{"id":"9d6e9667-e988-4b54-bf3e-e53831d22dda","arxiv_id":"2507.19387","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"An optimal transport protocol for a harmonically trapped Brownian particle, with work γΔλ²/tf, sets a lower time-energy bound for finite-time stochastic resetting via Eq. (5).","lead":"A particle in a movable harmonic trap is returned to a starting point using a simple optimal protocol: a linear sweep with two jumps. Applied to stochastic resetting, this protocol produces the best tradeoff between average search time and energy cost among the protocols tested.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal time-energy bound (Eq. 5) assumes no first-passage to the target during a resetting event, an assumption that is acknowledged but not tested as a function of target distance; this is the load-bearing gap.","rationale":"The reader identified exactly the weakest step: Eq. (5) is presented as a universal lower bound, but its derivation assumes the return protocol never lets the particle hit the target. This is not a minor technicality because the same absorption event that would shorten the MFPT also truncates the work integral, moving the measured tradeoff point off the predicted curve. The paper's own footnote [57] concedes the assumption, and Appendix E tests stiffness variations at one fixed target distance, which cannot probe the x0-dependence of the hit probability. The rest of the central construction is on solid ground: the OT protocol derivation in Appendix A is clean, the cost formula is verified by simulation, and the comparisons with SST and OC are informative. The conditional verdict is therefore appropriate: the protocol and its single-event optimality are likely correct, but the strong universal phrasing of Eq. (5) needs either a direct test varying target distance or a softened statement. My stress-test does not move the verdict; it reinforces the specific condition that should be satisfied before the bound is advertised as universal.","tokens_in":14779,"tokens_out":16038,"duration_ms":170283,"concrete_test":"Run the full Langevin resetting simulations with the OT return protocol for several target distances, e.g., x0/ζ ≈ 0.5, 1, and 2 (with r = 70 s⁻¹, fixed stiffness κ = 20 pN/µm), sweeping trst over the range used in Fig. 9. For each parameter set, record the measured mean first-passage time, the measured average work per resetting event, and the fraction of trajectories whose first passage occurs during a resetting event. If the (W, ⟨T⟩/⟨Tinst⟩) points deviate from Eq. (5) or a non-negligible fraction of first passages occur during returns, the no-hit assumption is the cause and the universal bound is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (5) is derived by combining the per-reset work bound W ≥ γ⟨x²⟩/tf with the finite-time MFPT formula ⟨T⟩ = ⟨Tinst⟩(1 + r trst). The second ingredient is exact only if the particle cannot hit the absorbing target during the return protocol. Footnote [57] explicitly states that 'the possibility that the particle reaches the target during the resetting event is ignored,' but this is not derived and is not isolated in the numerical checks. Appendix E varies the trap stiffness (σ/ζ) at fixed x0 = 0.1 µm and reports deviations in the work for low stiffness; it does not vary x0, which is the parameter controlling the distance of the searcher from the absorbing boundary during the return. If a trajectory hits x0 during a resetting event, the first-passage time is shortened relative to ⟨Tinst⟩(1 + r trst) and the work spent on that event is truncated, so the measured point (W, ⟨T⟩) can fall below the claimed Pareto curve and enter the gray 'excluded' region. The analytical α already accounts for absorption in the distribution of initial positions, so agreement at a single x0 does not validate the no-hit assumption for all target distances. The universal claim 'all protocols lie above this bound' therefore rests on an untested parameter regime.