{"id":"436d1a91-4e40-4454-8980-3b813f08207f","arxiv_id":"2506.10076","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Twisted Rényi-N correlators of the reduced density matrix exhibit long-range order along an artificial replica direction for SPT phases, mirroring the bulk strange correlator.","lead":"This paper shows that a nonlinear observable of a quantum system's reduced density matrix, the twisted Rényi-N correlator, can reproduce the bulk 'strange correlator' signature of symmetry-protected topological (SPT) phases from entanglement data alone. If correct, the result offers a practical probe of topological order that works from local reduced states and extends to thermal and open-system settings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact TRNC–strange-correlator equality is proven only for the fixed-point state built from the RDM, not for the original SPT wavefunction; for generic states the two can differ in value, so the paper establishes universal order, not exact reproduction.","rationale":"The reader's weakest_assumption already identifies the central gap: the replicated state built from the RDM in Eq. (4) is assumed to lie in the same SPT phase as the original wavefunction, but the exact correspondence is proven only for zero-correlation-length fixed-point states. My stress-test sharpens this into a concrete, checkable discrepancy: the TRNC limit in Eq. (18) is pinned to 1 by the degeneracy structure of the dominant entanglement subspace, while the bulk strange correlator of a generic Haldane chain need not saturate to the same value. Hence the claimed 'exact correspondence' in the abstract and Sec. II C is at best a universal-order statement for generic states, not an exact equality. This does not overturn the paper's main qualitative conclusions for 1d SPT detection, nor does it invalidate the 2d quasi-long-range prediction, but it does mean the central claim is overstated as written. The 2d extension inherits the same issue because the cut-and-glue entanglement Hamiltonian is an approximate model, and no direct computation of the bulk strange correlator is provided for comparison. The proposed test would settle whether the exact-correspondence language is defensible for generic states. Since the reader's CONDITIONAL verdict already reflects this weak spot, no change to the verdict is needed.","tokens_in":26580,"tokens_out":15172,"duration_ms":189513,"concrete_test":"On the infinite DMRG Haldane chain at a generic D (e.g., D=0.5J, same model as Eq. (14)), compute the bulk strange correlator of the original ground state: C_strange(r) = ⟨Ψ_trivial|Σ^x_0 Σ^x_r|ψ_D⟩/⟨Ψ_trivial|ψ_D⟩ with the same trivial state and Σ^x as in Eq. (13), extrapolate to r→∞, and compare with C_X(N→∞) from Eq. (16) evaluated on the RDM. If the two saturation values differ, the exact correspondence stated in the abstract and Sec. II C is false for generic states, and only the universal long-range-order claim survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section II C proves ⟨Ψ_trivial|O†_1 O_N|Ψ_SPT⟩/⟨Ψ_trivial|Ψ_SPT⟩ = C^{(N)} for the state |Ψ_SPT⟩ constructed in Eq. (4) by replicating ρ. This state is a zero-correlation-length fixed-point wavefunction built from the eigenvectors and eigenvalues of ρ; it is not the original SPT ground state. For a generic finite-correlation-length SPT state, the equality therefore does not follow. In 1d, the paper's Eq. (18) forces C_X(N→∞)=1 whenever the two leading entanglement levels are degenerate, independent of D. But the bulk strange correlator of the original Haldane chain at finite D (away from the AKLT point) is not generally equal to 1; its saturation value depends on the state and the choice of trivial state. The abstract and Sec. II C claim an 'exact correspondence between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted Rényi-N operator of the d-dimensional RDM,' but the proof only establishes this for the replica-constructed fixed-point state. The extension to realistic states relies on the unproven assumption that the replicated state lies in the same phase and that nonuniversal magnitudes are irrelevant; this is exactly the gap the reader flagged. The 2d chiral extension inherits the same issue via the cut-and-glue entanglement Hamiltonian (Eqs. (22)-(25)), which is an approximate, heuristic mapping rather than a theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a holographic framework