{"id":"343b3fbb-f5a8-4a18-abad-d261939cba5c","arxiv_id":"2607.12585","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Classical Krylov complexity is defined from phase-space Poisson structure and approximates quantum Krylov complexity until a Krylov-Ehrenfest time of order λ_K^{-1} log(1/ħ).","lead":"The authors define a classical Krylov complexity via the Lanczos algorithm on phase space, with Poisson brackets replacing commutators. This gives a practical early-time proxy for quantum complexity and a shell-by-shell tool for chaotic spin models.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limit already flagged by the Reader; the central semiclassical correspondence claim is coherent as stated.","rationale":"The Reader’s weakest_assumption is precisely the load-bearing condition required for the strongest_claim to hold. With only the abstract, that condition cannot be checked, nor can any concrete counter-example or missing bound be exhibited. The abstract’s logical structure is internally consistent: classical Krylov complexity is defined via symplectic geometry, the ħ\to0 limit is asserted to be smooth by standard phase-space QM, early-time agreement follows until the logarithmic Krylov-Ehrenfest scale, and microcanonical shells are used to resolve LMG’s saddle. No circularity or contradiction appears. Therefore the stress-test finds no additional concern that would alter the UNVERDICTED / LOW-confidence status. The concrete test above is the minimal analytic check that would settle the premise once equations are in hand; until then the Reader’s verdict stands unchanged.","tokens_in":2198,"tokens_out":565,"duration_ms":4755,"concrete_test":"Once the full text is available, re-derive the classical Lanczos coefficients a_n, b_n from the phase-space recursion (Poisson brackets + Liouville measure) and verify that they equal the ħ\to0 limit of the quantum coefficients order by order in ħ for the LMG Hamiltonian restricted to a single microcanonical shell away from the saddle; if the leading classical terms fail to match or if the first quantum correction grows faster than any power of ħ before n_*, the claimed smooth correspondence fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Reader already correctly isolates the load-bearing premise: that the ħ\to0 limit of the quantum Krylov/Lanczos framework is smooth and that the resulting classical construction (Poisson brackets + phase-space inner product) remains a faithful early-time proxy for quantum complexity up to the Krylov-Ehrenfest depth n∼n_*(ħ). From the abstract alone this premise is presented as following from “general methods of quantum mechanics in phase space,” with concrete support claimed for LMG and FP via microcanonical shells. No internal inconsistency is visible in the stated argument; the microcanonical resolution of LMG saddle instability by integrable structure away from the saddle is a modeling claim that cannot be audited without equations or figures. Because the full text is unavailable, no sharper technical fracture (e.g., a missing bound on the remainder of the ħ expansion, or an uncontrolled contribution from residual chaos in off-saddle shells) can be identified. The concern therefore remains exactly the one the Reader named, and does not yet rise to an independent objection that would move the verdict.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript develops a classical Lanczos algorithm that defines Krylov complexity from the symplectic structure of phase space, with Poisson brackets replacing commutators and phase-space integrals supplying the inner product. Using general methods of quantum mechanics in phase space, it claims that the ħ → 0 limit of the quantum Krylov/Lanczos framework passes smoothly into this classical construction. In systems with well-defined semiclassical limits, classical Krylov complexity is argued to approximate quantum Krylov complexity at early times, up to a Krylov-Ehrenfest depth n ∼ n_*(ħ) corresponding to t_* ∼ λ_K^{-1} log(1/ħ). Microcanonical (energy-shell) versions of both classical and quantum Krylov complexity are introduced. The framework is applied to the Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) collective spin models; the abstract states that LMG’s early-time saddle-dominated scrambling is resolved, shell by shell, by the integrable structure of the Hamiltonian away from the instability.","tokens_in":2436,"tokens_out":1118,"duration_ms":15282,"significance":"If the claimed smooth semiclassical correspondence and early-time approximation hold with controlled errors, the work supplies a practical classical diagnostic of early-time chaotic dynamics and a fine-grained, energy-shell probe of complexity growth. The notions of Krylov-Ehrenfest depth/time and microcanonical Krylov complexity are potentially useful conceptual tools for collective spin systems that classicalize in the thermodynamic limit. The LMG application, if substantiated, would clarify how saddle-dominated scrambling coexists with integrable structure off the saddle. These strengths are conditional on the derivations and numerical evidence that the abstract only announces.","major_comments":[{"comment":"The central load-bearing claim—that the ħ → 0 limit of the quantum Krylov/Lanczos framework is smooth and that the resulting classical construction (Poisson brackets + phase-space inner product) remains a faithful early-time proxy up to n ∼ n_*(ħ)—is asserted via “general methods of quantum mechanics in phase space” but cannot be audited from the abstract alone. A controlled remainder estimate or explicit comparison of Lanczos coefficients (or complexity growth) as ħ → 0 is required for the correspondence to support the paper’s main