{"id":"0b2284f4-2cbc-487e-bf43-96739a6742e5","arxiv_id":"2411.15954","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A symmetric exclusion process realizing Bernstein-polynomial diffusivities is constructed and claimed to satisfy the gradient condition, generalizing the Porous Media Model.","lead":"This paper constructs a new family of exclusion processes, the Bernstein model, whose bulk diffusion coefficient is a Bernstein polynomial, and claims it satisfies the gradient property that makes hydrodynamic limits tractable. The result fills a known gap, since such diffusivities cannot be obtained by superposing Porous Media Models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.7's stated H fails the gradient identity for L=1,n=0: explicit check gives b(e0,1-e1,0)=1 but -∇H=-1/2, so the main theorem is unproven as written.","rationale":"The reader's verdict was CONDITIONAL, and the reader's rationale did mention a suspected sign error in Proposition 2.7. Our independent small-case verification confirms this suspicion: the explicit potential H given in Proposition 2.7 does not satisfy the gradient identity in the minimal case L=1,n=0. This is a concrete, load-bearing problem because the gradient identity is the paper's central technical claim, not merely a step toward a hydrodynamic limit. The error appears to be a sign/indexing issue in the definition of g or in the proof's expansion, so it may be correctable; however, as written the main theorem is false. We also note that the reader's separate concern about Lemma 3.4 is valid: the stated h_{ℓ;L} contains a factor 1/C(L,ℓ), which gives H̄'(ρ)=ρ^ℓ(1-ρ)^{L-ℓ} rather than ρ^ℓ, contradicting the claimed diffusivity of the reduced PMM. These issues do not necessarily invalidate the underlying construction, but they show the manuscript requires substantive revision. Since the errors are potentially fixable, we do not move the verdict from the reader's CONDITIONAL assessment; we agree that the paper should not be accepted in its present form and that a corrected, fully verified version is needed.","tokens_in":81,"tokens_out":28353,"duration_ms":947916,"concrete_test":"Run an exhaustive check of Proposition 2.7 for L=1,n=0 and L=2,n=1 on a small torus (N=6): for every configuration, compute b(η)(e0,1-e1,0) and H(η)-H(τη) using the stated H, and list any mismatch. The L=1 configuration with a single particle at site 0 and all other relevant sites empty already violates the identity, so the test should detect this immediately. If L=2 also fails, the error is not confined to the minimal case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For L=1,n=0, the Bernstein constraint is b(η)=1/2(1{η(2)=0}+1{η(-1)=0}), the stated h is h(η)=1/2·1{η(0)+η(1)≥1}, and the stated g is g(η)=1/2·1{η(-2)=0}·(η(-1)-η(0)). Take η with η(0)=1 and all other relevant sites zero. Then b=1 and e0,1-e1,0=1, so the left side is 1. Meanwhile h(η)=1/2 and g(η)=-1/2, giving H(η)=0; for τη, h(τη)=0 and g(τη)=1/2·1{η(-1)=0}·(η(0)-η(1))=1/2, so H(τη)=1/2. Thus -∇H=H(η)-H(τη)=-1/2, not 1. The identity fails. The proof's expansion (3.4) also appears to carry the opposite sign: it rewrites b(e1,0-e0,1) with a plus gradient term, while the direct small-case computation gives the opposite sign. Hence the central gradient identity is not established as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces two families of symmetric exclusion processes on the one-dimensional torus: the Bernstein model B(n,L), whose exchange rate at a node is the fraction of length-(L+2) windows containing exactly n+1 particles that contain the node, and the reduced porous media model PMM_L(ℓ), whose rates are built from hypergeometric probabilities. The central claims are (i) the gradient property for B(n,L) with an explicit potential H_{n,L}=h_{n,L}+g_{n,L} (Proposition 2.7), (ii) an analogous gradient representation for PMM_L(ℓ) (Lemma 3.4), yielding diffusivities B_{n,L}(ρ) and