{"id":"cc0808a8-45a5-43d0-a0a3-07c9edc54774","arxiv_id":"2608.02536","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Committors and rates can be estimated from equilibrium samples and two state definitions by minimizing a boundary-free energy/fidelity ratio over ridge functions.","lead":"This paper derives a variational principle for committor functions that removes boundary conditions, replacing them with a measurable 'fidelity' quantity, so that trial functions violating the boundary values remain admissible. It then uses cheap one-dimensional projections to compute committors and rates for peptides, including folding rates from umbrella sampling alone.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The boundary-free variational identity is exact, but the practical estimator swaps the flux-weighted fidelity for an equilibrium basin-moment (Eq. 10); in high-dimensional projections this substitution is uncontrolled, so the chignolin rate claim—whose flux profile does not plateau—lacks the certifi","rationale":"The paper's exact variational identity (Eq. 4-5) is a genuine contribution and is carefully derived; the sliced construction is cheap and the code is available, which provides independent support. The reader's CONDITIONAL verdict already captures the key weakness: the exact theorem is not the estimator actually used. My stress-test agrees with that assessment and locates the load-bearing step in the replacement of the flux-weighted fidelity by the basin-moment difference (Eq. 10). In high dimensions the projected basins overlap for almost every direction, so the per-slice fidelity error can be large; the authors' weight-suppression argument is empirical and system-specific, not a guaranteed mechanism. The chignolin rate is the headline application of the method, and it is precisely there that the supplementary material reports the isocommittor flux plateau premise fails. Thus the reported rate rests on two uncontrolled approximations: the fidelity substitution and the non-flat flux profile. This does not refute the exact theory, nor does it invalidate the 2D and AIB9 committor validations, but it does mean the rate claim should remain conditional until the empirical fidelity and the plateau read-off are validated in a setting where both can be checked. Hence no change to the reader's verdict is warranted.","tokens_in":32117,"tokens_out":10300,"duration_ms":123170,"concrete_test":"Use the archived reproduction package to run a synthetic d-dimensional extension of the Wolfe-Quapp benchmark (e.g., embed the same potential in d=20 or d=52 with flat nuisance coordinates) so that projected state densities overlap as in the molecular systems. Compute exact F_j by finite-volume integration of the reactive current over ∂A and ∂B, then solve Eq. (13) with \\hat F_j. Check (i) whether E[\\bar q]/\\hat F[\\bar q]^2 ≥ ν_AB holds, i.e. whether the certified bound Eq. (S6) is preserved under the empirical fidelity, and (ii) whether the S16 plateau estimate \\hat ν_AB stays within that certified bound. If the empirical estimator violates the bound or the plateau read-off deviates by more than the paper's reported uncertainty, the chignolin rate claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (5) is correct as a statement about the exact flux fidelity F[u]. The implementation, however, solves Eq. (13) with \\hat f_j = b_j - a_j (Eq. 10), replacing the reactive-flux boundary average by an equilibrium basin average. This is not merely statistical noise; it is a different measure. The per-slice bias is ~0.015 on separated directions and ~0.15 on overlapping ones (supplement, 'Empirical fidelity'), and in the molecular feature spaces 'almost every direction overlaps', so the authors' defense rests on weight suppression (Fig. S1c), which is demonstrated on 2D exact ground truth and one peptide, not proven. The certified bound Eq. (S6) and the error identity Eq. (14) require F[\\bar q]=1; with \\hat f they become approximate. The weighted deficit 1 - F[\\bar q] can only be measured from reactive segments (Eq. S9); chignolin umbrella sampling provides none. The chignolin rate therefore has no check on this substitution, and its isocommittor flux profile is explicitly non-plateauing (flatness 0.71, band mass 27% vs 60%, window shifts 22%). The factor-2.5 agreement with two references carries an admitted factor 2-3 uncertainty and does not close this gap. The practical central claim—rates from umbrella sampling alone—rests on an uncontrolled approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new variational principle for committors and reaction rates in which the Dirichlet boundary conditions are replaced by a scalar normalization of a 'fidelity' functional F[u] that is measurable from state-labelled equilibrium samples. The exact identity E[q] = min_u E[u]/F[u]^2 (Eq. 5) is derived cleanly. The authors then propose a 'sliced committor' estimator built from 1D profiles along random projection directions, with a closed-form optimum (Eq. 13). They apply it to AIB9 and villin HP-35 committors in full torsion space and to chignolin folding/unfolding rates from umbrella sampling alone. The advertised practical contribution is that rates and committors can be obtained from equilibrium/reweighted samples without boundary-satisfying trial functions or dynamical trajectories. The exact mathematical identity is sound, but the estimator