in which the reduced density matrix (RDM) of a d-dimensional symmetry-protected topological (SPT) state is replicated along a new 'replica direction' to reconstruct a (d+1)-dimensional fixed-point SPT wavefunction. The central object is the twisted Rényi-N correlator (TRNC) of the RDM, Eq. (9), which is claimed to be exactly dual to the bulk strange correlator of the reconstructed SPT state, Eq. (10). The authors present a fixed-point construction (Sec. II C), numerical demonstrations in 1d Haldane and spin-2 SO(5) chains (Sec. III), analytical and numerical studies of 2d Chern and quantum spin Hall states via the cut-and-glue approach (Sec. IV), and corollaries for the Lieb-Schultz-Mattis theorem and mixed-state SPTs (Secs. V-VI).","tokens_in":26955,"tokens_out":11423,"duration_ms":128672,"significance":"If the central claim is appropriately qualified, the paper provides a useful new diagnostic: the TRNC is an entanglement-spectrum-derived quantity that can distinguish SPT order from trivial order using only the RDM, without the full wavefunction. The fixed-point identity in Sec. II C is clean, and the DMRG results for the Haldane chain, the SO(5) chain, and the ladder system are convincing evidence for universal long-range or quasi-long-range order of the TRNC. The extension to thermal states and mixed-state SPTs is suggestive. However, the strongest advertised statement--an exact correspondence between the TRNC and the bulk strange correlator of the original SPT wavefunction--is proven only for the fixed-point state built from the RDM, not for generic SPT ground states; the manuscript's own limitations in Sec. IV B confirm that the 2d extension rests on approximations. The paper is a solid contribution to the quantum-information characterization of SPT phases, but it needs careful revision of the claims' scope.","major_comments":[{"comment":"The abstract and Sec. II C claim an 'exact correspondence' between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted Rényi-N correlator of the d-dimensional RDM. The derivation in Eqs. (4)-(10) establishes this equality only for the fixed-point state |Ψ_SPT⟩ constructed by replicating the RDM, which is a zero-correlation-length state built from the eigenvectors and eigenvalues of ρ. For a generic finite-correlation-length SPT ground state, the bulk strange correlator is not generally equal to the TRNC. A concrete example is the Haldane chain away from the AKLT point: Eq. (18) gives C_X(N→∞)=1 purely from the degeneracy of the entanglement spectrum, while the actual strange correlator of the original wavefunction depends on D and on the choice of trivial reference. The manuscript therefore establishes universal long-range (or quasi-long-range) order of the TRNC, not the exact reproduction of the bulk strange correlator. Please either qualify the 'exact' claim throughout, including the abstract, or provide a proof that the replicated state lies in the same phase as the original wavefunction and that nonuniversal magnitudes do not affect the claimed duality.","section":"Sec. II C / Abstract"},{"comment":"The extension to 2d systems relies on the cut-and-glue entanglement Hamiltonian and on a phenomenological WZW mapping. Specifically, Eq. (23) introduces a coupling U between ket and bra fields, and the text then extends U m_L·m_R as (∂_x m)^2 locally in Eq. (25); this is an approximation, not a derivation from the original wavefunction. The manuscript itself states in Sec. IV B that 'our analysis relies on a simplified model of a non-interacting Chern insulator with zero correlation length' and that the chiral-boson form breaks down in more general settings, with a non-universal exponent η. Since the central 2d prediction of power-law decay along the replica direction rests on this approximate mapping, please provide a concrete criterion for when the free-boson answer is stable (e.g., under relevant perturbations, or with explicit verification that the exponent remains finite) or state clearly that the 2d quasi-long-range order is a conjecture supported by the ladder numerics.","section":"Sec. IV A-B / Sec. III B"},{"comment":"The 'strengthened LSM theorem' is not established by the arguments given. Corollary A asserts that the thermal density matrix e^{-H_{1d}} of a 1d spin-1/2 chain 'can always be viewed' as the RDM of a 2d SPT wavefunction, but this identification is not proven; a generic local Hamiltonian's