{"id":"9fb68f22-3143-4c97-b97f-212fedcf8fa3","arxiv_id":"2607.09408","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"In f(φ,R) modified-gravity inflation, the squeezed parameter r_k (and hence Krylov complexity) is suppressed after horizon exit while circuit complexity grows more than in canonical inflation.","lead":"This paper computes two quantum-complexity measures—circuit complexity and Krylov complexity—for the ripples produced during inflation in both ordinary scalar-field inflation and a modified-gravity inflation model f(φ,R). It claims the modified gravity changes how strongly the ripples are squeezed, making Krylov complexity smaller but circuit complexity larger.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (47) — the quadratic action for f(φ,R) perturbations — is asserted without derivation; all squeezed-parameter and complexity results inherit its validity, and the B=0 limit does not reproduce the canonical action (28).","rationale":"I agree with the reader's weakest_assumption. The paper's central claim cannot be evaluated until Eq. (47) is justified. The assertion of the quadratic action is the unique gate through which all subsequent results pass. I did not select the abstract/body contradiction (abstract says f(φ,R) enhances r_k and therefore suppresses K; body/Fig. 6 shows canonical K is far bigger) as the primary concern because the body's calculation is explicit; the contradiction is a framing error that could be fixed by editing the abstract. The missing derivation is a correctness risk that cannot be fixed by editing. The internal B=0 consistency check is part of the same concern: it suggests the authors' A/B notation conflates Fourier-space coefficients with action-level coefficients, reinforcing that Eq. (47) is an ansatz rather than a calculation. My proposed test would settle the issue: either the standard derivation reproduces Eq. (47), in which case the qualitative results may survive (though numerical reproducibility remains questionable), or it does not, in which case the central claim is unsupported. The paper does not provide machine-checked proofs or reproducible code; the parameter n/β and initial conditions are unspecified in the figures. These are secondary but compound the correctness risk.","tokens_in":22537,"tokens_out":6658,"duration_ms":67999,"concrete_test":"Independently derive the quadratic action for f(φ,R) from Eq. (44) using standard ADM/Mukhanov perturbation theory in the Jordan frame, keeping f(φ,R) general. Compare term-by-term with Eq. (47). Check (i) whether the sound speed is 1; (ii) whether the mass coefficient is B-A with B=n²a²R/(κ²β²); (iii) whether the n=0 (B=0) limit reproduces the canonical action Eq. (28) including the coefficient -8/3αμ²a². If any term differs, recompute r_k, φ_k, circuit complexity, and Krylov complexity from the corrected action and test whether the qualitative hierarchy in Figs. 3 and 6 survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the f(φ,R) coupling changes the two-mode squeezed state of curvature perturbations, suppressing r_k after horizon exit (Sec. III summary) and reversing the hierarchy between circuit complexity and Krylov complexity (Sec. IV). Every one of those outputs is generated by integrating the ODEs (49)–(53), which are obtained from the Hamiltonian (48), which in turn is obtained from the quadratic action (47). Equation (47) is not derived from the f(φ,R) action (44): the text says only 'we shall expand the correction for gravity to the second order.' No metric perturbations, gauge choice, or field redefinition are given. For a Jordan-frame f(φ,R) theory, the standard second-order action for curvature perturbations contains non-trivial sound-speed and non-minimal-coupling terms; a simple mass term B=n²a²R/(κ²β²) is not the generic result. The B=0 limit also fails an internal consistency check: setting n=0 in (47) gives -π²-(∂_i v)² - A v², but the canonical action (28) has -π²+(∂_i v)² - (8/3)αμ²a²v², and A was defined in Sec. III A as 4αμ²a²/(3k), a k-dependent Fourier coefficient, so the position-space action (47) mixes Fourier-space and position-space coefficients. Thus the seed action is unsupported and internally inconsistent; any extra terms or sign changes in the correct action would alter the ODEs and all complexity conclusions. The additional de Sitter background a=-1/(ηH0) is assumed without deriving it from f(φ,R) background equations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares the quantum-complexity history of primordial curvature perturbations in canonical single-field inflation with that in a Jordan-frame f(φ,R) modified-gravity inflation model. It derives evolution equations for the two-mode squeezed parameters (r_k, φ_k) from a claimed quadratic action, solves them numerically, and then computes circuit complexity, Krylov complexity K=sinh² r_k, Krylov entropy, Lanczos coefficients b_n, and a putative dissipative coefficient c_n. The central advertised result is that the f(φ,R) coupling changes the squeezing dynamics, enhancing r_k before horizon exit while suppressing it afterward, thereby reversing the hierarchy between