{"id":"f927bc8b-f041-423b-ba23-fd4b73d843a2","arxiv_id":"2509.07326","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Bayesian hyperparameter selection and ground-state-enforcing priors are compared for cluster expansions; standard posteriors rarely reproduce DFT ground states, and cone-restricted priors fix this.","lead":"This paper compares practical recipes for building Bayesian cluster expansion models of alloys, and shows how to force them to reproduce the correct ground states. The takeaway is a set of guidelines for uncertainty quantification in battery-relevant alloys such as Li-Mg and Li-Al.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Cone-search algorithm can falsely certify basis adequacy via the trivial zero-ECI minimizer of η, undermining the pruning workflow.","rationale":"The paper's core conceptual claim—that ECI space partitions into ground-state cones and that hard priors can restrict a posterior to a chosen cone—is sound and well-supported by standard linear-model geometry. The most load-bearing practical gap is the cone-search algorithm's reliability: it is used both to assess whether a truncation can represent the target ground states and to prune the basis, yet it has no convergence guarantee and admits a trivial spurious minimizer at w=0. This is precisely the secondary assumption the reader flagged. I do not fully agree with the reader's primary weakest assumption (that PBE ground states being wrong undermines the method), because the method is conditional on an externally supplied target set; if that set is wrong, the bias comes from the prior knowledge, not from the methodology. The cone-search concern is more central to the paper's claimed practical contribution. However, the concern does not change the verdict: the Li-Al demonstration includes a direct validation—the cone-restricted MAP voltage curve matches DFT steps—so the algorithm demonstrably worked in the showcased case, and the paper explicitly acknowledges the origin failure mode. Thus ACCEPT remains appropriate, with the caveat that the cone-search heuristic should be stress-tested and ideally replaced or supplemented by a rigorous feasibility method.","tokens_in":28080,"tokens_out":15576,"duration_ms":200922,"concrete_test":"For the 214-basis Li-Al set, independently verify the existence of a point in the strict target ground state cone using linear programming: express the conditions that each target configuration is a vertex of the lower convex hull and every other enumerated configuration lies strictly above it as linear inequalities in w, then solve a feasibility LP. Compare with the cone-search output: (a) does the cone-search return a w that satisfies all strict inequalities? (b) if the LP is feasible, does the cone-search find a feasible point? (c) if the LP is infeasible, does the cone-search incorrectly claim success by returning w=0 or a boundary point? Repeat for the 97-basis RVM set to confirm the reported insufficiency is real.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III E's central demonstration depends on the cone-search algorithm (Section II F) to (1) confirm that a finite basis can replicate the DFT-PBE ground state set, (2) assign removal penalties, and (3) prune from 505 to 214 basis functions. The masking function η(w) of Eq. 23 is positively homogeneous of degree 1: η(tw)=tη(w), and η(0)=0 for any target set because at w=0 all formation energies vanish, making f=g=0. Thus w=0 is a global minimizer of η for every target cone, yet it predicts not the target ground states but a degenerate state in which every configuration lies on the hull. Gradient descent can therefore converge to the origin, which the paper acknowledges as a possible failure, but no convergence proof or post-hoc check is given to ensure the returned point lies in the strict (non-degenerate) target cone. Without such a check, the algorithm may falsely certify a basis as capable or incapable of representing the target ground states, and the subsequent pruning decisions inherit this error. The central claim that a cone-restricted posterior guarantees qualitative consistency is thus only as reliable as this unproven heuristic for the practical workflow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops Bayesian strategies for cluster expansion (CE) surrogate models in first-principles statistical mechanics, focusing on hyperparameter selection and ground-state preservation. The authors compare cross-validation, Bayesian ridge, and the relevance vector machine for fitting BCC Li_xMg_1-x formation energies using LDA, PBE, and SCAN, and propagate posterior ECI samples to free energies and voltage curves. For Li_xAl_1-x, they introduce a masking function η(w) (Eq. 23) that measures deviation from a target ground-state cone, use gradient descent on η to locate points in the cone, and construct a γ→∞ cone-restricted posterior that by construction predicts the target ground states. They show that unconstrained likelihood and RVM posteriors populate many different ground-state cones (40,923 hulls in 500,000 samples), and they estimate the likelihood probability of the PBE cone as at most 3.2×10^-12%.","tokens_in":28309,"tokens_out":17297,"duration_ms":211620,"significance":"If the methods hold, the paper provides a practical framework for a recognized weakness in Bayesian CEs: standard posteriors assign weight to models with qualitatively different ground states, making downstream thermodynamic UQ ill-defined. The quantitative comparison of CV, Bayesian ridge, and