{"id":"99d7fcdd-1d21-47b5-8857-21219df9ff50","arxiv_id":"2608.12475","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The real-to-complex spectral threshold is set by the lowest exceptional-point threshold among all pairs of reference modes, λ_c = Δ_act / G_act.","lead":"This paper presents a general formula for when a non-Hermitian system's energy spectrum turns complex, the moment it starts amplifying. The authors show the threshold is set by a single activated pair of modes and verify the formula against exact numerics in three model systems.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower-envelope formula Eq. (6) is unproven for generic order-one λ_c: the paper's own Feshbach criterion Eq. (S13) controls remote-mode admixture only when λ_c is small, so the 'arbitrarily large disturbances' claim goes beyond the demonstrated regime.","rationale":"The reader's conditional verdict is appropriate. The framework is transparent, and the worked examples are convincing: exact diagonalization matches the closed-form thresholds in Figs. 2 and 3, and SM S1 explicitly benchmarks λ_c = O(0.1-1). However, the paper's stated scope is much broader than the regime tested. The proof of the selection of the (1,2) channel in SM S3 also relies on asymptotic scaling with fixed χ=κ(L+1) and an O(1) projected-coupling argument, not a complete proof for all finite L and κ; this is a second, related gap. Both gaps point to the same load-bearing assumption: that the first transition is always an isolated two-branch EP of the H0 eigenbasis. The lower envelope formula Eq. (6) is an ansatz, supported by examples, not a theorem; the Feshbach criterion Eq. (S13) quantifies when the ansatz is controlled but is not derived as an exact error bound. Therefore, the honest verdict is conditional until the two-branch assumption is either proven under stated conditions or its failure demonstrated.","tokens_in":29788,"tokens_out":4362,"duration_ms":45560,"concrete_test":"Take a small random ensemble of matrices H0 with a real, nearly degenerate triplet of eigenvalues and a non-Hermitian V of unit norm, normalize V so λ_c ∼ O(1), and compare the exact threshold from full diagonalization with the lower envelope Eq. (6) over all pairs. Any instance where the full spectrum develops complex eigenvalues at λ smaller than min_{i<j} λ_c^(i,j) (or where three eigenvalues coalesce before any pair) falsifies the general two-mode lower-envelope claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the exact threshold equals min_{i<j} λ_c^(i,j) from 2x2 projected blocks (Eq. 6). This requires the first real-to-complex transition to be a coalescence of exactly two reference branches, with all other modes contributing only a small Feshbach correction. The paper does not prove this; it asserts genericity and validates numerically on three models. Its own criterion for the two-mode reduction, SM S2 Eq. (S13), says the correction is small when λ_c C_pq / sqrt(-A_pq B_qp) << 1. In the worked examples λ_c is exponentially or algebraically small in L, so this condition is satisfied, and the numerics agree. But the abstract's claim of 'arbitrarily large disturbances' would require the same formula to hold at order-one λ_c, where Eq. (S13) gives no parametric suppression. A higher-order degeneracy (three or more coalescing levels) would escape the pair-resolved lower envelope entirely, and the paper's treatment of such cases is deferred to 'remote-mode corrections' rather than excluded. Thus the general principle, as opposed to the demonstrated examples, rests on an unproven two-branch assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an “exceptional activated mode” framework for the location of the real-to-complex spectral threshold in finite non-Hermitian families H(λ)=H0+λV. For each biorthogonal reference pair (i,j) it defines a 2×2 projected block, solves its exceptional-point condition exactly in λ, and declares the physical threshold to be the lower envelope of the pair-resolved thresholds, λ_c=min_{i<j} λ_c^(i,j), written in the transparent form λ_c=Δ_act/G_act. The framework is applied to three models: the critically non-Hermitian skin effect in coupled Hatano–Nelson chains under open boundary conditions, an impurity-closed version of the same ladder, and a Hermitian SSH chain with reflection-related local gain and loss. In each case the analytical thresholds are compared with full exact diagonalization; the agreement is excellent and no parameter is fitted to the numerical data.","tokens_in":30037,"tokens_out":5173,"duration_ms":52321,"significance":"If the central lower-envelope claim is valid in the stated generality, the paper provides a useful and non-perturbative