{"id":"9396ad9f-e421-49dd-a116-a3c68df2d9e8","arxiv_id":"2602.04961","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"CMI scaling function is monotonically non-increasing along RG flows in nonequilibrium systems, forbidding flows from lower to higher CMI fixed points.","lead":"The authors derive a new constraint on how correlations in non-equilibrium systems change under renormalization: conditional mutual information (CMI) can only decrease along the flow when it stays finite. This gives a rigorous way to rule out certain phase transitions driven by decoherence or symmetry-breaking noise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"UV-finiteness of CMI in nonequilibrium systems is asserted but not demonstrated; the abstract's promised 'anisotropic conserved dynamics' example is absent from the full text.","rationale":"The reader identified the UV finiteness of CMI as the weakest assumption, and I agree. My attack sharpens this by pointing to the specific missing support: the abstract promises a nonequilibrium example ('anisotropic conserved dynamics') that is absent from the text, which would have been the natural test of the UV-finiteness assumption in a genuinely driven system. The derivation of Eq. (2) from Eq. (1) is mathematically sound once the scaling form is granted; the examples are consistent; the convex-mixture bound (Eq. 3) and its application to classical SSB are also sound. The central weakness is therefore not an internal inconsistency but an unproven and partially unsupported premise. Because the paper explicitly flags the assumption and provides some evidence, and because the promised example could be supplied in a revision, the appropriate verdict remains conditional rather than reject. No change from the reader's CONDITIONAL is needed.","tokens_in":19861,"tokens_out":13080,"duration_ms":154766,"concrete_test":"Compute the CMI scaling function for a genuine nonequilibrium steady state with a local space-time action, e.g., the 1+1D asymmetric simple exclusion process or the 1D KPZ equation near its critical point, using exact/tensor-network methods. Test whether lim_{l→∞} I(l|l_B,t) equals a function f(t l_B^{1/ν}) that is finite and monotonic in |x|. If the data collapse fails or a residual l_B dependence persists, the UV-finiteness assumption is violated for this class and the central claim would not apply to nonequilibrium systems.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Eq. (2), is derived from the exact monotonicity Eq. (1) only after assuming that, in the scaling limit, I(∞|l_B,t) depends solely on x=t l_B^{1/ν} (UV finiteness). This assumption is load-bearing: if CMI retains separate l_B dependence, the derivative bound does not follow and the stability criterion collapses. The paper gives plausible arguments that local space-time actions produce UV-finite CMI, and supports this with exact/numerical examples (Illustrations 1–3). However, these examples are either simple mixed states arising from local channels or equilibrium pure states; none is a genuinely nonequilibrium steady state with a nonzero current. The abstract explicitly promises an example entitled 'area-law CMI in anisotropic conserved dynamics', but no such example appears anywhere in the full text. This is not merely a stylistic omission: it is the missing evidence that would substantiate the key assumption for the class of systems named in the title. The paper's own caveat that known UV-divergent CMI occurs only in fractonic/subsystem-symmetric systems is exculpatory but does not prove UV finiteness in general nonequilibrium settings. The stability claim 'a fixed point with smaller CMI cannot be destabilized toward one with larger CMI' is therefore as secure as this unverified assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives two information-theoretic constraints for mixed states and applies them to phase stability and RG flows. The first result starts from strong subadditivity (SSA): CMI in a slab geometry is monotonically decreasing in the separator length l_B, Eq. (1). Assuming CMI is UV finite in the scaling limit, the crossover function f(x) with x = t l_B^{1/ν} satisfies d f(x)/d|x| ≤ 0, Eq. (2), which is interpreted as a non-perturbative stability criterion: a fixed point with smaller CMI cannot flow to one with larger CMI. The second result, Eq. (3), bounds the CMI of a convex mixture by the averaged CMIs of its components plus a conditional entropy term, and is used to argue perturbative stability of classical symmetry-broken states under local decoherence. Applications include an exactly solvable one-dimensional asymmetric decoherence model, the transverse-field Ising model, and a two-dimensional trivial-to-SWSSB transition. Appendices A--C give the SSA proofs and the exact CMI calculation; Appendices D--E review SWSSB and provide additional numerics.","tokens_in":20197,"tokens_out":10972,"duration_ms":126747,"significance":"If the UV-finiteness assumption holds, Eq. (2) is a nontrivial, non-perturbative constraint on RG