{"id":"9b2e049f-e28c-48f7-8ff3-3194c7666dda","arxiv_id":"2606.27486","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"Introduces counterwork via reversed memory-filtered driving and derives Jarzynski equality for it with endpoint reversal implying reciprocal exponential averages and ⟨C⟩ + ⟨W⟩ ≥ 0.","lead":"The paper defines a counterwork functional from a sign-inverting memory-filtered effective protocol derived from an imposed driving λ(t). It shows Jarzynski equality applies directly to this counterwork, yielding ΔF_C = -ΔF_W and a bound on the product of exponential averages of work and counterwork.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"JE does not automatically apply to counterwork C because trajectories are generated under λ(t) not Λ(t)","rationale":"The identified concern is identical to the reader's weakest_assumption. Because the review was performed on the abstract, the paper's claim that the kernel 'preserves applicability of Jarzynski equality' remains unverified; the dynamical mismatch between driving protocol and work functional is the precise point that must be checked before the reciprocity and Jensen bound can be accepted.","tokens_in":1735,"tokens_out":397,"duration_ms":45441,"concrete_test":"Take the exactly solvable Brownian particle in a moving harmonic trap. Choose a linear λ(t) ramp from 0 to 1 over [0,T]. Construct the normalized kernel realizing exact endpoint reversal for Λ(t). Numerically integrate the Langevin equation under λ(t), compute the counterwork functional C along each trajectory, and test whether the sample average of e^{−βC} equals exp(−βΔF_C) to within sampling error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that Jarzynski equality holds for the counterwork C (defined via the Hamiltonian work for effective protocol Λ(t)) when averaged over trajectories driven by the imposed protocol λ(t). Standard JE derivations assume the measured work is performed by the actual time-dependent Hamiltonian that generates the dynamics, with the system starting in equilibrium at the protocol's initial value. Here the dynamics follow H(Γ,λ(t)) while C is computed as though the protocol were Λ(t) obtained by a memory kernel on ˙λ(t). Endpoint reversal via kernel normalization fixes ΔF_C = −ΔF_W but does not restore the required correspondence between the sampled ensemble and the protocol used to define C. Consequently ⟨e^{−βC}⟩ need not equal e^{−βΔF_C}, undermining the asserted reciprocity with the original Jarzynski average.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces a counterwork functional C obtained from an effective protocol Λ(t) generated by applying an active protocol-memory kernel to ˙λ(t) of an imposed protocol λ(t). It asserts that because C is the standard Hamiltonian work associated with H(Γ,Λ(t)), Jarzynski equality applies directly to trajectories driven by λ(t); when the kernel is normalized so that Λ(t) reverses the endpoints of λ(t), one obtains ΔF_C = −ΔF_W together with reciprocity of the exponential averages, and Jensen’s inequality yields ⟨C⟩ + ⟨W⟩ ≥ 0. The paper also treats counteroperations under incomplete thermodynamic information, recommending enforcement of endpoint reversal while minimizing dissipated counterwork.","tokens_in":1959,"tokens_out":510,"duration_ms":48120,"significance":"If the central claim is valid, the work would furnish a new reciprocity relation in nonequilibrium thermodynamics that bounds the sum of average work and counterwork and supplies a concrete strategy for counteroperations. The explicit construction of a kernel normalization that realizes exact endpoint reversal would be a useful technical contribution.","major_comments":[{"comment":"Abstract (and the section deriving the kernel normalization): the assertion that “Jarzynski’s equality applies directly to it” because C is the ordinary Hamiltonian work for H(Γ,Λ(t)) does not address the fact that the underlying trajectories are generated by the imposed protocol λ(t), not by Λ(t). Standard derivations of the Jarzynski equality require that the work functional be computed from the time-dependent Hamiltonian that actually drives the stochastic dynamics and that the initial ensemble be equilibrated at the initial value of that same protocol. The mismatch between driving protocol and work protocol therefore prevents the conclusion that ⟨e^{−βC}⟩ = e^{−βΔF_C} over the λ(t)-generated ensemble, undermining the claimed reciprocity with the original Jarzynski average.","section":"Abstract / kernel-normalization derivation"},{"comment":"Abstract: the statement that “the exponential average of the counterwork is the reciprocal of that of the original work” follows only if the Jarzynski equality holds for C; the preceding point shows that this step is not justified by the given construction. The subsequent claim that ⟨C⟩ + ⟨W⟩ ≥ 0 by Jensen likewise rests on the same unestablished equality.