RVM is useful, and the cone-restricted prior is a principled way to encode expert structural knowledge. Strengths include transparent reporting of the finite-δ upper bound and explicit acknowledgment of the trivial-origin failure in cone search. However, the cone-search algorithm of Sec. II F is load-bearing for basis certification and pruning, and it currently has an inconsistent descent-direction formula and no post-hoc verification excluding the zero-ECI minimizer. These issues are fixable and do not invalidate the conceptual framework, but they require revision before the practical workflow can be relied upon.","major_comments":[{"comment":"Eq. (27) as written gives +∇_w η(w), not the negative gradient stated in Eq. (24): the first sum is ∇_w∫f (from Eq. 26) and the second is ∇_w∫g, so the expression equals ∇_w(∫f − ∫g). The parenthetical 'primed (unprimed) quantities come from the integral of f (g)' is ambiguous and inconsistent with the notation of Eqs. (25)-(26). The same sign problem appears in Eq. (A2). Because the descent direction is central to the cone-search algorithm, please correct the signs and define primed/unprimed consistently.","section":"II F, Eq. (27) and Appendix A (A2)"},{"comment":"The masking function η in Eq. (23) is positively homogeneous of degree one, so η(0)=0 for every target set: at w=0 both f and g vanish. The zero-ECI vector is therefore a global minimizer of η that does not reside in the non-degenerate target cone. The text acknowledges that descent can 'go directly through the origin' but provides no convergence proof or post-hoc criterion to distinguish a valid cone point from this trivial solution. Since the same algorithm is used in Sec. III E to certify that the 505-function basis can replicate the PBE ground states and to assign ∞ removal penalties during pruning to 214 functions, a false positive would directly corrupt the reported basis-selection workflow. Please add a verification step (e.g., check η(w)=0 and enumerate the convex hull predicted by the returned w) and report its outcome.","section":"II F and III E"},{"comment":"The value 3.2×10^-12% for the probability of the target cone is an upper bound because the δ-integration in Eq. (30) is truncated, as the authors note. This is sufficient for the qualitative conclusion, but the number should be explicitly labeled 'upper bound' wherever it appears, and the truncation point should be stated. If feasible, provide a rough estimate of the neglected tail of ⟨η⟩(δ) to show that the upper bound is non-vacuous.","section":"III E / II G"}],"minor_comments":[{"comment":"The voltage-curve match of the cone-restricted model is built in by construction (γ→∞ prior forces the PBE cone); the real empirical content is that the OLS and RVM MAP models fail. Please state this explicitly to avoid overclaiming the agreement as independent validation.","section":"III E / Fig. 13(a)"},{"comment":"The abstract states that the paper 'provides methods to find and enforce ground-state-preserving models.' Given the heuristic nature of the cone search and the lack of convergence guarantee, consider softening 'find' or adding a caveat.","section":"Abstract / II F"},{"comment":"There is a typo: 'fails to predict a the correct set' should read 'fails to predict the correct set.'","section":"II D"},{"comment":"Equations (28)-(30): define the hypothesis M and clarify that P(M) is with respect to a particular posterior distribution (fixed β and λ). The notation is understandable but could be sharper.","section":"II G"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal's readership and the central idea is sound. The main risk is that the cone-search algorithm is used as a certifier in the pruning workflow; the sign inconsistency and lack of a trivial-solution check should be resolved before acceptance. The novelty relative to the authors' earlier work [37] should also be clarified for the editor: the practical comparisons (CV vs Bayesian ridge vs RVM) and the LiAl ground-state-enforcement workflow are new, but the cone concept and masking function are not."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. It's the clearest comparison I know of CV versus evidence-approximation hyperparameter selection for cluster expansions, and the ground-state-cone business is the part that matters. The core finding: for Li-Al, ordinary Bayesian posteriors (RVM included) essentially never reproduce the DFT-PBE ground-state set; a prior that zeroes out all ECI vectors outside the target cone fixes that, and the resulting voltage curves match DFT qualitatively. That's a real, transferable insight for anyone doing uncertainty quantification on cluster expansions.\n\nWhat's new: the systematic test of three basis-growth sequences, the cone-search algorithm (Eq. 27 / App. A), the pruning protocol, and the free-energy-integration estimate of cone probability. They credit the ground-state-cone idea to their own PRM 2024 and RVM to Aldegunde, so the novelty claim is honest. The math is standard and cleanly presented. The Li-Mg DFT-vs-CE uncertainty comparison is a nice sanity check: DFT functional spread dominates surrogate noise.