{"id":"d99c5510-4c17-4b77-a32c-4b996d6df236","arxiv_id":"2411.12577","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A measured response matrix at one driving frequency is algebraically continued into the complex-frequency plane, yielding a fingerprint that identifies non-Hermitian skin modes and complex-energy quasiparticle peaks.","lead":"The paper proposes a tool called the complex frequency fingerprint: measure how a driven lattice responds at each site, then invert the response matrix to read out the system's complex energy spectrum and a special kind of Green's function. The authors argue this can reliably detect the non-Hermitian skin effect and reveal quasiparticle features that ordinary real-frequency measurements miss.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Many-body CFF freezes the self-energy at ω0, so its complex-ω_c peaks are not necessarily quasiparticle poles of the interacting Green's function; Fig. 3's interpretation is unsupported.","rationale":"The paper's non-Hermitian CFF core is supported: under the stated dissipative linear-response assumptions, χ_∞ = G(ω0) and Eq. (8) follows from the algebraic identity (ω_c - ω0) + G(ω0)^{-1} = ω_c - H, and the numerical checks in Figs. 1, 2, and 6 are consistent with that argument. I do not object to that part. The many-body extension, however, has a more fundamental problem than the missing measurement protocol emphasized by the Reader. Eq. (13)/(S21) evaluates the self-energy at the real drive frequency ω0 and then scans ω_c in the resolvent, producing the resolvent of the fixed matrix H_eff(ω0) rather than the analytic continuation of the true interacting Green's function G^R(ω_c) = [ω_c - H_eff(ω_c)]^{-1}. The paper labels the resulting complex-plane peaks as quasiparticles in Fig. 3, but quasiparticle poles are defined by the self-consistent condition det(ω_c - H_eff(ω_c)) = 0. Since Σ_↓↓(ω) in Eq. (S27) has strong frequency dependence, the two fixed-ω0 peaks may not correspond to any actual pole of the retarded Green's function. This is a correctness risk in the central many-body claim, and it would not be cured by a better measurement protocol. The proposed numerical overlay of true poles onto Fig. 3 would settle the issue directly: if the peaks match, the frozen-self-energy shortcut is harmless in that regime; if not, the many-body section needs substantial reframing. The Reader's CONDITIONAL verdict remains appropriate, so I recommend no change to the verdict.","tokens_in":19080,"tokens_out":16450,"duration_ms":168933,"concrete_test":"For the model in Eq. (15), compute the true quasiparticle poles of the retarded Green's function by solving det(ω_c + iη - H_S - Σ_S(ω_c + iη) - H_MF) = 0 with the same self-consistent occupations, and overlay these solutions onto the CFF landscape ln |Tr G^CFF_S(ω_c, 0)| plotted in Fig. 3. If the two peaks in Fig. 3 coincide with the true pole positions, the frozen-self-energy construction is a valid quasiparticle diagnostic for this model; if the far-axis peak does not correspond to any true pole, that peak is an artifact of fixing ω0 = 0, and the claim that the CFF reveals quasiparticle peaks across the complex plane must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (13) of the main text and Eq. (S21) of the Supplementary Material define G^CFF_S(ω_c,ω0) = [ω_c + iη - H_S - Σ_S(ω0 + iη)]^{-1}. The self-energy is evaluated at the fixed real frequency ω0, while ω_c appears only in the resolvent denominator. The actual complex-frequency retarded Green's function from Eq. (11) is G^R_S(ω_c) = [ω_c + iη - H_S - Σ_S(ω_c + iη)]^{-1}. Poles of the former are eigenvalues of the fixed matrix H_eff(ω0), whereas poles of the latter solve det(ω_c - H_eff(ω_c)) = 0. These two sets coincide only when the self-energy is frequency-independent or when ω0 happens to be a self-consistent pole. Figure 3 and the surrounding text call the fixed-ω0 = 0 peaks \"quasi-particle peaks\" and use them to infer interaction effects. In the model of Eq. (15), Σ_↓↓(ω) is strongly frequency-dependent (Eq. S27), so this identification is not guaranteed and can be wrong, especially for the peak located far from the real axis. This is a correctness issue, not merely an experimental-accessibility issue: it would survive even if every matrix element of G^R(ω0) were perfectly measurable. The complete-quasiparticle-information claim is therefore unsupported as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the complex frequency fingerprint (CFF), a protocol in which the full