{"id":"6b295d8b-aa92-4ad5-af9e-3616f5f62a7f","arxiv_id":"2511.10236","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The claimed exact fluctuation relations reduce to Hamiltonian-of-mean-force free-energy-perturbation identities; their trajectory forms require the final state to be exactly canonical, and the advertised variational/Bessel law is absent from the text.","lead":"This paper derives identities that express free-energy differences of a strongly coupled open system as exponential averages over its final equilibrium marginal, corrected by the chi-squared overlap of the endpoint marginals. These identities are exact but are standard free-energy-perturbation formulas rewritten with the Hamiltonian of mean force; the nonequilibrium dynamics enters only through a postulate that the final marginal is canonical.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Asymptotic equilibration postulate (Eq. 19) is unproven and restrictive; without it the trajectory identities fail, so the claimed generality to arbitrary dynamics and active/biological systems is unsupported.","rationale":"The reader identified the same weakest assumption: the asymptotic equilibration postulate Eq. (19). I agree that this is the load-bearing point for the claimed trajectory-level results and the advertised applications. The endpoint equalities are mathematically correct but are standard FEP identities; the only dynamical content is the substitution under Eq. (19). The paper neither proves nor characterizes this condition, and the validation enforces it by construction with an underdamped Langevin bath, so the agreement in Fig. 1 is a consistency check rather than a test of generality. I also note the abstract promises a Bessel-form variational law and Hessian uncertainty estimates that do not appear in the full text, but I do not rest my objection on that because the core logical gap is Eq. (19). Since the reader's verdict was already REJECT and my concern supports it, I recommend no change to the verdict.","tokens_in":12981,"tokens_out":7948,"duration_ms":86393,"concrete_test":"Run the same two-stage protocol with the non-Liouvillian ramp (parameters from Table I) but replace the underdamped Langevin relaxation by purely dissipative dynamics (set eta_x=eta_y=0, gamma=2, same ramp and durations). The final marginal is then a non-canonical collection of fixed points, violating Eq. (19). Compute the RHS of Eq. (25) over N_traj=30000 and compare with the exact HMF Delta F*_S from Eq. (35). If the estimator deviates beyond sampling error, this confirms that Eq. (19) is load-bearing and that the 'arbitrary dynamics' claim fails when the postulate is not enforced.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 (Eqs. 15-16) is an algebraic FEP identity: given the HMF canonical marginals P0^S and Pteq^S, Eq. (16) is the standard forward perturbation formula and Eq. (15) is the reverse/overlap formula; no dynamics appear. The central claim to go 'beyond JE' rests on Theorem 2 (Eqs. 21-25), which replaces the final marginal by the pushforward of the initial ensemble. This replacement is valid only under the asymptotic equilibration postulate (19): P_t^S -> P_teq^S. The paper states this as a postulate, defers justification to a companion paper [49], and provides no characterization of the protocols or dynamics for which it holds. It is not a consequence of 'arbitrary dynamics.' In the advertised settings—active matter, biological processes—the stationary marginal is generally not the HMF canonical marginal; for example, active Ornstein-Uhlenbeck particles have non-Boltzmann stationary distributions. In such cases Eqs. (21)-(25) will not hold. The numerical validation builds Eq. (19) in by using underdamped Langevin relaxation whose stationary state is exactly canonical; hence Fig. 1 checks only internal consistency of the algebra, not the generality claim. Without Eq. (19) the remaining content is a known equilibrium identity, not a nonequilibrium fluctuation theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives identities for the Hamiltonian of mean force (HMF) free energy difference ΔF*_S between two equilibrium endpoints of an open system, claiming they hold for arbitrary driven dynamics without microscopic reversibility, detailed balance, fluctuation-dissipation, or local detailed balance. Theorem 1 (Eqs. 15–16) expresses e^{±βΔF*_S} as exponential moments of the HMF shift averaged over the final system