{"id":"98674d3a-85a5-4b40-95b3-b3aedfe329c8","arxiv_id":"2610.21574","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Applying Krylov complexity to two-field inflation shows two squeezing channels with different onset times and oscillatory dynamics after horizon crossing.","lead":"This paper studies Krylov complexity in two-field inflation, solving coupled curvature and entropy perturbations to compute quantum information spreading. It finds that two squeezing channels develop at different times and exhibit distinct oscillatory behavior, providing a new probe of multi-field interactions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Off-diagonal pair-production coefficient A_12 is not computed or bounded; the central factorization assumption is unverified for the chosen parameters.","rationale":"The reader identified the subdominance of off-diagonal pair production as the weakest assumption, and I concur. This is precisely the condition that must hold for the factorized squeezed state (Eq. 26) to be accurate. The paper states this condition but provides no verification. While the paper acknowledges it as a scope condition, the lack of any numerical bound leaves the central construction unvalidated. The proposed concrete test is directly implementable from the authors' existing mode-function data and would settle whether the assumption is justified. The reader's verdict of CONDITIONAL is appropriate, with the condition being the verification of A_12 subdominance. I see no other concern of comparable weight: the numerical methodology, while lacking convergence checks, would be secondary if the factorization itself is invalid; the parameter sensitivity is a quantitative robustness issue, not a foundational one. Therefore I agree with the reader and recommend no change to the verdict.","tokens_in":21032,"tokens_out":2037,"duration_ms":22047,"concrete_test":"Compute the off-diagonal coefficient A_12(τ) from the solved mode functions using the same expression as A_11 and A_22 (Eqs. 17, 19) but with index 1 replaced by 2 in the first factor and vice versa (i.e., using ξ_1k and ψ_1k in one factor and ξ_2k and ψ_2k in the other). Evaluate |A_12|(τ) / sqrt(|A_11(τ)||A_22(τ)|) over the full evolution interval τ_i to τ_f. If this ratio exceeds, e.g., 0.1 at any time after horizon crossing, the diagonal assumption is quantitatively unsupportable and the factorized state should not be trusted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central analysis assumes the two squeezing channels are independent (factorized squeezed state, Eq. 26), which requires the off-diagonal pair-production coefficient A_12 to be subdominant relative to the diagonal channels. The authors explicitly state this condition (Sec. II: 'The two-channel description is adopted when the off-diagonal contribution remains subdominant') but never compute A_12 from the solved mode functions or compare its magnitude to A_11 and A_22. Without this check, the validity of the factorized squeezed state and the resulting squeezing parameters r_1k, r_2k is unknown. If A_12 is comparable to the diagonal coefficients, the evolution equations (30–33) derived under the factorized ansatz are incorrect, and the subsequent Krylov complexity, entropy, and Lanczos results are unsupported. This is the most load-bearing concern because it underpins the entire two-channel description; if the assumption fails, the qualitative and quantitative claims about distinct onset times and oscillatory dynamics may be artifacts of an invalid approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The manuscript studies Krylov-space diagnostics for the coupled curvature and entropy perturbations of two-field inflation. Starting from a quadratic action, the authors solve the coupled mode equations numerically with Bunch-Davies-type initial conditions, construct the quadratic Hamiltonian, and represent the Gaussian state as a product of two two-mode squeezing operators acting on the two sectors (Eqs. 24-26). They derive first-order evolution equations for the two squeezing strengths and angles (Eqs. 30-34), then compute the state-space spread complexity K_k = sinh^2 r_1 + sinh^2 r_2 (Eq. 53), the Krylov entropy (Eqs. 59-62), and channel-resolved hopping coefficients b_n^(1), b_m^(2) (Eq. 66). The main numerical claim is that the two squeezing channels have different onset times after horizon exit, that the second channel dominates the intermediate evolution, and that both channel complexities oscillate across horizon crossing with the oscillations amplified by the nonlinear relation K_ik = sinh^2 r_ik. A