circuit and Krylov complexity relative to canonical inflation.","tokens_in":23023,"tokens_out":4239,"duration_ms":44060,"significance":"The question of whether modified gravity leaves distinctive signatures in quantum-complexity observables of inflationary perturbations is a legitimate and timely one. The paper usefully reviews and applies standard squeezed-state and Krylov formalism, and the algebraic step K=sinh² r_k from the two-mode squeezed state is correct. However, the significance of the claimed result is not realized because every downstream conclusion rests on a quadratic action that is asserted rather than derived, and the manuscript contains direct contradictions about the sign of the f(φ,R) effect. The numerical outputs are therefore uncontrolled: a correct derivation of the f(φ,R) second-order action would in general change the ODEs, the squeezed parameters, and all complexity diagnostics. As it stands, the paper does not provide a reliable comparison between the two inflationary models.","major_comments":[{"comment":"The quadratic action S^(2)=∫dτ d³x[-π²-(∂_i v)²+(B-A)v²] is the load-bearing input for the entire paper, but it is not derived from the f(φ,R) action (44). The sentence 'we shall expand the correction for gravity to the second order' is not a derivation: no metric perturbation, gauge choice, field redefinition, or sound-speed term is provided. For a Jordan-frame f(φ,R) theory the standard second-order action is known to contain non-minimal coupling and sound-speed terms; a simple mass term B=n²a²R/(κ²β²) is not the generic result. Moreover, the B=0 limit fails an internal consistency check: setting n=0 in (47) gives -π²-(∂_i v)²-A v², whereas the canonical action (28) has +(∂_i v)² and -(8/3)αμ²a²v²; and A was defined in §III A as the Fourier-space coefficient 4αμ²a²/(3k), so (47) mixes position-space and Fourier-space objects. Any change in the correct action propagates directly into OD","section":"§III B, Eq. (47)"},{"comment":"The claimed physical effect is internally contradictory. The abstract states that the f(φ,R) coupling 'enhances the squeezed strength relative to the canonical scalar field inflation' and that 'this enhancement leads to a smaller growth in Krylov complexity.' But Eq. (70) gives K=sinh² r_k, which is strictly increasing in r_k; enhancing r_k cannot reduce K. The main text, in contrast, says r_k is suppressed after horizon exit (§III summary) and that canonical Krylov complexity is 'far bigger' than f(φ,R) (Fig. 6 discussion). The concluding bullet in §V says Krylov complexity is 'significantly amplified when the f(φ,R) coupling is included.' These statements cannot all be true. This contradicts the central quantitative message and prevents the reader from identifying what the paper actually claims.","section":"Abstract; §IV B 2; §V"},{"comment":"The Lanczos and dissipation coefficients are presented as new physical diagnostics, but b_n and c_n are algebraic functions of the same A, B, a′/a inputs that already determine r_k; they add no independent information. The plots do not state which n is used, even though b_n ∝ n and c_n ∝ (2n+1); without this the numerical values in Figs. 4 and 5 are not reproducible. The coefficient c_n is called an 'effective dissipative contribution' and an 'open-system extension,' but no Lindblad/master equation is introduced; Eq. (57) is just the Liouvillian in tridiagonal form with a complex diagonal. The physical interpretation of c_n as energy exchange with the environment is therefore unsupported.","section":"§IV B 1, Eqs. (63), (69)"},{"comment":"The numerical section omits essential data. No initial conditions r_k(y_i), φ_k(y_i) are given; the parameter n/β is only constrained as ≥1, not specified; and the de Sitter background a=-1/(ηH0) is assumed without derivation from the f(φ,R) background equations. The horizon exit is placed at y=-2 in Fig. 1 but at y=0 in Fig. 6. These omissions make the figures irreproducible and complicate any assessment of the claimed differences between the two models.","section":"§III, Eqs. (51)–(53); Figs. 1, 6"}],"minor_comments":[{"comment":"The caption says 'The numerical solutions of r_k' but the plot shows φ_k; this should be corrected.","section":"Fig. 2 caption"},{"comment":"Two consecutive paragraphs are nearly verbatim duplicates ('Prior to the exit of the horizon...' and 'Prior to the horizon exit...'). The redundancy should be removed.","section":"§IV A"},{"comment":"The text alternates between 'compression strength' and 'squeezed strength' for r_k. Use one consistent term.","section":"§III and throughout"},{"comment":"The notation for creation/annihilation operators is inconsistent: c_k vs c_{⃗k} and c†_{−⃗k}. Please standardize.","section":"Eq. (33)"},{"comment":"The approximation from f(φ)=1+ξ(e^{-βnφ}+e^{-β^{-1}nφ}) to f(φ)≈1+e^{-nφ/β} is not explained; the condition ξ=1 and the symmetry β↔1/β are stated but the approximation step needs justification.","section":"Eqs. (45)–(46)"},{"comment":"The vertical axes are log-scaled but the axes labels still use 'y(Linear)' for the horizontal axis; the label is confusing and should be