{"id":"6feb6f9e-3540-4cb3-a824-424b8a8c1d31","arxiv_id":"2606.29770","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Krylov complexity in non-inertial frames equals the average number of Rindler pairs, with dynamics falling into hyperbolic, critical, or bounded regimes depending on detuning versus pair-production parameters.","lead":"The paper formulates Krylov complexity for uniformly accelerating observers by identifying the Rindler pair-number sector as the natural Krylov basis via the SU(1,1) structure of the Klein-Gordon field. It derives that this complexity equals the mean number of particle pairs created by acceleration-induced Bogoliubov mixing.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Equality of Krylov complexity to mean Rindler pair number follows by construction from taking pair-number states as the Krylov basis.","rationale":"The reader's weakest_assumption isolates exactly the step at which the basis choice renders the equality definitional. This matches the load-bearing concern; the abstract already exhibits the construction, so the full-text absence does not alter the diagnosis. The three dynamical regimes remain potentially interesting but do not rescue the central equality.","tokens_in":1749,"tokens_out":376,"duration_ms":18183,"concrete_test":"Extract the explicit form of the generalized Bogoliubov Hamiltonian from the SU(1,1) construction; apply the Lanczos algorithm starting from the inertial vacuum to generate the first 10 Krylov vectors; test whether they coincide with the Rindler number states |n> up to phase. If they differ, recompute complexity in the true Krylov basis and check whether equality to <n_Rindler> still holds.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The strongest claim asserts an explicit derivation that Krylov complexity equals the mean number of correlated Rindler pairs. This identification is made after positing that the Rindler pair-number sector 'naturally forms the Krylov basis' because of the SU(1,1) sector emerging from the Klein-Gordon symplectic form. In standard Krylov complexity, once the basis is fixed to number states |n>, the complexity reduces to the expectation value of the level index (or a linear function thereof) for states generated by Bogoliubov transformations. No independent verification is indicated that the Lanczos orthogonalization procedure applied to the time-evolution operator or Hamiltonian yields precisely these states rather than a different chain. Consequently the reported equality is tautological under the stated basis choice rather than a non-trivial dynamical result.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript formulates Krylov complexity in non-inertial quantum systems for uniformly accelerating observers. It posits that the SU(1,1) sector arising from the Klein-Gordon symplectic form makes the Rindler pair-number states the natural Krylov basis, generalizes Bogoliubov coefficients via the SU(1,1) group structure, and claims an explicit derivation that Krylov complexity equals the mean number of correlated Rindler pairs produced by Bogoliubov mixing. The work further identifies three regimes (hyperbolic spreading, critical growth, bounded motion) governed by competition between detuning and pair-production parameters, with exponential confinement and localization of complexity in the detuning-dominated regime.","tokens_in":1896,"tokens_out":450,"duration_ms":21301,"significance":"If the equality between Krylov complexity and mean pair number is established as a non-trivial dynamical result rather than a direct consequence of the basis definition, the result would link a standard complexity measure to Unruh-type particle production, offering a concrete bridge between Krylov spreading and quantum field theory in accelerated frames. The regime classification provides a parameter-based taxonomy of spreading behavior that could be tested in analogous oscillator models.","major_comments":[{"comment":"Abstract: The claim of an 'explicit derivation' that Krylov complexity equals the mean number of correlated Rindler pairs is load-bearing for the central result, yet the text states that the Rindler pair-number sector 'naturally forms the Krylov basis' because of the SU(1,1) emergence from the symplectic form. This raises the possibility that the equality follows by construction once the basis is fixed to number states |n>, as the complexity then reduces to a linear function of the mean occupation for Bogoliubov-generated states; the manuscript must show that the Lanczos orthogonalization procedure applied to the time-evolution operator or Hamiltonian independently yields precisely these states.","section":"Abstract"}],"minor_comments":[{"comment":"The opening sentence of the abstract contains the apparent typographical error 'non-In an inertial quantum system'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive feedback. We address the major comment below, agreeing that explicit verification of the Lanczos procedure is warranted to strengthen the central claim.","responses":[{"response":"We agree that the presentation would benefit from an explicit demonstration that the Lanczos algorithm independently generates the Rindler pair-number basis. The SU(1,1) structure arising from the Klein-Gordon symplectic form ensures the Hamiltonian is a linear combination of generators that act as number-changing operators, rendering it tridiagonal in the |n> basis. Consequently, repeated application of H to the initial vacuum, followed by Gram-Schmidt orthogonalization, produces precisely the |n> states. In the revised manuscript we will add a new subsection with the explicit computation of the first few Lanczos vectors b_n and coefficients a_n, confirming they match the number basis. This establishes the equality K(t) = <n(t)> as a dynamical consequence rather than a definitional artifact. We will also update the abstract to emphasize this verification. Revision will be made.","revision_made":"yes","referee_comment":"[Abstract] Abstract: The claim of an 'explicit derivation' that Krylov complexity equals the mean number of correlated Rindler pairs is load-bearing for the central result, yet the text states that the Rindler pair-number sector 'naturally forms the Krylov basis' because of the SU(1,1) emergence from the symplectic form. This raises the possibility that the equality follows by construction once the basis is fixed to number states |n>, as the complexity then reduces to a linear function of the mean occupation for Bogoliubov-generated states; the manuscript must show that the Lanczos orthogonalization procedure applied to the time-evolution operator or Hamiltonian independently yields precisely these states."