{"id":"4c3cf606-3718-4cdc-9a7a-816fe723f4e9","arxiv_id":"2607.16639","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Feedback control that maintains a target steady state requires an information rate at least equal to the passive entropy rate of the uncontrolled dynamics, and a time-reversal protocol attains this rate for state-independent systems.","lead":"This paper derives a mathematical limit on how much information a controller needs to keep a noisy system near a target: the required information rate is at least the rate at which the uncontrolled system's randomness spreads its state. It also gives an optimal control rule for a broad class of systems and applies the bound to particle localization, microbial navigation, and information engines.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Pareto-frontier curves in state-dependent applications are lower bounds, not proven achievable; Eq. (12) itself is sound.","rationale":"The central theorem — the inequality Eq. (12) — is carefully derived and I could not find a genuine flaw in the chain-rule/DPI/stationarity argument. The saturation result for state-independent passive dynamics is also proven in detail in App. C: the induction argument showing Term 2 vanishes is valid, and the explicit protocol Eq. (17) does yield an optimal controller under full observability. The reader's weakest assumption correctly identifies that saturation, not the inequality itself, carries the extra premises. My concern is narrower: the paper's applications label lower-bound curves as 'Pareto frontiers' even when the passive dynamics are state-dependent and no saturating protocol has been proven to exist. The authors do state a conjecture and provide a gap bound, so this is an overstatement rather than a mathematical error. This does not undermine the main theorem, and it is already flagged in the reader's weakest-assumption analysis. I therefore recommend no change to the verdict; the issue could be addressed editorially by saying 'lower bounds' where tightness is unproven.","tokens_in":26013,"tokens_out":30652,"duration_ms":328322,"concrete_test":"For the Mexican–hat stable-target example (Fig. 3b), fix a target ρ on the reported frontier and numerically minimize Term1+Term2 in the exact gap expression Eq. (14) over a flexible class of Markovian controllers with full state observability (e.g., optimizing state-dependent drift and diffusion/jump kernels at fixed ρ). If the infimum of the gap is strictly positive, Eq. (12) is not tight for that target, so the black curve is an unattained lower bound rather than an achievable Pareto frontier.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (12) is well supported; the derivation via chain rules, DPI, and stationarity is sound under the stated assumptions. The load-bearing soft spot is the step from the inequality to an exact performance–information Pareto frontier in applications with state-dependent passive dynamics. Saturation is proven only under Eq. (15) (state-independent F and λ, full observability, no reflecting boundaries). For general F(x) — e.g., the Mexican–hat potential in Section III.B and the information engine in Appendix H — the time-reversal protocol (C16) is a conjecture, and the gap bound (C17) upper-bounds Tdot−Sdot_p but does not show that it vanishes. Consequently, the black 'Pareto frontier' curves in Figs. 3–5 are rigorous lower bounds on the information required, but not demonstrated achievable trade-offs. If the gap is positive for the optimal achievable protocol, the true frontier lies above these curves, and the claim that Sdot_p alone yields Pareto frontiers is not established outside the state-independent class. The paper honestly labels general near-optimality as a conjecture, but the 'Pareto frontier' terminology in the applications overstates the proven content.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies information-limited feedback control of Markovian stochastic systems maintained at a target steady state. The main theoretical result is Eq. (12): for any feedback controller that maintains a stationary distribution ρ(x), the transfer-entropy rate from the state history to the control action, ˙T_{x→dx^c}, is bounded below by the passive entropy rate ˙S_p of the uncontrolled dynamics evaluated at ρ(x). The paper then shows that when the passive dynamics are state-independent (Eq. (15)), an explicit probabilistic time-reversal protocol (Eq. (17)) saturates the bound, establishing exact information-optimality within that class. Applications are given to bit control, Brownian localization in a Mexican-hat potential, microbial navigation, and a feedback information engine; the latter three use the bound to derive performance–information trade-off curves, with the navigation case also proving that a memoryless Run-Reverse strategy is asymptotically optimal under limited observability. The paper also derives a Schrödinger-equation