marginal, with a chi-squared endpoint-overlap term in one form. Theorem 2 (Eqs. 21–22) gives trajectory versions under an 'asymptotic equilibration' postulate (Eq. 19). Frozen-coupling corollaries (Eqs. 24–25) and a heat-work-feedback decomposition (Eqs. 26–34) are presented, and the paper claims the Jarzynski equality (JE) and its feedback extension (GJE) are recovered as limiting cases. A numerical validation on a non-Liouvillian ramp followed by underdamped Langevin relaxation is reported. The abstract also promises a maximum-entropy variational construction, which does not appear in the main text.","tokens_in":13213,"tokens_out":8610,"duration_ms":88187,"significance":"If the claims were fully established, the framework would be a significant step beyond traditional fluctuation theorems, because it would allow free-energy reconstruction for strongly coupled, non-reversible dynamics without detailed balance. The algebraic identities in Theorem 1 are correct, and the numerical demonstration that the naive JE estimator fails under a non-Liouvillian ramp while the endpoint identities reproduce the exact partition ratio is a useful consistency check. However, the central scientific claim is not supported. Theorem 1 is essentially a rewriting of known free-energy perturbation identities with no dynamical content; Theorem 2 depends on an unproven and restrictive equilibrium-relaxation postulate; and the numerical validation builds that postulate into the simulation protocol. The advertised applications to active matter and biological systems, whose stationary marginals are generally non-canonical, are therefore not justified by anything in the paper.","major_comments":[{"comment":"The trajectory counterparts (Eqs. 21–22) are valid only under the asymptotic equilibration postulate (19), which asserts that the final system marginal equals the HMF canonical marginal. This is not a consequence of 'arbitrary dynamics'; it is a strong equilibrium assumption. The paper does not prove sufficient conditions or characterize the protocols and dynamics for which it holds, and it defers to a companion paper [49]. For active-matter and biological systems, the stationary marginal is generally non-canonical (e.g., active Ornstein–Uhlenbeck particles), so the advertised generality is unsupported. Without Eq. (19), only Theorem 1 remains, which is an equilibrium identity with no dynamics.","section":"§3, Eq. (19) and Theorem 2"},{"comment":"The endpoint equalities are algebraic identities: Eq. (16) is the standard free-energy perturbation formula ⟨e^{β(H*_f−H*_i)}⟩_{P_f} = Z_i/Z_f = e^{βΔF*}, and Eq. (15) is the corresponding overlap form. The trajectory map T_t and the dynamics do not enter these relations. Claiming that they constitute 'exact fluctuation relations' for nonequilibrium processes overstates their content: JE is a trajectory-level statement about work under a specified dynamical generator, while Theorem 1 requires sampling the exact equilibrium final marginal. The paper should explicitly state that Theorem 1 is an equilibrium identity, not a nonequilibrium fluctuation theorem.","section":"§2, Eqs. (15)–(16)"},{"comment":"Eqs. (35)–(40) do not derive JE as a limiting case of Theorem 1 or Theorem 2. They instead re-derive the conventional JE by assuming Hamiltonian dynamics, Liouville's theorem, and a work definition. Eq. (40) is simply the standard JE written in HMF notation. The statement that the framework 'includes the JE and its feedback extensions (GJE) as limiting cases' is unsupported; in particular, GJE is never derived or shown to reduce to a limit of the present identities, only mentioned qualitatively in the 'GJE connection' paragraph.","section":"§4, JE as limiting case"},{"comment":"The numerical validation is a consistency check, not a test of the trajectory theorem. The 'exact' reference is the partition ratio in Eq. (35), and the estimators in Eqs. (24)–(25) are evaluated after an underdamped Langevin relaxation that was deliberately chosen to satisfy Eq. (19) and converge to the same canonical ensemble used to compute the exact reference. Thus the agreement in Fig. 1 is built into the protocol. The failure of the JE estimator is expected because W_λ is not the proper work for the non-Liouvillian ramp; this does not demonstrate that the new identities are useful for arbitrary nonequilibrium processes. A meaningful test would need to verify Eq. (19) independently or use a protocol