clarified.","section":"Figs. 4–6"}],"recommendation":"reject","confidential_remarks":"The manuscript's central equation (47) is asserted, not derived, and the B=0 limit does not reproduce the canonical action (28). In addition, the abstract, main text, and conclusions give opposite statements about whether f(φ,R) enhances or suppresses r_k and Krylov complexity. These are not presentation issues but load-bearing defects: the numerical results and their physical interpretation cannot be trusted as they stand. A correct derivation of the f(φ,R) quadratic action and a consistent statement of the claim would be needed before this could be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a routine application of the authors' established squeezed-state complexity machinery to f(phi,R) inflation. The new equations — the squeezed-parameter ODEs (49)–(53) and the Lanczos/dissipation coefficients (69) — are new for this action, and the paper is honest about borrowing the machinery from earlier work. The problem is that the quadratic action it all rests on, Eq. (47), is asserted in a single sentence, and the B=0 limit doesn't reduce to the canonical action (28): the gradient term has the wrong sign and A is a Fourier-space coefficient sitting in a position-space action. That's not a minor gap; the ODEs and all complexity outputs inherit whatever is wrong with that seed. The numerical section also omits initial conditions, the exact n/beta, and which Lanczos index is plotted, so the figures are not reproducible as presented. And the claims are not internally consistent: the abstract says f(phi,R) enhances r_k, the Sec. III summary says it suppresses r_k after horizon exit, and Sec. IV says Krylov complexity is amplified while Fig. 6 shows the opposite. The open-system 'negative growth' conclusion is not supported by the computed exponential growth either.\n\nWhat the paper does well: it frames the question clearly, engages the existing literature (including the authors' own prior work, which is a legitimate foundation), and the formal definition of Krylov complexity for the two-mode squeezed state is standard and correctly recalled. If the quadratic action were derived properly, the subsequent steps would be straightforward. That's why this is fixable in principle.\n\nWho is this for? Someone working in the quantum-information-in-cosmology subfield who wants to see another Lagrangian put through the complexity machine. Not a general hep-th audience. I would send it to a referee because the question is legitimate and the flaws are specific enough to be addressed, but the referee should demand the full derivation of (47) from the action (44), and until that happens the paper should not be cited as evidence about f(phi,R) complexity.","headline":"New ODEs for the f(phi,R) squeezed parameters, but the quadratic action they come from is asserted without derivation and fails the B=0 consistency check against the canonical result.","tokens_in":23496,"tokens_out":3595,"would_cite":false,"duration_ms":38335,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83F05","81P45","83D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that in f(φ,R) modified-gravity inflation, the squeezed strength of primordial curvature perturbations is suppressed after horizon exit, making Krylov complexity much smaller while circuit complexity becomes larger than in","keywords":["Krylov complexity","circuit complexity","two-mode squeezed state","inflation","modified gravity","f(φ,R) gravity","primordial perturbations","Lanczos coefficients"],"falsifier":"Re-derive the quadratic action for scalar perturbations in f(φ,R) inflation from the action (44) by explicitly expanding the metric and scalar field to second order in a fixed gauge (e.g., comoving curvature perturbation). If the resulting equation of motion differs from Eq. (47) in the coefficient (B−A), then the predicted suppression of r_k after horizon exit—and hence the complexity hierarchy—does not hold for generic f(φ,R) theories.","tokens_in":22391,"feed_emoji":"🌌","tokens_out":5194,"duration_ms":54243,"temperature":0.7,"pith_summary":"This paper compares two inflationary models—canonical scalar-field inflation and a modified-gravity f(φ,R) inflation—through the lens of quantum complexity of primordial perturbations. The perturbations are modeled as a two-mode squeezed state, whose evolution is governed by the squeezed strength r_k and angle φ_k. The central claim is that the f(φ,R) coupling suppresses r_k after horizon exit, so Krylov complexity (K = sinh² r_k) grows far less than in canonical inflation, while the circuit complexity grows substantially more. If true, this means different gravitational theories leave distinct quantum-complexity fingerprints on the seeds of cosmic structure, offering a new way to probe modified gravity in the early universe.","feed_headline":"Modified gravity shrinks Krylov complexity of cosmic perturbations","feed_subtitle":"While circuit complexity grows larger, Krylov complexity falls because the squeezed strength of curvature perturbations is suppressed after","key_machinery":"The central object is the two-mode squeezed