PREreview of A New Understanding of Einstein-Rosen Bridges

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\title{\vspace{-2cm}PRE-REVIEW REFEREE REPORT: ``A NEW UNDERSTANDING OF EINSTEIN-ROSEN BRIDGES''}

\author{Anik Chakraborty}

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\maketitle

\noindent \textbf{Manuscript:} A New Understanding of Einstein-Rosen Bridges\\

\noindent \textbf{Authors:} Enrique Gazta\~naga, K. Sravan Kumar, Jo\~ao Marto\\

\noindent \textbf{Manuscript ID:} preprints202410.0190.v2\\

\noindent \textbf{Recommendation:} Major Revision Required

\section{Summary}

The manuscript proposes a novel framework termed ``direct-sum quantum theory'' (DQFT) to address long-standing unitarity issues in quantum field theory on curved spacetimes, particularly concerning the black hole information paradox and observational anomalies in the cosmic microwave background (CMB).

The authors synthesize historical concepts: Einstein-Rosen bridges (1935), Schr\"odinger's antipodal identification in de Sitter space (1956), and Berry-Keating's inverted harmonic oscillator quantization (1999)—within a unified mathematical structure.

The core proposal involves decomposing the Hilbert space as a direct sum $\mathcal{H} = \mathcal{H}_+ \oplus \mathcal{H}_-$, where components correspond to parity-conjugate spatial regions evolving with opposite arrows of time. This framework is applied to black hole physics, inverted harmonic oscillator quantization, and inflationary cosmology, with the authors claiming their ``direct-sum inflation'' (DSI) model exhibits a Bayes factor 650 times superior to standard inflation when compared to Planck CMB data.

While the manuscript demonstrates impressive interdisciplinary breadth and addresses genuinely important problems in quantum gravity and cosmology, it suffers from several critical deficiencies: insufficient mathematical rigor in constructing the direct-sum Hilbert space, inadequate engagement with recent (2019-2024) developments in the black hole information paradox (particularly the island formula and Page curve calculations), and statistical claims requiring independent verification. The proposed geometric superselection sector structure appears inconsistent with established results in algebraic quantum field theory, notably the Reeh-Schlieder theorem. The CMB analysis, while potentially significant, lacks transparency in methodology and does not adequately address alternative explanations for observed anomalies. These issues must be resolved before the manuscript meets publication standards for a high-tier physics journal.

\section{Major Issues Requiring Correction}

\subsection{Mathematical Rigor and Consistency of Direct-Sum Quantum Theory}

\textbf{Issue 1 - Hilbert Space Construction:} The fundamental claim that the Hilbert space decomposes as $\mathcal{H} = \mathcal{H}_+ \oplus \mathcal{H}_-$ (Section 5) with components evolving under opposite time directions ($i\partial_t$ versus $-i\partial_t$) lacks rigorous mathematical justification. Standard quantum field theory on a spacetime with causally disconnected regions $U$ and $U'$ yields the tensor product structure $\mathcal{H} = \mathcal{H}_U \otimes \mathcal{H}_{U'}$, not a direct sum. The manuscript acknowledges (p.~27) potential tension with the Reeh-Schlieder theorem, which establishes that the vacuum state $|0\rangle$ is cyclic and separating for local operator algebras, implying \emph{entanglement} between spatial regions rather than orthogonal decomposition. This contradiction is noted but not resolved.

The authors should provide either: (a) a proof that their geometric superselection sectors (SSS) satisfy conditions exempting them from Reeh-Schlieder, or (b) an explicit resolution showing how DQFT remains consistent with the theorem. The current presentation leaves the framework's mathematical consistency in doubt.

\textbf{Issue 2 - Commutation Relations:} Equations 99, 102, and 103 (pp.~23-24) introduce operator decompositions such as $\phi = \phi_+ \oplus \phi_-$ with commutation relations $[\phi_+(x), \phi_+(y)] = iG_+(x,y)$ and $[\phi_+, \phi_-] = 0$. These relations are \emph{asserted} without derivation from canonical quantization procedures or verification of consistency with the equal-time commutation relations $[\phi(t,\mathbf{x}), \pi(t,\mathbf{y})] = i\delta^3(\mathbf{x} - \mathbf{y})$. For operators living in orthogonal subspaces, the vanishing commutator $[\phi_+, \phi_-] = 0$ is expected, but the manuscript does not demonstrate how measurements in one sector relate to the other given that the authors claim these sectors represent \emph{the same spacetime} with opposite time flows, not distinct parallel universes.