whose final stationary marginal is non-canonical.","section":"§5, Validation and Fig. 1"},{"comment":"The abstract and title promise a 'maximum-entropy construction' yielding a 'Bessel-form scalar-action law' and 'Hessian uncertainty estimates,' but no such construction appears in the main text or the appendix. If this content is deferred to the companion paper [49], the present Letter is incomplete relative to its stated scope; if it is intended to be part of this work, it is missing. This is a substantive omission, as the 'variational' aspect of the claimed result is absent.","section":"Abstract and main text"}],"minor_comments":[{"comment":"The notation ⟨•⟩^S for the final-marginal average and ⟨•⟩_{X0} for the initial full-system average is easy to confuse; consider using distinct bracket symbols or subscripts.","section":"Notation, Eqs. (15)–(20)"},{"comment":"The value C=1.0 is listed twice under 'System–environment coupling'; one entry appears to be a typo.","section":"Table I"},{"comment":"The chi-squared divergence is written as −1 + ∫ (P_teq^S)^2 / P_0^S; please state explicitly that this is χ²(P_teq^S ∥ P_0^S) and define the orientation clearly.","section":"Eq. (18)"},{"comment":"The label 'feedback' for the term II (Eq. (31)) is misleading; this term is a projection of the driven velocity onto the initial force field. The connection to information-theoretic feedback in the Sagawa–Ueda sense is not established, and using the same word may cause confusion.","section":"Corollary 2"},{"comment":"The phrase 'exact trajectory forms' may suggest the identities hold for any trajectory ensemble; in fact they require the final pushed-forward marginal to equal the equilibrium HMF marginal. Please add an explicit caveat beneath Theorem 2.","section":"Theorem 2 wording"}],"recommendation":"reject","confidential_remarks":"The manuscript is mathematically correct in its algebraic identities, but the central novelty is a reformulation of known free-energy perturbation expressions with an extra equilibrium-relaxation assumption. The overclaims in the abstract and conclusions are substantial: the trajectory theorem is not a general nonequilibrium fluctuation theorem, and the validation is tautological. The missing variational section further weakens the paper's fit as a Letter. If the authors were to reframe the contribution as a diagnostic tool based on overlap measures, the algebraic identities might be publishable in a more specialized venue, but in the present form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a clean algebraic repackaging of free-energy perturbation in Hamiltonian-of-mean-force variables, with a trajectory extension that rests on an unproven and restrictive relaxation postulate, and a numerical validation that is essentially tautological. The abstract advertises a maximum-entropy variational construction, a Bessel-form scalar-action law, and Hessian uncertainty estimates; none of that appears in the full text.\n\nWhat's good: the endpoint identities in Theorem 1 are derived correctly, the chi-squared overlap is a genuinely useful diagnostic for endpoint mismatch, and the heat-work-feedback decomposition in Corollary 2 is a neat algebraic separation, even if it's bookkeeping. The writing is clear, and the authors are honest that Theorem 1 coincides with their own prior endpoint equality [50]. The algebra checks out.\n\nThe problems are in the framing. Theorem 1 is exactly the standard Zwanzig FEP identity and its reverse/overlap form in HMF language—no dynamics, no fluctuation theorem. Theorem 2, the supposed 'beyond JE' content, requires Eq. (19), the asymptotic equilibration postulate: the final marginal must be exactly the canonical HMF marginal. That is not proven or characterized; it is deferred to a companion paper. For many advertised applications—active matter, biology—stationary marginals are non-Boltzmann, so the postulate fails. The validation builds the postulate in: the ramp is followed by underdamped Langevin relaxation whose stationary state is canonical, so comparing Eqs. (24)-(25) to the 'exact' partition ratio is checking the same identity against itself. The JE failure in their figures is a nice illustration of phase-space compression breaking Liouville, but it does not test the generality claim. The 'JE as limiting case' statement is true only in the trivial sense that Eq. (36) is the partition ratio; the appendix just recapitulates Jarzynski's own derivation, and the framework is not needed.