ρ^ℓ respectively, (iii) binomial inversion formulas relating the two families, and (iv) structural properties such as partition of unity, mobile clusters, and blocked configurations. The hydrodynamic limit is explicitly deferred to a companion paper.","tokens_in":13129,"tokens_out":16690,"duration_ms":138281,"significance":"The construction is well motivated and, if correct, would fill a genuine gap: the known superposition of PMMs cannot produce Bernstein-basis diffusivities with two high-multiplicity roots, and the paper proposes a natural combinatorial mechanism. The manuscript is self-contained and contains no fitted parameters; the inversion formulas (2.5)-(2.6) are clean and potentially useful. However, the main algebraic theorem (Proposition 2.7) is false as stated: the proposed H fails even for L=1,n=0. Since the gradient property is the paper's core contribution and the companion hydrodynamic paper would rely on it, the result cannot be accepted without a substantially corrected derivation and a correct potential.","major_comments":[{"comment":"Proposition 2.7 cannot be correct as stated. Take L=1, n=0, a torus with N≥5, and a configuration η with η(0)=1 and all other sites in {-1,0,1,2} empty. From (2.3), b_{0,1}(η)=1/2(1{η(-1)=0}+1{η(2)=0})=1. Since e_{0,1}(η)=1 and e_{1,0}(η)=0, the left-hand side b_{0,1}(η)(e_{0,1}(η)-e_{1,0}(η)) equals 1. On the other hand, Eq. (2.8) gives h_{0,1}(η)=1/2 and g_{0,1}(η)=-1/2, so H_{0,1}(η)=0; for τη one gets h_{0,1}(τη)=0 and g_{0,1}(τη)=1/2, so H_{0,1}(τη)=1/2. Hence -∇H = H(η)-H(τη) = -1/2, not 1. The same example shows that the expansion (3.4) carries the opposite sign: for b(e_{1,0}-e_{0,1}) the left-hand side is -1, while the first bracket equals 1/2 and (1/2)∇g equals 1/2, so the right-hand side of (3.4) is +1. This is a load-bearing error, not a typo.","section":"Section 2.2, Proposition 2.7, Eq. (2.8); Section 3.2, Eq. (3.4)"},{"comment":"The stated potential in Lemma 3.4 is incompatible with the claimed diffusion coefficient. With h_{ℓ;L}(η)=(1/((L+1) binom(L,ℓ))) 1{⟨η⟩_L ≥ (ℓ+1)/(L+1)}, the canonical average under the Bernoulli measure ν_ρ is H̄(ρ)=P(Bin(L+1,ρ)≥ℓ+1)/((L+1) binom(L,ℓ)), whose derivative is ρ^ℓ(1-ρ)^{L-ℓ}, not ρ^ℓ for ℓ<L. This contradicts the identification D(ρ)=H̄'(ρ) from Section 2.1 together with Proposition 2.9(vi), which directly gives ∫ p_{ℓ;L} dν_ρ = ρ^ℓ. Thus the gradient potential for the reduced PMM is not established; a correct h obtained from the inversion formula (2.6) would be a sum of threshold terms over n, not the single threshold with a binomial denominator.","section":"Section 3.2, Lemma 3.4"}],"minor_comments":[{"comment":"The displayed generator in Eq. (2.1) is written as a sum plus an identical sum; this is presumably a typo, and the two sums should involve the two different exclusion factors e_{0,1} and e_{1,0}.","section":"Section 2.1, Eq. (2.1)"},{"comment":"The sentence beginning \"Throughout this work, for simplicity we fix Let N ∈ N+ be fixed\" is garbled and needs rewriting.","section":"Section 1, Introduction"},{"comment":"In the proof of Proposition 2.8, the sentence \"Regarding (iv), an example is...\" should refer to property (v); moreover, Proposition 2.8(iv) first states b_{n,L}(η)∈[0,1) and then correctly observes that the maximum 1 is attained, so the interval should be [0,1].","section":"Section 3.3, proof of Proposition 2.8"},{"comment":"The notation ⟦a,b⟧ is used without definition; it should be defined as the integer interval {a,a+1,...,b} or replaced by standard notation.","section":"Definition 2.3"}],"recommendation":"reject","confidential_remarks":"The counterexample in the report is elementary and should have been caught by small-case testing; the central gradient identity is false as stated. If the author can supply a correct potential H and a valid proof, the underlying model may still be valuable, but the current manuscript cannot be published