\textbf{Issue 3 - Opposite Time Arrows:} The claim that $|\psi_+\rangle$ and $|\psi_-\rangle$ evolve with opposite time directions while representing the same positive-energy quantum state requires clarification regarding causality and observer experience. In standard physics, all local observers measure the same time flow dictated by the causal structure. The authors cite Donoghue \& Menezes (2019) to argue that $i \leftrightarrow -i$ equivalence implies time-arrow ambiguity, but that work demonstrates global arrow choice equivalence, not that \emph{local opposite arrows} coexist causally within a single spacetime. Without additional structure (e.g., CPT-twisted boundary conditions explicitly shown to preserve causality), this raises unresolved paradoxes.

\textbf{Required Action:} Provide rigorous mathematical proofs or extensive derivations in an appendix addressing: (1) Hilbert space construction compatible with AQFT axioms, (2) canonical quantization yielding the proposed commutation relations, and (3) resolution of the opposite-time-arrow causality structure. If these cannot be provided, acknowledge these as open mathematical problems requiring future work.

\subsection{Insufficient Engagement with Recent Black Hole Information Paradox Literature}

\textbf{Issue:} The manuscript motivates DQFT by asserting that standard quantum field theory in curved spacetimes leads to unitarity violation via mixed states for observers restricted to one side of a horizon (Section 6). While historically accurate as the information paradox formulation, this presentation ignores major developments from 2019-2024 that have substantially altered the landscape:

\begin{enumerate}[label=(\alph*)]

\item \textbf{Island Formula and Page Curve:} Almheiri et al. (2019), Penington (2020), and subsequent works introduced the quantum extremal surface prescription, showing that semi-classical gravity with quantum corrections reproduces the unitary Page curve for black hole evaporation without requiring modifications to quantum mechanics. Recent calculations extend this to Schwarzschild, Kerr (rotating), charged, and accelerating black holes, with explicit demonstrations that entanglement entropy follows the expected Page curve at late times when island contributions are included.

\item \textbf{Replica Wormholes and Non-Perturbative Effects:} The island formula relies on replica wormhole saddles in the gravitational path integral, representing non-perturbative contributions. This framework has been validated in numerous models (JT gravity, Schwarzschild-de Sitter, AdS black holes) and is now widely considered to resolve the information paradox within semi-classical gravity.

\item \textbf{Observer Complementarity Formalization:} Black hole complementarity, proposed by Susskind et al. (1993), has been further formalized in holographic contexts. Observers inside and outside horizons have complementary descriptions that do not conflict due to causal inaccessibility. The manuscript claims DQFT provides observer complementarity (p.~32) but does not compare with these established frameworks or demonstrate advantages.

\end{enumerate}

The manuscript's dismissal of these developments appears limited to brief statements that ``decades of effort have not yielded a fully unitary formulation'' (p.~2). This characterization was more accurate circa 2015 but does not reflect the post-2019 consensus that semi-classical gravity with quantum extremal surfaces achieves unitarity.

\textbf{Required Action:} Add a dedicated subsection in Section 6 engaging with the island formula literature. The authors must either: (1) acknowledge that semi-classical approaches restore unitarity and clarify what additional problems DQFT solves, or (2) identify specific shortcomings of the island approach that DQFT addresses. Simply omitting this literature is not acceptable for a manuscript claiming to resolve the information paradox.

\subsection{Inverted Harmonic Oscillator and Berry-Keating Quantization}

\textbf{Issue:} Section 5.2 claims DQFT resolves ambiguities in quantizing the inverted harmonic oscillator (IHO) relevant to the Berry-Keating Hamiltonian $H_{BK} = (xp + px)/2$, which conjecturally relates to Riemann zeta zeros. However, no explicit calculation of the spectrum is provided. Berry \& Keating's original work and subsequent studies (Sierra 2007, Giordano et al. 2023) perform detailed spectral analysis using boundary conditions, Weyl quantization, and asymptotic matching to Riemann-Siegel $\theta$-functions. The manuscript asserts that DQFT's four-sector decomposition (Figure 1, corresponding to phase space regions I-IV) provides a natural quantization but does not:

\begin{itemize}

\item Derive energy eigenvalues $E_n$ for the DQFT-quantized IHO.