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for identifying a key subtlety in the applicability of the Jarzynski equality. We respond to each major comment below and will implement the indicated revisions.","responses":[{"response":"We agree that the standard Jarzynski equality requires the work functional to be evaluated along trajectories driven by the same time-dependent Hamiltonian that governs the dynamics, with the initial ensemble equilibrated at the initial value of that protocol. In the present construction the trajectories evolve under λ(t) while C is defined via Λ(t), so the equality ⟨e^{-βC}⟩ = e^{-βΔF_C} does not follow directly. We will revise the abstract and the kernel-normalization section to remove the claim of direct applicability and will either supply a self-contained derivation of the appropriate equality for C or qualify the reciprocity statements accordingly.","revision_made":"yes","referee_comment":"[Abstract / kernel-normalization derivation] Abstract (and the section deriving the kernel normalization): the assertion that “Jarzynski’s equality applies directly to it” because C is the ordinary Hamiltonian work for H(Γ,Λ(t)) does not address the fact that the underlying trajectories are generated by the imposed protocol λ(t), not by Λ(t). Standard derivations of the Jarzynski equality require that the work functional be computed from the time-dependent Hamiltonian that actually drives the stochastic dynamics and that the initial ensemble be equilibrated at the initial value of that same protocol. The mismatch between driving protocol and work protocol therefore prevents the conclusion that ⟨e^{−βC}⟩ = e^{−βΔF_C} over the λ(t)-generated ensemble, undermining the claimed reciprocity with the original Jarzynski average."},{"response":"We concur that both the reciprocity of the exponential averages and the subsequent Jensen bound rest on the validity of the Jarzynski equality for C. Because that equality does not hold by the standard argument, we will revise the abstract and all related passages to excise or suitably qualify these claims until a rigorous derivation is provided.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the statement that “the exponential average of the counterwork is the reciprocal of that of the original work” follows only if the Jarzynski equality holds for C; the preceding point shows that this step is not justified by the given construction. The subsequent claim that ⟨C⟩ + ⟨W⟩ ≥ 0 by Jensen likewise rests on the same unestablished equality."}],"tokens_in":1493,"tokens_out":533,"duration_ms":47122,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper constructs a counterwork C from an effective protocol Λ(t) obtained by applying a memory kernel to the derivative of the imposed protocol λ(t). When the kernel is normalized to reverse the endpoints, it claims ΔF_C = -ΔF_W and that the exponential average of C is the reciprocal of the Jarzynski average for W, plus the inequality ⟨C⟩ + ⟨W⟩ ≥ 0 from Jensen.\n\nThe construction itself is new: an active kernel on ˙λ(t) rather than a passive response, with an explicit normalization condition for reversal. The Jensen step is straightforward once the reciprocity is assumed. The discussion of strategy under incomplete information is a reasonable extension.\n\nThe central problem is exactly the one in the stress-test note. The system is driven by H(Γ, λ(t)), so the sampled trajectories and their statistics match the original protocol and its initial equilibrium. C is computed as the Hamiltonian work under Λ(t), but the ensemble is not the one that would arise from driving with Λ(t) from equilibrium at Λ(0). Jarzynski equality therefore does not transfer to C, and the reciprocity does not follow. Endpoint reversal fixes the free-energy difference but leaves the sampling mismatch untouched. No argument in the abstract or the described derivation appears to restore the required correspondence.\n\nThe paper is aimed at researchers who track extensions of fluctuation theorems to filtered or memory-dependent protocols. A specialist might extract the kernel normalization or the inequality as a technical point, but the main functional relation rests on an assumption that does not hold.