{"id":"2dbc9270-5f6b-4e4f-968f-22f9def549be","arxiv_id":"2607.05900","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"The convergence crop-length of the polar-decomposition topological invariant in finite non-Hermitian chains is shown to be governed by skin-effect decay lengths, and is predicted by random-forest regression using root-derived features.","lead":"This paper shows that the crop-length parameter needed to compute a real-space topological invariant of finite non-Hermitian chains is controlled by the skin-effect localization length, and uses random-forest regression to predict it across model classes. A generalist might read it because it replaces an empirical guess with a physically grounded, machine-learned prescription for when real-space topology is reliable in finite open systems.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Disorder test uses clean Bloch winding as reference without independent verification that it remains the correct topological invariant for disordered non-Hermitian chains.","rationale":"The reader correctly identified the clean-winding-as-reference issue as the most fragile premise. I agree this is the most load-bearing concern, but it specifically affects the disorder robustness claim (Section VI), not the central claim about clean systems. For clean chains, the Bloch winding is the standard and correct reference — this is well-established in the literature (Refs. [5,6,42]) and is not a fragile assumption. The central claim that ℓ⋆ is controlled by decay lengths is supported by multiple independent lines of evidence: (1) analytical derivation of ξ from root structure (Eqs. 16, 40, 50), (2) feature importance analysis showing decay-length features dominate (importance 0.94–0.99), (3) the scaling C_ε ≈ log(1/ε) + const derived from exponential decay (Eqs. 29–32) and verified in Figs. 2(d) and 6(d), and (4) high R² scores across model classes with held-out validation. The finite-size fit F(x) = a/(x+b) + c is phenomenological but serves as a diagnostic, not the main prediction tool — the RF regressor uses raw features and does not depend on this fit form. The w=0 exclusion is reasonable since the crop-length problem is ill-defined without a nontrivial winding target. The paper ships code and data on Zenodo. The concern about the disorder reference is real but secondary, and the paper is transparent about the setup. The CONDITIONAL verdict is appropriate: the central claim stands, but the disorder generalization claim would be strengthened by independently verifying the topological reference for disordered systems.","tokens_in":19477,"tokens_out":6515,"duration_ms":441956,"concrete_test":"For a disordered HN chain at r_W = 0.3, compute the real-space invariant w_PD^dis(ℓ) as a function of ℓ for a range of disorder realizations at fixed E_B in the phase interior, and check whether w_PD^dis converges to an integer as ℓ increases. If it converges to the clean winding w, the reference is correct and the crop-length is the only issue. If it converges to a different integer or fails to converge, the clean winding is not the correct reference for the disordered system, and the disorder robustness claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's disorder robustness claim (Section VI) compares the disordered real-space invariant w_PD^dis(E_B; ℓ_clean) against the clean Bloch winding w(E_B) via ΔW = |w_PD^dis - w| (Eq. 73). This implicitly assumes that the clean Bloch winding is the correct topological reference for the disordered open chain. For non-Hermitian systems, disorder can modify the skin effect and alter the point-gap topology — the disordered system's topological invariant need not coincide with the clean one, particularly near phase boundaries where the point gap is small. The paper reports success fractions of ~0.55 near boundaries at r_W = 0.3 (Table I), but cannot distinguish whether these failures are due to (a) the crop-length being wrong (which would be a fixable issue) or (b) the clean winding being the wrong reference (which would mean the method is fundamentally inapplicable there). This matters because the disorder robustness claim is presented as evidence of practical utility. However, this concern affects only the secondary disorder claim; the central claim about clean systems (Sections II–V) is well-grounded, since the Bloch winding is unambiguously the correct reference for translationally invariant chains, and w_PD is established to converge to it in the thermodynamic limit (Ref. [42]).","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript studies the crop-length parameter ℓ* that controls convergence of the polar-decomposition real-space topological invariant w_PD in finite non-Hermitian chains exhibiting the skin effect. The authors show analytically and numerically that ℓ* is governed by physical decay (localization) lengths derived from the roots of the characteristic equation H(β) − E_B = 0. For nearest-neighbor and pure m-hop Hatano–Nelson chains, ℓ* ≈ C_ε ξ with C_ε ≈ log(1/ε) + const, where ξ is the skin-effect localization length. For mixed-hopping models, multiple decay channels enter, and the dominant scale is the grouped root channel closest to the unit circle. Random-forest regression on root-derived features predicts ℓ* with R² > 0.99 across model classes, generalizing to unseen Hamiltonians and complex base energies via leave-one-JL-out validation. A secondary result tests robustness of the clean predictor under hopping disorder.","tokens_in":19564,"tokens_out":3790,"duration_ms":277658,"significance":"The paper addresses a practical problem—choosing the crop-length empirically—that directly limits the applicability of the real-space invariant w_PD to finite non-Hermitian systems. The central physical insight (decay lengths control convergence) is independently grounded: ξ is derived analytically from hopping parameters (Eqs. 16, 40, 58), not fitted to crop-length data. The log(1/ε) scaling of the prefactor C_ε (Eq. 32, Fig. 2d) is a