mapping for constructing optimal steady-state distributions at fixed ˙S_p and provides numerical transfer-entropy computations via path weight sampling.","tokens_in":26231,"tokens_out":9426,"duration_ms":98622,"significance":"If the results hold, Eq. (12) is a valuable and broadly applicable lower bound: it depends only on the target distribution and the local passive dynamics, and it yields a simple information-theoretic obstruction to control performance. The explicit time-reversal saturating protocol for state-independent passive dynamics is an elegant constructive result, and the navigation application, including the proof of asymptotic optimality of Run-Reverse in a restricted-observability limit, is of independent interest. The paper is also commendable for providing Python code for reproducing the figures and for making the gap analysis explicit rather than hiding it. The main caveat, discussed below, is that the term 'Pareto frontier' is used for curves that, in the state-dependent applications, are rigorously established only as lower bounds, not as achievable trade-offs.","major_comments":[{"comment":"The curves labeled 'Pareto frontier' in the Mexican-hat example and the information-engine example are rigorous lower bounds on the information rate required for a given performance, but they are not shown to be achievable. Exact saturation is proven only for state-independent passive dynamics (Eq. (15)); the Mexican-hat dynamics (Eq. (25)) and the information-engine dynamics (Eq. (29)) have state-dependent drift, so the time-reversal protocol's optimality is not established for them. The gap bound in Eq. (C17) is an upper bound on ˙T−˙S_p for the time-reversal protocol, and it does not vanish except in limiting regimes. Thus the black curves in Figs. 3 and 5 overstate the proven content. I recommend either proving achievability (or at least showing the gap vanishes for the plotted regimes) or relabeling these curves as 'lower-bound frontiers' and qualifying the abstract's claim that the","section":"III.B, III.D; Figs. 3, 5"},{"comment":"The proof that the time-reversal protocol sets Term 2 of Eq. (14) to zero uses, in an essential way, the translation-invariance of the passive kernel K^p(x'|y)=q(x'−y) for state-independent passive dynamics (Eq. (C12)–(C14)). For general state-dependent F(x) and λ(x,Δx), this argument fails, and the paper's 'near-optimality' conjecture for Eq. (C16) is not supported by a quantitative statement. The gap bound (C17) involves ⟨F·D^{-1}F⟩_ρ and a jump log-term; while these vanish when ρ becomes very narrow, the paper does not provide a precise condition or rate. Since the high-information near-optimality claim is used to justify the application curves, this is a load-bearing gap. I ask that the authors either prove the relevant gap vanishes for the specific examples (with explicit bounds), or clearly restrict the optimality claims to the state-independent class and present the state-dependen","section":"II.B, App. C (Eqs. C12–C17)"}],"minor_comments":[{"comment":"The notation p(y_t) in Eq. (16) is undefined; it should be the post-control distribution, denoted ρ'(y) in App. C. Please define it explicitly.","section":"Eq. (16)"},{"comment":"The sum over λ_i Re(λ_i) is ambiguous when some eigenvalues of A=−H_* have negative real parts (as for a stable fixed point). Please specify whether the sum runs over all eigenvalues or only those with positive real part, and discuss the sign.","section":"Eq. (22)"},{"comment":"The caption says parameters are 'from the experimental setup in Ref. [10]', but the protocol is from Ref. [43]. Please correct the reference.","section":"Fig. 5 caption"},{"comment":"The Schrödinger mapping is presented for the case without jumps; the jump case is said to lead to a nonlinear/nonlocal equation. Since jumps appear in the bit-control example, it would be helpful to state explicitly how the linear mapping fails there and what is done instead.","section":"App. B"},{"comment":"The measurement-noise restoration works only when inequality (H20) holds. The main text should mention this limitation in the discussion of the information-engine application, especially because experimental implementations inevitably have measurement noise.","section":"App. H, Eq. (H20)"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a correct-looking central inequality and a well-executed saturation proof for a nontrivial class. My main concern is not the mathematics of Eq. (12) but the scope of the Pareto-frontier claims for state-dependent applications. The paper is honest about the conjecture in App. C, but the abstract, figures, and discussion present these lower-bound curves as exact trade-offs. I believe the paper can be made acceptable by qualifying those claims and, ideally, by adding explicit gap estimates for the state-dependent examples. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. The core inequality Eq. (12) is the real thing: the proof is transparent, self-contained, and the bound depends only on the target steady state and passive dynamics. The genuinely new piece is Eq. (17), the explicit probabilistic time-reversal protocol that saturates the bound when the passive dynamics are state-independent. That is the result worth citing. The bit-flip demonstration is a nice worked example, and the navigation application extends previous work with a memory-free optimality proof in the low-information limit. The numerics use TE-PWS and the code is available; that counts.\n\nWhere I part company with the reader's headline 'Pareto frontiers' is in the state-dependent applications. The inequality is sound, but the step from bound to an exact Pareto frontier requires achievability. Saturation is proven only under Eq. (15) — state-independent F and lambda, full observability, no reflecting boundaries. For the Mexican-hat potential and the information engine, the time-reversal protocol is a well-motivated conjecture, and the gap bound (C17) does not show that the gap vanishes. In Figs. 3 and 5, the black curves are therefore rigorous lower bounds on information (or upper bounds on performance) but not demonstrated achievable trade-offs. The paper labels general near-optimality as a conjecture, which is honest, but the 'Pareto frontier' terminology overstates the proven content. In the navigation case, the passive dynamics are state-independent rotational diffusion, so the Eq. (28) frontier with full observability is on much firmer ground.\n\nOther soft spots are minor. The TE-PWS numerical points have no error bars. Measurement noise is handled only under condition (H20). These do not affect Eq. (12).\n\nBottom line: the main theorem is sound, the explicit optimal protocol for state-independent dynamics is a genuine advance, and the applications deserve referee time. The revision should qualify 'Pareto frontier' in the state-dependent examples and state clearly which curves are proven achievable. This paper deserves a serious referee, and I would bring it to reading group.","headline":"Equation (12) is a real, clean bound; the explicit time-reversal protocol for state-independent passive dynamics is the genuinely new result, but the state-dependent 'Pareto frontiers' are proven lower bounds, not achieved trade-offs.","tokens_in":26728,"tokens_out":2210,"would_cite":true,"duration_ms":25436,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J70","93E20","94A17","82C31"],"pacs":["05.40.-a","89.70.Cf"],"model":"deepseek-v4-flash","headline":"This paper establishes that any stationary feedback controller must receive information from the controlled system at a rate no smaller than the entropy rate of the uncontrolled dynamics.","keywords":["information-limited control","feedback control","transfer entropy rate","passive entropy rate","Pareto frontiers","time-reversal protocols","stochastic dynamics","biological navigation"],"falsifier":"Run the bit-control example with passive flip rate γ=1 and target ρ0=0.7 using a controller whose control error rate is set below the time-reversal value λ0=γ(1−ρ0)/ρ0, then compute the transfer-entropy rate numerically by path-weight sampling; if it falls below (2ρ0−1)γ ln(ρ0/(1−ρ0)), Eq. (12) would be refuted.","tokens_in":25907,"feed_emoji":"🎛️","tokens_out":5937,"duration_ms":62531,"temperature":0.7,"pith_summary":"The paper asks how much information a controller must receive to hold a noisy stochastic system near a target steady state. It proves a lower bound: the steady-state transfer-entropy rate from system to controller is never less than the entropy growth rate of the uncontrolled passive dynamics evaluated at the target distribution. This bound depends only on the target distribution and the passive drift, diffusion, and jumps, so it converts a hard history-dependent information calculation into a local, computable quantity. For passive dynamics that do not depend on the state, the paper constructs an explicit information-optimal protocol that probabilistically time-reverses the passive dynamics, and shows the same time-reversal idea is near-optimal more generally. The result yields performance-information Pareto frontiers for examples ranging from bit control and particle localization to microbial navigation and feedback information engines.","feed_headline":"Feedback control needs at least the passive entropy rate","feed_subtitle":"A new inequality turns performance-versus-information limits into Pareto frontiers, with time-reversal as the optimal strategy.","key_machinery":"The argument revolves around two quantities: the passive entropy rate (the entropy growth of the uncontrolled dynamics, expressed as a sum of diffusion, drift, and jump terms evaluated at the target distribution) and the transfer-entropy rate, which measures the causal information flow from the system's past to the controller's