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The central dynamical equation for the condensate (Eq. (1) and/or Eq. (3): $\dot N=-H+\gamma Q_{\rm mem}/N^2$) is not dimensionally consistent as written if $N$ and $Q_{\rm mem}$ are dimensionless occupation numbers (Sec. 2.1; Sec. 3.1). In that case $\dot N$ and $H$ have units of time$^{-1}$, but $\gamma Q_{\rm mem}/N^2$ is dimensionless unless $\gamma$ carries units, contradicting the text’s description of $\gamma$ as “order-one constant”. This ambiguity propagates into stability, timescales, and numerical integration.
Recommendation: Make the time variable and normalization explicit (e.g., specify whether derivatives are with respect to cosmic time $t$, conformal time, or e-fold time $N_e\equiv\ln a$). If $t$ is cosmic time, modify the backreaction term to include an explicit rate scale (typically $H$ or another microscopic rate), or state that $\gamma$ has dimensions and define it. Then re-check all downstream relations that depend on Eq. (1)/(3) (nullcline condition, eigenvalues in Sec. 3.1, and numerical units in Figs. 1–3).
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The dynamical system is not fully specified/closed: the paper asserts $N=M_{\rm Pl}^2/H^2$ (Sec. 2.1) while also treating $N$ as a dynamical variable evolved by Eq. (1)/(3). It is unclear whether $H$ is (i) held fixed, (ii) updated algebraically at each time step via $H(N)=M_{\rm Pl}/\sqrt{N}$, or (iii) evolved by an additional equation. Several claims require consistent $H(t)$ evolution (e.g., “slow drift causes gradual decrease in $H$” in Sec. 3.1; and $N_e=\int H\,dt$ in Sec. 3.3).
Recommendation: State explicitly (in Sec. 2.1 and in the numerical-method description) which variables are independent and what equations are solved. If $H$ is derived from $N$, rewrite the ODE(s) in a manifestly closed form (e.g., $\dot N=f(N,Q)$, $\dot Q=g(N,Q)$ with $H(N)$ substituted). If $H$ is treated as approximately constant, state the approximation regime and quantify the allowed drift. Ensure the definitions of $N_e$, tilt, and quantum-breaking time use the same choice.
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The stability/attractor analysis in Sec. 3.1 appears mathematically inconsistent with the stated evolution laws and with the terminology used. With $\dot Q_{\rm mem}\approx N_s H$ (Sec. 2.1), the full 2D system $(N,Q_{\rm mem})$ generically has no fixed point because $\dot Q_{\rm mem}\neq 0$. What can exist is an attracting nullcline/slow manifold (e.g., $\dot N\simeq 0$) with drift along it as $Q_{\rm mem}$ grows. Relatedly, the reported eigenvalue $\lambda=-2H$ (Sec. 3.1, p.5) does not follow from the written equations under standard interpretations (even before accounting for $H(N)$).
Recommendation: Revise Sec. 3.1 to analyze the correct object: attraction transverse to the $\dot N=0$ nullcline (or a slow manifold if a timescale hierarchy is assumed), rather than a fixed-point Jacobian unless an actual fixed point is introduced. Write the explicit Jacobian of the system you actually integrate, specify whether derivatives are taken at constant $H$ or with $H(N)$, and show the derivation of the transverse eigenvalue(s). If $\lambda=-2H$ is obtained only after nondimensionalization (e.g., using e-fold time), state that clearly and adjust notation accordingly.
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Ambiguity between the quasi-static/nullcline condition and the quantum-breaking condition: the paper defines quasi-static equilibrium/attractor behavior via $\dot N\approx 0$ (Sec. 2.1), but later uses “equilibrium condition $Q_{\rm mem}\sim N$” to derive the dynamical selection equation (Sec. 2.4; Sec. 3.2). Earlier, $Q_{\rm mem}=N$ is the termination (quantum breaking) criterion (Secs. 2.1–2.3, 3.4), not the condition defining the inflationary attractor.
Recommendation: Disambiguate and consistently name: (i) the attractor/nullcline condition from $\dot N=0$ (which gives a relation between $Q_{\rm mem}$ and $N$ given $H$), versus (ii) the breaking/saturation criterion $Q_{\rm mem}=N$. If the selection mechanism relies on the breaking condition rather than the inflationary attractor, explain why an end-of-phase condition determines the inflationary scale during the phase, and under what dynamical assumptions this is valid.
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The derivation of cosmological perturbations is currently too heuristic to assess observational viability. The key asserted chain $\zeta\sim \delta N/N\Rightarrow \Delta_\zeta^2\sim 1/N\sim H^2/M_{\rm Pl}^2$ (Sec. 2.2; Sec. 3.1) is not derived in a gauge-invariant perturbation framework, and it is unclear how graviton-occupation fluctuations map to curvature perturbations (in standard inflation, $\zeta$ is related to fluctuations in the local expansion history $\delta N_{\rm efolds}$, not directly to a graviton number). Additionally, the numerical statement that $A_s\simeq 2.1\times 10^{-9}$ is reproduced for $H/M_{\rm Pl}\sim 10^{-5}$ is inconsistent with $\Delta_\zeta^2\sim (H/M_{\rm Pl})^2$ without an extra $\mathcal{O}(10)$ prefactor (Sec. 3.1). Claims about tilt, tensors $r$, and non-Gaussianity are also not backed by explicit formulas (Secs. 3.1, 3.4; Sec. 4).