conclusions.","section":null},{"comment":"The claimed scaling t_* ∼ λ_K^{-1} log(1/ħ) and the associated Krylov-Ehrenfest depth n_*(ħ) are stated as results. Without the derivation of how n_* is extracted from the classical/quantum Krylov chains, and without quantitative evidence that classical and quantum complexities track until that scale and diverge thereafter, the early-time approximation claim remains unverified.","section":null},{"comment":"For LMG, the abstract asserts that microcanonical Krylov complexity resolves the saddle instability because off-saddle shells are controlled by integrable structure, both at early and late times. This is a modeling claim that is load-bearing for the LMG application; it needs explicit shell-resolved classical and quantum data (and a clear definition of the microcanonical ensembles) to show that residual chaos or saddle contamination does not dominate the reported shells.","section":null},{"comment":"The FP application is said to feature spectral chaos for some couplings, yet the abstract does not indicate how classical vs. quantum microcanonical Krylov complexity distinguishes chaotic from non-chaotic regimes, nor what quantitative diagnostics (e.g., growth rates, late-time plateaus) are used. Without those comparisons the claim that classical Krylov complexity is a useful early-time characteristic of chaos cannot be assessed.","section":null}],"minor_comments":[{"comment":"The abstract is dense and packs several new definitions (classical Lanczos recursion, Krylov-Ehrenfest time/depth, microcanonical Krylov complexity) without a one-sentence roadmap of the paper’s section structure; a clearer outline sentence would help readers.","section":null},{"comment":"Notation for the classical inner product and the precise replacement of the quantum Liouvillian by the classical Liouville operator should be fixed early and used consistently once the full text is available.","section":null},{"comment":"When the full manuscript is supplied, figures comparing classical and quantum Krylov complexity (and shell-resolved versions) with explicit ħ or large-spin scaling would be essential; the abstract alone cannot convey error bars or the quality of the early-time match.","section":null}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review: the full text of arXiv:2607.12585 was not available. The stated program is coherent and the circularity risk appears low (the classical object is defined from symplectic structure rather than fitted to quantum data), but every load-bearing claim—smooth ħ → 0 limit, early-time fidelity up to n_*(ħ), and the LMG microcanonical resolution of the saddle—requires the derivations, bounds, and figures that are missing here. I recommend obtaining the full manuscript before any accept/reject decision; with only the abstract I cannot responsibly move beyond “uncertain.” Scope appears appropriate for cond-mat.stat-mech / quantum chaos venues if the technical claims hold."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing you need to know is that this is an abstract-only read. What they claim is a classical Krylov complexity built from the symplectic structure—Poisson brackets instead of commutators, phase-space integrals as the inner product—plus a smooth ħ\to0 limit of the usual quantum Lanczos setup, a named Krylov-Ehrenfest time t_* ~ λ_K^{-1} log(1/ħ), and microcanonical (shell-by-shell) versions applied to LMG and FP.\n\nWhat looks new and useful is the packaging: a classical Lanczos recursion that is meant to track early-time operator growth, the explicit Krylov-Ehrenfest depth where classical and quantum diverge, and the microcanonical resolution of LMG’s saddle instability by integrable structure away from the saddle. That last point is the most concrete payoff if it holds. The construction is presented as definitional rather than fitted, which is the right posture for this kind of diagnostic. Within quantum chaos and operator growth, a clean classical proxy that organizes early-time scrambling and gives shell-resolved control on collective spins is worth having.\n\nThe soft spot is exactly the load-bearing premise the reader flagged: that the ħ\to0 limit is smooth and that the classical object remains a faithful early-time proxy up to n_*(ħ) in the models they study. From the abstract alone you cannot check the remainder of the expansion, the quality of the early-time match, or whether off-saddle LMG shells are truly controlled by integrability rather than residual chaos. No equations, figures, or data are here, so soundness stays provisional. Nothing in the stated argument looks internally inconsistent; the stress-test did not find a sharper fracture.\n\nThis is for people who already work on Krylov complexity, Lanczos algorithms, and semiclassical chaos diagnostics—especially anyone who wants a classical counterpart or a shell-resolved tool for large-spin models. It is not a field-redefining result, but it is a methodological contribution that deserves a serious referee once the full text is available. I would send it to peer review rather than desk-reject; the claims are sharp enough and the framing is clean enough to warrant checking the math and the LMG/FP numerics. Bring it to reading group only after the PDF is out.","headline":"Abstract-only: classical phase-space Krylov via Poisson brackets looks coherent and useful for early-time chaos, but the ħ\to0 claim and LMG shell results cannot be audited yet.","tokens_in":3096,"tokens_out":576,"would_cite":false,"duration_ms":4795,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Classical Krylov complexity defined on phase space approximates quantum Krylov complexity at early times until a ħ-dependent Krylov-Ehrenfest depth.","keywords":["Krylov