}],"tokens_in":1390,"tokens_out":374,"duration_ms":23018,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's central result is that Krylov complexity equals the average number of Rindler pairs once the pair-number states are taken as the Krylov basis via the SU(1,1) algebra from the Klein-Gordon symplectic form. They then use a generalized Bogoliubov transformation under that group structure and split the dynamics into three regimes set by the relative size of the detuning and pair-production parameters.\n\nWhat is new is the explicit framing of Krylov complexity for uniformly accelerating observers and the identification of hyperbolic spreading, critical growth, and bounded motion with low-level localization in the detuning-dominated case. The regime analysis is a concrete addition that could be checked numerically in simple oscillator models.\n\nThe soft spot is the main equality. The abstract states that the Rindler pair-number sector naturally forms the Krylov basis, after which the complexity reduces to the mean pair number by the usual counting. No steps are given showing that the Lanczos algorithm applied to the time-evolution operator independently produces exactly those states rather than a different chain. If that verification is missing in the full text, the result is tautological under the stated construction.\n\nThe work is aimed at readers already working on Krylov complexity in quantum field theory or relativistic quantum information. Someone tracking applications to the Unruh effect or analogs of curved-spacetime dynamics might find the three-regime picture useful for further modeling. The paper is coherent enough on its own terms to merit referee time, mainly to settle whether the basis construction is derived or imposed.\n\nI would send it to review.","headline":"The equality of Krylov complexity to mean Rindler pair number follows from the basis choice rather than an independent derivation.","tokens_in":2462,"tokens_out":385,"would_cite":false,"duration_ms":21422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Krylov complexity equals the mean number of correlated Rindler pairs in accelerated quantum systems.","keywords":["Krylov complexity","Rindler pairs","Bogoliubov transformations","non-inertial frames","SU(1,1) algebra","Unruh effect","quantum field theory in curved spacetime"],"falsifier":"An explicit computation of the Krylov complexity operator expectation value and the expectation value of the Rindler number operator in the same accelerated state that shows the two quantities differ by a nonzero amount.","tokens_in":2595,"feed_emoji":"","tokens_out":666,"duration_ms":19436,"temperature":0.7,"pith_summary":"The paper shows that for uniformly accelerating observers, Krylov complexity of the quantum state is identical to the average number of Rindler particle pairs created by Bogoliubov transformations. This equality holds because the pair-number operator sector forms the natural Krylov basis once the SU(1,1) structure is extracted from the Klein-Gordon symplectic form. The Hamiltonian is written in SU(1,1) generators, allowing exact solution of the time evolution and direct identification of complexity with pair production. Three dynamical regimes appear depending on the relative size of detuning versus pair-production strength: unbounded hyperbolic growth, critical linear growth, and bounded oscillatory motion with localization at low levels.","feed_headline":"Krylov complexity counts Rindler pairs exactly","feed_subtitle":"In accelerated frames the complexity measure equals the average number of particle pairs created by Bogoliubov mixing.","key_machinery":"The Rindler pair-number sector serving as the Krylov basis under the SU(1,1) algebra extracted from the Klein-Gordon symplectic form.","core_discovery":"Within the SU(1,1) group-structured Hamiltonian obtained by generalizing the Bogoliubov coefficients, the Krylov complexity is exactly equal to the mean number of correlated Rindler pairs generated via Bogoliubov mixing. The Rindler pair-number sector supplies the Krylov basis, and the competition between detuning and pair-production parameters partitions the dynamics into hyperbolic Krylov spreading, critical growth, and bounded Krylov-space motion, with exponential confinement to low levels in the detuning-dominated regime.","pith_inferences":["The equality supplies a route to measure Krylov complexity through Unruh-like particle detectors.","The localization transition may serve as a diagnostic for the crossover from inertial to accelerated dynamics in analog systems."],"forward_implications":["Krylov complexity acquires a direct particle-counting interpretation in non-inertial frames.","The three regimes (hyperbolic, critical, bounded) are controlled solely by the ratio of detuning to pair-production parameters.","In the detuning-dominated regime the wave packet remains exponentially localized at low Krylov levels.","Bounded Krylov motion corresponds to suppressed pair production and observable localization of the complexity measure."],"fun_headline_variants":["Krylov complexity equals Rindler pair count under acceleration","Rindler pairs set Krylov complexity via SU(1,1) Hamiltonian","Detuning localizes Krylov complexity in accelerating frames","Three regimes split Krylov dynamics by detuning versus pairs","Krylov basis matches Rindler pair sector for observers"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Rindler pair-number sector naturally forms the Krylov basis once the SU(1,1) sector emerges directly from the Klein-Gordon symplectic form in inertial systems.","fun_headline_variants_meta":{"raw":{"variants":["Krylov complexity equals Rindler pair count under acceleration","Rindler pairs set Krylov complexity via SU(1,1) Hamiltonian","Detuning localizes Krylov complexity in accelerating frames","Three regimes split Krylov dynamics by detuning versus pairs","Krylov basis matches Rindler pair sector for observers"]},"model":"grok-4.3","cost_usd":0.004451,"raw_usage":{"total_tokens":2219,"prompt_tokens":662,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":44512000,"prompt_tokens_details":{"text_tokens":662,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1475,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":662,"tokens_out":82,"duration_ms":11934,"temperature":1.0,"reasoning_tokens":1475,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T06:21:48.840110+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation of the Krylov complexity operator expectation value and the expectation value of the Rindler number operator in the same accelerated state that shows the two quantities differ by a nonzero amount.","supporting_citations":[],"review_version":1}