unification: the cNHSE crossover from exponential to algebraic scaling, the impurity-driven switching of the active mode pair, and the edge-to-bulk activation without a topological transition are all described by the same pair-resolved exceptional-point minimization. The strengths of the paper are its explicit exact algebra in Eqs. (2)–(7), the absence of any fitted parameter, and the systematic benchmarking against full spectra in Figs. 2–3 and S1. The main weakness is that the general claim for “arbitrarily large disturbances” is not supported by a proof of the two-mode reduction at order-one λ_c; the paper’s own validity criterion is small-λ_c controlled, and the order-one regime is only validated a posteriori in one example. This gap is load-bearing for the paper’s universality claim, though not for the specific examples presented.","major_comments":[{"comment":"The general claim λ_c=min_{i<j} λ_c^(i,j) for “arbitrarily large disturbances” is not proven for λ_c=O(1). The paper’s own validity criterion, SM S2 Eq. (S13), states that the two-mode reduction is controlled when λ_c C_pq / sqrt(-A_pq B_qp) << 1, which is automatically satisfied only when λ_c itself is small. In the analytic examples λ_c is exponentially or algebraically small in L, so Eq. (S13) supplies a parametric control. The order-one regime in Sec. S1 is explicitly described by the text as “not covered by this small-λc scaling argument” and is validated a posteriori against full diagonalization for a few small sizes. That is numerical evidence, not a proof that the lower envelope of two-mode exceptional points persists for arbitrary λ_c. Either supply a bound on the remote-mode correction that is valid for λ_c=O(1), or revise the abstract and the surrounding statements to state that the general principle is proved in the small-λ_c regime and otherwise is a numerically verified conjecture.","section":"Abstract, Eq. (6), and SM S2"},{"comment":"The framework assumes that the first real-to-complex transition is an isolated second-order EP of exactly two reference branches. This is stated rather than proved: the text says “Generically, this first collision EP is a two-branch coalescence; higher-order degeneracies correspond to remote-mode corrections,” and End Matter Sec. I adds that violation of the admissibility conditions is a “diagnostic that the activated sector exceeds two dimensions.” That is a definitional classification, not a mathematical exclusion. Three or more coalescing levels can produce a real-to-complex transition while every pair-resolved discriminant remains positive, in which case Eq. (6) is not the correct threshold. The manuscript should either prove a genericity statement under explicit assumptions on H0 and V, or clearly restrict the central theorem to the case of an isolated pairwise coalescence and state that higher-order EPs are outside the demonstrated scope.","section":"End Matter Sec. I and SM S2"},{"comment":"The derivation of the closed-form cNHSE threshold Eq. (11) relies on proving that the minimizing pair is (1,2) (together with its reflected partner), but the proof in SM S3 is incomplete. Equation (S41) shows that among fixed low-lying indices the pair (1,2) has the smallest detuning factor, and the text states that “direct finite-sum minimization of Eq. (S24) confirms this selection for the parameter regime used in the numerical comparison.” This does not provide a global analytic inequality over all L(L−1)/2 pairs, so the claim that Eq. (11) is the exact threshold for all L and κ is not fully established. The numerical agreement is strong, but if the closed-form expression is presented as an exact result, a rigorous global bound on min_{i<j} λ_c^(i,j) is needed, or the statement should be softened to an empirically verified selection in the displayed parameter regime.","section":"SM S3 and Eq. (11)"}],"minor_comments":[{"comment":"The abstract contains the typo “atallsystem sizes”; it should read “at all system sizes.”","section":"Abstract"},{"comment":"There is an unresolved cross-reference “Eq. (??)” in the sentence about the reflected partner (L−1,L); this should be replaced with the actual equation number for the reflection symmetry, Eq. (S34).","section":"SM S3"},{"comment":"Since the order-one λ_c benchmark is the main evidence for the non-perturbative character of the method, it would be helpful to state the values of L used in Fig. S1 directly in the caption and to report the maximum relative deviation between λ_c^num and the lower-envelope prediction.","section":"SM S1 and Fig. S1"},{"comment":"The notation λ^(p,q)_c in Eq. (29) is introduced only in the End Matter; for readability it should be explicitly tied to Eq. (6), or the pair