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives an optimal-transport (OT) protocol λ_OT(t) that moves a harmonically trapped overdamped Brownian particle between two equilibrium positions in finite time tf while minimizing the mean work. The protocol is obtained from an Euler-Lagrange minimization in Appendix A and takes the simple form λ_OT(t)=λ_i+Δλ(ω0 t+1)/(ω0 tf), with cost ΔW_OT=γ Δλ²/tf. The authors compare this protocol with constrained-but-unoptimized SST protocols, unconstrained optimal-control protocols, and an abrupt STEP protocol, showing that the OT cost is a lower bound for the former and an upper bound for the latter. They then apply the OT protocol to stochastic resetting, in which each resetting event returns the particle to the origin in a finite time trst. Combining the per-event work cost with the finite-time MFPT formula ⟨T⟩=⟨Tinst⟩(1+r trst), they obtain Eq. (5), ⟨T⟩=⟨Tinst⟩(1+2α kBT/W_rst^OT), and claim that this is a universal minimal time-energy bound for finite-time stochastic resetting. Langevin simulations are used to verify the protocol, the work formula, and the resulting tradeoff relation.","tokens_in":15006,"tokens_out":6146,"duration_ms":65677,"significance":"If the main claim holds, the paper provides a clean, analytically tractable, and experimentally relevant result: a stiffness-independent optimal transport protocol with a sharply defined time-energy cost, and a simple closed-form tradeoff for finite-time stochastic resetting. The derivation is self-contained, the cost formula is checked against Langevin simulations, and the contrast with SST, OC, and STEP protocols clarifies the thermodynamic landscape. The connection between optimal transport in stochastic thermodynamics and the stochastic-resetting literature is timely and likely to be of interest to the readership of cond-mat.stat-mech. The paper would be strengthened if the universality of the bound were supported by a proof or by a clearly delimited regime of validity.","major_comments":[{"comment":"The no-hit-during-return assumption is load-bearing and is not tested in the relevant parameter regime. The derivation of Eq. (5) uses both ⟨T⟩=⟨Tinst⟩(1+r trst) and the averaged work W_rst^OT, and both quantities assume that the particle does not reach the absorbing target during a resetting event. If a trajectory hits x0 during the return, the true MFPT is shorter than the formula and the measured point (W,⟨T⟩) can fall below the claimed bound. Footnote [57] explicitly states that this possibility is ignored, but Appendix E tests only the trap stiffness κ at fixed x0=0.1 μm and fixed r=70 s⁻¹, so it does not isolate the assumption as a function of the target distance. Since the analytical α already accounts for absorption only in the distribution of starting positions and not in the return dynamics, agreement at one x0 does not validate the assumption for all target distances. Please either derive a quantitative condition under which hits during resetting are negligible, add simulations varying x0 (and trst) to confirm the bound, or qualify the universality of the bound.","section":"Application and Appendix E"},{"comment":"The claim that Eq. (5) is a bound for 'any finite-time implementation of SR' is not established. The optimality proven in Appendix A is for the mean trajectory under the constraint that the final distribution is in equilibrium at tf; the SR application requires an additional statement that no other finite-time resetting implementation can achieve a lower ⟨T⟩ at the same W. The paper compares with SST, OC, and STEP protocols, but for unconstrained protocols the resetting duration is defined through a threshold (Appendix F) and the comparison is numerical. Consequently, the gray 'excluded' region in Fig. 4(b) is not rigorously excluded for all possible protocols. The authors should either provide a general lower-bound argument for the full class of resetting implementations or reformulate the claim as a bound within the class of equilibrium-constrained finite-time protocols.","section":"Lower bound, upper bound; Fig. 4; Appendix F"}],"minor_comments":[{"comment":"The phrase 'direct comparaison' in the text before Fig. 2 should be 'direct comparison'.","section":"Thermodynamics of optimal transport"},{"comment":"The caption contains an incomplete phrase: 'a by of the equilibrium standard deviation' should read 'a fraction of the equilibrium standard deviation'.","section":"Appendix F"},{"comment":"The formula for α is typeset ambiguously: α = 1−(1−e^z)^2 + z²/4 (1−cosh z/√(1−e^{−z})) needs parentheses and an explicit definition of z = x0√(r/D).","section":"Application section"},{"comment":"The statement 'deviations are only observed for slow protocols' should be made quantitative, since it is used to support the validity of the main tradeoff relation.","section":"Appendix E"}],"recommendation":"major_revision","confidential_remarks":"This is a solid paper with a clean derivation and convincing numerical checks for the tested parameters. The main issue is the gap between the universal claim and the actual proof: the no-hit-during-return assumption is acknowledged but not tested as a function of target distance, and the bound for 'all protocols' is only illustrated for