action. Eq. (12) connects these two rates. Tightness is achieved by the probabilistic time-reversal protocol of Eq. (17), which draws the post-control state from the Bayesian posterior of the passive kernel; for state-independent passive dynamics this forces both gap terms in Eq. (14) to vanish. A Schrödinger-equation mapping then converts fixed-information performanc","core_discovery":"The central discovery is Eq. (12): for any stationary feedback controller, the transfer-entropy rate from state to control action is bounded below by the passive entropy rate, the rate at which the Shannon entropy of the target distribution would grow if control were suddenly stopped. The inequality follows from stationarity, the data-processing inequality, and the Markovianity of the passive dynamics. When the passive dynamics are state-independent, the explicit protocol in Eq. (17), which probabilistically time-reverses the passive kernel, saturates the bound and is therefore information-optimal among controllers with full state observability. The paper further maps fixed-information perfo","pith_inferences":["Because the bound depends only on the passive dynamics and the target distribution, an observer could estimate the minimal information rate of an unobserved biological controller by measuring how fast the uncontrolled system relaxes, without monitoring the controller's internal decisions.","If a Landauer-style relation between transfer-entropy rate and the controller's energy consumption holds, this bound would also imply energetic lower bounds on feedback control; the paper raises this as an open conjecture rather than a theorem.","The paper proves optimality of time reversal only for state-independent passive dynamics and full observability; extending the framework to time-dependent targets, finite-time control, or first-passage objectives is a natural next step the paper itself flags."],"forward_implications":["Any feedback controller that maintains a given steady-state distribution must receive information at a rate at least as large as the passive entropy rate, so performance improvements beyond that bound are impossible without more information.","For state-independent passive dynamics, probabilistic time reversal is exactly information-optimal; no controller using memory of past states or actions can do better under full observability.","Performance-information Pareto frontiers can be obtained from the ground state of a Schrödinger-like operator, yielding analytic frontiers in the bit, particle-localization, and navigation examples.","In microbial navigation with full state observability, memory cannot improve up-gradient velocity; with only heading measurements available, a memoryless Run-Reverse strategy is the unique optimal strategy at low information rates.","For an experimentally realized information engine, approaching the theoretical maximum rate of heat extraction from a thermal bath requires a diverging information rate, as confirmed by simulations."],"fun_headline_variants":["Feedback control info floor: passive entropy rate","Minimum info for feedback control? Passive entropy rate","Time-reversal yields info-optimal control for passive dynamics","Passive entropy rate sets control info lower bound","Pareto frontiers from info-limited feedback control bound"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Tightness of the bound and the optimality of the time-reversal protocol rest on the passive dynamics being state-independent, the controller observing the full state without delay and being able to implement arbitrary stochastic actions including exact passive noise, and the state space having no reflecting boundaries—if any of these fails, the inequality still holds but the constructed protocol need not be optimal.","fun_headline_variants_meta":{"raw":{"variants":["Feedback control info floor: passive entropy rate","Minimum info for feedback control? Passive entropy rate","Time-reversal yields info-optimal control for passive dynamics","Passive entropy rate sets control info lower bound","Pareto frontiers from info-limited feedback control bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001385,"raw_usage":{"total_tokens":5373,"prompt_tokens":601,"completion_tokens":4772,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":345,"completion_tokens_details":{"reasoning_tokens":4699}},"tokens_in":345,"tokens_out":4772,"duration_ms":34408,"temperature":1.0,"reasoning_tokens":4699,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:24:44.153273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the bit-control example with passive flip rate γ=1 and target ρ0=0.7 using a controller whose control error rate is set below the time-reversal value λ0=γ(1−ρ0)/ρ0, then compute the transfer-entropy rate numerically by path-weight sampling; if it falls below (2ρ0−1)γ ln(ρ0/(1−ρ0)), Eq. (12) would be refuted.","supporting_citations":[],"review_version":1}