thermal state is a specific mixed state that need not coincide with the RDM obtained from any 2d SPT wavefunction. Similarly, Corollary B for mixed states assumes that any short-range-correlated state with weak symmetries maps to the same 2d WZW description. These are strong assumptions that go beyond the fixed-point identity. Please either supply a precise class of Hamiltonians or states for which the identification holds, or present the LSM statements as conjectures rather than theorems.","section":"Sec. V, Corollary A-B"},{"comment":"The order-of-limits issue is load-bearing for the 2d claim. The paper correctly notes that one must take l_x→∞ before N→∞ to obtain power-law decay, and that for finite l_x the correlator decays exponentially as e^{-N/l_x}. Since any numerical or experimental probe has finite system size, the practical content of the 2d fingerprint depends on the separation of scales N≪l_x. The manuscript should state this limitation explicitly in the main text of Sec. IV and provide guidance on how large l_x must be relative to N for the power-law to be observable, given that the ladder numerics in Fig. 9 use N≤L.","section":"Sec. IV A / Sec. IV B"}],"minor_comments":[{"comment":"The abstract contains a typo ('colloary') and a grammatical error ('we generalized'); please correct these.","section":"Abstract"},{"comment":"The definition of M_O is unclear: the phrase 'M_O = O_{ij} δ_{ij}' seems to have an index mismatch. Please define M_O explicitly as a tensor with its indices and specify the contraction with ρ in Eq. (9).","section":"Eq. (8) and surrounding text"},{"comment":"The notation Σx and Σy in the strange-correlator expression is ambiguous, as x and y appear both as spatial labels and as operators. Please use distinct symbols for the operators and the positions.","section":"Eq. (13)"},{"comment":"The figures for the Haldane and SO(5) chains do not show convergence of C_X(N) with respect to the MPS bond dimension χ; please add a convergence check analogous to Fig. 9(b), or state that the values are converged at χ=60 (Fig. 5) and χ=50 (Fig. 7).","section":"Figs. 5 and 7"},{"comment":"The text says that the algebraic decay 'is in quantitative agreement' with the strange correlator of the QSH state, but the exponent is stated to be non-universal. Please clarify which quantity is in quantitative agreement, given that both exponents are non-universal.","section":"Sec. IV B"},{"comment":"The independent study mentioned in the conclusion should be discussed in the introduction or conclusion to clarify the incremental contribution of this work relative to Ref. [106].","section":"Sec. VII / Ref. [106]"}],"recommendation":"major_revision","confidential_remarks":"The core identity in Eqs. (9)-(10) is imported from Ref. [50], by the same group, and the manuscript does not provide a fully self-contained derivation. The editors may wish to verify that the novelty relative to Ref. [50] and to the independent study [106] is clearly delineated. The numerical work is solid, but the abstract's 'exact correspondence' claim is stronger than what is proven."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth your time if you care about entanglement-based diagnostics for SPT phases. The genuinely new content: a twisted Rényi-N correlator (TRNC) built from the reduced density matrix, used as a phase discriminator in 1d chains, with clean quantized plateaus across the Haldane and SO(5) transitions, plus analytic predictions for Chern/QSH power laws and an extension to mixed-state SPTs via surgery operators. The DMRG on the ladder (Fig. 9) shows a power law decaying as N^-0.52 that matches the Luttinger-liquid expectation, which is a real check.\n\nWhat the paper does well: it makes a strong practical case that an RDM-only quantity, accessible experimentally via Rényi measurements, can detect SPT order and distinguish distinct phases. The degeneracy argument in Eq. (18) is the load-bearing piece: for any RDM with a two-fold degenerate leading eigenspace in different parity sectors, C_X(N→∞)=1; that is rigorous and does not depend on the fixed-point construction. The SO(5) result showing C=2 vs C=1 vs 0 across phases is a nice demonstration that the correlator sees the dimension of the projective representation.