\n\nThe abstract mismatch is the most serious issue: it promises a variational construction with Bessel-form and Hessian outputs that do not exist in the manuscript. That is not a minor editing slip.\n\nWho is this for? A reader working on strong-coupling free energy estimation might take away the chi-squared overlap diagnostic. But as a Letter claiming to go beyond the Jarzynski equality, it does not deliver. My verdict: the math is right but the central claim is unsupported. A serious referee should see it because the issues are substantive and worth catching, but my own recommendation would be reject as a standalone Letter unless the variational results are actually included and the postulate is characterized.","headline":"Correct but known HMF/FEP identities are dressed up as a new fluctuation theorem; the abstract promises variational results that the text never delivers.","tokens_in":13764,"tokens_out":4662,"would_cite":false,"duration_ms":47326,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Exact free-energy identities for open systems that need no reversibility, detailed balance, or fluctuation-dissipation structure, with the Jarzynski equality as a limiting case.","keywords":["Hamiltonian of mean force","Jarzynski equality","fluctuation theorems","nonequilibrium free energy","strong coupling","chi-squared divergence","non-Liouvillian dynamics","open quantum systems"],"falsifier":"Run a driven open system with the same endpoints but a protocol that demonstrably relaxes to a non-canonical non-equilibrium steady state (e.g., an active-matter bath or an underdamped Langevin system with a non-conservative force). If the trajectory equality e^{+βΔF*_S}=⟨e^{+βΔH*_β}⟩^S still holds numerically, the asymptotic equilibration postulate is not necessary; if it fails, the postulate is confirmed as the limiting condition. A direct check is to measure the final marginal P^S_t at large t and compare it to the HMF canonical form via the chi-squared divergence — nonzero divergence signa","tokens_in":12784,"feed_emoji":"⚡","tokens_out":2263,"duration_ms":26270,"temperature":0.7,"pith_summary":"This paper derives exact identities for the free energy difference between two equilibrium endpoints of an open system, valid for arbitrary driven dynamics that connect those endpoints. The exponential of the free-energy difference is expressed as an exponential moment of the Hamiltonian-of-mean-force shift divided by a chi-squared overlap between the endpoint marginals, and a complementary identity holds for the negative exponent. These endpoint identities require no microscopic reversibility, detailed balance, fluctuation-dissipation structure, or local detailed balance. A trajectory-level version is proven under an asymptotic equilibration postulate, and in the frozen-coupling regime the identities reduce to bare-system expressions with a heat-work-feedback decomposition. The significance is that free energy differences can in principle be reconstructed from nonequilibrium trajectories in regimes where the standard Jarzynski equality fails, such as strongly coupled, active, or biological systems.","feed_headline":"Free-energy identities that work when the Jarzynski equality fails","feed_subtitle":"Open-system free energies from endpoint overlap and HMF moments, with no detailed-balance or reversibility assumptions.","key_machinery":"The central objects are the Hamiltonian of mean force (HMF) H*_β, which defines the open system's canonical marginal and free energy F*_S, and the chi-squared divergence χ²(P^S_teq∥P^S_0) between the endpoint marginals. The asymptotic equilibration postulate — that the time-evolved marginal converges to the final canonical HMF marginal — is the mechanism that converts the algebraic endpoint identities into trajectory-level equalities. In the frozen-coupling regime, the HMF shift reduces to the bare-system Hamiltonian shift, yielding a heat-work-feedback decomposition (W_S, Q_S, and a reference functional II) that compensates dynamical asymmetry.","core_discovery":"The central claim is that for an open system S coupled to an environment E, with equilibrium endpoints at inverse temperature β described by Hamiltonian-of-mean-force (HMF) free energies F*_S, the identities e^{-βΔF*_S} = ⟨e^{-βΔH*_β}⟩^S / (1+χ²(P^S_teq∥P^S_0)) and e^{+βΔF*_S} = ⟨e^{+βΔH*_β}⟩^S hold, where the average is over the final HMF marginal and χ² is the chi-squared divergence