single-field limit is discussed as a consistency check in Sec. IV.B.","tokens_in":21446,"tokens_out":6912,"duration_ms":327776,"significance":"If the factorized two-channel description is valid for the chosen parameters, the paper provides a clean and useful extension of the single-field Krylov-complexity formalism to a multi-field setting. The derivation from the quadratic action through the mode functions to the squeezing equations is transparent, and the closed-form results K_k = sinh^2 r_1 + sinh^2 r_2 and S_K = sum_i [(1+K_ik) ln(1+K_ik) - K_ik ln K_ik] are simple and testable. The authors also correctly note that their two squeezing eigenchannels are not separately the curvature and entropy fields, which avoids a potentially serious misinterpretation. However, the central factorization ansatz (Eq. 26) is adopted under an explicitly stated but unverified dominance assumption for the off-diagonal pair-production coefficient A_12, and the Lanczos-coefficient part of the paper lacks a derivation. For these reasons the qualitative claims about channel-dependent onset times and oscillatory Krylov behavior are currently conditional on an unquantified approximation.","major_comments":[{"comment":"The factorized squeezed state (Eq. 26) and the resulting evolution equations (30)-(34) are valid only when the off-diagonal pair-production coefficient A_12 is subdominant. The authors state this condition (Sec. II: \"The two-channel description is adopted when the off-diagonal contribution remains subdominant\") and later state that the validity of the reduction \"is checked by the relative size of the off-diagonal Hamiltonian coefficients,\" but no such check is reported anywhere in the manuscript. Because A_12 can be computed from the same mode functions via Eq. (15), this is a straightforward quantitative task. Please report |A_12| relative to sqrt(|A_11 A_22|), or a comparable measure, over the full integration range, and show explicitly that the off-diagonal channel is subdominant for the parameters H=1, epsilon=0.002, eta_SR=0.01, mu=0.5H, lambda=0.1, k=0.01. Without this check, the subsequent claims in Eqs. (53), (60), and (66) are supported only conditionally.","section":"Sec. II, Eqs. (14)-(20), and Sec. III, Eq. (26)","authors_acknowledge":"The two-channel description is adopted when the off-diagonal contribution remains subdominant.","authors_acknowledge_p":0.64},{"comment":"The quantities b_n^(1) = |A_11| sqrt(n)/2 and b_m^(2) = |A_22| sqrt(m)/2 are introduced as \"channel-resolved Lanczos coefficients,\" but no derivation from the Lanczos recursion is given. Moreover, the pair-operator matrix elements written in Eqs. (64)-(65) are n and n+1, not sqrt(n) and sqrt(n+1), so Eq. (66) does not follow from those matrix elements in an obvious way. If these coefficients are intended as hopping amplitudes in a two-dimensional occupation lattice, the definition should be stated explicitly and connected to the Hamiltonian in the occupied-sector basis; if they are intended as actual Lanczos coefficients, the Lanczos recursion should be applied to the two-channel state and compared with Eq. (66) in the one-channel limit. As written, the quantitative Lanczos results shown in Fig. 6 are unsupported.","section":"Sec. V.B, Eq. (66)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The numerical procedure is not specified sufficiently for reproducibility. The paper states that two linearly independent initial modes are considered with Bunch-Davies initial conditions, but it does not give the explicit initial conditions for the four mode functions, the form of a(tau) used in the quasi-de Sitter background, the differential equations actually solved, the numerical integrator, or error tolerances. The later statement that the angle equations are replaced by equivalent complex squeezing variables is also not accompanied by the explicit equations used. Since the central qualitative claims, especially the different onset times of r_1 and r_2, rest entirely on the numerical integration, please provide the complete numerical setup and a convergence or accuracy test.","section":"Sec. II, numerical setup","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The discussion of A_12 contains a repetitive sentence that should be rewritten for clarity: \"The off-diagonal coefficient A_12 characterizes pair production involving both channels and the present analysis focuses on...