in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the model construction is new and worth knowing about, but the main theorems as written do not hold. I checked Proposition 2.7 for L=1, n=0. With η(0)=1 and all other relevant sites empty, the left side b(e0,1-e1,0) is 1, while -∇H is -1/2. So the identity fails. This is not a sign typo in the proof; the proof's expansion carries the same problem. Lemma 3.4 also looks off: the stated h gives derivative ρ^ℓ(1-ρ)^{L-ℓ} under the Bernoulli measure, not ρ^ℓ. Since the g term has zero mean by symmetry, the full H̄' cannot equal the claimed diffusivity. So the reduced PMM gradient identity is also unproven.\n\nThe good part: the Bernstein model B(n,L) is a legitimately new dynamics, and the binomial inversion relating it to the reduced PMM is clean and correct (Proposition 2.6 and Lemma 3.1). The elementary properties in Propositions 2.8 and 2.9—partition of unity, symmetry, interpolation, mobile clusters—are clearly argued and appear sound. The paper also honestly states that the hydrodynamic limit is deferred to a companion paper, so the macroscopic equation is not at issue here.\n\nThe soft spots are load-bearing: the gradient property is the paper's main contribution, and it is contradicted by an explicit computation. The errors are specific and might be repairable—perhaps the correct H is different, or the intended range is n≥1—but as written the claims are false. A referee could help sort this out, because the construction itself is promising and fills a real gap in attainable diffusivities.\n\nThis paper is for people working on gradient exclusion processes and hydrodynamic limits. I would not cite it in its current form, but I would not dismiss the underlying idea. Recommendation: send it to a serious referee, with the expectation of major revision; the author needs to fix the gradient identity for all n and the reduced PMM potential.","headline":"The Bernstein model is a genuinely new construction, but the central gradient identity in Proposition 2.7 fails on a small example and Lemma 3.4 gives the wrong diffusivity, so the paper's main theorem is false as stated.","tokens_in":13692,"tokens_out":7227,"would_cite":false,"duration_ms":61172,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"By introducing the Bernstein model, a symmetric exclusion process whose jump rates average to the Bernstein polynomial $B_{n,L}(\\rho)$, this paper proves the gradient property for these rates and closes the gap of attainable diffusivities…","keywords":["gradient model","Bernstein polynomial basis","exclusion process","kinetically constrained lattice gas","porous media model","diffusion coefficient","binomial transform","hydrodynamic limit"],"falsifier":"Enumerate all particle configurations in a box of length $L+2$ for a small case such as $n=1,L=2$ and check the identity $b_{n,L}(\\eta)(e_{0,1}(\\eta)-e_{1,0}(\\eta))=-\\nabla H_{n,L}(\\eta)$ with $H$ as defined; any violation refutes Proposition 2.7. Separately, for the reduced Porous Media Model, compute $\\bar H'(\\rho)$ from the stated $h_{\\ell;L}$ and compare with $\\rho^\\ell$; the two must agree if the gradient potential integrates to the claimed diffusivity.","tokens_in":12617,"feed_emoji":"🔄","tokens_out":7905,"duration_ms":69977,"temperature":0.7,"pith_summary":"This paper constructs a symmetric exclusion process, the Bernstein model, whose exchange rates are local box densities, and proves that it satisfies the gradient property: the microscopic current is the discrete gradient of an explicitly given potential. The consequence is a diffusion coefficient $D(\\rho)=B_{n,L}(\\rho)$, a Bernstein basis polynomial, which the paper argues cannot be obtained by superposing Porous Media Models. A companion reduced Porous Media Model attains $D(\\rho)=\\rho^\\ell$ on the same interaction range and is tied to the Bernstein model by a binomial inversion formula. If the promised