongest claim, Theorem 2.13 for the interpolating model, depends on the Regime-I conditions in Definition 2.7(ii), and the manuscript's verification of those conditions for Definition 2.3 is a sketch rather than a proof. In particular, the coefficient δ_N in the decomposition of c_N^m depends on ℓ_N and can be small, so a uniform positive lower rate r⋆ is not automatic. Lemma 4.6 then relies on r⋆, and its proof also contains a garbled auxiliary set Ω^{p,w,y} in (4.11), making the two-block estimate impossible to follow as written. These are internal proof gaps, not disagreements with the surrounding consensus; the entropy method strategy is standard and the model is plausible. The novelty is moderate and there is no machine-checked or reproducible formalization. The appropriate verdict remains conditional: the central claim is plausible and likely fixable, but it is not fully established in the present manuscript.","tokens_in":26951,"tokens_out":16983,"duration_ms":175118,"concrete_test":"For fixed n and m (e.g., n=1, m=1/2), implement c_N^m as in (2.3) on a finite interval, using the Bernstein expansion in Definition 2.4 to disambiguate p_{n,k}. Enumerate all configurations of a box of length κ=n+3 containing at least n+2 particles and at least one vacancy, and all adjacent pairs •◦/◦• inside the box, and compute the rate for the exchange with the outside configuration set to the worst case among empty/full/alternating fillings. Take the infimum of these rates for ℓ_N = 10, 100, 1000. If the infimum decays to zero as ℓ_N grows, the asserted Regime-I lower bound r⋆ does not exist; if it stabilizes at a positive value, the mobile-cluster claim is internally consistent and the remaining obstruction is the garbled Lemma 4.6, which should then be redrafted with a corrected definition of Ω.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem's Regime-I branch (Theorem 2.13, Definition 2.7(ii)) requires mobile clusters of length κ⋆ with all required jumps at rates at least r⋆, independent of N. The only verification for Definition 2.3 is a paragraph in Section 2.1 asserting that boxes with at least n+2 particles and at least one vacancy are mobile because removing a •◦ pair leaves a state with c_N^m > 0. No quantitative lower bound is computed. This matters because, in the decomposition c_N^m = δ_N p_n + m p_{n,1} + Σ_{k≥2}|C(m,k)|p_{n,k}, the coefficient δ_N is a tail of the binomial series and depends on ℓ_N; it can tend to zero as ℓ_N → ∞. A uniform r⋆ must instead come from a lower bound on the remaining finite-sum terms for every cluster configuration, and that bound is not established. Lemma 4.6 then uses r⋆ in the two-block estimate, specifically in the path count after (4.17) and in the assertion that r⋆(Q_n η⋆) ≥ r0, so if r⋆ fails the replacement lemma collapses. Independently, the auxiliary set Ω^{p,w,y} in (4.11) is garbled: its two clauses are identical, so the subsequent case split and the bound that at most κ⋆+1 values of r contribute outside Ω are not justified. The proof of Lemma 4.6 is therefore incomplete as written. Both gaps are likely fixable, but the proof of Theorem 2.13 is not currently complete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new one-dimensional kinetically constrained symmetric exclusion process, the 'interpolating process' (Definition 2.3), which is constructed via a Bernstein-polynomial/generalized-binomial expansion and is intended to interpolate between the Porous Media Models PMM(n) and PMM(n+1). The associated hydrodynamic limit is claimed to be ∂tρ = ∂u(ρ^{n+m}∂uρ). The paper then states a general hydrodynamic-limit theorem (Theorem 2.13) for perturbed gradient exclusion KCMs satisfying Assumption 2.6 and either a mobile-cluster condition (Regime I) or a uniform positivity condition (Regime II). The proof follows the standard entropy method: tightness of the empirical measures, replacement lemmas (one-block and two-block), and an energy estimate for uniqueness of the limiting weak solution. The general framework is intended to be basis-independent and to cover the interpolating model as well as superpositions of Bernstein models.","tokens_in":27276,"tokens_out":5017,"duration_ms":47761,"significance":"The paper's central ambition is significant: it proposes a basis-independent set of sufficient conditions for