\" should be split into a definition and a statement of the adopted approximation.","section":"Sec. II, around Eq. (14)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are several typographical and formatting issues, including \"HAMIL TONIAN\" in the Sec. II heading, non-rendered characters in the Schrödinger equation and in several formulas, and inconsistent use of phi versus varphi for the squeezing angle. These should be corrected in the final version.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the single-field consistency check, the expression for B_k in terms of the Mukhanov variable is stated without showing that it reduces to the standard Hamiltonian coefficient associated with the v action. A brief derivation or a comment on the absence of the z''/z term would make the consistency check more convincing.","section":"Sec. IV.B, Eq. (56)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The notation b_n^(1) conflicts with the conventional Lanczos coefficient b_n used in Sec. IV. Please use a different symbol, such as h_n^(1), for the channel-resolved hopping amplitudes to avoid confusion.","section":"Sec. V.B","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper takes the next logical step: Krylov complexity in two-field inflation with coupled curvature and entropy perturbations. The main new result is that the two squeezing eigenchannels show different onset times and oscillatory dynamics after horizon exit. The derivation from the quadratic action to the squeezing equations is clean, and the single-field limit checks out. The paper is well-structured and cites the relevant literature.\n\nThe load-bearing issue is the off-diagonal pair-production coefficient A_12. The authors state explicitly that they focus on diagonal channels and assume subdominant off-diagonal contributions (Sec. II, b119–b121), but they never compute A_12 or bound its size for their chosen parameters. Without that check, the factorized squeezed state (Eq. 26) and the resulting two-channel evolution equations (30–33) rest on an unverified premise. If A_12 is comparable to the diagonal coefficients, the whole two-channel picture collapses, and the quantitative claims about distinct onset times, oscillatory patterns, and channel hierarchy could be artifacts. The paper acknowledges the limitation, but acknowledging it does not substitute for checking it.\n\nOther soft spots: no error estimates or convergence checks on the numerical solutions; the parameter choices (epsilon=0.002, mu=0.5H, lambda=0.1) are arbitrary, and their effect on the oscillatory structures is unexplored. These are fixable but need attention.\n\nWho is this for? Researchers working on quantum information measures in cosmology. The paper extends a known framework to a slightly richer setting; the qualitative claim that two channels behave differently is plausible, but the quantitative details require the missing verification.\n\nRecommendation: I would send this to peer review. The idea is solid and the derivation is sound, but the authors must compute or bound A_12 and demonstrate that the factorized description is valid for their parameter range. Without that, the paper is incomplete.","headline":"Extends Krylov complexity to two-field inflation with a clear two-channel structure, but the central factorization assumption is stated, not verified, leaving the quantitative results unsupported.","tokens_in":21900,"tokens_out":932,"would_cite":false,"duration_ms":11114,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Two-field inflation reveals distinct Krylov complexity oscillations across two squeezing channels.","keywords":["two-field inflation","Krylov complexity","squeezed states","curvature perturbation","entropy perturbation","quantum information in cosmology","Lanczos coefficients","oscillatory dynamics"],"falsifier":"Compute the full Sp(4,R) evolution including off-diagonal pair production (A₁₂ ≠ 0) for the same parameter values (H=1, ε=0.002, μ=0.5H, λ=0.1). If the resulting Krylov complexity shows only a single oscillatory pattern or if the two-channel hierarchy disappears, then the diagonal approximation fails and the central claim about distinct channel dynamics is restricted to the subdominant-A₁₂ regime.","tokens_in":22475,"feed_emoji":"🌌","tokens_out":2555,"duration_ms":26020,"temperature":0.7,"pith_summary":"This paper extends Krylov complexity—a measure of how quantum states spread in an abstract operator space—from single-field inflation to two-field inflation, where curvature and entropy perturbations couple. The authors show that the four-mode squeezed state describing such a system contains two independent squeezing channels. Solving the coupled mode equations numerically, they find that the two channels develop different squeezing histories after