between endpoint marginals. These endpoint equalities are proven without any dynamical generator assumptions; they follow from equilibrium partition-function ratios. Under the asymptotic equilibration postulate — that the driven final marginal equals the canonical HMF marginal — the equalities acquire exact tra","pith_inferences":["The endpoint equalities (Theorem 1) are essentially equilibrium free-energy perturbation identities rewritten in HMF variables; their content is algebraic rather than dynamical. The real dynamical claim is the trajectory form (Theorem 2), which stands or falls on the asymptotic equilibration postulate.","A testable extension: for protocols that relax to a non-canonical non-equilibrium steady state, the trajectory identities should fail exactly when Eq. (19) is violated; quantifying the discrepancy would measure how far the final marginal is from the HMF canonical form.","The chi-squared overlap term can be viewed as a sampling-efficiency pre-factor: its divergence signals the regime where exponential averaging becomes dominated by rare trajectories, connecting to known difficulties with Jarzynski estimators in high-dimensional systems.","The heat-work-feedback decomposition suggests a practical protocol-design criterion: minimizing the reference functional II relative to the realized heat Q_S should reduce the variance of the exponential estimator, which could be tested in simulation."],"forward_implications":["If correct, the identities provide a route to reconstruct free energy differences from nonequilibrium trajectories in strongly coupled, active, or biological systems where detailed balance and local detailed balance are violated.","The standard Jarzynski equality and the Sagawa-Ueda generalized Jarzynski equality become special cases, so the framework unifies and enriches them by explicitly exposing endpoint overlap and system-environment interaction effects.","In the frozen-coupling regime, the trajectory equalities reduce to bare-system expressions, allowing practical estimation using only system coordinates and a heat-work-feedback correction.","The chi-squared overlap term acts as a diagnostic: large overlap burden signals poor endpoint overlap, which explains common failures of the Jarzynski equality in simulations and experiments.","The variational scalar-action law (maximum-entropy construction) is claimed to provide three outputs from the same sampled configurations — the HMF free energy difference, the endpoint-overlap burden, and a Hessian uncertainty estimate — independent of the microscopic system and environment at the level of the variational reconstruction."],"fun_headline_variants":["Exact free energies without reversibility or detailed balance","Open-system free energies from endpoint overlap","When Jarzynski fails, try endpoint identities","Free energy identities that work when Jarzynski fails"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is Eq. (19): that after the driving stops and the composite relaxes, the final marginal of the open system is exactly the canonical Hamiltonian-of-mean-force marginal at the final control parameters.","fun_headline_variants_meta":{"raw":{"variants":["Exact free energies without reversibility or detailed balance","Open-system free energies from endpoint overlap","When Jarzynski fails, try endpoint identities","Free energy identities that work when Jarzynski fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000819,"raw_usage":{"total_tokens":3462,"prompt_tokens":822,"completion_tokens":2640,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":2581}},"tokens_in":566,"tokens_out":2640,"duration_ms":18436,"temperature":1.0,"reasoning_tokens":2581,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:29:00.714540+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a driven open system with the same endpoints but a protocol that demonstrably relaxes to a non-canonical non-equilibrium steady state (e.g., an active-matter bath or an underdamped Langevin system with a non-conservative force). If the trajectory equality e^{+βΔF*_S}=⟨e^{+βΔH*_β}⟩^S still holds numerically, the asymptotic equilibration postulate is not necessary; if it fails, the postulate is confirmed as the limiting condition. A direct check is to measure the final marginal P^S_t at large t and compare it to the HMF canonical form via the chi-squared divergence — nonzero divergence signa","supporting_citations":[],"review_version":1}