replaces the flux-weighted fidelity by a basin-moment difference (Eq. 10), and the chignolin rate read-off uses an isocommittor flux plateau that the paper's own diagnostics show to be absent.","tokens_in":32514,"tokens_out":5538,"duration_ms":62311,"significance":"If the numerical estimator were as reliable as the exact variational identity, this would be a significant advance: it removes the principal construction cost in variational committor methods and allows trial spaces that violate boundary conditions, including cheap 1D slices. The paper is commendably open with code, data, and extensive diagnostics; the supplementary material explicitly quantifies biases, reports negative gaps, and flags the chignolin plateau failure. These features raise the quality of the report. However, the significance of the practical claims is currently limited because the substitution in Eq. (10) is uncontrolled in high-dimensional molecular feature spaces, and the one rate application, chignolin, rests on a flux read-off that the authors themselves show to be non-plateauing. The exact identity and the algorithmic framework are publishable, but the molecular rate and error-bound claims are not yet supported at the level asserted.","major_comments":[{"comment":"The exact variational bound Eq. (5) and the certified cap Eq. (S6) are stated for the flux-weighted fidelity F[u]. The implementation replaces it by the equilibrium basin-moment difference \\hat f_j = b_j - a_j. This is a different measure, not just statistical noise. The supplement ('Empirical fidelity on a molecular system') states that in the d=52–350 feature spaces 'almost every direction overlaps' and that the per-slice bias saturates near 0.15 on overlapping directions. The defense is weight suppression (Fig. S1c), demonstrated on two 2D systems and on AIB9, not on chignolin. For chignolin no reactive segments exist to evaluate the weighted deficit of Eq. (S9), so the central rate estimate has no check on this substitution. This is load-bearing and should be addressed by either supplying a reactive-flux fidelity estimate for chignolin or explicitly qualifying the rate as uncontrolle","section":"Main text, §'Fidelity from samples', Eq. (10)"},{"comment":"The rate is read from the isocommittor flux plateau, but Supplement 'Chignolin: the plateau premise fails' reports that the profile decays monotonically, with flatness 0.71 (four times the largest of the other systems), band mass 27% versus the 60% expected from flux conservation, and 22% shifts between nested windows. The flatness-selected sub-band gives ehat = 1.76, and the paper states that the reported ehat should be read 'as a diagnostic ... and not as a calibrated error bar.' Despite this, the main text reports chignolin folding/unfolding rates of 2.2 and 0.20 µs^{-1} and claims agreement within a factor of 2.5. This is an internal inconsistency: the rate claim is not backed by the method's own certification.","section":"Chignolin rate read-off, Eq. (S16)"},{"comment":"The supplementary analysis of AIB9 shows that the plateau read-off overestimates the flux by at least 9.7% (cap E/F^2 = 0.01819 versus plateau µν_AB = 0.01995), and the M=1024 rung gives r ≥ 1.29 and e ≥ 0.175. This demonstrates that the isocommittor plateau estimator of Eq. (S16) can be substantially biased high even when the pointwise committor RMSE is good. Since the same read-off is used for the chignolin rate, this is a direct concern for the paper's central numerical claim, independent of the chignolin plateau failure.","section":"AIB9 diagnosis in Supplement, 'Diagnosing the negative gap'"},{"comment":"Equation (14) is called an 'error bound without the answer,' but with the estimated \\hat\\nu_AB it is an estimate, not a bound; the supplement's Table S1 and the AIB9 analysis make this clear. The certified bound Eq. (S6) is only certified for F, not for \\hat f. The paper oscillates between 'bound' and 'estimate' language. This is not merely a wording issue: it obscures the fact that the central error statement is uncontrolled in the chignolin application. Please either prove a bound for the empirical estimator under explicit assumptions or consistently present Eq. (14) as a diagnostic.","section":"Main text, §'An error bound without the answer', Eq. (14)"}],"minor_comments":[{"comment":"The symbol F is used both for the fidelity functional F[u] and for the projected free energy F_j(s) in Eq. (8) and Fig. 1. This is confusing and should be disambiguated, e.g., \\Phi_j(s) for the free energy.","section":"Notation throughout"},{"comment":"The abstract states that rates are obtained 'from umbrella sampling alone,' but the method also requires a diffusion constant estimate D0 (main text, chignolin section). Please mention this input in the abstract to avoid overstatement.","section":"Abstract and Introduction"},{"comment":"The villin RMSE of 0.21 is said to be limited by the reference (only ≈15 committed folding events). The caption should state the statistical uncertainty of the reference explicitly, since the reader cannot otherwise interpret the quality of the sliced committor.","section":"Fig. 2 caption"},{"comment":"The held-out quotient used to select hyperparameters is a reasonable model-selection rule, but the text should clarify that the folds are contiguous blocks within states; this is mentioned only in passing. Also, the grid over ε is extensive in