hydrodynamic limits of gradient kinetically constrained exclusion processes, and it exhibits a genuinely new continuously parametrized model covering non-integer exponents. The construction is explicit and parameter-free in the sense that the target diffusivity emerges from the rate definition rather than being fitted; the diffusivity computation for the interpolating model is a clear strength. If the proof gaps are repaired, the framework would unify and extend the companion works [10] and [13]. However, the manuscript as written does not fully verify the mobile-cluster hypothesis for the new model, and the two-block estimate contains a defective auxiliary set; both points are load-bearing for Theorem 2.13. The paper is therefore not yet complete, but the issues appear local and fixable within the manuscript's scope.","major_comments":[{"comment":"The claim that every box of length at least n+2 containing at least n+2 particles and at least one vacancy is a mobile cluster with all required jump rates bounded below by a constant r⋆ independent of N is asserted but not proved. In the decomposition cm_N = δ_N p_n + m p_{n,1} + Σ_{k≥2} |C(m,k)| p_{n,k}, the coefficient δ_N = 1 − Σ_{k=1}^{ℓ_N} |C(m,k)| is a binomial tail that depends on ℓ_N and may vanish as ℓ_N → ∞; a uniform lower rate r⋆ must be derived from the remaining finite-sum terms (for example, from the coefficient m of p_{n,1} and from the structure of p_{n,k} for 2 ≤ k ≤ ℓ_N), but no such quantitative bound is computed. This is load-bearing because Lemma 4.6 invokes the uniform lower bound r⋆(Q_n η⋆) ≥ r0 after (4.17) and in the path-count argument; if r⋆ is not established independently of N, the two-block estimate in Regime I collapses and Theorem 2.13 is not proved for the interpolating model.","section":"Section 2.1, after Definition 2.3; Definition 2.7(ii)"},{"comment":"The auxiliary set Ω^{p,w,y} is garbled: its two clauses are identical, both reading (⟨τ_{p+w}η⟩_L ≥ (k⋆+1)/L or ⟨τ_yη⟩_L ≥ (k⋆+1)/L). Consequently, the assertion that outside Ω^{p,w,y} at most k⋆+1 values of r ∈ B_L give nonzero exchange rates is not justified (a block could be nearly full while the other is nearly empty without satisfying the displayed condition as written), and the subsequent case split into cases (1) and (2) is not exhaustive. The proof of the two-block estimate is therefore incomplete as written; the definition needs to be corrected (e.g., requiring one block to be high and the other low) or the counting argument must be revised accordingly.","section":"Lemma 4.6, Eq. (4.11)"}],"minor_comments":[{"comment":"The text states that EνNα[cm_N] → mα^{m−1} uniformly, but the computation immediately below it yields mα^n Σ_{k=0}^{ℓ_N} (−1)^k C(m,k)(1−α)^k, which converges to mα^{n+m}; this is also the diffusivity needed for the stated equation ∂tρ = ∂u(ρ^{n+m}∂uρ). The displayed limit should be corrected.","section":"Section 2.1, diffusivity display"},{"comment":"The statement lists '(2.11), (2.7),(2.8), (2.10) and (2.11)' with (2.11) repeated; presumably one of these should be the verification of (2.9) or a renumbering is needed.","section":"Lemma 2.9"},{"comment":"The auxiliary constraints p_{n,k} and p_j_{n,k} use a notation that is hard to follow; in particular, the superscript j appears in 'pj n,k' but the definition of p_j_{n,k} is not explicitly separated from p_{n,k}. Please clarify the indexing.","section":"Definition 2.3"},{"comment":"In the decomposition of the product difference, the maps φ^{εN,L}_m, φ^L_m, and φ^L_m are introduced; two of them are printed with the same symbol φ^L_m. If they are different objects, they should be renamed; if they are identical, the decomposition is redundant.","section":"Section 3.3, Eqs. (3.13)–(3.16)"},{"comment":"The letter r is used both for an index in B_L and for the rate lower bound r⋆ (and r0), which makes the displays around (4.17) hard to read. Please use a different symbol for the spatial index.","section":"Lemma 4.6 proof"},{"comment":"There are several typos and inconsistencies: 'interacing' and 'condtition' in the Introduction, 'Riez' for Riesz in Section 5, 'arrodingly' in Section 3.3, and the varying use of 'Porous Media Model' versus 'Porous Medium Equation'. A thorough proofreading is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on the author's