Recommendation: Provide a concrete perturbation derivation: specify the “clock”/single-degree-of-freedom controlling adiabatic perturbations; relate $\delta N$ (occupation) to $\delta H/H$, $\delta\rho$, or $\delta N_{\rm efolds}$ in a separate-universe/$\delta N$-formalism sense; and show how the two-point function gives $\Delta_\zeta^2$. State the statistical assumption for $\delta N$ (Poissonian? condensate correlator?) and the evaluation/freeze-out condition (is it still $k=aH$? what is $c_s$ and how is it computed?). Give explicit expressions (even parametric) for $A_s$, $n_s-1$, $r$, and a leading estimate of $f_{\rm NL}$, and reconcile the stated $H$ value with $A_s$ by including the missing prefactors or a model-specific suppression/enhancement mechanism.
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The information-theoretic bound $N_e N_s\lesssim M_{\rm Pl}^2/H^2$ (Sec. 3.3; Eq. (6)) is plausible as a scaling estimate but its derivation assumes approximations that conflict with other statements in the paper. The step $N_e=H t_{\rm qb}$ with $t_{\rm qb}\sim N/(N_s H)$ assumes $H$ and $N$ are roughly constant while $Q_{\rm mem}$ grows, whereas elsewhere the model relies on a slow drift in $H$ along the attractor to generate a red tilt (Sec. 3.1). The relation to the species bound Eq. (5) is also asserted rather than clearly separated (are these independent constraints or two faces of the same cutoff logic?).
Recommendation: Either (i) justify the constant-$H$/constant-$N$ approximation quantitatively (e.g., show drift is small over the quantum-breaking time in the viable corridor), or (ii) re-derive the bound using $N_e=\int H(t)dt$ with the explicit $H(N(t))$ relation and the actual $\dot N,\dot Q$ equations. Clarify which assumptions are required for Eq. (5) (species cutoff) versus Eq. (6) (memory/quantum-breaking) and whether one implies the other in your setup.
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The “dynamical selection” of $H$ from $N_s$ depends critically on an ad hoc ansatz $Q_{\rm mem}(H)\sim N_s F(M_{\rm Pl}/H)$ and on using $Q_{\rm mem}\sim N$ to infer Eq. (4) (Secs. 2.4, 3.2). While algebraically consistent (e.g., power-law $F(x)=x^\delta$), the model does not provide a microscopic argument for why $Q_{\rm mem}$ should scale as a chosen function of the horizon size, or why the linear case $\delta=1$ is preferred. The observation that holographic scaling $\delta=2$ forces $N_s\sim 1$ further underscores that predictions are dominated by the saturation criterion and the choice of $F$, not derived dynamics.
Recommendation: Add a dedicated justification (or at least a controlled model) for $Q_{\rm mem}(H)$: what degrees of freedom are being counted, what sets their density of states, and why does their cumulative “memory” scale as $F(M_{\rm Pl}/H)$? If a first-principles derivation is not available, demonstrate robustness: show which qualitative outcomes (viable $N_e$, predicted $H$, corridor existence) persist across broad families of $F$ and which are artifacts of specific $\delta$. Clearly separate what is a postulate vs. a derived consequence (suggested in Sec. 2.4 and Sec. 3.2).
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Conceptual consistency and EFT/energy-budget questions are underdeveloped. The “memory backreaction pressure” is introduced without an effective stress-energy description or an explicit modification of Einstein’s equations, making it unclear how energy conservation and backreaction are implemented consistently (Secs. 2.1–2.2; Sec. 3.4). The reheating claim—efficient energy transfer to $N_s$ species at quantum breaking—is asserted without a mechanism or parametric estimate of transfer/thermalization and homogeneity.
Recommendation: Clarify the regime of validity (large $N$, weak coupling, adiabaticity) and provide an effective description of how the memory term sources background evolution (e.g., an effective $\rho$ and $p$ contribution, or a coarse-grained equation consistent with Bianchi identities). For reheating, provide at least parametric estimates of the energy available at breaking, the rate into species, and whether it yields a thermal radiation bath; state any conditions under which reheating is prompt/slow and how this impacts $N_e$ and observables.
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Numerical results and the “stability corridor” in $(N_s,H)$ space (Secs. 2.3, 3.1–3.3; Figs. 1–3) are not reproducible from the description. Key missing items include: the exact ODEs integrated (and whether $H$ is fixed or updated from $N$), initial conditions for $N(t_0)$ as well as $Q_{\rm mem}(t_0)$, the definition of e-folds used in the code, solver choice/tolerances, stopping criteria for quantum breaking, and how a scan over “input $H$” is performed if $H$ is determined by $N$.
Recommendation: Add a concise “Numerical methods” subsection (or an appendix) listing the full dynamical system, variable definitions, parameter values (including $\gamma$), initial conditions, solver (e.g., RK45/LSODA), tolerances, time/e-fold integration range, event detection for $Q_{\rm mem}=N$, and how $N_e$ is computed. Ensure the figure captions contain enough detail to reproduce each panel. Consider providing code or pseudocode.