complexity","Lanczos algorithm","phase space","Poisson brackets","semiclassical limit","microcanonical complexity","Lipkin-Meshkov-Glick model","Feingold-Peres model"],"falsifier":"A direct numerical comparison of classical and quantum Krylov complexities in the LMG or FP model at small but finite ħ that shows divergence at a depth substantially different from the predicted n_*(ħ) ∼ log(1/ħ), or that shows the microcanonical classical complexity already failing to track the quantum one inside an integrable energy shell at early times.","tokens_in":3054,"feed_emoji":"🌀","tokens_out":950,"duration_ms":18238,"temperature":0.7,"pith_summary":"The paper constructs a classical Lanczos algorithm on phase space in which Poisson brackets replace quantum commutators and phase-space integrals supply the inner product, thereby defining a classical Krylov complexity. It shows that this object is recovered as the smooth ħ → 0 limit of the ordinary quantum Krylov framework. In systems that possess a well-defined semiclassical limit the classical complexity tracks its quantum counterpart at early times, remaining faithful until a characteristic Krylov-Ehrenfest depth n ∼ n_*(ħ) that corresponds to the time scale t_* ∼ λ_K^{-1} log(1/ħ). Microcanonical versions of both the classical and quantum complexities are introduced so that growth can be examined energy shell by energy shell. Applied to the LMG and FP collective-spin models, the construction shows that the early-time instability of the LMG model is confined to the saddle region; away from it the microcanonical complexity is controlled by the integrable structure of the Hamiltonian at both early and late times.","feed_headline":"Classical phase-space Krylov matches quantum until log(1/ħ)","feed_subtitle":"A Poisson-bracket Lanczos algorithm supplies an early-time proxy and resolves LMG scrambling shell by shell.","key_machinery":"The classical Lanczos algorithm on phase space (Poisson brackets in place of commutators, phase-space integrals as the inner product) is the central object; it is the ħ → 0 limit of the quantum Krylov construction and furnishes the early-time approximation together with the definition of the Krylov-Ehrenfest scale.","core_discovery":"In theories with well-defined semiclassical limits, classical Krylov complexity obtained from the phase-space Lanczos algorithm accurately approximates quantum Krylov complexity at early times, until a Krylov-Ehrenfest depth n ∼ n_*(ħ) that translates into the time scale t_* ∼ λ_K^{-1} log(1/ħ).","pith_inferences":["The same phase-space Lanczos construction can be applied to classical field theories with a well-defined Poisson structure, giving a purely classical diagnostic of operator growth before any quantization.","Extracting the Krylov-Ehrenfest scale from complexity growth and comparing it with the ordinary Ehrenfest time obtained from wave-packet spreading would test whether the two notions of semiclassical breakdown coincide.","Restricting spectral form factors or out-of-time-order correlators to the same microcanonical shells that show integrable Krylov complexity should likewise reveal integrable rather than chaotic signatures away from the LMG saddle."],"forward_implications":["Classical phase-space Krylov complexity can serve as a practical early-time proxy for quantum complexity growth in any system that classicalizes.","The Krylov-Ehrenfest time t_* ∼ λ_K^{-1} log(1/ħ) supplies a universal scale at which classical and quantum Krylov complexities must diverge in chaotic systems.","Microcanonical Krylov complexities diagnose, shell by shell, whether operator growth is chaotic or integrable.","In the LMG model the saddle-dominated scrambling is confined to a narrow spectral window; outside that window microcanonical complexity remains integrable-like at all times."],"fun_headline_variants":["Classical Krylov matches quantum until Krylov-Ehrenfest log(1/ħ)","Phase-space Poisson Lanczos tracks quantum Krylov early times","Shell-by-shell Krylov proxies chaos till n ~ log(1/ħ) depth","Microcanonical Krylov resolves LMG instability shell by shell","Classical phase-space complexity approximates until t_* scale"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"That the ħ → 0 limit of the quantum Krylov framework is smooth and that the resulting classical construction remains a faithful early-time proxy for quantum complexity in the models studied.","fun_headline_variants_meta":{"raw":{"variants":["Classical Krylov matches quantum until Krylov-Ehrenfest log(1/ħ)","Phase-space Poisson Lanczos tracks quantum Krylov early times","Shell-by-shell Krylov proxies chaos till n ~ log(1/ħ) depth","Microcanonical Krylov resolves LMG instability shell by shell","Classical phase-space complexity approximates until t_* scale"]},"model":"grok-4.5","effort":"low","cost_usd":0.007178,"raw_usage":{"total_tokens":1819,"prompt_tokens":886,"num_sources_used":0,"completion_tokens":96,"cost_in_usd_ticks":71780000,"prompt_tokens_details":{"text_tokens":886,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":837,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":886,"tokens_out":96,"duration_ms":6573,"temperature":1.0,"reasoning_tokens":837,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T04:56:14.179658+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A direct numerical comparison of classical and quantum Krylov complexities in the LMG or FP model at small but finite ħ that shows divergence at a depth substantially different from the predicted n_*(ħ) ∼ log(1/ħ), or that shows the microcanonical classical complexity already failing to track the quantum one inside an integrable energy shell at early times.","supporting_citations":[],"review_version":1}