companion works [10] and [13]; in particular, the gradient property and the Bernstein-model decomposition are imported from [13] without complete proofs. The editor should ensure that [13] is publicly available and contains the quoted results. The interpolation construction is attractive, but the final version should either prove the mobile-cluster lower bound explicitly or state it as a lemma with a complete derivation from the coefficients in (2.3). The two-block-estimate defect in (4.11) is likely repairable, but it is central and cannot be left as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a promising paper with a genuinely new model, but the proof of the main theorem has holes that need filling before the result is fully established.\n\nThe interpolating process in Definition 2.3 is a real contribution. It uses the Bernstein basis to define a one-parameter family of exclusion processes with diffusivity ρ^{n+m} for integer n and m∈[0,1], extending the earlier monomial construction in [10] beyond exponent 2. The decomposition c^m_N = δ_N p_n + m p_{n,1} + Σ_{k≥2}|C(m,k)|p_{n,k} is clean, and the gradient property is inherited from the Bernstein models. The general conditions in Assumption 2.6 are also useful: they separate algebraic growth conditions from the dynamics, and the two regimes are a sensible organizing device.\n\nThe problems are in the proof. The verification that the interpolating model belongs to Regime I is essentially a sketch. The claim that boxes with at least n+2 particles and one vacancy are mobile clusters (which, for length exactly n+2, is internally inconsistent — presumably n+1 particles is meant) is asserted, but δ_N tends to 0 as ℓ_N→∞, and the p_{n,k} terms for large k also have coefficients going to 0. A uniform r⋆ needs a quantitative argument that, for a fixed cluster length κ⋆, some term with coefficient bounded away from 0 is always active. That is not supplied. Lemma 4.6 also has a clear typo: the auxiliary set Ω^{p,w,y} in (4.11) repeats the same condition twice, so the case split and the counting argument that follow do not work as written. The energy estimate in (5.2) has liminf where limsup is needed; that is likely a simple fix. Regime II is deferred with a 'proof is identical' remark, which is probably acceptable but not fully written out.\n\nNone of this looks fatal. The strategy is the standard entropy method, the structure is right, and the gaps appear fixable with work rather than new ideas. But as it stands, Theorem 2.13 is not completely proved. The paper deserves a serious referee: the model alone justifies referee time. I would send it out with a request for major revision and a careful check of the two-block estimate and the Regime I verification. For my reading group, I would wait for a revised version.","headline":"New model, right strategy, but two concrete gaps in the proof mean Theorem 2.13 is not yet established as written.","tokens_in":27821,"tokens_out":13205,"would_cite":true,"duration_ms":119268,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","35K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a continuously parametrised family of kinetically constrained exclusion processes has a hydrodynamic limit given by the porous medium equation, covering every positive power-law exponent.","keywords":["hydrodynamic limit","kinetically constrained exclusion processes","porous medium equation","Bernstein polynomial basis","gradient condition","replacement lemma","nonlinear diffusion","empirical measure"],"falsifier":"Take a configuration on a large torus with $n+2$ consecutive particles followed by one vacancy in a box of length $n+2$, arranged so that no smaller aligned block of $n$ particles appears in the box; if the rate $c_N^m$ for a jump inside that box decays with $N$ or vanishes, the mobile-cluster lower bound fails and the Regime-I replacement lemmas cannot yield the hydrodynamic limit. A direct computation of $c_N^m$ on such a configuration settles the point.","tokens_in":26686,"feed_emoji":"💧","tokens_out":7544,"duration_ms":71282,"temperature":0.7,"pith_summary":"The paper aims to show that a wide class of one-dimensional symmetric exclusion processes with kinetic constraints still obeys a deterministic hydrodynamic equation after diffusive rescaling, provided the constraints satisfy