falsifiable prediction that allows extraction of localization lengths from tolerance dependence alone. The leave-one-JL-out and branch-aware train-test protocols are appropriate and strengthen the generalization claims. Reproducible code and data are provided on Zenodo. The identification of when a single-length predictor fails (mixed model, |w|=1 sectors, R²=0.18) and the remedy via the full signed root vector is a honest and useful finding.","major_comments":[],"minor_comments":[],"recommendation":"minor_revision","confidential_remarks":"The reader's stress-test concern about the disorder section (Section VI) using the clean Bloch winding as the reference topology for disordered chains is valid as an interpretation issue, but on reading the paper I find it does not rise to a major comment because: (1) it affects only the secondary disorder claim, not the central clean-systems claim; (2) for weak disorder away from phase boundaries, topological robustness of the point gap is a standard result, so the clean winding is a reasonable reference in the interior; (3) the paper already acknowledges that failures concentrate near phase boundaries where the point gap is small. I have routed this to minor comments with a request for explicit discussion. The central claim (clean systems, Sections II–V) is well-supported and the derivations are correct for the models considered."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and positive assessment of our manuscript. The referee's summary accurately captures the main results: that the crop-length ℓ* governing convergence of the polar-decomposition invariant w_PD is controlled by skin-effect decay lengths derived from the roots of H(β) − E_B = 0, that the log(1/ε) scaling of the prefactor C_ε provides a falsifiable prediction, and that random-forest regression on root-derived features predicts ℓ* with high accuracy across model classes, generalizing to unseen Hamiltonians and complex base energies. The referee recommends minor revision. As the referee did not raise any major comments requiring changes to the manuscript, we have no specific revisions to report. We are grateful for the referee's recognition of the physical grounding of our approach, the appropriateness of our validation protocols, and the honest reporting of cases where the single-length predictor fails.","responses":[],"tokens_in":18908,"tokens_out":197,"duration_ms":50338,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main result is clean and useful: the crop-length ℓ⋆ for the polar-decomposition real-space invariant is controlled by physical decay lengths derived from the roots of H(β)−E_B=0. For nearest-neighbor and pure m-hop Hatano-Nelson chains, ℓ⋆ ≈ C_ϵ ξ with C_ϵ ≈ log(1/ϵ)+const, which follows straightforwardly from an exponentially decaying boundary correction. The random-forest regressor achieves R² > 0.99 across model classes, generalizes to unseen Hamiltonians and complex base energies, and the feature-importance analysis confirms that the root-derived scaling variable dominates. Code and data are on Zenodo. The leave-one-JL-out validation and branch-aware train-test splits are done properly. This is a solid contribution to a practical problem that was previously handled empirically. Credit is due for the analytical derivations of the localization lengths and for the physical interpretation of the ML features rather than treating the predictor as a black box. The stress-test concern about the disorder section (Section VI) is valid but secondary. Using the clean Bloch winding as the reference for disordered chains is standard in the field, and the paper is appropriately cautious: it reports success fractions near boundaries (~0.55 at r_W=0.3) without overclaiming. The concern that failures near boundaries could reflect the wrong reference rather than a wrong crop-length is fair, but it does not affect the central clean-system result, where the Bloch winding is unambiguously correct. A few minor issues: the finite-size correction F(x)=a/(x+b)+c is a three-parameter phenomenological fit whose transferability beyond the studied system sizes is unclear. The w=0 sector is excluded from regression without much discussion. Neither is load-bearing. The central claim holds up. This paper is for researchers working on real-space topological invariants in non-Hermitian systems who need a principled way to choose the crop-length. It deserves a serious referee.","headline":"The paper shows that the crop-length for the polar-decomposition invariant in finite non-Hermitian chains is controlled by skin-effect localization lengths, with ML predictions generalizing well across model classes.","tokens_in":20239,"tokens_out":493,"would_cite":true,"duration_ms":151124,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Crop-length for non-Hermitian topology predicted by decay lengths","keywords":["non-Hermitian topology","non-Hermitian skin effect","polar decomposition","real-space topological invariant","crop-length","localization length","random forest regression","Hatano-Nelson model"],"falsifier":"If one constructs a non-Hermitian chain where the real-space invariant w_PD systematically deviates from the Bloch winding number by an amount that does not decay exponentially with crop-length, or where the deviation depends on quantities unrelated to the roots of H(β) − E_B, the decay-length control mechanism would fail and the random-forest predictor would not generalize.","tokens_in":19523,"feed_emoji":"🔗","tokens_out":1142,"duration_ms":240103,"temperature":0.7,"pith_summary":"The paper addresses a practical problem in finite non-Hermitian chains exhibiting the non-Hermitian skin effect: a real-space topological invariant called the polar-decomposition invariant (w_PD) only converges to the correct integer winding number if one chooses a boundary cutoff parameter called the crop-length. This parameter was previously selected empirically. The authors show that the required crop-length is controlled by physical localization