superscript should be defined in the sentence preceding Eq. (29).","section":"End Matter Sec. IV and Eq. (29)"}],"recommendation":"major_revision","confidential_remarks":"The paper is competently executed and the examples are convincing, but the gap between the advertised universal principle and the demonstrated regime is substantive. I would accept a revised version that either proves the two-mode lower-envelope formula under conditions that include λ_c=O(1), or honestly restricts the general claim and presents the cNHSE crossover as the precisely characterized small-λ_c application. The unresolved issues are fixable within the scope of the manuscript, so reject is not warranted. I saw no evidence of citation or novelty problems; the self-citations are background references rather than the basis of the claimed prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper gives a clean way to compute real-to-complex thresholds as the minimum over pair-resolved exceptional-point conditions, and in the three examples it works very well. The cNHSE crossover formula Eq. (11) is new and is the strongest piece: it connects the exponential skin-overlap regime to the algebraic band-edge regime with one closed expression, and the numerics match across the full crossover. The impurity-closed channel sequence (1,2)->(2,3)->(4,5) and the SSH edge-to-bulk activation switch are also genuinely new, and both are supported by exact diagonalization. No parameters are fitted. That is real work and it deserves a serious referee.\n\nWhere I would push back: the abstract says the closed-form condition holds for arbitrarily large 'disturbances.' That is not established. The paper's own Feshbach criterion, Eq. (S13), controls remote-mode admixture only when lambda_c is small; the lambda_c = O(1) cases are validated numerically, not parametrically. More importantly, the lower-envelope equality lambda_c = min lambda_c^(i,j) assumes the first transition is a two-branch coalescence. A higher-order degeneracy would escape the pair-resolved envelope, and the paper defers that case to 'remote-mode corrections' rather than excluding it. So the general principle is a well-motivated conjecture, not a theorem, and the abstract should say so.\n\nThe other soft spots are minor. The (1,2) selection in the cNHSE ladder is proven for the parameter regime shown, not for all L and kappa; that is probably fixable. The switching sequence for L=10 is explained convincingly. The citation pattern is fine: self-citations are background, not the source of the prediction.\n\nNet: the demonstrated results are solid, the framework is useful, and the overclaim is a scope problem, not a fatal flaw. The paper would benefit from rewriting the abstract, adding an explicit statement of the two-branch assumption, and either proving or numerically probing a case where three levels coalesce.\n\nRecommendation: send to peer review. I would expect conditional acceptance after the generality claims are scaled back and the validity criterion is discussed honestly.","headline":"A genuinely useful lower-envelope EP framework that nails three worked examples with exact numerics, but the 'arbitrarily large disturbances' generality claim outruns the paper's own two-mode validity criterion.","tokens_in":30577,"tokens_out":1145,"would_cite":true,"duration_ms":13686,"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":"The paper claims that the real-to-complex threshold of a non-Hermitian family is set by the cheapest two-mode exceptional point among all reference pairs.","keywords":["exceptional points","real-to-complex transitions","non-Hermitian physics","critical non-Hermitian skin effect","PT symmetry","activated mode theory","spectral detuning","topological edge states"],"falsifier":"Compute the exact spectrum of a finite family $H(\\lambda)=H_0+\\lambda V$ engineered so that the first complex eigenvalue arises from a third-order exceptional point, and compare the numerical onset with the pairwise lower envelope $\\min_{i<j}\\lambda^{(i,j)}_c$; any mismatch at the onset falsifies the two-mode reduction. A broader check is to scan many sparse anti-Hermitian perturbations $V$ on a fixed real-spectrum $H_0$ and search for a case where the first collision involves non-adjacent or strongly admixed reference branches, whose exact threshold would test the boundary of the paper's validity criterion.","tokens_in":29586,"feed_emoji":"⚡","tokens_out":10550,"duration_ms":89703,"temperature":0.7,"pith_summary":"The paper advances a general principle: for any finite family $H(\\lambda)=H_0+\\lambda V$ whose reference spectrum is real, the first real-to-complex transition is set by one selected pair of reference modes