specific protocol families. I think the paper is fixable, but the authors need to either provide the missing validation or appropriately circumscribe the claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the moving-trap OT protocol is derived cleanly and the cost expression ΔW=γΔλ²/tf is verified in simulations, so that part is solid. The genuinely new piece—applying that protocol to stochastic resetting and claiming a universal time-energy lower bound—is more fragile. The bound in Eq. (5) inherits an explicit 'no hitting the target during return' assumption (footnote [57]) that is acknowledged but never tested as a function of target distance x0. Appendix E varies trap stiffness at fixed x0 and shows agreement there, but that does not probe the regime where a resetting trajectory starting near x0 could cross the absorbing boundary mid-return. If such crossings occur, the measured MFPT is shorter than ⟨Tinst⟩(1+r t_rst) and the work per reset is truncated, so the point can fall below the claimed Pareto curve. The reader's conditional verdict is right: the core protocol stands, but the universal 'all protocols lie above' phrasing is not fully supported.\n\nWhat the paper does well: the Euler-Lagrange derivation in Appendix A is transparent, the comparison with SST/OC/STEP is informative, and the double-bounding role (lower for SST, upper for OC) is a nice unifying picture. The stiffness-independence of the OT cost is a useful experimental handle. The simulations are consistent with the analytics for the parameters shown, though symbols lack error bars and code is not shared—minor issues in a theory paper.\n\nWho benefits: experimental groups working on optical-trap resetting and finite-time thermodynamics get a practical protocol and a benchmark. The OT cost formula alone is worth citing. But the SR bound should be treated as a plausible conjecture with a stated assumption, not a proven theorem.\n\nRecommendation: send it to peer review with a request for either a softened claim or additional simulations varying x0 (and maybe a quantitative estimate of the no-hit probability). A serious referee can sort this out; it is not a desk reject.","headline":"Solid OT protocol, but the universal SR bound rests on an untested no-hit assumption—referee it with a request for x0-dependent checks.","tokens_in":15576,"tokens_out":3042,"would_cite":true,"duration_ms":30273,"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":"A straight trap ramp achieves the optimal time-energy tradeoff in stochastic resetting.","keywords":["stochastic resetting","optimal transport","stochastic thermodynamics","mean first-passage time","time-energy tradeoff","optimal control","harmonic trap","Brownian particle"],"falsifier":"A concrete test is to fix the trap stiffness $\\kappa$ and resetting rate $r$, implement returns with the optimal-transport protocol, and sweep the target distance $x_0$ from values much larger than $\\zeta=(2D/r)^{1/2}$ down to $x_0\\approx\\zeta$, measuring the mean first-passage time. Equation (5) predicts all data fall on the universal curve $\\langle T\\rangle=\\langle T_{\\mathrm{inst}}\\rangle(1+2\\alpha k_B T/W)$; any data point below that curve — most plausibly at small $x_0$ or long reset durations, where mid-return target hits matter — would falsify the claimed lower bound.","tokens_in":14540,"feed_emoji":"🎯","tokens_out":11229,"duration_ms":99915,"temperature":0.7,"pith_summary":"Stochastic resetting speeds up a search, but every return of the particle to the origin costs work and takes time. This paper asks whether one can prescribe the return motion so that the search is as fast as possible for a given energy budget. It derives an optimal-transport protocol, $\\lambda_{\\mathrm{OT}}(t)=\\lambda_i+\\Delta\\lambda(\\omega_0 t+1)/(\\omega_0 t_f)$, which moves the mean position at constant velocity along a straight line, lands exactly in the final equilibrium state after time $t_f$, and spends work $\\Delta W_{\\mathrm{OT}}=\\gamma\\Delta\\lambda^2/t_f$ that is independent of trap stiffness. When every resetting event follows this protocol, the mean first-passage time obeys $\\langle T\\rangle=\\langle T_{\\mathrm{inst}}\\rangle(1+2\\alpha k_B T/W^{\\mathrm{rst}}_{\\mathrm{OT}})$, and the paper's central claim is that this curve is a lower bound for any finite-time implementation of stochastic resetting: all other protocols lie above it in the time-energy plane. The practical consequence is that optical-trap experiments can implement near-optimal resetting with a simple, prescribable trap motion and a single calculable speed-versus-energy