horizon exit: the second channel grows strong earlier and dominates the intermediate evolution, while the first channel shows a delayed rise. The resulting Krylov complexity oscillates across horizon crossing, with oscillations originating from phase-sensitive squeezing dynamics and amplified by the nonlinear relation K = sinh²r. These results provide a quantum-information signature of genuine multi-field effects, going beyond what power spectra or bispectra alone can reveal.","feed_headline":"Two-field inflation splits Krylov complexity into distinct oscillations","feed_subtitle":"A four-mode squeezed state reveals two independent squeezing channels with different onset times and phase-sensitive oscillations across…","key_machinery":"The central object is the four-mode squeezed state (Eq. 26), factorized into two independent two-mode squeeze operators each acting on a (k, −k) pair for one of the two field sectors. The squeezing parameters r₁ₖ and r₂ₖ evolve according to first-order equations (34) driven by the diagonal pair-production coefficients A₁₁ and A₂₂, which are explicit functions of the coupled curvature–entropy mode functions. The Krylov complexity then follows directly as Kₖ = sinh² r₁ₖ + sinh² r₂ₖ, and the Krylov entropy factorises similarly. The phase-sensitive source term sin(2φᵢₖ − arg Aᵢᵢ) is the mechanism that generates oscillatory modulations of the squeezing growth rates, which become strongly amplified in the superhorizon regime.","core_discovery":"In two-field inflation, the diagonal squeezing basis of the four-mode squeezed state yields two independent squeezing channels, each governed by its own Hamiltonian coefficients constructed from coupled curvature–entropy mode functions. The two channels exhibit different onset times for strong squeezing: the second channel develops rapidly after horizon exit and dominates the intermediate stage, while the first channel shows a delayed but steep enhancement. Both channels produce oscillatory Krylov complexity across horizon crossing and pronounced fluctuations in the superhorizon regime. The oscillations arise because the instantaneous squeezing rate is proportional to sin(2φᵢₖ − arg Aᵢᵢ), and the resulting modulations are strongly amplified through the nonlinear relation Kᵢₖ = sinh² rᵢₖ. The total Krylov complexity, Krylov entropy, and channel-resolved Lanczos coefficients all reflect this two-channel hierarchy, which is absent in single-field models.","pith_inferences":["If the off-diagonal pair production coefficient A₁₂ is not subdominant for realistic parameter values (e.g., stronger curvature–entropy coupling λ > 0.1), the factorized squeezed state (Eq. 26) may be inaccurate, and one would need to treat the full Sp(4,R) structure. The oscillations and onset hierarchy reported here could then be modified or washed out by channel mixing.","The phase-sensitive oscillations in Kₖ might be imprinted on some observer-level correlator if the state is subsequently decohered or if particle production couples to a detector, potentially providing a testable signature beyond the power spectrum.","The two-channel Krylov construction could be mapped onto a two-dimensional Krylov lattice; the hopping amplitudes bₙ⁽¹⁾ and bₘ⁽²⁾ grow differently, suggesting that operator spreading becomes anisotropic in the two-mode occupation space after horizon crossing."],"forward_implications":["Krylov complexity can serve as a quantum-information diagnostic that distinguishes single-field from multi-field inflationary dynamics, even when isocurvature perturbations are observationally suppressed.","The oscillatory features in Krylov complexity offer a new theoretical handle on the role of phases during horizon crossing; these phases are not erased by the usual squeezing process in multi-field settings.","The two-channel hierarchy identified here implies that at intermediate times one mode carries most of the quantum information spreading, affecting predictions for entropy and Lanczos growth.","Because the Krylov complexity is directly tied to the squeezing strengths, future extensions to N-field inflation will produce N independent squeezing channels, leading to even richer lattice dynamics in Krylov space."],"supporting_citations":[{"why":"Establishes the universal operator growth hypothesis and the Lanczos-algorithm definition of Krylov complexity, which is the core framework used in this paper.","marker":"[1]"},{"why":"First application of Krylov complexity to inflationary cosmology using a two-mode squeezed state; provides the single-field baseline that this paper extends to two