\item Compare these eigenvalues to Riemann zeta zeros $\zeta(1/2 + i\gamma_n) = 0$.

\item Show that DQFT reproduces known results from other quantization schemes.

\end{itemize}

Without these calculations, the claim that DQFT ``resolves'' IHO quantization is unsupported. The connection to phase space horizons ($p = \pm q$) is conceptually interesting but does not constitute a quantization procedure.

\textbf{Required Action:} Either (1) perform explicit spectral calculations showing DQFT yields zeta zeros or a meaningful approximation thereof, or (2) soften the language to state that DQFT provides a \emph{framework} for IHO quantization that remains to be fully developed. Cite and compare with existing quantization approaches (particularly Giordano et al. 2023 on the Generalized Born oscillator).

\subsection{CMB Analysis: Statistical Claims and Methodological Transparency}

\textbf{Issue 1 - Bayes Factor Magnitude:} The claim of a Bayes factor of 650 in favor of DSI over standard inflation (Section 7, Figure 8, Table 2) is extraordinary and requires extraordinary scrutiny. Bayesian model comparison penalizes additional parameters and structural complexity. While the authors state DSI introduces ``no additional free parameters,'' the fundamental change from standard quantum mechanics to DQFT represents a \emph{framework-level modification} that should incur a prior penalty. The manuscript does not discuss:

\begin{itemize}

\item Prior probability assignments for DSI versus standard inflation.

\item Whether the likelihood function properly accounts for cosmic variance at low $\ell$ ($\ell \lesssim 30$), where the claimed improvement occurs.

\item Sensitivity analysis showing robustness to prior choices.

\end{itemize}

\textbf{Issue 2 - Alternative Explanations:} CMB parity asymmetry has been studied extensively, with proposed explanations including:

\begin{itemize}

\item Pre-inflationary anisotropies or superhorizon perturbations.

\item Finsler spacetime geometry (Li \& Wang 2018).

\item Galactic foreground contamination and systematic errors in component separation.

\item Late-time integrated Sachs-Wolfe effects.

\end{itemize}

The manuscript acknowledges hemispherical asymmetry but does not quantitatively compare DSI's Bayes factor against these alternatives. A complete analysis would compute Bayes factors for multiple competing models.

\textbf{Issue 3 - Look-Elsewhere Effect:} Multiple CMB anomalies (low quadrupole, octopole alignment, lack of large-angle correlations, hemispherical asymmetry, parity asymmetry, cold spot) have been investigated over two decades. The statistical significance of any single anomaly decreases when accounting for multiple hypothesis testing. The manuscript does not address this concern.

\textbf{Issue 4 - Reproducibility:} The computational methods for computing the DSI power spectrum $C_\ell^{\text{DSI}}$ and fitting to Planck data are described qualitatively but lack sufficient detail for reproduction. Code availability is not mentioned.

\textbf{Required Action:}

\begin{enumerate}

\item Provide full details of the Bayesian analysis in an appendix: prior specifications, likelihood construction, MCMC sampling methods (if used), convergence diagnostics.

\item Compare DSI quantitatively to at least two alternative explanations for parity asymmetry, computing Bayes factors or BIC/AIC scores.

\item Address cosmic variance limitations and multiple testing corrections.

\item Make analysis code publicly available (e.g., GitHub repository) or provide it as supplementary material.

\item Consider independent statistical review before publication, given the extraordinary claim magnitude.

\end{enumerate}

\subsection{Superselection Sectors and Algebraic Quantum Field Theory}

\textbf{Issue:} The manuscript introduces ``geometric superselection sectors'' (SSS) based on spatial location (Sections 4-5), asserting that regions with opposite parity are superselected. In algebraic quantum field theory (AQFT), superselection sectors are rigorously defined via the DHR (Doplicher-Haag-Roberts) framework, which requires:

\begin{itemize}

\item Haag duality: $\mathcal{A}(\mathcal{O})' = \mathcal{A}(\mathcal{O}')$ relating algebras in a region and its causal complement.