flows that does not require Lorentz invariance or detailed balance, and it has no equilibrium counterpart of comparable generality. The SSA derivations in Appendices A and B are correct, the exact CMI calculation in Illustration 1 is a useful controlled test, and the SWSSB numerics use a publicly available tensor-network implementation. These are genuine strengths. However, the advertised scope is broader than what is established: the key scaling assumption is not proved for general nonequilibrium steady states, and one promised nonequilibrium example is missing. The paper is therefore a valuable conditional contribution, but the central claim needs either additional support or a more carefully stated scope.","major_comments":[{"comment":"Eq. (2), d f(x)/d|x| ≤ 0, follows from Eq. (1) only after assuming CMI is UV finite, so that I(∞|l_B,t) depends solely on x = t l_B^{1/ν}. This assumption is load-bearing and is not proven for generic nonequilibrium steady states. The locality argument based on MSRJD actions is plausible but not a proof, and none of the examples is a genuine NESS with nonzero current. The stability conclusions derived from Eq. (2) inherit this conditionality.","section":"Derivation of Eq. (2), paragraph beginning \"Let's now consider the limit\""},{"comment":"The abstract promises an example entitled \"area-law CMI in anisotropic conserved dynamics\", but no such example appears in the full text or in the appendices. This is not a cosmetic omission: it is the only advertised illustration involving genuinely nonequilibrium conserved dynamics, and it would have directly supported the UV-finiteness assumption for the class named in the title. Either the example should be supplied or the promise removed.","section":"Abstract and Introduction"},{"comment":"The statement \"if the IR fixed point has infinite Markov length, the UV fixed point cannot be a Gibbs state of a local Hamiltonian\" does not follow from Eq. (2). Infinite Markov length means only that CMI does not decay exponentially; it may decay algebraically to zero. In that case monotonicity I_IR ≤ I_UV is satisfied by a Gibbs UV fixed point with I_UV = 0. To exclude a Gibbs UV fixed point one needs a positive fixed-point CMI in the IR, not merely an infinite Markov length. The same issue affects the inference that a zero fixed-point CMI implies a Gibbs state via Hammersley–Clifford: the theorem requires conditional independence at finite l_B, not merely I → 0 as l_B → ∞.","section":"Consequences of Eq. (2), paragraph beginning \"Conversely, if the IR fixed point\""},{"comment":"The abstract states that monotonicity \"implies that the CMI scaling exponent cannot increase along the RG flow,\" but no theorem about a scaling exponent is derived in the body. The only exponent inequality obtained is η ≥ 0 for the finite-l correction I ∼ [(αl − l_B)/l_B]^η f(x), which is not the fixed-point CMI scaling exponent. Either derive the claimed exponent monotonicity or delete/soften the sentence.","section":"Abstract"}],"minor_comments":[{"comment":"The differentiability assumption on f(x) is stronger than needed for Eq. (2); the inequality is naturally one-sided in |x|. The later discussion of a discontinuous limit x→0 in Illustration 1 should be reconciled with the continuity assumption stated just before Eq. (2).","section":"Paragraph containing Eq. (2)"},{"comment":"The text says f(x) is monotonically increasing for x∈(−∞,0), which is consistent with df/d|x| ≤ 0 but may confuse readers reading Eq. (2) literally. Please state explicitly that monotonicity in |x| corresponds to f increasing toward x=0^- in this example.","section":"Illustration 1, Eq. (5)"},{"comment":"The captions do not state whether l/l_B and l|p−p_c|^ν are large enough to satisfy the limits in which Eq. (2) is claimed. Adding these values would help the reader judge the apparent violations in Fig. 4.","section":"Figure captions, Figs. 2-4"},{"comment":"Ref. [39] points to \"Supplemental Material\" for Eq. (3), the SWSSB review, and the finite-r calculation, but these appear in the visible Appendices C and D. Please update the cross-references so the reader is directed to the correct appendices.","section":"References to Supplemental Material"}],"recommendation":"major_revision","confidential_remarks":"The SSA-derived inequalities are correct as mathematical statements, and the paper contains useful exact and numerical checks. My main reservation is that the title-level claim about nonequilibrium systems is not supported by proof or example of UV finiteness for a genuine steady state with current, and the promised anisotropic conserved-dynamics example is absent. The converse claim involving infinite Markov length is also logically too strong. These issues are fixable by either adding the missing example and a careful treatment of UV finiteness, or by reframing the paper as a conditional result for states with UV-finite CMI. I do not see grounds for rejection, but the present version overstates its scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the