\n\nSoft spots: the cone-search algorithm has a trivial failure mode—η(w) is positively homogeneous and vanishes at w=0, so gradient descent can collapse to the zero vector. The paper acknowledges this and suggests multiple starts, but gives no convergence guarantee or explicit post-hoc check that the returned point lies in the strict target cone. Since that algorithm is used to certify basis capability and drive pruning, the pruning step inherits whatever the search finds. For the actual systems it worked—the cone-restricted model indeed reproduces the voltage steps—but the method as described could falsely certify a basis. The 3.2e-12% cone probability is explicitly an overestimate from finite-δ integration, which is fine. The bigger practical issue is that no code or data are shipped, so replication requires re-implementation. That's a genuine weakness for a methods paper.\n\nOverall, the central claim holds: a cone-restricted posterior guarantees consistent ground states by construction, and the comparison with unrestricted posteriors is convincing. The paper should go to peer review. I'd ask the authors to add a post-hoc cone-membership check in the cone-search, or at least report the η value of returned vectors, and to release the data and code if at all possible.","headline":"A genuinely useful Bayesian cluster-expansion practice paper; the cone-restricted prior is the real contribution, but the cone-search algorithm's zero-ECI failure mode needs stricter handling.","tokens_in":28864,"tokens_out":2494,"would_cite":true,"duration_ms":29305,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that consistent uncertainty quantification for cluster-expansion alloy models requires restricting the Bayesian posterior to a single ground-state cone in coefficient space.","keywords":["Bayesian cluster expansion","uncertainty quantification","ground-state cones","relevance vector machine","effective cluster interactions","alloy thermodynamics","DFT functional uncertainty","lithium alloys"],"falsifier":"Take 100 ECI vectors sampled from the cone-restricted posterior for Li-Al, build each model's zero-temperature convex hull, and compare the predicted ground states with the DFT-PBE set. If any sample predicts a spurious or missing ground state, the claim that the γ→∞ prior guarantees a single ground-state set fails. Alternatively, re-estimate the reported 3.20×10−12% probability of the PBE cone with a longer free-energy integration; if the value is not negligibly small, the paper's characterization of the unrestricted posterior is contradicted.","tokens_in":27865,"feed_emoji":"⚛️","tokens_out":8774,"duration_ms":109785,"temperature":0.7,"pith_summary":"Cluster expansion models are fast stand-ins for DFT in statistical-mechanics calculations, but their weights are uncertain. The paper argues that uncertainty in downstream thermodynamics is only meaningful if every model drawn from the Bayesian posterior predicts the same ground states, because otherwise sampled models disagree qualitatively about which ordered phases are stable. It shows that the space of expansion coefficients divides into cones, each cone containing models with the same ground-state set, and builds a prior—a masking function that is zero inside the target cone and grows outside—that restricts the posterior to one cone. Applied to BCC Li-Al, the restricted posterior reproduces every step of the DFT voltage curve, while ordinary likelihood and relevance-vector-machine posteriors straddle thousands of cones and add or miss voltage steps. The paper also compares cross-validation and evidence-based hyperparameter choice, finding the relevance vector machine effective for sparse Bayesian cluster expansions.","feed_headline":"One prior pins Bayesian alloy models to one ground-state set","feed_subtitle":"All sampled models reproduce the same ground states, so voltage curves stay qualitatively consistent.","key_machinery":"The central object is a ground-state cone in ECI space: the set of coefficient vectors w that predict the same set of lowest-energy configurations (ground states). The carrying mechanism is the masking function η(w), defined as the integral over composition of the area between the target hull f(w,x) and the hull g(w,x) predicted by w. Zero inside the target cone and monotonically increasing outside it, η is inserted into the prior as exp(−γη), so γ→∞ gives zero probability to models outside the cone. A gradient-descent cone finder, using the gradient of η, locates a starting ECI vector in the target cone and also tests whether a given truncation can realize the target ground states at all.","core_discovery":"The central discovery is that ground-state replication can be enforced as a prior rather than left to chance. ECI space divides into ray-bounded cones, one per ground-state set, and the masking function η(w)=∫[f(w,x)−g(w,x)]dx measures the area between the target and predicted ground-state hulls. The prior P(w)∝exp(−γη(w)), with γ→∞, restricts the posterior to the target cone. For BCC Li-Al, the RVM and least-squares models fail to reproduce the DFT-PBE ground states and their voltage curves gain spurious or missing steps; the cone-restricted MAP reproduces every DFT step. Sampling 500,000 unrestricted ECI vectors yielded 40,923 different ground-state sets, none the DFT set, and the estimate","pith_inferences":["Beyond the paper: cone restriction makes posterior uncertainty conditional on the target ground-state set being true; total uncertainty should therefore also include the choice of DFT functional, which the paper shows moves predictions more than surrogate noise.","The free-energy integration used to estimate the probability of one cone could be reused as a hypothesis test to rank competing candidate ground-state sets, not just to estimate a single cone's probability.","A soft, finite-γ version of the masking prior could serve as a diagnostic: increasing the conjugate parameter δ and watching the mean masking function ⟨η⟩ decline shows how strongly the data pull the posterior away from the imposed ground states.","The same cone construction could be extended beyond exact ground-state matching, for example to enforce relative stability rankings among a handful of competing phases, which is the qualitative constraint most relevant for machine-learned interatomic potentials."],"forward_implications":["A cone-restricted posterior guarantees that every sampled cluster expansion predicts the same target ground states, so downstream free-energy and voltage uncertainties reflect quantitative spread rather than qualitative disagreement.","For Li-Al, unrestricted sampling essentially never lands in the DFT-PBE ground-state cone, so reliable uncertainty quantification for ordered alloys requires an explicit ground-state prior.","The relevance vector machine is a practical hyperparameter scheme for Bayesian cluster expansions: it sparsifies the basis automatically, keeps the posterior Gaussian and analytical, and produces physically reasonable short-range few-body interactions.","Uncertainty across DFT approximations is larger than the cluster-expansion surrogate uncertainty, so honest uncertainty quantification for first-principles thermodynamics should span multiple exchange-correlation functionals as well as surrogate models.","The same cone-and-masking construction transfers to other linear surrogate models, including atomic cluster expansions, to enforce qualitative stability predictions."],"supporting_citations":[{"why":"Introduces the ground-state cones and masking-function prior that this paper applies and extends to a cone-restricted posterior.","marker":"[37]"},{"why":"Provides the complete orthonormal cluster basis that defines ECI space and its ground-state cone geometry.","marker":"[5]"},{"why":"Supplies the Bayesian regression, evidence approximation, and RVM machinery used to set hyperparameters.","marker":"[36]"},{"why":"Defines the relevance vector machine, the sparse evidence-approximation method whose posterior straddles many ground-state cones.","marker":"[40]"},{"why":"First applied the RVM to cluster expansions; its posterior is the baseline for the hyperparameter and uncertainty comparisons.","marker":"[41]"},{"why":"Supplies the cross-validation selection of truncation and hyperparameters that the paper compares with evidence-based selection.","marker":"[38]"},{"why":"Formulates the Bayesian treatment of cluster expansions including correlated ECI priors, grounding the posterior construction.","marker":"[48]"},{"why":"Defines the atomic cluster expansion, the transfer target for the ground-state-cone prior strategy.","marker":"[12]"}],"fun_headline_variants":["Prior that forces Bayesian alloys to match DFT ground states","Bayesian prior locks alloy models onto exact ground-state cones","Ground-state prior: no more spurious voltage steps in alloys","Cone-restricted prior reproduces every DFT ground state","Bayesian cluster expansions get a ground-state enforcement prior"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the DFT-PBE ground-state set is the true set every acceptable model must reproduce; if that set is wrong or functional-dependent, the entire cone-restricted posterior is biased, and the gradient-descent cone search that must find a point in that cone is presented without a convergence proof.","fun_headline_variants_meta":{"raw":{"variants":["Prior that forces Bayesian alloys to match DFT ground states","Bayesian prior locks alloy models onto exact ground-state cones","Ground-state prior: no more spurious voltage steps in alloys","Cone-restricted prior reproduces every DFT ground state","Bayesian cluster expansions get a ground-state enforcement prior"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1433,"prompt_tokens":826,"completion_tokens":607,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":527}},"tokens_in":570,"tokens_out":607,"duration_ms":7317,"temperature":1.0,"reasoning_tokens":527,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T22:24:47.814305+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take 100 ECI vectors sampled from the cone-restricted posterior for Li-Al, build each model's zero-temperature convex hull, and compare the predicted ground states with the DFT-PBE set. If any sample predicts a spurious or missing ground state, the claim that the γ→∞ prior guarantees a single ground-state set fails. Alternatively, re-estimate the reported 3.20×10−12% probability of the PBE cone with a longer free-energy integration; if the value is not negligibly small, the paper's characterization of the unrestricted posterior is contradicted.","supporting_citations":[{"cited_title":"Batzner, A","cited_arxiv_id":null,"evidence_quote":"Introduces the ground-state cones and masking-function prior that this paper applies and extends to a cone-restricted posterior."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the relevance vector machine, the sparse evidence-approximation method whose posterior straddles many ground-state cones."},{"cited_title":"Aldegunde, N","cited_arxiv_id":null,"evidence_quote":"First applied the RVM to cluster expansions; its posterior is the baseline for the hyperparameter and uncertainty comparisons."},{"cited_title":"van de Walle and G","cited_arxiv_id":null,"evidence_quote":"Formulates the Bayesian treatment of cluster expansions including correlated ECI priors, grounding the posterior construction."}],"review_version":1}