a few explicit growth and algebraic conditions. To demonstrate the scope, it constructs a family of gradient models, continuously parametrised by $m\\in[0,1]$ on top of an integer $n$, whose diffusion coefficient is $m\\rho^{n+m-1}$ and whose hydrodynamic limit is the porous medium equation $\\partial_t\\rho=\\partial_u(\\rho^{n+m}\\partial_u\\rho)$. Previously such power-law diffusivities were available only for integer exponents, or for $m\\in(0,2)$ with restrictions; the new Bernstein-basis construction removes the obstruction and yields all positive exponents. The hydrodynamic limit is proved for a macroscopically negligible perturbation of the dynamics by the symmetric simple exclusion process, which guarantees local mixing without changing the limiting equation.","feed_headline":"Particle systems reach every power-law porous media equation","feed_subtitle":"One continuous family of exclusion processes converges to ∂tρ = ∂u(ρ^{n+m}∂uρ) for every n ≥ 1, m ∈ [0,1].","key_machinery":"The load-bearing object is the interpolating constraint $c_N^m=p_n+\\sum_{k=1}^{\\ell_N}\\binom{m}{k}(-1)^k(p_n-p_{n,k})$, a superposition of the integer PMM constraint $p_n$ with auxiliary constraints $p_{n,k}$; it is built so that the generator splits into Bernstein-model generators, which forces the gradient condition $j_{0,1}=\\nabla H_N$. The proof's quantitative engine is the entropy method together with three replacement estimates: a one-block lemma and a two-block lemma control the difference between local density and particle occupation in a box, and an energy estimate gives the $H^1$ regularity of $\\Phi(\\rho)$ needed for uniqueness. In Regime I the mixing that makes the replacement lemmas work comes from mobile clusters: boxes of length at least $n+2$ containing at least $n+2$ particles and one vacancy can be reorganised by nearest-neighbour exchanges at rates bounded below independently of $N$.","core_discovery":"The central statement is Theorem 2.13: for any sequence of initial measures in local equilibrium with a profile $\\rho_{\\mathrm{ini}}$, the empirical measure of the $N^2$-speed process converges in probability to the unique weak solution $\\rho$ of $\\partial_t\\rho=\\partial_u^2\\Phi(\\rho)$, whenever the constraints satisfy Assumption 2.6 and belong to either Regime I (mobile clusters with uniformly positive rates) or Regime II (uniformly positive rates everywhere). For the interpolating process of Definition 2.3 the diffusivity satisfies $\\mathbb{E}_{\\nu^N_\\alpha}[c_N^m]\\to m\\alpha^{n+m-1}$ uniformly in $\\alpha$, so the limiting equation is the porous medium equation $\\partial_t\\rho=\\partial_u(\\rho^{n+m}\\partial_u\\rho)$. This covers the integer Porous Media Models, the Bernstein models, and their superpositions, and the proof is designed to be independent of the choice of polynomial basis: it uses only the stated rate bounds, the gradient condition with its explicit primitive $H_N=h_N+g_N$, and the one-block and two-block replacement lemmas.","pith_inferences":["The mobile-cluster condition could be tested numerically: simulate the interpolating model at large $N$ on a block with $n+2$ particles and one vacancy and measure the smallest jump rate; if it stays bounded below by a positive constant independent of $N$, the missing assertion in Section 2.1 is likely true for these models.","The same Assumption 2.6 could be checked for other finite-range gradient constraints, for example multi-species or higher-dimensional symmetric exclusion processes, potentially yielding hydrodynamic limits beyond the one-dimensional PMM setting.","The decomposition into Bernstein generators suggests a design principle: particle systems whose hydrodynamic equation is a prescribed polynomial or Bernstein diffusivity can be assembled as superpositions of elementary Bernstein models, and the proof here indicates which superpositions are tractable."],"forward_implications":["The hydrodynamic limit provides a weak law of large numbers for the empirical measure, so local particle densities are well approximated by a deterministic PDE for large system sizes.","The interpolating process gives a constructive route to any power-law nonlinear diffusion $\\partial_t\\rho=\\partial_u(\\rho^p\\partial_u\\rho)$ for $p>0$, including fast diffusion for $p<1$ and slow diffusion