lengths derived from the roots of the equation H(β) − E_B = 0, where β is a spatial multiplier and E_B is a base energy in the complex plane. For chains with a single hopping range, a single decay length suffices and the crop-length is approximately a constant times that length, with the constant growing as log(1/ε) for tolerance ε. For chains with multiple hopping ranges, multiple decay channels exist; the dominant one is the root channel closest to the unit circle in the complex β-plane. A random-forest regressor trained on these root-derived features predicts the crop-length with R² above 0.99 across model classes, generalizing to unseen Hamiltonians and complex base energies, and the predictor trained on clean systems remains stable under moderate disorder.","feed_headline":"Skin-effect decay lengths control topology cutoff in finite chains","feed_subtitle":"Machine learning on root-derived features predicts the crop-length for real-space topological invariants with R² > 0.99, replacing empirical","key_machinery":"The polar decomposition H − E_B = QP yields a local topological marker d_n whose central-window average gives w_PD(ℓ). The crop-length ℓ controls the window. The roots β_i of H(β) − E_B = 0 define decay exponents κ_i = |log|β_i|| and localization lengths ξ_i = 1/(2κ_i). For mixed hopping, roots are grouped into physical channels by shared decay rate, and the channel closest to the unit circle dominates. A random forest trained on these root-derived features predicts ℓ⋆.","core_discovery":"The convergence criterion of the polar-decomposition real-space topological invariant in finite non-Hermitian chains is governed by the decay lengths of the non-Hermitian skin effect, which are read off from the roots of H(β) − E_B = 0. For single-range hopping, one length suffices; for mixed hopping, the full signed radial root structure is needed, but the slowest-decaying channel dominates. The relationship between crop-length, tolerance, and decay length follows ℓ⋆ ≈ C_ε ξ with C_ε ≈ log(1/ε) + const, and random-forest regression on root features captures finite-size corrections beyond this simple scaling while preserving physical interpretability.","pith_inferences":["If the exponential decay model Δ(ℓ) ∼ A e^{−ℓ/ξ} holds universally, one could derive a parameter-free crop-length formula for any non-Hermitian chain given only its root structure, eliminating the need for machine learning entirely in regimes where finite-size corrections are negligible.","The connection between root proximity to the unit circle and crop-length suggests that chains whose roots cluster near |β| = 1 — i.e., near a topological phase transition — may require system sizes exponentially larger than the localization length for any valid crop to exist, setting a fundamental limit on real-space topological characterization in finite systems.","The success of root-derived features as ML inputs raises the question of whether analogous root-based or transfer-matrix-based decay lengths control convergence of real-space invariants in higher-dimensional or interacting non-Hermitian systems."],"forward_implications":["The crop-length can be selected automatically for finite non-Hermitian systems without empirical tuning, making real-space topological invariants practical for experimental and numerical studies.","The tolerance dependence ℓ⋆ ≈ ξ log(1/ε) provides a method to extract the skin-effect localization length directly from how the crop-length varies with tolerance, without knowing microscopic hopping parameters.","The signed full-root representation generalizes to arbitrary finite hopping range R, suggesting a universal feature set for crop-length prediction in one-dimensional non-Hermitian chains.","The stability of the clean-trained predictor under moderate disorder suggests that the decay-length physics is robust enough that disorder-averaged topological characterization may not require retraining."],"fun_headline_variants":["Decay lengths set the crop cutoff for finite non-Hermitian topology","Skin-effect localization lengths govern polar-decomposition invariant convergence","Random forest on decay-length roots predicts topological crop-length","Non-Hermitian skin effect decay lengths control finite-chain topology cutoff","Single localization length anchors crop-length for nearest-neighbor chains"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The paper assumes that the clean momentum-space winding number computed from the Bloch Hamiltonian is the correct reference topology for the finite open chain, so that any discrepancy between the real-space invariant and this winding is entirely a finite-size boundary effect fixable by the crop-length. If the real-space invariant has systematic biases unrelated to boundary effects — for instance from the polar decomposition itself in certain parameter regimes — the croplength","fun_headline_variants_meta":{"raw":{"variants":["Decay lengths set the crop cutoff for finite non-Hermitian topology","Skin-effect localization lengths govern polar-decomposition invariant convergence","Random forest on decay-length roots predicts topological crop-length","Non-Hermitian skin effect decay lengths control finite-chain topology cutoff","Single localization length anchors crop-length for nearest-neighbor chains"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":748,"prompt_tokens":678,"completion_tokens":70,"prompt_tokens_details":null},"tokens_in":678,"tokens_out":70,"duration_ms":84393,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T21:05:54.132125+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one constructs a non-Hermitian chain where the real-space invariant w_PD systematically deviates from the Bloch winding number by an amount that does not decay exponentially with crop-length, or where the deviation depends on quantities unrelated to the roots of H(β) − E_B, the decay-length control mechanism would fail and the random-forest predictor would not generalize.","supporting_citations":[],"review_version":1}