time-dependent response matrix of a driven-dissipative system is measured under sequential harmonic drives and then inverted to form G_{ω0}(ω_c; t) = 1 / ((ω_c − ω0) + [χ_{ω0}(t)]^{-1}). For non-Hermitian systems governed by Eq. (1), the paper shows that this object converges to 1/(ω_c − H_nH) as t → ∞, and uses this to access complex-frequency Green's functions, complex spectra, and non-Hermitian skin eigenstates, including a distinction between Bloch and non-Bloch responses. The paper then generalizes the construction to quantum many-body systems by defining G_CFF_S(ω_c, ω0) = 1/(ω_c − H_eff(ω0)), where the self-energy is evaluated at a fixed real frequency ω0, and illustrates the idea on a mean-field two-level model coupled to a thermal bath, claiming that peaks of this object in the complex plane constitute complete quasiparticle information.","tokens_in":19366,"tokens_out":5203,"duration_ms":55913,"significance":"The non-Hermitian part of the paper is a clean and potentially useful contribution: the derivation of the convergence in Eq. (8) from Eq. (S16) is correct under the stated dissipative assumption, and the numerical demonstrations in Figs. 1, 2, and 6 support the claim that the CFF provides a steady-state route to complex-frequency Green's functions, spectra, and eigenstates, including the non-Hermitian skin effect. The CFF is explicitly a measurement reconstruction rather than a new theoretical object, which is a strength from an experimental perspective. However, the many-body extension, which is a central advertised novelty, currently overclaims. The object in Eq. (13) freezes the self-energy at a real frequency and therefore its poles are not generally the quasiparticle poles of the interacting Green's function. Because this issue affects the core interpretation of Fig. 3 and the 'complete quasiparticle information' claim, the manuscript requires substantial revision before its main claims can be accepted.","major_comments":[{"comment":"The definition G_CFF_S(ω_c, ω0) = 1/(ω_c + iη − H_S − Σ_S(ω0 + iη)) evaluates the self-energy at a fixed real frequency ω0, so that ω_c appears only in the resolvent denominator. As a result, the poles of G_CFF_S are the eigenvalues of the fixed matrix H_eff(ω0), while the poles of the true retarded Green's function in Eq. (11) solve det[ω_c − H_S − Σ_S(ω_c + iη)] = 0. These two sets of poles coincide only when the self-energy is frequency-independent or when ω0 happens to satisfy a self-consistent pole condition. In the example of Eq. (15), the self-energy Σ_↓↓(ω) is strongly frequency dependent, as shown explicitly in Eq. (S27). Therefore the claim that Eq. (13) contains 'complete quasi-particle information' across the complex frequency plane is not supported by the stated equations and requires either a proof of the pole correspondence under well-defined conditions or a substantial reframing of what G_CFF_S actually measures.","section":"Eq. (13) and Supplementary Eq. (S21)"},{"comment":"Because of the issue in Eq. (13), the identification of the two peaks in Fig. 3 as 'quasiparticle peaks' of the interacting system is not justified. The peaks are poles of the fixed effective Hamiltonian H_eff(ω0 = 0), not necessarily poles of the interacting Green's function G^R_S(ω_c). In a model with strong frequency dependence of the self-energy, a pole of the fixed matrix can be located arbitrarily far from the true quasiparticle pole on the complex plane. The statement that the two peaks 'reveal two distinct quasi-particle peaks' and the conclusion that their repulsion provides an intuitive picture of many-body interactions are therefore unsupported without an additional argument establishing the physical meaning of the fixed-ω0 poles.","section":"Fig. 3 and surrounding text"},{"comment":"The claimed experimental accessibility of the many-body CFF is not established. Supplementary Section V states only that one should 'measure all matrix elements of the Green's function' and then defines G_CFF_ω0(ω_c) by inverting G^R_S(ω0). For an interacting quantum many-body system, it is not shown how the full single-particle Green's function matrix in a fixed real-space basis can be measured with the required completeness, including off-diagonal elements and phase information. Without a concrete protocol analogous to Steps 1–3 of the non-Hermitian case, the assertion that the double-frequency Green's function is 'a physical observable' remains a postulate rather than a demonstrated property.","section":"Supplementary Section V and Eq. (S20)"},{"comment":"The