\n\nSoft spots. The abstract and Sec. II C claim an 'exact correspondence' between the bulk strange correlator of the (d+1)-dimensional SPT wavefunction and the TRNC of the d-dimensional RDM. The proof in Sec. II C establishes that for the state built by replicating ρ, not for the original generic SPT ground state. For a generic finite-correlation-length state, the TRNC saturates to a value set by the ES degeneracy, which is a phase diagnostic but not an exact reproduction of the original strange correlator. The authors should soften that language; the reader flagged this, and the stress test is right. The practical diagnostic survives, because Eq. (18) does not rely on equality with the bulk correlator.\n\nThe 2d extension is visibly heuristic: cut-and-glue plus a phenomenological WZW mapping. The authors themselves note in Sec. IV B that the exponent η is non-universal and depends on microscopic details, so the 'quantitative agreement' claim is looser there. The ladder simulation is encouraging but not a proof.\n\nMinor: no code/data release, and the DMRG figures lack error analysis (though bond-dimension convergence is addressed). The LSM corollary in Sec. V is suggestive, built on the same WZW mapping; it is a corollary of an approximate argument, not a new theorem.\n\nOverall: the 1d results are solid and worth publishing; the 2d/mixed-state parts are speculative but interesting. I would send it to a serious referee, asking the authors to rescope the 'exact' language to the fixed-point construction and to separate rigorous from heuristic claims more carefully. I would cite the 1d diagnostic and the ladder data.","headline":"RDM-only SPT diagnostic is real in 1d, plausible in 2d; the 'exact correspondence' claim overshoots the proof.","tokens_in":27468,"tokens_out":3455,"would_cite":true,"duration_ms":41702,"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":"By replicating the reduced density matrix of a symmetry-protected topological state along an extra dimension, the twisted Rényi-N correlator of that matrix reproduces the bulk strange correlator and reveals long-range (or…","keywords":["symmetry-protected topological order","entanglement Hamiltonian","twisted Rényi-N correlator","strange correlator","replica direction","reduced density matrix","mixed-state SPT phases"],"falsifier":"Take a 2d Chern-insulator reduced density matrix, fix a finite spatial length l_x, and increase the replica number N beyond l_x. The paper predicts exponential decay C(N) ∼ $e^{{-N/l_x}}$ in this order of limits; observing a power-law instead would falsify the order-of-limits claim. Conversely, for a genuinely finite-correlation-length interacting SPT state, compute C(N) after sending l_x → ∞ first: if the correlator decays exponentially rather than algebraically, the phase-correspondence assumption behind the holographic mapping fails.","tokens_in":26379,"feed_emoji":"⚛️","tokens_out":11788,"duration_ms":114279,"temperature":0.7,"pith_summary":"The paper claims that, in symmetry-protected topological (SPT) phases with area-law entanglement, the reduced density matrix of a spatial cut can be treated as a lower-dimensional mixed state, and that replicating this density matrix along an extra 'replica direction' reconstructs the original higher-dimensional SPT wavefunction. The central discovery is an exact correspondence: the twisted Rényi-N correlator built from the reduced density matrix equals the bulk strange correlator of the reconstructed SPT state. As a result, the reduced density matrix alone carries a universal signature of topological order — its twisted Rényi-N correlator shows long-range (or quasi-long-range) order along the replica direction, even though ordinary correlation functions of the RDM are short-ranged. This gives a practical tool to detect SPT phases, locate phase transitions, and distinguish different SPT phases from entanglement data alone, with extensions to thermal states, open quantum systems, and mixed-state SPT phases.","feed_headline":"Replicating a density matrix reveals hidden topological order","feed_subtitle":"Replicating the reduced density matrix turns its strange correlator into an entanglement signature.","key_machinery":"The central object is the dimensional-extension (replica) construction: each copy of the reduced density matrix ρ = Σ_ν λ_ν |ν⟩⟨ν| is vectorized to an entangled pair |ν⟩_L|ν⟩_R, and a rank-four tensor T_{ijkl} = δ_{ij}δ_{kl} inserts an intra-unit-cell coupling between the bra of copy i and the ket of copy i+1. This produces a matrix-product-state-like fixed-point