for $p>1$, from exclusion-type microdynamics.","Because the proof is basis-independent, the same theorem covers any gradient constraint family that satisfies the uniform rate bounds, the uniform Cauchy condition on $h_N$, and one of the two mixing regimes.","The $H^1$ energy regularity of $\\Phi(\\rho)$ (or of $\\rho$ itself in Regime II) yields uniqueness of the weak solution, so every subsequential limit of the empirical measures coincides with the same macroscopic profile."],"supporting_citations":[{"why":"introduces the Porous Media Model and proves its hydrodynamic limit, the baseline process this paper interpolates and generalises.","marker":"[8]"},{"why":"constructs non-integer PMM for exponents in (0,2) and supplies the truncation and two-block strategy that the present proof adapts.","marker":"[10]"},{"why":"introduces the Bernstein polynomial models and proves their gradient property, which the interpolating constraint inherits through its generator decomposition.","marker":"[13]"},{"why":"defines the gradient condition and the explicit diffusion-coefficient formula used in Assumption 2.6.","marker":"[15]"},{"why":"originates the entropy method by which the hydrodynamic limit is proved.","marker":"[11]"},{"why":"supplies the martingale, relative-entropy, Doob and Feynman-Kac tools used in tightness and the replacement lemmas.","marker":"[12]"},{"why":"gives conditions for superpositions of PMM constraints, showing the kind of coefficient family the general theorem is designed to cover.","marker":"[4]"},{"why":"provides the Oleinik uniqueness argument used to identify the weak solution in the slow-diffusion regime.","marker":"[2]"}],"fun_headline_variants":[],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Regime-I verification rests on an unproved mobile-cluster assertion: every box of length at least $n+2$ containing at least $n+2$ particles and one vacancy is mobile, and jumps inside it occur at rates bounded below by a constant independent of the system size.","fun_headline_variants_meta":{"error":"DeepSeek 429: {\"error\":{\"message\":\"Too many requests. Your current concurrency is 119, which exceeds your concurrency limit of 117 based on your remaining balance. Please top up your balance to restore your concurrency.\",\"type\":\"rate_limit_error\",\"param\":null,\"code\":\"invalid_request_error\"}}"},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:31:18.090152+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a configuration on a large torus with $n+2$ consecutive particles followed by one vacancy in a box of length $n+2$, arranged so that no smaller aligned block of $n$ particles appears in the box; if the rate $c_N^m$ for a jump inside that box decays with $N$ or vanishes, the mobile-cluster lower bound fails and the Regime-I replacement lemmas cannot yield the hydrodynamic limit. A direct computation of $c_N^m$ on such a configuration settles the point.","supporting_citations":[{"cited_title":"Papanicolaou, and S","cited_arxiv_id":null,"evidence_quote":"originates the entropy method by which the hydrodynamic limit is proved."},{"cited_title":"Kipnis and C","cited_arxiv_id":null,"evidence_quote":"supplies the martingale, relative-entropy, Doob and Feynman-Kac tools used in tightness and the replacement lemmas."},{"cited_title":"Bonorino, R","cited_arxiv_id":null,"evidence_quote":"provides the Oleinik uniqueness argument used to identify the weak solution in the slow-diffusion regime."},{"cited_title":"Gon¸ calves, C","cited_arxiv_id":null,"evidence_quote":"introduces the Porous Media Model and proves its hydrodynamic limit, the baseline process this paper interpolates and generalises."},{"cited_title":"Nahum, and Marielle Si mon","cited_arxiv_id":null,"evidence_quote":"constructs non-integer PMM for exponents in (0,2) and supplies the truncation and two-block strategy that the present proof adapts."},{"cited_title":"A gradient model for the Bernstein polynomial basis","cited_arxiv_id":"2411.15954","evidence_quote":"introduces the Bernstein polynomial models and proves their gradient property, which the interpolating constraint inherits through its generator decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the gradient condition and the explicit diffusion-coefficient formula used in Assumption 2.6."}],"review_version":1}