hydrodynamic limit goes through, these are the first kinetically constrained lattice gases with these diffusivities, and they give a microscopic underpinning for nonlinear PDEs with such polynomial diffusivities.","feed_headline":"Bernstein diffusivities realized by gradient exclusion process","feed_subtitle":"A local exchange dynamics reaches Bernstein-polynomial diffusion coefficients, filling a gap in attainable PDEs.","key_machinery":"The load-bearing object is the gradient identity for the Bernstein constraint. The proof converts the average over boxes $W_j^L$ into a sum over subsets $P$ of the enlarged box $\\{0,\\ldots,L+1\\}$, rewrites the difference of occupation products as a discrete gradient using $1-\\tau^{-j}=\\sum_{i=1}^j \\nabla\\tau^{-i}$, and telescopes the boundary terms into an indicator that the density in $\\{0,\\ldots,L\\}$ is at least $(n+1)/(L+1)$. The same combinatorial bookkeeping, together with binomial inversion, carries the gradient property to the reduced Porous Media Model and connects the two families of constraints.","core_discovery":"The central claim is Proposition 2.7: for any $n\\le L$, the Bernstein constraint $b_{n,L}(\\eta)=\\frac{1}{L+1}\\sum_{j=0}^L \\mathbf{1}\\{\\langle\\eta\\rangle_{W_j^L}=\\frac{n}{L}\\}$ satisfies $b_{n,L}(\\eta)(e_{0,1}(\\eta)-e_{1,0}(\\eta))=-\\nabla H_{n,L}(\\eta)$, where $H_{n,L}=h_{n,L}+g_{n,L}$, $h_{n,L}(\\eta)=\\frac{1}{L+1}\\mathbf{1}\\{\\langle\\eta\\rangle_L\\ge\\frac{n+1}{L+1}\\}$, and $g_{n,L}$ is a sum of shifted antisymmetric terms. Since the constraint averages to $B_{n,L}(\\rho)$ under the invariant Bernoulli measure, the model is a gradient model with diffusion coefficient $B_{n,L}(\\rho)$, and it coincides with the Porous Media Model when $n=L$. The paper further identifies the reduced Porous Media Model with constraint $p_{\\ell;L}$ and diffusivity $\\rho^\\ell$, and proves the binomial inversion formulas linking $b_{n,L}$ and $p_{\\ell;L}$.","pith_inferences":["One implicit direction: because Bernstein polynomials are dense in continuous functions on $[0,1]$, a construction of this kind could, in the large-$L$ limit, approximate arbitrary nonnegative diffusivities by superposing Bernstein models with gradient property.","The paper's root-multiplicity obstruction is specific to the monomial/PMN basis; in the Bernstein basis, nonnegativity of coefficients is automatic for nonnegative polynomials, so the Bernstein model may be the natural canonical gradient model for polynomial diffusivities.","A testable extension: replace boxes of fixed length $L$ with boxes of length growing with the system size; the gradient identity may still hold, but the hydrodynamic limit would then involve spatially dependent or nonlocal diffusivities.","The claimed obstruction to long-range jumps suggests the gradient property is sensitive to the locality of the constraint, so one could test whether convolutions of local Bernstein rates with a kernel of finite width preserve the gradient structure."],"forward_implications":["If the companion hydrodynamic limit is completed, the Bernstein model yields the macroscopic equation $\\partial_t\\rho=\\partial_u^2 \\bar H(\\rho)$ with diffusion coefficient $D(\\rho)=B_{n,L}(\\rho)$.","Superposition over $n$ gives a gradient model for every polynomial with nonnegative Bernstein coefficients, and since $\\sum_n B_{n,L}=1$, the generators add to the SSEP generator (partition of unity).","The reduced Porous Media Model is gradient for each $0\\le\\ell\\le L$ and has $D(\\rho)=\\rho^\\ell$ while keeping interaction range $2L+2$, offering a fixed-range alternative to the PMM at each diffusivity.","The inversion formulas show the Bernstein constraints are a binomial transform of reduced-PMM constraints, so a target polynomial diffusivity can be translated into local rates by coefficient comparison.","The mobile-cluster structure (boxes of length $L+2$ with exactly $n+1$ particles) is