wavefunction |Ψ_SPT⟩ whose unit cells run along the replica index. The twisted Rényi-N correlator $C^{{(N)}}$ = Tr[ρ^N M_O ρ^N M_O^†]/Tr[$ρ^{{2N}}$] is the observable that carries the argument: it is shown to be the overlap of |Ψ_SPT⟩ with a trivial product state with operators inserted at positions separated by N, i.e., the bulk strange correlator. The mechanism for long-range order is that the projective representation of the symmetry group forces a degenerate leading eigenspace of ρ; in the large-N limit only that degenerate space survives, so a charge-flipping insertion M_O yields a non-vanishing (and quantized) correlator. In 2d, ρ^N maps to a chiral-boson path integral on a torus, so the correlator becomes a temporal two-point function with algebraic decay.","core_discovery":"Working at the level of a spatial bipartition, the authors start with the reduced density matrix ρ of an SPT wavefunction and build a pure state by taking N copies of ρ, vectorizing each copy into an entangled pair, and gluing the pairs together with a fixed rank-four tensor. The resulting state |Ψ_SPT⟩ is the fixed-point SPT wavefunction in one higher dimension. The paper's central identity is that the twisted Rényi-N correlator $C^{{(N)}}$ = Tr[ρ^N M_O ρ^N M_O^†]/Tr[$ρ^{{2N}}$] equals the strange correlator ⟨Ψ_trivial|O_1^† O_N|Ψ_SPT⟩/⟨Ψ_trivial|Ψ_SPT⟩, with the operator insertions separated by N copies along the replica axis. Because the SPT strange correlator is (quasi) long-ranged, the twisted correlator of the RDM must be (quasi) long-ranged in the replica direction: in 1d it saturates to a quantized value determined by the degeneracy of the entanglement spectrum, while in 2d Chern and quantum spin Hall systems it decays algebraically with an exponent controlled by the interaction parameter K, provided the spatial length is taken to infinity before N. The authors verify the correspondence numerically for 1d spin chains and a coupled-ladder model, and use it to reformulate the half-integer-spin constraint for closed and open systems and to propose diagnostics for mixed-state SPT order.","pith_inferences":["If the correspondence holds beyond the fixed point, the quantized large-N value of the twisted correlator becomes a direct readout of the entanglement-spectrum degeneracy, which could be measured in experiments through randomized-measurement schemes for nonlinear functionals of the density matrix.","The replica construction can be read as a tensor-network prescription: it converts a mixed-state preparation into a higher-dimensional pure-state preparation, suggesting a route to synthesize SPT states from lower-dimensional mixed states on near-term quantum devices.","The thermal-state corollary implies that the anomaly constraint leaves a fingerprint not just in the energy spectrum but in the replica-direction correlations of the Gibbs state, which might be probed by measuring Rényi-N correlation functions in ultracold atomic gases.","For mixed-state SPT order, the surgery-operator correlator may be less sensitive to the choice of trivial reference state than existing diagnostic correlators, since it directly constructs the fixed-point state whose strange correlator is the target; this is a conjecture to be tested."],"forward_implications":["The twisted Rényi-N correlator of the half-chain reduced density matrix is a stand-alone SPT diagnostic: it saturates to a finite quantized value in an SPT phase, falls to zero in the trivial phase, and numerically tracks the spin-1 SPT-to-trivial transition as the anisotropy D is tuned.","The same correlator distinguishes distinct SPT phases: in the spin-2 chain with (Z2 × Z2)^2 symmetry it takes the values 2, 1, and 0 in the SO(5)-symmetric, intermediate, and trivial phases respectively.","For 2d Chern and quantum spin Hall states the correlator decays algebraically in the replica index, with the exponent set by the interaction parameter K, reproducing the quasi-long-range strange correlator of the bulk.","As a corollary, the thermal density matrix of a half-integer-spin chain with translation and spin-rotation symmetries must exhibit (quasi) long-range order in its twisted Rényi-N correlator, giving a strengthened form of the half-integer-spin constraint for both closed and open systems.","In mixed-state SPT phases, the twisted Rényi-N correlator built from the surgery operator ρ_v ρ_t is proposed as a diagnostic that detects and