\n\nI would not send this to peer review. The gap is load-bearing and would need to be addressed before the reciprocity claim can be evaluated.","headline":"The reciprocity claim fails because trajectories evolve under λ(t) while C is defined under Λ(t), so Jarzynski does not apply to the counterwork.","tokens_in":2431,"tokens_out":421,"would_cite":false,"duration_ms":34355,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"When a memory-filtered effective protocol reverses the endpoints of the original driving, the Jarzynski exponential average of the resulting counterwork equals the reciprocal of the original work average.","keywords":["Jarzynski equality","counterwork","memory-filtered driving","nonequilibrium thermodynamics","free energy difference","fluctuation theorems","endpoint reversal","work functionals"],"falsifier":"Construct the normalized kernel for a concrete driving λ(t), implement the resulting effective protocol Λ(t) in a simulation or experiment, compute the exponential averages of work and counterwork, and check whether they are reciprocals and whether their ordinary averages sum to a nonnegative value.","tokens_in":2633,"feed_emoji":"","tokens_out":729,"duration_ms":28232,"temperature":0.7,"pith_summary":"The paper defines a counterwork by generating an effective protocol through an active memory kernel applied to the derivative of an imposed driving protocol. Jarzynski equality applies directly to this counterwork because it is ordinary Hamiltonian work with respect to the effective protocol. When the kernel is normalized to reverse the endpoints exactly, the associated free-energy difference is the negative of the original, making the exponential averages reciprocals of each other. Jensen's inequality applied to the product of these exponentials then shows that the sum of the two average works is nonnegative. The paper also identifies endpoint reversal with minimal dissipation as the robust choice when thermodynamic information is incomplete.","feed_headline":"Reversed memory filter yields reciprocal Jarzynski averages for work and counterwork","feed_subtitle":"When the effective protocol reverses endpoints, free-energy differences negate and the sum of the two average works is nonnegative by Jensen","key_machinery":"The sign-inverting memory-filtered effective protocol Λ(t) generated by applying an active protocol-memory kernel directly to the time derivative of the imposed protocol.","core_discovery":"Given an imposed protocol λ(t), the effective protocol Λ(t) is obtained by applying an active protocol-memory kernel to λ̇(t). The counterwork is the ordinary Hamiltonian work associated with H(Γ,Λ(t)), so Jarzynski equality applies directly. When Λ(t) reverses the endpoints of λ(t), ΔF_C = -ΔF_W and the exponential average of the counterwork is the reciprocal of that of the original work. The kernel normalization realizing this reversed displacement is derived, and Jensen's inequality implies ⟨C⟩ + ⟨W⟩ ≥ 0.","pith_inferences":["The explicit separation between active kernel application and any passive bath response opens the possibility of engineering the reversal independently of the system's natural relaxation.","The same kernel construction could be inserted into other nonequilibrium identities that rely on work functionals, such as Crooks' fluctuation theorem.","Numerical verification would require only standard nonequilibrium sampling of trajectories under both the original and the filtered protocols."],"forward_implications":["The product of the exponentials of the average work and the average counterwork is bounded above by unity.","A negative average counterwork can occur only when it is compensated by a positive average work in the original operation.","Under incomplete thermodynamic information the robust counteroperation strategy is to enforce endpoint reversal while minimizing dissipated counterwork."],"fun_headline_variants":["Reversed memory filter gives reciprocal Jarzynski averages","Counterwork obeys Jarzynski under reversed memory kernel","Endpoint reversal negates free energy for work and counterwork","Jensen requires nonnegative sum of average work and counterwork"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A kernel normalization exists that achieves exact endpoint reversal while keeping the counterwork eligible for Jarzynski equality and while the kernel acts actively rather than through passive bath response.","fun_headline_variants_meta":{"raw":{"variants":["Reversed memory filter gives reciprocal Jarzynski averages","Counterwork obeys Jarzynski under reversed memory kernel","Endpoint reversal negates free energy for work and counterwork","Jensen requires nonnegative sum of average work and counterwork"]},"model":"grok-4.3","cost_usd":0.005277,"raw_usage":{"total_tokens":2569,"prompt_tokens":701,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":52774500,"prompt_tokens_details":{"text_tokens":701,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1812,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":701,"tokens_out":56,"duration_ms":22683,"temperature":1.0,"reasoning_tokens":1812,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T00:59:59.670455+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Construct the normalized kernel for a concrete driving λ(t), implement the resulting effective protocol Λ(t) in a simulation or experiment, compute the exponential averages of work and counterwork, and check whether they are reciprocals and whether their ordinary averages sum to a nonnegative value.","supporting_citations":[],"review_version":1}