\item Localized charges: sectors differ from the vacuum only in bounded regions.

\item Statistics: superselection sectors in 3+1D obey statistics determined by the relevant symmetry group.

\end{itemize}

Standard DHR theory addresses \emph{charge} superselection (electric charge, baryon number) arising from global internal symmetries, not geometric/spatial superselection. M\"uger (1998) proved that in massive QFT with Haag duality and the split property, the vacuum sector is unique—no non-trivial DHR sectors exist. This appears contradictory to the manuscript's framework unless the authors can show their SSS:

\begin{enumerate}

\item Do not satisfy DHR axioms (in which case, what axioms \emph{do} they satisfy?).

\item Exploit a loophole in M\"uger's theorem (e.g., massless fields, violation of split property).

\item Represent a fundamentally different structure not captured by DHR.

\end{enumerate}

The manuscript does not cite DHR theory or address this literature at all.

\textbf{Required Action:} Add discussion of superselection sectors within AQFT, citing DHR and related work. Clarify the relationship (or distinction) between geometric SSS and standard superselection theory. If DQFT's sectors are incompatible with DHR, this should be explicitly stated and mathematically justified.

\section{Minor Issues}

\subsection{Notation and Presentation}

\begin{itemize}

\item \textbf{Notation Inconsistency:} The symbol $\phi_\pm$ sometimes denotes field operator components in the direct-sum decomposition (Eq.~99), and other times denotes fields related by time reversal (Section 4). Consistent notation would improve clarity.

\item \textbf{Redundant Background:} Sections 2-3 review standard QFT in Minkowski space, IHO classical mechanics, and spacetime symmetries at textbook level. For a research manuscript targeting expert audiences, this material could be condensed or moved to appendices, reducing the manuscript from 54 pages to a more focused 35-40 pages.

\item \textbf{Figure 3 Interpretation:} Figure 3 (p.~22) depicts IHO wavefunctions at $x > 0$ and $x < 0$ with opposite time evolution arrows. This risks confusion—are these separate particles, or one particle with position-dependent time flow? Clarify in the caption.

\end{itemize}

\subsection{Thermofield Double States}

The manuscript criticizes thermofield double (TFD) states as introducing ``fictitious dual Fock spaces'' (p.~20). However, TFD is a standard purification technique in quantum statistical mechanics, used extensively in AdS/CFT, black hole thermodynamics, and quantum information. The doubling is a mathematical tool analogous to the authors' own geometric doubling via $\mathcal{H}_+ \oplus \mathcal{H}_-$. The distinction claimed is unclear. Clarify why DQFT's doubling is physically motivated while TFD's is ``fictitious.''

\subsection{Literature Coverage}

Several relevant areas are underrepresented in citations:

\begin{itemize}

\item \textbf{Superselection Sectors:} No citation of Doplicher-Haag-Roberts, Haag duality, or M\"uger's work on massive QFT sectors.

\item \textbf{CMB Anomalies:} Missing recent (2023-2024) papers on Planck PR4 data and hemispherical asymmetry reassessments.

\item \textbf{Inflationary Perturbations:} Standard references on Mukhanov-Sasaki variables and quantum-to-classical transition are absent.

\item \textbf{Elliptic de Sitter Space:} Parikh \& Savonije (2003) on elliptic de Sitter should be cited for detailed analysis of Schr\"odinger's antipodal identification.

\end{itemize}

\textbf{Required Action:} Expand references to cover these areas. Specific suggestions:

\begin{itemize}

\item DHR: Doplicher, S., Haag, R., \& Roberts, J. E. (1971). \textit{Commun. Math. Phys.}, 23, 199-230.

\item M\"uger: M\"uger, M. (1998). \textit{Rev. Math. Phys.}, 10, 1147-1170.

\item Elliptic dS: Parikh, M. K., \& Savonije, I. (2003). \textit{Phys. Rev. D}, 67, 064005.

\item Islands: Almheiri, A., et al. (2019). \textit{JHEP}, 12, 063; Penington, G. (2020). \textit{JHEP}, 03, 205.

\item CMB: Planck Collaboration (2020). \textit{Astron. Astrophys.}, 641, A6.