state |ψ⟩_sq = (1/cosh r_k) Σ (−1)^n e^{2in φ_k} tanh^n r_k |n;n⟩, parameterized by the squeezed strength r_k and squeezed angle φ_k. Its evolution is determined by first-order differential equations (49)–(53) derived from the quadratic action. The Krylov complexity reduces exactly to K = sinh² r_k, so all Krylov diagnostics (complexity, entropy, Lanczos coefficients) are controlled by r_k alone. The modified-gravity contribution B enters the evolution equations and is the mechanism that suppresses r_k after horizon exit, thereby reducing Krylov complexity while increasing circuit complexity.","core_discovery":"Working in the Jordan frame, the paper derives a quadratic action S^(2) = ∫dτ d³x [−π² − (∂v)² + (B−A)v²] for curvature perturbations in f(φ,R) inflation, where B = n² a² R/(κ² β²) encodes the modified-gravity correction and A encodes the potential. From this action it obtains evolution equations for the squeezed strength r_k and squeezed angle φ_k. The central result is that the f(φ,R) correction changes these parameters dramatically: before horizon exit it enhances the entanglement and causes oscillations, but after horizon exit it suppresses r_k relative to canonical inflation. Since the Krylov complexity of the two-mode squeezed state equals K = sinh² r_k, the paper finds that canonical","pith_inferences":["A proper second-order perturbation of the f(φ,R) action, with explicit gauge fixing, might reveal additional terms (e.g., sound-speed or non-minimal coupling) that could alter or even invert the hierarchy between canonical and f(φ,R) complexity; this is worth testing.","Because K = sinh² r_k equals the mean particle number per mode, the suppression of r_k suggests f(φ,R) inflation produces fewer particles after horizon exit; this quantitative prediction could be checked against the amplitude of the curvature power spectrum.","The open-system treatment with dissipation c_n implies that environmental effects during inflation could reduce observable complexity growth; extending the model to compute the resulting power spectrum would give a testable cosmological signature.","The opposite behaviors of circuit complexity (higher in f(φ,R)) and Krylov complexity (lower in f(φ,R)) indicate that these two measures are not interchangeable probes of early-universe dynamics, and future work should clarify which one, if either, connects to observables."],"forward_implications":["In f(φ,R) inflation, Krylov complexity K is much smaller than in canonical inflation because r_k is suppressed after horizon exit, implying slower operator growth of curvature perturbations.","Circuit complexity is significantly larger in f(φ,R) inflation, and the Lyapunov exponent is nearly unchanged while post-exit oscillations persist, suggesting more intricate quantum evolution.","Lanczos coefficients b_n grow monotonically and are greatly amplified by f(φ,R), indicating enhanced chaos in the modified-gravity model.","The dissipative coefficient c_n, introduced in an open-system extension, also grows after horizon exit; when dissipation is included, the paper infers that Krylov complexity can decrease rather than grow monotonically.","The paper connects larger Krylov complexity to larger curvature perturbations, implying that canonical inflation produces stronger late-time structure seeds than f(φ,R) inflation."],"fun_headline_variants":["Modified gravity slows quantum complexity growth in inflation","f(φ,R) inflation: less Krylov complexity, more circuit complexity","Krylov complexity drops in modified gravity inflation","Modified gravity's squeezed states shrink Krylov complexity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything hinges on the asserted quadratic action (47) for f(φ,R) perturbations, which is written down without derivation—no metric perturbation, gauge choice, or sound-speed term is given; if the correct action contains extra terms, the squeezed parameters and all complexity outputs change.","fun_headline_variants_meta":{"raw":{"variants":["Modified gravity slows quantum complexity growth in inflation","f(φ,R) inflation: less Krylov complexity, more circuit complexity","Krylov complexity drops in modified gravity inflation","Modified gravity's squeezed states shrink Krylov complexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000543,"raw_usage":{"total_tokens":2454,"prompt_tokens":780,"completion_tokens":1674,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":1610}},"tokens_in":524,"tokens_out":1674,"duration_ms":11363,"temperature":1.0,"reasoning_tokens":1610,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:34:29.342240+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-derive the quadratic action for scalar perturbations in f(φ,R) inflation from the action (44) by explicitly expanding the metric and scalar field to second order in a fixed gauge (e.g., comoving curvature perturbation). If the resulting equation of motion differs from Eq. (47) in the coefficient (B−A), then the predicted suppression of r_k after horizon exit—and hence the complexity hierarchy—does not hold for generic f(φ,R) theories.","supporting_citations":[],"review_version":2}