M; please state the exact range used for each system in the tables.","section":"Supplement, Eq. (S14)"},{"comment":"There is a typo: 'at the the Gauss Centre' should read 'at the Gauss Centre.'","section":"Acknowledgments"}],"recommendation":"major_revision","confidential_remarks":"The exact variational identity in Eq. (5) is clean and likely correct, and the open code/data policy is a strength. The practical claims, however, outrun the evidence: the empirical fidelity substitution is uncontrolled in high dimensions, and the chignolin rate is read from a non-plateauing flux profile that the authors themselves diagnose as invalid for error calibration. A revised version that either validates the substitution with reactive-trajectory data for chignolin or substantially tempers the rate/error-bound claims would be publishable. I would not accept the current version because the central advertised rate result is not supported by the method's own diagnostics."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: Eq. (5) is a real advance. The authors show the committor can be characterized by minimizing E[u]/F[u]^2 over trial functions that need not satisfy the boundary conditions, replacing them with a scalar fidelity measurable from labelled equilibrium samples. That is not in the prior literature; the discrete-time bound in [64] is the closest antecedent, and the continuous identity plus the sliced-ridge construction are new. The closed-form optimum (one Cholesky) and the linear scaling in N and d at fixed M make it practical. The paper also ships code, data, and unusually transparent supplementary diagnostics. On AIB9 the committor matches a forward-tracing reference well; on villin the accuracy is limited mostly by the reference. This deserves serious referee attention.\n\nNow the soft spot, and it is not minor. The exact bound in Eq. (5) holds for the true flux-weighted fidelity F. The implementation swaps in the basin-moment difference \\hat F = b - a (Eq. 10), which is a different measure. The per-slice bias is ~0.015 on separated directions and ~0.15 on overlapping ones; in d=52-350 feature space almost every direction overlaps. The authors' defense is weight suppression: the constrained solve puts weight on the less biased directions, and they demonstrate this on 2D exact ground truth and on AIB9 where reactive segments are available. That is credible as an empirical argument, but it is not a controlled approximation for the general case.\n\nThe chignolin rate is where this bites. The rate read-off assumes the isocommittor flux profile plateaus (Eq. S16). Their own supplement says chignolin's profile does not plateau; flatness 0.71, the band carries 27% of the Dirichlet mass instead of 60%, and the read-off shifts 22% across nested windows. There are no unbiased reactive segments in umbrella sampling to measure F, so the fidelity substitution is unchecked. The reported factor-2.5 agreement with two references carries an admitted factor 2-3 uncertainty and does not close the gap. I would not call the method invalid — the math is right and the AIB9 committor is convincing — but the main-text claim that rates come from umbrella sampling alone is not fully supported. The paper itself contains the tools to see this; that honesty is to its credit.\n\nMy recommendation: send it to peer review. A good referee will ask for (a) the chignolin flux plateau to be addressed or the rate claim demoted to preliminary, and (b) more evidence on the fidelity substitution in high dimensions. The exact identity and the general construction are strong enough to warrant that work.","headline":"Genuinely new boundary-free variational principle, clean math, honest limitations; but the chignolin rate claim rests on an uncertified plateau and should be labeled preliminary.","tokens_in":32967,"tokens_out":2733,"would_cite":true,"duration_ms":29984,"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":"Boundary conditions can be dropped from the committor variational principle, making trial functions that violate them admissible.","keywords":["committor","transition path theory","reactive flux","Dirichlet energy","variational principle","reaction coordinate","rare events","umbrella sampling"],"falsifier":"Run the estimator on a high-dimensional system with known committor (e.g., a harmonic barrier with many overlapping projections) and compare the predicted ν_AB against a long-trajectory forward-flux estimate. If the variational quotient E/F^2 ever falls below the true reactive flux, the identity is wrong; if the convergence in the number of directions stalls while the plateau diagnostic fails, the rate read-off needs revisiting.","tokens_in":32017,"feed_emoji":"⚛️","tokens_out":4657,"duration_ms":49617,"temperature":0.7,"pith_summary":"The committor—the probability that a trajectory reaches product before reactant—is the ideal reaction coordinate for rare transitions, but it is normally defined as the minimizer of a Dirichlet energy over functions that vanish on one state and equal one on the other. This paper proves that these boundary conditions can be replaced by a single scalar normalization, the fidelity, which is measurable from state-labelled equilibrium samples. Because the boundary conditions are gone, trial functions that cannot satisfy them—such as one-dimensional profiles along random projections—are admissible, and the optimal combination has