distinguishes mSPT topology."],"supporting_citations":[{"why":"Introduces the dimensional-extension mapping that reconstructs a higher-dimensional SPT wavefunction from replicated copies of the density matrix, and first defines the twisted Rényi-N correlator.","marker":"[50]"},{"why":"Provides the cut-and-glue approach for the entanglement Hamiltonian of a Chern insulator, the starting point for the 2d analysis.","marker":"[43]"},{"why":"Derives the explicit chiral-boson form of the entanglement Hamiltonian for Chern and quantum spin Hall states used in Sec. IV.","marker":"[45]"},{"why":"Defines the bulk strange correlator and its (quasi) long-range order, the target object that the twisted correlator reproduces.","marker":"[38]"},{"why":"Gives the power-law strange correlator of quantum spin Hall states with non-universal exponent, the quantitative benchmark for the replica-direction decay.","marker":"[42]"},{"why":"Supplies the fixed-point spin-1 SPT wavefunction used as the canonical example of the replica construction.","marker":"[64]"},{"why":"Establishes the projective-representation classification of 1d SPT phases that underlies the degeneracy of the RDM spectrum.","marker":"[65]"},{"why":"Provides the nonlinear-sigma-model description of 2d SPT strange correlators used in the field-theory argument for quasi-long-range order.","marker":"[75]"}],"fun_headline_variants":["Twisted Rényi-N correlators reveal SPT order holographically","Replicating density matrices exposes hidden topological structure","Entanglement holography via twisted Rényi-N operators","Strange correlators from Rényi-N replicas in SPT phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The replication is exact only for zero-correlation-length fixed-point states; for generic finite-correlation-length SPT wavefunctions the paper assumes that the replicated state still lies in the same SPT phase as the original, so that its strange correlator keeps the universal (quasi) long-range order, an assumption supported by heuristic cut-and-glue and field-theory arguments rather than by a proof.","fun_headline_variants_meta":{"raw":{"variants":["Twisted Rényi-N correlators reveal SPT order holographically","Replicating density matrices exposes hidden topological structure","Entanglement holography via twisted Rényi-N operators","Strange correlators from Rényi-N replicas in SPT phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000436,"raw_usage":{"total_tokens":2268,"prompt_tokens":1048,"completion_tokens":1220,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":1146}},"tokens_in":664,"tokens_out":1220,"duration_ms":9211,"temperature":1.0,"reasoning_tokens":1146,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:37:28.615938+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a 2d Chern-insulator reduced density matrix, fix a finite spatial length l_x, and increase the replica number N beyond l_x. The paper predicts exponential decay C(N) ∼ $e^{{-N/l_x}}$ in this order of limits; observing a power-law instead would falsify the order-of-limits claim. Conversely, for a genuinely finite-correlation-length interacting SPT state, compute C(N) after sending l_x → ∞ first: if the correlator decays exponentially rather than algebraically, the phase-correspondence assumption behind the holographic mapping fails.","supporting_citations":[{"cited_title":"Lundgren, Y","cited_arxiv_id":null,"evidence_quote":"Introduces the dimensional-extension mapping that reconstructs a higher-dimensional SPT wavefunction from replicated copies of the density matrix, and first defines the twisted Rényi-N correlator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the cut-and-glue approach for the entanglement Hamiltonian of a Chern insulator, the starting point for the 2d analysis."},{"cited_title":"Wu, Y.-Y","cited_arxiv_id":null,"evidence_quote":"Derives the explicit chiral-boson form of the entanglement Hamiltonian for Chern and quantum spin Hall states used in Sec. IV."},{"cited_title":"Lundgren, J","cited_arxiv_id":null,"evidence_quote":"Defines the bulk strange correlator and its (quasi) long-range order, the target object that the twisted correlator reproduces."},{"cited_title":"Mollabashi, N","cited_arxiv_id":null,"evidence_quote":"Gives the power-law strange correlator of quantum spin Hall states with non-universal exponent, the quantitative benchmark for the replica-direction decay."}],"review_version":1}