construction of the CFF requires the inverse of the measured response matrix χ_{ω0}(t). The paper does not discuss when this inverse exists. At finite times, χ_{ω0}(t) may be singular for certain parameter values and times, and in the long-time limit invertibility requires that ω0 is not an eigenvalue of H_nH. If ω0 lies in the spectrum, G(ω0) is not invertible and Eq. (8) is not well defined without a regularization or generalized inverse. This condition should be stated and its practical consequences, such as which driving frequencies are admissible, should be addressed.","section":"Eqs. (7) and (8)"}],"minor_comments":[{"comment":"The notation [χ_{ω0}(t)]_{j1} is slightly ambiguous because the second index labels the drive site while the first labels the measured site; a more explicit notation such as [χ_{ω0}(t)]_{j1} defined with a sentence identifying the column drive would improve readability.","section":"Eq. (5)"},{"comment":"The panels (c) and (d) lack axis labels and color-bar labels; adding these would make the convergence demonstration easier to evaluate quantitatively.","section":"Fig. 1"},{"comment":"Substituting ω_c exactly equal to an eigenvalue E_n into the resolvent is formally singular; the numerical detection of eigenstates necessarily uses a finite-time or finite-η regularization. The text should state the regularization procedure used in the numerical examples.","section":"Eq. (20)"},{"comment":"The notation 'Sign[x ≥ 0] = 1 and Sign[x < 0] = 0' is unconventional; this is the Heaviside step function, and using θ(x) instead would avoid confusion with the usual sign function.","section":"Supplementary Eq. (S27)"},{"comment":"The terms 'Bloch response' and 'non-Bloch response' are defined only for the Green's function example; since the paper later argues about responses more broadly, a more general definition of what counts as a physical response would make the table more precise.","section":"Introduction and Table I"}],"recommendation":"major_revision","confidential_remarks":"The non-Hermitian CFF result appears sound and publishable after revision. The main risk is the many-body interpretation: the paper currently identifies poles of a fixed-ω0 effective Hamiltonian with quasiparticle poles, which is not generally correct. If the authors can either prove the correspondence under precise conditions or substantially scale back the completeness claim, the manuscript could become acceptable. I would also suggest the editor check whether the many-body experimental protocol is sufficiently developed for the claims made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core identity (χ(t)=G(ω0)-G(ω0)e^{-i(H-ω0)t}) is standard linear response, but the post-processing step—using the inverse response matrix to define G(ω_c;t)—is an elegant and, as far as I know, new construction. It gives a practical way to reconstruct the resolvent across the complex plane from a single real-frequency measurement after transients decay. The paper's distinction between Bloch and non-Bloch responses, and its argument that real-frequency data cannot detect NHSE, is clear and useful. The numerical demonstrations (1D skin modes, the 2D geometry-dependent skin effect, point gap bound states) back up the claims.\n\nThe soft spots are real but uneven. The invertibility of χ(t) is never addressed; the definition (7) just assumes it. That may be harmless in practice but should at least be flagged. The no-go theorem is close to tautological—if non-Bloch response is defined by nonzero spectral winding, real frequencies trivially have zero winding—though it does sharpen the problem statement.\n\nThe bigger problem is the many-body section. Equation (13) defines G_CFF(ω_c,ω0) with the self-energy frozen at ω0. That is not the interacting Green's function G_R(ω_c)=1/(ω_c-H-Σ(ω_c)). The peaks in Fig. 3 are eigenvalues of H_eff(ω0), not solutions to det(ω_c-H_eff(ω_c))=0. For the model in Eq. (15), Σ_↓↓(ω) is strongly frequency-dependent (Eq. S27), so the identification of the CFF peaks with quasiparticle poles is unsupported. The \"complete quasiparticle information\" claim is therefore wrong. This survives even if every matrix element were perfectly measurable. The many-body example could still be useful if reframed as \"here is a different object, with its own interesting structure,\" but as written it overclaims.