relation.","feed_headline":"One straight trap ramp sets the best resetting tradeoff","feed_subtitle":"Returning the searcher at constant speed sets the work-versus-search-time curve for every competing protocol.","key_machinery":"The load-bearing mechanism is a variational optimal-transport construction. The functional $J=\\int_0^{t_f}[\\dot W+\\mu(\\dot u-\\omega_0(\\lambda-u))]dt$ combines the mean work rate with the overdamped equation of motion enforced by a Lagrange multiplier; Euler-Lagrange minimization gives $\\ddot u=0$, so the mean position follows the straight geodesic $u_{\\mathrm{OT}}(t)=\\lambda_i+\\Delta\\lambda\\,t/t_f$. The protocol $\\lambda_{\\mathrm{OT}}(t)=u_{\\mathrm{OT}}(t)+\\Delta\\lambda/(\\omega_0 t_f)$ is the geodesic plus a counterdiabatic jump term, so the trap exerts a constant force $\\kappa\\Delta\\lambda/(\\omega_0 t_f)$ that exactly cancels viscous drag and yields the stiffness-independent work $\\gamma\\Delta\\lambda^2/t_f$. Geometrically this is the geodesic of the $L^2$-Wasserstein metric on Gaussian states, giving minimal entropy production $\\Sigma=\\gamma\\Delta\\lambda^2/(T t_f)$ and hence $\\Delta W_{\\mathrm{OT}}=T\\Sigma$. For resetting, the absorbing boundary at the target modifies the average return work through the factor $\\alpha$, obtained from the second moment of the quasi-stationary solution of the resetting Fokker-Planck equation.","core_discovery":"The central discovery is that the constrained optimal-transport return protocol — a linear ramp plus two instantaneous jumps of amplitude $\\Delta\\lambda/(\\omega_0 t_f)$ — saturates a minimal time-energy relation for finite-time stochastic resetting. For a single transport between equilibrium states its cost is $\\Delta W_{\\mathrm{OT}}=\\gamma\\Delta\\lambda^2/t_f$; this sits between two protocol families: it is a lower bound on the cost of swift state-to-state transformations that enforce final equilibrium, and an upper bound on unconstrained optimal-control protocols that stop driving before relaxation is complete. In the resetting application, the per-event work is $W^{\\mathrm{rst}}_{\\mathrm{OT}}=2\\alpha k_B T/(r t_{\\mathrm{rst}})$, where the factor $\\alpha<1$ accounts for the absorbing target at $x_0$ by removing return trajectories that start beyond it. The resulting relation $\\langle T\\rangle=\\langle T_{\\mathrm{inst}}\\rangle(1+2\\alpha k_B T/W^{\\mathrm{rst}}_{\\mathrm{OT}})$ is claimed as the optimal time-energy tradeoff: it defines the excluded low-cost/fast-search region, and SST, unconstrained optimal-control, and step-protocol implementations of resetting all fall above the curve at equal resetting duration (Fig. 4b).","pith_inferences":["The same Lagrangian optimization should carry over to resetting protocols that modulate trap stiffness or temperature rather than trap position, producing analogous $\\alpha$-modified time-energy tradeoffs for thermal shortcuts; the moving-trap case is the simplest member of that family.","The paper validates the bound by varying stiffness, not target distance. A dedicated experiment sweeping $x_0$ down to the resetting length scale would map the regime where the no-hit-during-return assumption holds and where it breaks.","Because the OT protocol contains instantaneous jumps, real traps with finite feedback bandwidth will smear them; measuring the added work versus bandwidth would quantify how close practical implementations sit to the ideal curve."],"forward_implications":["Any finite-time resetting protocol that forces return to equilibrium in a fixed time must spend at least $\\gamma\\Delta\\lambda^2/t_f$; the paper's explicit polynomial SST example spends exactly 20% more.","Equation (5) excludes a region of the time-energy plane for finite-time resetting: no implementation can simultaneously have a shorter mean first-passage time and a lower per-event work than the OT curve.","Unconstrained optimal-control protocols can reduce the work per return only by leaving the particle out of equilibrium, which lengthens the effective return time; plotted against the search MFPT they still lie above the OT bound.","The OT work cost does not depend on trap stiffness, so the same tradeoff curve is attainable across the experimental stiffness range (the numerics span $\\kappa\\in[1,100]\\,\\mathrm{pN}/\\mu\\mathrm{m}$).","In the fast-return limit $t_{\\mathrm{rst}}\\to0$, Eq. (5) reduces to the instantaneous-resetting mean first-passage time $\\langle T_{\\mathrm{inst}}\\rangle$, recovering the original stochastic-resetting speedup."],"supporting_citations":[{"why":"Defines instantaneous stochastic resetting and gives the mean