\end{itemize}

\subsection{Historical Context}

The connection drawn between Einstein-Rosen bridges, Schr\"odinger's work, and modern ER=EPR is intellectually stimulating but occasionally overstates causal relationships. ER=EPR (Maldacena \& Susskind 2013) specifically concerns \emph{non-traversable wormholes} connecting entangled black holes in holographic theories, distinct from the direct-sum Hilbert space structure proposed here. Clarify that DQFT offers an alternative interpretation rather than a realization of ER=EPR as originally formulated.

\section{Strengths}

Despite the issues noted, the manuscript has several commendable aspects:

\begin{itemize}

\item \textbf{Interdisciplinary Synthesis:} Connecting quantum gravity, operator theory, the Riemann hypothesis, and observational cosmology within one framework is ambitious and intellectually valuable.

\item \textbf{Historical Scholarship:} The detailed review of Einstein-Rosen's original intent and Schr\"odinger's elliptic de Sitter space provides useful historical context often overlooked in modern literature.

\item \textbf{Testable Predictions:} The claim that DSI predicts parity-asymmetric B-mode polarization (mentioned in Section 7) is a concrete, falsifiable prediction distinguishable from standard inflation. With upcoming CMB-S4 and LiteBIRD missions, this could be tested within a decade.

\item \textbf{CMB Phenomenology:} Focusing on \emph{parity} (discrete symmetry breaking) rather than continuous anisotropy is conceptually clearer and may inspire alternative phenomenological models even if DQFT's foundations require revision.

\item \textbf{Computational Work:} The numerical computation of DSI power spectra and comparison to Planck data appears carefully executed (though requiring transparency improvements as noted).

\end{itemize}

\section{Assessment of Novelty}

The direct-sum quantum theory framework appears novel within quantum gravity literature, though elements resemble:

\begin{itemize}

\item Schr\"odinger's antipodal identification (acknowledged).

\item Two-Hilbert-space formalisms in quantum measurement theory (Zurek, decoherence program).

\item PT-symmetric quantum mechanics (Bender et al.), though that work addresses non-Hermitian Hamiltonians rather than opposite time flows.

\end{itemize}

The application to CMB parity asymmetry is, to my knowledge, unique. The 650$\times$ Bayes factor, if validated, would represent a significant observational anomaly requiring explanation—whether by DQFT or alternative mechanisms.

\section{Reproducibility and Data Availability}

\textbf{Computational Reproducibility:} The manuscript references Planck 2018 data but does not provide:

\begin{itemize}

\item Analysis code for computing $C_\ell^{\text{DSI}}$.

\item MCMC chains or posterior samples (if Bayesian analysis was performed).

\item Preprocessing steps for Planck data (masking, component separation choices).

\end{itemize}

\textbf{Recommendation:} Deposit code in a public repository (Zenodo, GitHub, or journal supplementary material) with a DOI. Provide a README documenting dependencies, data sources, and steps to reproduce Figure 8 and Table 2.

\section{Recommendations for Revision}

This manuscript addresses important open problems and presents intriguing ideas, but substantial revisions are required before it meets publication standards for a peer-reviewed journal. I recommend \textbf{Major Revision} with the following actions:

\subsection{Critical (Must Address Before Acceptance)}

\begin{enumerate}

\item \textbf{Rigorous Mathematical Foundations (Section 5):} Provide detailed derivations or proofs addressing:

\begin{itemize}

\item Hilbert space construction consistent with Reeh-Schlieder theorem.

\item Canonical quantization yielding proposed commutation relations.

\item Causality structure for opposite time arrows.

\end{itemize}

Alternatively, explicitly acknowledge these as open problems and reframe DQFT as a heuristic framework pending rigorous formulation.

\item \textbf{Engage Island Formula Literature (Section 6):} Add a subsection discussing quantum extremal surfaces, Page curve calculations (2019-2024), and clarifying what DQFT provides beyond semi-classical unitarity restoration. Cite at least 10 key papers from this literature.

\item \textbf{CMB Analysis Transparency (Section 7):} Provide full methodological details in an appendix:

\begin{itemize}

\item Prior specifications.

\item Likelihood function and cosmic variance treatment.

\item Sensitivity analysis for Bayes factor.

\item Comparison to alternative explanations (Finsler geometry, pre-inflationary anisotropies, systematics).

\end{itemize}

Make analysis code publicly available.