\n\nWho is this for? Experimentalists working on NHSE in classical wave platforms will find the CFF scheme plausible and directly implementable. Theorists should be cautious about the many-body part until it is revised. The paper deserves a serious referee—the non-Hermitian core is solid and the CFF is a distinctive protocol—but the referee should require the many-body claims to be substantially pulled back or properly justified (e.g., by showing how scanning ω0 could reconstruct Σ(ω)).","headline":"The CFF protocol for non-Hermitian systems is a genuinely useful steady-state trick for accessing complex-frequency Green's functions from a real-frequency drive; the many-body extension overreaches because freezing the self-energy at ω0 replaces the actual quasiparticle poles with eigenvalues of H_eff(ω0).","tokens_in":19899,"tokens_out":3674,"would_cite":false,"duration_ms":35409,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single real-frequency drive exposes the entire complex Green's function after transients decay, including non-Hermitian spectra, skin eigenstates, and many-body quasiparticles.","keywords":["complex frequency fingerprint","complex frequency Green's function","non-Hermitian skin effect","non-Bloch response","spectral winding number","driven-dissipative systems","quasiparticle resolution","double-frequency Green's function"],"falsifier":"Set up the driven-dissipative lattice model of Eq. (2) with $\\gamma_1\\neq\\gamma_2$ (parameters of Fig. 1: $N=100$, $t_1=1.5$, $t_2=1$, $\\mu=0.3$, $\\lambda=-1$, $\\gamma_1=2$, $\\gamma_2=1$): drive only site $x_0=60$, assemble $\\chi_{\\omega_0}(t)$ at $\\omega_0=0$, and evaluate the CFF at the in-gap frequency $\\omega_2=1.3-1.35i$ at times $t=1,5,10$. If the matrix elements $[G_{\\omega_0}(\\omega_2;t)]_{xx_0}$ fail to approach the directly computed resolvent $1/(\\omega_2-H_{\\mathrm{nH}})$ as $t$ grows, or if the converged result shows no exponential spatial growth and no bulk difference from the periodic-boundary Green's function despite $\\nu(\\omega_2)\\neq 0$, the central convergence claim fails. Symmetrically, the same protocol on the $\\gamma_1=\\gamma_2$ nonreciprocal-without-skin-effect model must show decaying, boundary-insensitive CFF at every complex frequency; an apparent skin signature there would refute the diagnostic.","tokens_in":3130,"feed_emoji":"📡","tokens_out":3749,"duration_ms":147698,"temperature":0.7,"pith_summary":"The paper proposes the complex frequency fingerprint (CFF): drive the system harmonically at one real frequency $\\omega_0$, site by site, record the full response matrix $\\chi_{\\omega_0}(t)$, and form $G_{\\omega_0}(\\omega_c;t)=1/((\\omega_c-\\omega_0)+[\\chi_{\\omega_0}(t)]^{-1})$. The central claim is that as $t\\to\\infty$ this object converges to the complex-frequency Green's function $G(\\omega_c)=1/(\\omega_c-H_{\\mathrm{nH}})$, because the transient term in $\\chi_{\\omega_0}(t)$ decays whenever all eigenvalues of $H_{\\mathrm{nH}}$ have negative imaginary parts. This matters because it makes the non-Bloch response experimentally accessible: inside regions of nonzero spectral winding the open-boundary Green's function grows exponentially in one spatial direction and is boundary-sensitive, and the paper claims this is the unique steady-state signature that separates the non-Hermitian skin effect from ordinary nonreciprocity. The same construction extends to quantum many-body systems, where the double-frequency Green's function $1/(\\omega_c-H_S^{\\mathrm{eff}}(\\omega_0))$ is claimed to be a measurable object carrying complete single-particle quasiparticle information across the complex frequency plane.","feed_headline":"One drive frequency exposes the entire complex spectrum","feed_subtitle":"After transients decay, one response matrix becomes the complex Green's function and certifies the skin effect.","key_machinery":"The load-bearing object is the complex frequency fingerprint itself, Eq. (7): $G_{\\omega_0}(\\omega_c\\in\\mathbb{C};t)=1/((\\omega_c-\\omega_0)+[\\chi_{\\omega_0}(t)]^{-1})$, built from the measured response matrix $\\chi_{\\omega_0}(t)$ obtained by sequential single-site harmonic driving at real frequency $\\omega_0$. Its convergence is carried by the identity $\\chi_{\\omega_0}(t)=G(\\omega_0)-G(\\omega_0)e^{-i(H_{\\mathrm{nH}}-\\omega_0)t}$: the transient $e^{-i(H_{\\mathrm{nH}}-\\omega_0)t}$ decays because the dissipative Hamiltonian's eigenvalues have negative imaginary parts, leaving $G(\\omega_0)$ dominant and making the long-time limit independent of $\\omega_0$. Two companion notions organize the argument: the spectral winding number $\\nu(\\omega_c)=\\frac{1}{2\\pi i}\\oint