core results are clean and mostly correct: strong subadditivity gives CMI monotonicity in the separator thickness (Eq. 1), which becomes the scaling-function monotonicity df/d|x| ≤ 0 (Eq. 2) once you assume UV finiteness, and the convex-mixture bound (Eq. 3) with the extra conditional-entropy term is a genuinely useful new tool. Appendix A is correct, and Illustration 1 is an exact analytic check that displays the claimed monotonicity nicely. Second, the paper's reach exceeds its evidence. The stability corollary — smaller CMI fixed point cannot be destabilized toward larger CMI — is exactly as secure as the UV-finiteness assumption, and that assumption is argued, not proven, for general nonequilibrium systems. The examples are equilibrium pure states, classical mixtures, or states produced by local channels; none is a genuine nonequilibrium steady state with a nonzero current. Worse, the abstract I have promises an example titled “area-law CMI in anisotropic conserved dynamics,” but no such example appears anywhere in the full text. This is not a stylistic omission; it is the missing evidence for the title's central claim.\n\nWhat is new deserves credit. The CMI scaling monotonicity applied to RG flows of mixed states, and the convex-mixture bound Eq. (3), are not in the cited SSA/RG literature. The application to SWSSB stability (e.g., Z2 SWSSB cannot flow to Z4 SWSSB) is a nice, concrete payoff, and the TFIM and SWSSB data collapses support the scaling form as far as they go. The honest discussion of finite-size deviations in the SWSSB numerics is also to the authors' credit — the deviations are there, and they say so.\n\nSoft spots, in proportion: the UV-finiteness assumption is load-bearing and not proven; the missing anisotropic-conserved-dynamics example is a real gap; and the numerical confirmation of Eq. (2) is partial. One minor issue: Illustration 1 has a discontinuity at x = 0 (f → log 2 from below, zero at x = 0), which sits uneasily with the earlier continuity assumption; the authors handle it by putting the UV fixed point at 0^-, but it deserves a clearer sentence. Also, the convex-mixture bound's usefulness depends on exponential decay of Sσ(D|B), which requires the p-bounded-noise condition — a restriction that could be stated more prominently.\n\nWho is this for? People working on mixed-state phases, decoherence transitions, and information-theoretic RG constraints. It is a serious paper, clearly written, with correct SSA arguments and a promising new constraint. It deserves a serious referee, but that referee should insist that the abstract match the content and that the UV-finiteness assumption be either proven for a genuine nonequilibrium example or substantially narrowed. I would send it to peer review with expectation of major revision.","headline":"Useful SSA-based CMI monotonicity for mixed-state RG flows, but the main stability claim leans on an unproven UV-finiteness assumption, and the abstract promises an example the text does not deliver.","tokens_in":20685,"tokens_out":2257,"would_cite":true,"duration_ms":27322,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B28","81P45","82B26"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives information-theoretic constraints on renormalization group flows in nonequilibrium systems, showing that conditional mutual information (CMI) can only remain constant or decrease along a flow, and that CMI bounds the stab","keywords":["conditional mutual information","renormalization group","nonequilibrium steady states","strong subadditivity","Markov length","decoherence","spontaneous symmetry breaking","mixed-state phases"],"falsifier":"A concrete counterexample would be a local nonequilibrium steady state, expressible by a local space-time action, where numerical or analytic computation shows the CMI scaling function f(x) increases with |x| over some range while l/l_B and l|t|^ν are large. Alternatively, if a system with conventional thermodynamic limit is found where CMI in the scaling limit depends separately on l_B and x, the UV-finiteness premise fails.","tokens_in":19751,"feed_emoji":"⚛️","tokens_out":3904,"duration_ms":42565,"temperature":0.7,"pith_summary":"This paper derives two information-theoretic constraints on out-of-equilibrium phases. First, when conditional mutual information (CMI)—a measure of correlations between two regions that are not mediated by the region between them—is finite in the scaling limit, its crossover function must be monotonically non-increasing as the relevant scaling variable grows. Thus, under RG flow, a fixed point with smaller CMI cannot be destabilized toward one with larger CMI. Second, the CMI of a convex mixture of states is bounded by the weighted sum of component CMIs plus a classical decoding-error term; this implies that spontaneous symmetry-broken states remain in the same phase under sufficiently weak symmetry-breaking local noise. The authors demonstrate these constraints in classical and quantum examples and use