colliding at an exceptional point, and the threshold is the minimum over all pairs of a pair-resolved exceptional-point condition. Each candidate pair is handled by projecting the full Hamiltonian onto its two-mode biorthogonal block and demanding that the block's discriminant vanish, which yields a closed threshold without expanding in $\\lambda$. The physical threshold takes the competition form $\\lambda_c = \\Delta_{\\mathrm{act}}/G_{\\mathrm{act}}$, where $\\Delta_{\\mathrm{act}}$ is the reference detuning of the winning pair and $G_{\\mathrm{act}}$ its projected non-Hermitian coupling. This recasts amplification onset as a variational mode-selection problem that is independent of any particular symmetry. The authors demonstrate it on a coupled Hatano-Nelson ladder at all system sizes, on boundary-impurity-closed ladders, and on a topological SSH chain with local gain and loss.","feed_headline":"One mode pair decides when non-Hermitian spectra turn complex","feed_subtitle":"Threshold equals the ratio of reference detuning to projected gain-loss coupling, solved without expansion","key_machinery":"The load-bearing object is the exceptional activated mode: a candidate pair $(i,j)$ of reference eigenmodes whose projected $2\\times2$ block $H^{\\mathrm{EP}}_{ij}(\\lambda)$ becomes degenerate under the perturbation $\\lambda V$. The machinery is the discriminant condition of that block, which balances the reference detuning $\\Delta E_{ij}$, the projected diagonal shift $\\Delta D_{ij}$, and the biorthogonal off-diagonal product $A_{ij}B_{ji}=\\mathrm{Tr}[P_i V P_j V]$, with $P_i$ the biorthogonal projector onto mode $i$. Each pair yields EP branches $\\lambda^{(i,j)}_{c,\\sigma}$, and the physical threshold is their lower envelope, equivalent to the ratio form $\\lambda_c=\\Delta_{\\mathrm{act}}/G_{\\mathrm{act}}$. A companion validity criterion controls the neglected admixture of remote modes: their second-order Feshbach correction must be small compared with the active scale at the EP, which is automatic when $\\lambda_c$ itself is exponentially or algebraically small.","core_discovery":"The central claim is that a generic real-to-complex transition is governed by a single activated mode pair, rather than by the full spectrum. For $H(\\lambda)=H_0+\\lambda V$ with biorthogonal reference modes and real energies, each pair $(i,j)$ defines a $2\\times 2$ active block whose off-diagonal entries are the projected couplings $\\lambda V_{ij}$ and $\\lambda V_{ji}$. The exceptional-point condition is the vanishing of the discriminant, $(\\Delta E_{ij}+\\lambda\\Delta D_{ij})^2+4\\lambda^2 V_{ij}V_{ji}=0$, producing two pair-resolved threshold branches $\\lambda^{(i,j)}_{c,\\sigma}$. The physical real-to-complex threshold is the smallest positive branch over all pairs, $\\lambda_c=\\min_{i<j}\\lambda^{(i,j)}_c=\\lambda^{(p,q)}_c$, which equals the ratio $\\Delta_{\\mathrm{act}}/G_{\\mathrm{act}}$. The activated pair is selected by the smallest ratio of detuning to projected coupling, not by the smallest detuning. This is not a weak-coupling expansion: the active block's EP condition is solved exactly in $\\lambda$, and the paper verifies the prediction against full numerical diagonalization at thresholds of order one.","pith_inferences":["Beyond the paper, the lower-envelope principle suggests a practical design rule: spatial shaping of $V$ can switch the activated channel by changing $G_{\\mathrm{act}}$ without moving reference energies, allowing controlled switching of amplification channels in finite non-Hermitian devices.","Beyond the paper, the same ratio logic implies that threshold predictions for new platforms reduce to a finite enumeration over reference pairs, so the method could be exported to disordered or networked systems where a real-spectrum reference is available.","Beyond the paper, a natural stress test is to engineer a family whose first complex eigenvalue comes from a simultaneous three-branch coalescence; the paper's spectral-adjacency condition already flags such cases, but an exact higher-order activated-mode extension would be needed there."],"forward_implications":["For the open coupled Hatano-Nelson ladder, the closed-form threshold Eq. (11) holds for every system size and connects the conventional exponentially small skin-overlap law, $\\kappa L\\gg1$, to the algebraic band-edge law, $\\kappa L\\ll1$, through the same activated pair $(1,2)$.","Closing the ladder by boundary impurity bonds modifies only the reference data inside the same exceptional-point condition; as the reference approaches its own exceptional point, the threshold collapses linearly to zero because the active detuning vanishes as a square root while the