dk\\,\\partial_k\\ln\\det[H_{\\mathrm{nH}}(k)-\\omega_c]$, which partitions the complex plane into Bloch and non-Bloch response regions, and the $\\beta$-root scaling of the Green's function, which turns a nonzero winding number into the exponential spatial growth $[G^{\\mathrm{OBC}}]_{i\\alpha,i_0\\beta}\\sim\\beta_3^{-(i_0-i)}$ that defines non-Bloch response. For many-body systems the same formula is transplanted onto the effective non-Hermitian Hamiltonian $H_S^{\\mathrm{eff}}(\\omega_0)=H_S-i\\eta+\\Sigma_S(\\omega_0+i\\eta)$, so that the double-frequency Green's function $1/(\\omega_c-H_S^{\\mathrm{eff}}(\\omega_0))$ is the complex-frequency resolvent of a frequency-dependent non-Hermitian problem.","core_discovery":"The paper's thesis is that complex-frequency Green's functions, not real-frequency ones, carry the unique physics of the non-Hermitian skin effect, and that these functions can be measured in steady state. Concretely, the CFF obeys $\\lim_{t\\to\\infty} G_{\\omega_0}(\\omega_c;t)=G(\\omega_c)=1/(\\omega_c-H_{\\mathrm{nH}})$, independent of the driving frequency $\\omega_0$, so one real-frequency experiment maps the resolvent over the entire complex plane. The paper proves a no-go complement: in any purely dissipative system all poles of the open-boundary Green's function lie below the real axis, so the spectral winding number $\\nu(\\omega_r)$ vanishes for real $\\omega_r$, and real-frequency Green's functions always belong to the Bloch response region — meaning ordinary nonreciprocal correlations, seen at real frequencies, cannot certify the skin effect. By contrast, at complex frequencies inside the point gap the open-boundary Green's function scales as $\\beta_3^{-(i_0-i)}$ for $i<i_0$ with $|\\beta_3|<1$, an exponential growth in real space that encodes the non-Bloch response and the skin eigenstates. For quantum many-body systems the analogous object, $G^{\\mathrm{CFF}}_S(\\omega_c,\\omega_0)=1/(\\omega_c-H_S^{\\mathrm{eff}}(\\omega_0))$ with $H_S^{\\mathrm{eff}}(\\omega_0)=H_S-i\\eta+\\Sigma_S(\\omega_0+i\\eta)$, is argued to be an observable containing all single-particle excitation information; in the paper's two-level fermion example it reveals two quasiparticle peaks in the complex plane where real-frequency spectroscopy shows one.","pith_inferences":["Editorial inference: the method's practical reach is set by the convergence time of the transient $e^{-i(H_{\\mathrm{nH}}-\\omega_0)t}$; in systems with eigenvalues whose imaginary parts are near zero, the readout time needed for the key limit to hold may exceed the dephasing or coherence time, so the scheme's range is tied to the dissipative gap of $H_{\\mathrm{nH}}$.","Editorial inference: because $G^{\\mathrm{CFF}}_S(\\omega_c,\\omega_0)$ is a double-frequency object, sweeping $\\omega_0$ amounts to a two-dimensional spectroscopy that traces the frequency dependence of the self-energy $\\Sigma_S(\\omega_0+i\\eta)$; this suggests a concrete program — measuring the CFF at many real driving frequencies — that goes beyond the single-$\\omega_0$ examples shown.","Editorial inference: the many-body claim would be fully realized by an explicit experimental protocol for measuring all matrix elements of the retarded Green's function of an interacting quantum system; the paper provides that protocol for driven-dissipative and classical-wave systems, but for the interacting case it states the measurability without giving the step-by-step procedure.","Editorial inference: the CFF should also detect spectral degeneracies such as exceptional points and point-gap bound states without extra apparatus, since the complex-frequency DOS divergence and the eigenstate reconstruction in Supplementary Section IV are generic; testing this on a two-band model with an exceptional point would be a direct extension."],"forward_implications":["The non-Hermitian skin effect becomes certifiable by a steady-state measurement: at complex frequencies with nonzero spectral winding the open-boundary CFF shows exponential growth in one spatial direction and obeys $|G^{\\mathrm{OBC}}|\\neq|G^{\\mathrm{PBC}}|$ in the bulk, while systems without the skin effect show decaying, boundary-insensitive responses at all complex frequencies.","Non-Hermitian spectra are read out directly from the complex-frequency density of states $D_{\\omega_0}(\\omega_c;t)=-\\frac{1}{N\\pi}\\mathrm{Im}\\,\\mathrm{Tr}\\,G_{\\omega_0}(\\omega_c;t)$, which diverges as $\\omega_c$ approaches any eigenvalue $E_n$, including eigenvalues far from the real axis.","Near an eigenvalue, the