fields.","marker":"[12]"},{"why":"Studies Krylov complexity for curvature perturbations during inflation with closed- and open-system descriptions; supplies the single-field consistency check.","marker":"[14]"},{"why":"Provides the four-mode squeezed state description for two-field systems using the symplectic group Sp(4,R); the theoretical foundation for the state decomposition used here.","marker":"[52]"},{"why":"Develops the linear perturbation theory for multi-field inflation and separates adiabatic and entropy modes; the quadratic action and mode equations in this paper follow that framework.","marker":"[26]"},{"why":"Describes inflationary perturbations as squeezed states; the squeezing formalism (two-mode squeeze operator) is directly adopted in the present work.","marker":"[69]"}],"fun_headline_variants":["Two-field inflation yields two-timing Krylov oscillations","Dual squeezing channels split Krylov complexity into two rhythms","Phase-sensitive oscillations drive two-field Krylov complexity","Four-mode squeezed state reveals separate Krylov complexity channels","Inflation with two fields creates dual oscillatory Krylov dynamics"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analysis assumes that off-diagonal pair production between the two squeezing channels (coefficient A₁₂) is subdominant, so the two channels can be treated independently and the factorized squeezed state (Eq. 26) remains accurate.","fun_headline_variants_meta":{"raw":{"variants":["Two-field inflation yields two-timing Krylov oscillations","Dual squeezing channels split Krylov complexity into two rhythms","Phase-sensitive oscillations drive two-field Krylov complexity","Four-mode squeezed state reveals separate Krylov complexity channels","Inflation with two fields creates dual oscillatory Krylov dynamics"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"cost":0.00014115654,"is_byok":false,"cost_details":{"upstream_inference_cost":0.00014115654,"upstream_inference_prompt_cost":4.049094e-05,"upstream_inference_completions_cost":0.0001006656},"total_tokens":2021,"prompt_tokens":901,"completion_tokens":1120,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":1043}},"tokens_in":901,"tokens_out":1120,"cost_source":"openrouter_usage","duration_ms":11589,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":1043,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-16T14:08:48.730243+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the full Sp(4,R) evolution including off-diagonal pair production (A₁₂ ≠ 0) for the same parameter values (H=1, ε=0.002, μ=0.5H, λ=0.1). If the resulting Krylov complexity shows only a single oscillatory pattern or if the two-channel hierarchy disappears, then the diagonal approximation fails and the central claim about distinct channel dynamics is restricted to the subdominant-A₁₂ regime.","supporting_citations":[{"cited_title":"The second channel becomes larger than the first, consistent with the difference between|A 22|and|A 11|","cited_arxiv_id":null,"evidence_quote":"Establishes the universal operator growth hypothesis and the Lanczos-algorithm definition of Krylov complexity, which is the core framework used in this paper."},{"cited_title":"Krylov complexity in quantum field theory, and beyond","cited_arxiv_id":"2212.14429","evidence_quote":"First application of Krylov complexity to inflationary cosmology using a two-mode squeezed state; provides the single-field baseline that this paper extends to two fields."},{"cited_title":"Krylov Complexity in Quantum Field Theory","cited_arxiv_id":"2204.02250","evidence_quote":"Studies Krylov complexity for curvature perturbations during inflation with closed- and open-system descriptions; supplies the single-field consistency check."},{"cited_title":"Geometrical Destabilization of Inflation","cited_arxiv_id":"1510.01281","evidence_quote":"Provides the four-mode squeezed state description for two-field systems using the symplectic group Sp(4,R); the theoretical foundation for the state decomposition used here."},{"cited_title":"Metric-torsion preheating: cosmic dynamo mechanism?","cited_arxiv_id":"1412.8248","evidence_quote":"Develops the linear perturbation theory for multi-field inflation and separates adiabatic and entropy modes; the quadratic action and mode equations in this paper follow that framework."},{"cited_title":"Adiabatic and entropy perturbations from inflation","cited_arxiv_id":"astro-ph/0009131","evidence_quote":"Describes inflationary perturbations as squeezed states; the squeezing formalism (two-mode squeeze operator) is directly adopted in the present work."}],"review_version":1}