projected activation strength diverges with the inverse square root.","In a Hermitian SSH chain with reflection-related local gain and loss, the activated channel can switch from the topological edge pair to a bulk pair as the defect position changes, with no change in any topological invariant; the edge-channel threshold scales as $\\lambda_c\\sim |w| r^{L-2m+2}$.","Since the threshold is a ratio of detuning to projected coupling, different microscopic mechanisms produce the same threshold law whenever their detuning and coupling exponents differ by the same amount, so scaling laws are organized by ratios rather than by individual exponents.","The whole construction requires no parity-time symmetry or any other symmetry: a real-spectrum reference and weak remote-mode admixture at onset suffice."],"supporting_citations":[{"why":"Supplies the critical non-Hermitian skin effect scenario and the conventional asymptotic law that the paper's closed-form threshold supersedes at all system sizes.","marker":"[29]"},{"why":"Provides the universal competitive spectral scaling context for coupled skin-effect chains that the activated-mode threshold reproduces and extends.","marker":"[32]"},{"why":"Gives the biorthogonal eigenmode formalism used to define the projected matrix elements and active blocks.","marker":"[39]"},{"why":"Defines the perturbative eigenvalue theory against which the non-perturbative EP condition is contrasted.","marker":"[56]"},{"why":"Introduces the Hatano-Nelson chain whose oppositely nonreciprocal coupling forms the reference ladder of the first two examples.","marker":"[57]"},{"why":"Defines the SSH model used as the Hermitian reference chain in the third example.","marker":"[64]"},{"why":"Is the closest precedent of a reflection-symmetric gain-loss defect pair in an SSH chain, repurposed here to study activated-channel switching.","marker":"[68]"}],"fun_headline_variants":["Activated mode pair sets non-Hermitian spectral threshold","Single mode pair triggers real-to-complex transition","Exceptional activation: one pair controls spectrum breakdown","Minimal mode pair decides when spectra go complex","One mode pair governs non-Hermitian transition threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the first real-to-complex transition is always a collision of exactly two reference branches, with all other modes contributing only a small correction; if a higher-order degeneracy or strong admixture from remote modes triggers the onset, the lower-envelope formula would not hold.","fun_headline_variants_meta":{"raw":{"variants":["Activated mode pair sets non-Hermitian spectral threshold","Single mode pair triggers real-to-complex transition","Exceptional activation: one pair controls spectrum breakdown","Minimal mode pair decides when spectra go complex","One mode pair governs non-Hermitian transition threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2993,"prompt_tokens":992,"completion_tokens":2001,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":1927}},"tokens_in":608,"tokens_out":2001,"duration_ms":13379,"temperature":1.0,"reasoning_tokens":1927,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:07:20.122392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact spectrum of a finite family $H(\\lambda)=H_0+\\lambda V$ engineered so that the first complex eigenvalue arises from a third-order exceptional point, and compare the numerical onset with the pairwise lower envelope $\\min_{i<j}\\lambda^{(i,j)}_c$; any mismatch at the onset falsifies the two-mode reduction. A broader check is to scan many sparse anti-Hermitian perturbations $V$ on a fixed real-spectrum $H_0$ and search for a case where the first collision involves non-adjacent or strongly admixed reference branches, whose exact threshold would test the boundary of the paper's validity criterion.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the critical non-Hermitian skin effect scenario and the conventional asymptotic law that the paper's closed-form threshold supersedes at all system sizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the universal competitive spectral scaling context for coupled skin-effect chains that the activated-mode threshold reproduces and extends."},{"cited_title":"Kato,Perturbation Theory for Linear Operators, Clas- sics in Mathematics (Springer, Berlin, 1995)","cited_arxiv_id":null,"evidence_quote":"Defines the perturbative eigenvalue theory against which the non-perturbative EP condition is contrasted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the closest precedent of a reflection-symmetric gain-loss defect pair in an SSH chain, repurposed here to study activated-channel switching."}],"review_version":1}