CFF matrix elements reconstruct the right and left eigenstates ($G_{ij_0}\\propto\\langle i|\\psi^R_n\\rangle$, $G_{j_0 i}\\propto\\langle\\psi^L_n|i\\rangle$), so skin-mode profiles — including the geometry-dependent 2D skin effect — are experimentally recoverable.","In quantum many-body systems, the CFF resolves quasiparticle peaks across the complex frequency plane, remains meaningful even when quasiparticle weight $Z\\to 0$, and in the two-level fermion model shows an intuitive repulsion between quasiparticles as the chemical potential rises.","Because the linear dissipative equation (1) also governs classical wave systems, the protocol transfers directly to photonic, acoustic, electric-circuit, and mechanical platforms where amplitude and phase of the field are measurable."],"supporting_citations":[{"why":"The paper's own supplementary material: derives Eq. (1) from Lindblad and gives the identity chi(t)=G(omega0)-G(omega0)e^{-i(HnH-omega0)t} that underlies Eq. (8).","marker":"[102]"},{"why":"Defines the spectral winding number used in the paper's classification of Bloch versus non-Bloch response regions.","marker":"[30]"},{"why":"Provides the biorthogonal expansion G(omega_c)=sum_m |psiR_m><psiL_m|/(omega_c-E_m) used to reconstruct eigenstates from CFF matrix elements.","marker":"[15]"},{"why":"Defines the point-gap bound state that the CFF detects in Supplementary Section IV.","marker":"[113]"},{"why":"Coupled-mode equations showing that Eq. (1) also governs classical wave systems, supporting the experimental accessibility claim.","marker":"[103–107]"}],"fun_headline_variants":["One steady state drive maps the complex plane","Complex fingerprint reveals skin effect and quasiparticles","Single probe exposes entire complex frequency space","Complex Green's function from one real-frequency drive","Mapping all complex spectral responses with one drive"],"cache_read_input_tokens":22016,"weakest_assumption_plain":"The central premise is that the full $N\\times N$ response matrix can be measured by driving each site one at a time, that the system obeys the linear dissipative equation (1), and that $\\chi_{\\omega_0}(t)$ remains invertible at the readout times — and for the many-body extension, that every matrix element of the retarded Green's function is a directly measurable observable, for which the paper gives a protocol in classical-wave settings but not for interacting quantum systems.","fun_headline_variants_meta":{"raw":{"variants":["One steady state drive maps the complex plane","Complex fingerprint reveals skin effect and quasiparticles","Single probe exposes entire complex frequency space","Complex Green's function from one real-frequency drive","Mapping all complex spectral responses with one drive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000342,"raw_usage":{"total_tokens":1927,"prompt_tokens":1038,"completion_tokens":889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":822}},"tokens_in":654,"tokens_out":889,"duration_ms":9280,"temperature":1.0,"reasoning_tokens":822,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:23:13.814725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set up the driven-dissipative lattice model of Eq. (2) with $\\gamma_1\\neq\\gamma_2$ (parameters of Fig. 1: $N=100$, $t_1=1.5$, $t_2=1$, $\\mu=0.3$, $\\lambda=-1$, $\\gamma_1=2$, $\\gamma_2=1$): drive only site $x_0=60$, assemble $\\chi_{\\omega_0}(t)$ at $\\omega_0=0$, and evaluate the CFF at the in-gap frequency $\\omega_2=1.3-1.35i$ at times $t=1,5,10$. If the matrix elements $[G_{\\omega_0}(\\omega_2;t)]_{xx_0}$ fail to approach the directly computed resolvent $1/(\\omega_2-H_{\\mathrm{nH}})$ as $t$ grows, or if the converged result shows no exponential spatial growth and no bulk difference from the periodic-boundary Green's function despite $\\nu(\\omega_2)\\neq 0$, the central convergence claim fails. Symmetrically, the same protocol on the $\\gamma_1=\\gamma_2$ nonreciprocal-without-skin-effect model must show decaying, boundary-insensitive CFF at every complex frequency; an apparent skin signature there would refute the diagnostic.","supporting_citations":[{"cited_title":"Ghatak, M","cited_arxiv_id":null,"evidence_quote":"The paper's own supplementary material: derives Eq. (1) from Lindblad and gives the identity chi(t)=G(omega0)-G(omega0)e^{-i(HnH-omega0)t} that underlies Eq. (8)."},{"cited_title":"Haus and W","cited_arxiv_id":null,"evidence_quote":"Defines the point-gap bound state that the CFF detects in Supplementary Section IV."}],"review_version":1}