preserved, so the dynamics remains non-cooperative and has the same type of blocked configurations."],"supporting_citations":[{"why":"supplies the gradient-property definition and the variational formula that gives the diffusion coefficient as the averaged constraint.","marker":"[11]"},{"why":"states the criterion that the gradient condition is equivalent to the vanishing of the dynamical part of the Green-Kubo formula, which is the standard used to identify attainable diffusivities.","marker":"[9]"},{"why":"introduces the Porous Media Model, its mobile clusters, and its hydrodynamic limit, which the new models extend.","marker":"[6]"},{"why":"extends the Porous Media Model to slow and fast diffusion and fixes the polynomial-superposition calculus that the paper shows cannot reach Bernstein basis elements.","marker":"[7]"},{"why":"provides the inclusion-exclusion identities used in the binomial transformation and inversion formula relating the Bernstein and reduced-PMM constraints.","marker":"[12]"}],"fun_headline_variants":["Gradient exclusion process realizes Bernstein diffusivities","Bernstein diffusion coefficients from gradient exclusion","Gradient process matches Bernstein polynomial basis","Bernstein-based diffusivity achieved by gradient dynamics","New gradient model fills gap in attainable diffusivities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction's physical conclusion depends on the standard gradient-model identification $D(\\rho)=\\int c\\,d\\nu_\\rho=\\bar H'(\\rho)$, whose hydrodynamic-limit proof is deferred to a companion paper.","fun_headline_variants_meta":{"raw":{"variants":["Gradient exclusion process realizes Bernstein diffusivities","Bernstein diffusion coefficients from gradient exclusion","Gradient process matches Bernstein polynomial basis","Bernstein-based diffusivity achieved by gradient dynamics","New gradient model fills gap in attainable diffusivities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000751,"raw_usage":{"total_tokens":3322,"prompt_tokens":906,"completion_tokens":2416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":2350}},"tokens_in":522,"tokens_out":2416,"duration_ms":16245,"temperature":1.0,"reasoning_tokens":2350,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:46:36.842119+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all particle configurations in a box of length $L+2$ for a small case such as $n=1,L=2$ and check the identity $b_{n,L}(\\eta)(e_{0,1}(\\eta)-e_{1,0}(\\eta))=-\\nabla H_{n,L}(\\eta)$ with $H$ as defined; any violation refutes Proposition 2.7. Separately, for the reduced Porous Media Model, compute $\\bar H'(\\rho)$ from the stated $h_{\\ell;L}$ and compare with $\\rho^\\ell$; the two must agree if the gradient potential integrates to the claimed diffusivity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the gradient-property definition and the variational formula that gives the diffusion coefficient as the averaged constraint."},{"cited_title":"On the green–kubo formula and the gradient con dition on currents","cited_arxiv_id":null,"evidence_quote":"states the criterion that the gradient condition is equivalent to the vanishing of the dynamical part of the Green-Kubo formula, which is the standard used to identify attainable diffusivities."},{"cited_title":"Gon¸ calves, C","cited_arxiv_id":null,"evidence_quote":"introduces the Porous Media Model, its mobile clusters, and its hydrodynamic limit, which the new models extend."},{"cited_title":"Gon¸ calves, G","cited_arxiv_id":null,"evidence_quote":"extends the Porous Media Model to slow and fast diffusion and fixes the polynomial-superposition calculus that the paper shows cannot reach Bernstein basis elements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the inclusion-exclusion identities used in the binomial transformation and inversion formula relating the Bernstein and reduced-PMM constraints."}],"review_version":1}