them to argue that certain nonequilibrium fixed points cannot be Gibbs states of local Hamiltonians.","feed_headline":"Correlations cannot grow along RG flows, even out of equilibrium","feed_subtitle":"A quantum-information bound makes fixed points with smaller correlation measure stable against flows to larger ones.","key_machinery":"The central object is conditional mutual information I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC), whose non-negativity follows from strong subadditivity. The key identity is the monotonicity of CMI as the separating slab B thickens (Eq. 1), which after assuming UV finiteness becomes a monotonicity constraint on the crossover scaling function f(x). The second tool is a bound Iρ(A:C|B) ≤ Σ pα Iρα(A:C|B) + Sσ(D|B), which lets the authors control CMI of mixtures via error-decoding entropy. These are used to infer stability of phases and to rule out certain RG flows.","core_discovery":"The central claim is that strong subadditivity of von Neumann entropy, applied to a slab geometry, forces the scaling function of conditional mutual information to satisfy df(x)/d|x| ≤ 0 along a renormalization-group flow, provided CMI is UV finite (i.e., depends only on x = t l_B^{1/ν} in the scaling limit). Since CMI measures correlations not mediated by the intermediate region, this says a fixed point with lower CMI is stable against perturbations that would drive it to one with higher CMI. The authors also prove a convex-decomposition bound on CMI and use it to show that classical symmetry-broken states retain finite Markov length under weak symmetry-breaking channels, meaning they remai","pith_inferences":["The monotonicity of CMI could serve as a general 'c-theorem-like' irreversibility principle for nonequilibrium RG flows, complementing the F-theorem without requiring Lorentz invariance.","The convex-decomposition bound suggests a practical numerical diagnostic: measuring CMI scaling near decoherence-driven transitions can test whether a mixed state is a Gibbs state.","The UV-finiteness assumption may fail in systems with fractonic or subsystem symmetries; a concrete test would be to search for counterexamples where df/d|x| > 0 appears exactly in such systems.","The paper's application to flocking predicts that the Markov length diverges at continuous flocking transitions; this is testable in numerical simulations of active-matter models."],"forward_implications":["A fixed point with zero CMI (e.g., a Gibbs state of a local Hamiltonian, or a product state) cannot be destabilized toward a state with nonzero CMI.","A SWSSB fixed point with CMI log 2 cannot be driven to a Z4 SWSSB state with CMI log 4; similarly, SWSSB states cannot destabilize toward a Z2×Z2 SPT phase.","If the IR fixed point has infinite Markov length (e.g., 1+1D directed percolation), the UV fixed point cannot be a Gibbs state of a local Hamiltonian.","In 2D, a continuous flocking transition fixed point cannot be a Gibbs state due to Mermin-Wagner and CMI monotonicity.","The anomalous dimension of CMI in 1D with finite l must be non-negative.","The convex-decomposition bound implies that classical symmetry-broken states remain in the same phase under sufficiently weak p-bounded local noise, with a threshold that can be estimated from decoding error probabilities."],"fun_headline_variants":["Quantum info bound blocks correlation growth in RG flows","No increase in correlations along renormalization group flow","Strong subadditivity sets direction of RG flow in nonequilibrium","Information bound stabilizes symmetry-broken states against quantum channels"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"CMI is UV finite in the scaling limit—that is, I(∞|l_B,t) depends only on the combination x = t l_B^{1/ν} and not separately on the cutoff, l_B, or l—so that the monotonicity in l_B translates into monotonicity in x.","fun_headline_variants_meta":{"raw":{"variants":["Quantum info bound blocks correlation growth in RG flows","No increase in correlations along renormalization group flow","Strong subadditivity sets direction of RG flow in nonequilibrium","Information bound stabilizes symmetry-broken states against quantum channels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000501,"raw_usage":{"total_tokens":2262,"prompt_tokens":697,"completion_tokens":1565,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":1498}},"tokens_in":441,"tokens_out":1565,"duration_ms":15390,"temperature":1.0,"reasoning_tokens":1498,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:09:42.919942+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a local nonequilibrium steady state, expressible by a local space-time action, where numerical or analytic computation shows the CMI scaling function f(x) increases with |x| over some range while l/l_B and l|t|^ν are large. Alternatively, if a system with conventional thermodynamic limit is found where CMI in the scaling limit depends separately on l_B and x, the UV-finiteness premise fails.","supporting_citations":[],"review_version":1}