\item \textbf{Superselection Sector Theory (New Subsection):} Discuss relationship between geometric SSS and DHR superselection sectors. Address M\"uger's theorem and clarify whether DQFT satisfies, violates, or circumvents standard AQFT axioms.

\item \textbf{Expand References:} Add citations for:

\begin{itemize}

\item DHR theory and algebraic QFT (Haag, Doplicher, Roberts, M\"uger).

\item Island formula and Page curve (Almheiri, Penington, Engelhardt, Marolf, et al.).

\item Recent CMB analyses (Planck 2020, hemispherical asymmetry reassessments).

\item Berry-Keating spectral calculations (Sierra, Giordano et al.).

\item Elliptic de Sitter space (Parikh \& Savonije).

\end{itemize}

Target 40-50 total references.

\end{enumerate}

\subsection{Strongly Recommended}

\begin{enumerate}[resume]

\item \textbf{Calculate IHO Spectrum (Section 5.2):} Perform explicit spectral calculations for DQFT-quantized IHO or state clearly that this remains future work.

\item \textbf{Condense Background Material (Sections 2-3):} Reduce textbook-level reviews to 2-3 pages or move to appendices. Target 35-40 pages total.

\item \textbf{Clarify TFD Criticism (Section 6):} Explain distinction between TFD doubling and DQFT's $\mathcal{H}_+ \oplus \mathcal{H}_-$ decomposition more carefully, or acknowledge they serve analogous purification roles.

\item \textbf{Quantitative B-Mode Prediction (Section 7):} Provide numerical prediction for B-mode parity asymmetry amplitude with error bars, facilitating future observational tests.

\end{enumerate}

\subsection{Optional Improvements}

\begin{enumerate}[resume]

\item Consider splitting into two papers: (1) Theoretical framework (DQFT, IHO, black holes) for quantum gravity audience, and (2) Cosmological application (DSI, CMB) for astrophysics audience. Each would be more focused and accessible.

\item Add a brief discussion of potential experimental tests beyond CMB (e.g., gravitational wave echoes, black hole shadows, laboratory analogs).

\item Discuss philosophical implications of opposite time arrows more thoroughly, potentially in connection with CPT theorem and cosmological arrow of time.

\end{enumerate}

\section{Conclusion}

This manuscript tackles profound questions at the intersection of quantum mechanics, general relativity, and cosmology. The direct-sum quantum theory represents a bold attempt to resolve unitarity paradoxes through a novel mathematical structure, and the claimed CMB observational support is potentially significant. The interdisciplinary synthesis and historical scholarship are valuable contributions.

However, the work suffers from critical deficiencies in mathematical rigor, outdated characterization of the information paradox literature, and insufficient transparency in statistical claims. The proposed framework's consistency with fundamental QFT theorems (Reeh-Schlieder) and established superselection theory remains unclear. The CMB analysis requires independent verification given the extraordinary Bayes factor claimed.

With substantial revisions addressing the issues outlined above—particularly providing rigorous mathematical foundations, engaging island formula literature, and ensuring full transparency and reproducibility of the CMB analysis—this manuscript could make a valuable contribution to quantum gravity and cosmology. In its current form, it is not ready for publication in a high-tier journal but shows sufficient promise to warrant major revision rather than rejection.

I recommend submission to a specialized journal such as \textit{Classical and Quantum Gravity}, \textit{General Relativity and Gravitation}, or \textit{Journal of Cosmology and Astroparticle Physics} after revisions. The theoretical sections may also suit \textit{Foundations of Physics} or \textit{International Journal of Modern Physics D}.

\vspace{1em}

\noindent \textbf{Reviewer's Note:} This review is provided constructively to strengthen the manuscript. The authors have undertaken an ambitious project addressing important open problems, and with appropriate revisions, this work has the potential to stimulate valuable discussions in the quantum gravity and cosmology communities. I encourage the authors to address the mathematical consistency issues carefully and to engage fully with recent literature before resubmission.

\vspace{2em}

\noindent \textbf{Pre-Reviewer:} Anik Chakraborty\\

\noindent \textbf{Affiliation:} Research Scholar, Department of Mathematics, University of Delhi, India\\

\noindent \textbf{Date:} November 9, 2025

\end{document}

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.