first-passage time $\\langle T_{\\mathrm{inst}}\\rangle$ that the finite-time bound extends.","marker":"[1]"},{"why":"Supplies the review results and the quasi-stationary distribution with absorbing boundary used to compute the correction factor $\\alpha$.","marker":"[2]"},{"why":"Establishes the optimal resetting rate that minimizes $\\langle T_{\\mathrm{inst}}\\rangle$, the rate used in the numerical benchmark.","marker":"[5]"},{"why":"Provides the swift state-to-state transformation protocol family whose finite-time cost is compared with the OT cost.","marker":"[28]"},{"why":"Provides the unconstrained optimal-control protocol family whose work and duration are plotted against the OT bound.","marker":"[32]"},{"why":"Demonstrates optimal-transport protocols and a universal time-energy bound for a harmonically trapped Brownian particle, the predecessor of Eq. (4).","marker":"[37]"},{"why":"Connects entropy production to the $L^2$-Wasserstein distance, identifying the geodesic structure the OT protocol realizes.","marker":"[44]"},{"why":"Explains the counterdiabatic decomposition of optimal-transport protocols, including the jump discontinuities of $\\lambda_{\\mathrm{OT}}(t)$.","marker":"[45]"},{"why":"Gives a prior thermodynamic trade-off relation between work and first-passage time under resetting, against which the OT relation is positioned.","marker":"[54]"},{"why":"Gives a cost-time Pareto front for resetting with stochastic return, used as a comparison target in the time-energy plane.","marker":"[55]"}],"fun_headline_variants":["A straight ramp is the optimal resetting move","One trap ramp locks in the resetting tradeoff","Optimal resetting: one ramp, two jumps, best cost","The cheapest fastest reset is a constant-speed return","Resetting's best deal: a linear return path"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bound assumes that during a finite-time resetting event the particle never reaches the absorbing target at $x_0$; the paper states this assumption rather than proving it, and checks it only indirectly by varying the trap stiffness. If a particle can hit the target mid-return, the true mean first-passage time is shorter and the claimed lower bound could fail.","fun_headline_variants_meta":{"raw":{"variants":["A straight ramp is the optimal resetting move","One trap ramp locks in the resetting tradeoff","Optimal resetting: one ramp, two jumps, best cost","The cheapest fastest reset is a constant-speed return","Resetting's best deal: a linear return path"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1390,"prompt_tokens":970,"completion_tokens":420,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":343}},"tokens_in":586,"tokens_out":420,"duration_ms":4182,"temperature":1.0,"reasoning_tokens":343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:54:06.781266+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to fix the trap stiffness $\\kappa$ and resetting rate $r$, implement returns with the optimal-transport protocol, and sweep the target distance $x_0$ from values much larger than $\\zeta=(2D/r)^{1/2}$ down to $x_0\\approx\\zeta$, measuring the mean first-passage time. Equation (5) predicts all data fall on the universal curve $\\langle T\\rangle=\\langle T_{\\mathrm{inst}}\\rangle(1+2\\alpha k_B T/W)$; any data point below that curve — most plausibly at small $x_0$ or long reset durations, where mid-return target hits matter — would falsify the claimed lower bound.","supporting_citations":[{"cited_title":"Time-energy tradeoff in stochastic resetting using optimal control","cited_arxiv_id":"2507.19387","evidence_quote":"Provides the swift state-to-state transformation protocol family whose finite-time cost is compared with the OT cost."},{"cited_title":"Raynal, T","cited_arxiv_id":null,"evidence_quote":"Provides the unconstrained optimal-control protocol family whose work and duration are plotted against the OT bound."},{"cited_title":"Proesmans, J","cited_arxiv_id":null,"evidence_quote":"Demonstrates optimal-transport protocols and a universal time-energy bound for a harmonically trapped Brownian particle, the predecessor of Eq. (4)."},{"cited_title":"Ito, Stochastic thermodynamic interpretation of infor- mation geometry, Physical Review Letters 121, 030605 (2018)","cited_arxiv_id":null,"evidence_quote":"Connects entropy production to the $L^2$-Wasserstein distance, identifying the geodesic structure the OT protocol realizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives a cost-time Pareto front for resetting with stochastic return, used as a comparison target in the time-energy plane."}],"review_version":2}