a closed form. The resulting estimator consumes only configurations reweightable to equilibrium, two state definitions, and a diffusion constant, and the paper demonstrates it on peptides in full torsion space and on chignolin folding rates from umbrella sampling alone. The cost is roughly linear in sample count and dimension at fixed number of directions.","feed_headline":"Boundary conditions dropped from committor computation","feed_subtitle":"Fidelity-normalized variational principle recovers committors and rates from samples alone, up to 350 dimensions.","key_machinery":"The load-bearing object is the flux–fidelity identity: for any trial function u of finite Dirichlet energy, the inner product ⟨u,q⟩_D equals the reactive flux ν_AB times the fidelity F[u], defined as the flux-weighted average of u over the product boundary minus the flux-weighted average over the reactant boundary. This turns the Dirichlet error expansion into a ratio that needs no boundary conditions. The paper pairs it with a ridge-function ansatz: one-dimensional reaction-diffusion profiles q_j along Fisher-discriminant-broadened random projections, combined by the closed-form solve of a rank-one generalized eigenvalue problem, with regularization selected by a held-out variational score.","core_discovery":"The central claim is the identity ν_AB = min_u E[u]/F[u]^2, where E[u] is the Dirichlet energy and F[u], the fidelity, is the difference between the flux-weighted boundary averages of u. The identity follows from the flux-fidelity relation ⟨u,q⟩_D = ν_AB F[u], obtained by integrating the trial function against the conserved reactive current. The quotient is scale- and offset-free, the minimum is attained by u = λq + const, and no boundary condition is imposed. In a finite trial space the same principle yields an upper bound on the reactive flux that tightens monotonically as directions are added. Using basin-sample moments as empirical fidelities, the paper obtains high-dimensional committor","pith_inferences":["The same flux–fidelity identity should transfer to other reversible processes with two absorbing sets, such as nucleation, allele fixation, or climate transitions, where equilibrium-like samples exist but no clean boundaries can be imposed.","One testable extension is to position-dependent diffusion and non-reversible steady states: the paper's closed-form Gram matrix assumes constant diffusion, but the flux–fidelity identity itself does not.","The empirical-fidelity substitution (basin moments for flux-weighted averages) could be validated more generally by comparing against reactive-trajectory estimators on systems with many more committed events than villin's fifteen.","The plateau assumption in the rate read-off is the fragile step; a search for a flat isocommittor-flux region (or lack thereof) should become a standard diagnostic in any application."],"forward_implications":["Committors can be estimated from equilibrium or biased (umbrella-sampled, metadynamics) configurations alone, with no time-lagged pairs, shooting trajectories, or iterative sampling.","The variational bound survives without boundary conditions, so any trial function yields a certified upper bound on the reactive flux that can be tightened monotonically.","Rates can be recovered from biased data by reading the isocommittor flux plateau, as demonstrated for chignolin folding and unfolding from umbrella sampling alone.","The method scales linearly in sample count and dimension at fixed direction count, making full-torsion-space committors feasible for peptides of 52–350 dimensions.","The fidelity measurement provides a boundary diagnostic that can be evaluated for any variational committor method, not just this one."],"fun_headline_variants":["Committors without boundary conditions","Fidelity normalization replaces boundary conditions","Reactive flux from unconstrained trial functions","High-dimensional committors from samples alone"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The exact theorems hold for the true flux-weighted fidelity, but the practical estimator substitutes the difference of basin sample averages, which turns the bound into an approximation; the rate step additionally assumes a flat isocommittor-flux plateau that is explicitly violated in the chignolin application.","fun_headline_variants_meta":{"raw":{"variants":["Committors without boundary conditions","Fidelity normalization replaces boundary conditions","Reactive flux from unconstrained trial functions","High-dimensional committors from samples alone"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":1894,"prompt_tokens":730,"completion_tokens":1164,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":1113}},"tokens_in":474,"tokens_out":1164,"duration_ms":9312,"temperature":1.0,"reasoning_tokens":1113,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T05:08:54.608410+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the estimator on a high-dimensional system with known committor (e.g., a harmonic barrier with many overlapping projections) and compare the predicted ν_AB against a long-trajectory forward-flux estimate. If the variational quotient E/F^2 ever falls below the true reactive flux, the identity is wrong; if the convergence in the number of directions stalls while the plateau diagnostic fails, the rate read-off needs revisiting.","supporting_citations":[],"review_version":1}