{"id":"d927215a-8777-4b28-82ba-3acafe0c9635","arxiv_id":"2505.09337","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A single small-scale perturbation can form ultralight black holes during a break in inflation and heavier black holes after inflation, linking reheating to dark matter.","lead":"This paper proposes a two-stage inflation model where one small-scale perturbation falls back inside the horizon twice, first forming ultralight black holes that evaporate and reheat the universe, then forming heavy black holes that survive as dark matter. A reader might care because the model predicts two distinct gravitational wave peaks, one in the LISA band and one in the DECIGO band, that could be tested.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"All-CDM claim rests on an undocumented transfer-function computation; the derived k* > 8.24 k0 threshold is exponentially sensitive to transition details and needs a reproducible numerical check.","rationale":"The reader's weakest assumption identifies the suppression mechanism as the load-bearing element, and I agree that the robustness of the suppression window is the critical issue. However, the reader's specific complaint that the AR = 1 example 'over-suppresses' PBH2 is not fully accurate: using Eq. (20) with AR = 1, Delta = 0.5, delta_th = 0.4 and k* = 8.24 k0 gives beta2/beta1 ≈ 6e-17, i.e., exactly the Condition 2 bound for M1 = 1e8 g, M2 = 1e17 g, so that parameter point can place PBH2 at the full-CDM abundance. The deeper problem is that this point is exponentially sensitive: the exponent in Eq. (20) is large, so modest uncertainties in the transfer amplitude or threshold change the PBH2 abundance by orders of magnitude. The numerical transfer function is computed for an unspecified smooth transition and is not provided in reproducible form, and the analytic lognormal approximation leading to Eq. (20) is not derived in enough detail to verify the threshold. There is also an internal calibration issue between the AR and beta1 values in Fig. 6 and the Press-Schechter formula, which undermines the plotted GW amplitudes. These issues do not invalidate the scenario, but they mean the central claim is conditional on supplying a reproducible transfer-function calculation and a consistent parameter set. The reader's CONDITIONAL verdict is therefore appropriate, and my stress-test does not change it.","tokens_in":13740,"tokens_out":39792,"duration_ms":376778,"concrete_test":"Make the transfer-function code available or re-implement Eq. (16) with explicit transition profiles (e.g., tanh forms with widths 0.5, 1, and 2 e-folds) and Delta N = 9 e-folds; directly integrate the density-contrast variances for PBH1 and PBH2 with a top-hat window function from the numerical P_R(k;t_f). Then compute beta2/beta1 via Press-Schechter with delta_th = 0.4 and 0.45 for the AR = 1, Delta = 0.5 peak. Check whether the value k* = 8.24 k0 (and a more realistic AR ~ 0.02, beta1 = 10^-2 point) reproduces beta2/beta1 within a factor of 3 of the Condition 2 bound. If the required k*/k0 shifts by more than 30%, or beta2/beta1 by more than a factor of 10, under transition-width variation, the all-CDM claim is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires beta2/beta1 ≈ 6e-17 (M1/1e8 g)^{5/2} (M2/1e17 g)^{-1} (Condition 2, Eq. 13). In the Press-Schechter estimate, Eq. (20), this ratio contains an exponential whose argument is O(40) at the CDM point; a 10% change in the transferred amplitude A(t_re,2), or in delta_th, changes beta2/beta1 by a factor of order e^4 ≈ 50 or more. The only input that sets A(t_re,2) is the numerical transfer function T(k) of Sec. II.B/Fig. 4, computed for one undocumented 'smooth' transition profile (w from -0.999 to 1/3 and back, c_s^2 from 1 to 1/3 and to 1, duration O(1) e-fold). The analytic finite-width estimate leading to Eq. (20) approximates the transferred spectrum by a lognormal with peak value A_R(k0/k*2)^4; the derivation of the prefactor (k0/k*)^2 and the 4 Delta^2 term is not shown, and Eq. (19)'s e^{8 Delta^2} factor is not obviously consistent with directly integrating the numerical P_R(k;t_f) to obtain the density-contrast variance. If the variance is instead computed numerically from P_R(k;t_f), the threshold k* > 8.24 k0 could shift significantly. In addition, the AR = 0.05, beta1 = 10^-2 values used in Fig. 6 are not connected to the Press-Schechter prediction from Eq. (18): for AR = 0.05, delta_th = 0.4, Delta = 0.5, sigma_1^2 = (16/81)*0.05*e^2 = 0.073, giving beta1 ≈ 0.14 with gamma = 0.2, not 0.01. Thus the all-CDM parameter point is a fine-tuned, not-yet-reproducible example, and the plotted GW amplitudes are not calibrated to a consistent abundance calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that in a two-stage inflationary scenario with an intermediate non-inflationary break, a single comoving scale can cross the Hubble horizon twice: first during the break stage and again after the end of inflation. The first crossing produces ultralight PBHs (PBH1) whose Hawking evaporation can reheat the universe, while the second produces heavier PBHs (PBH2) that can constitute cold dark matter. The authors derive mass and abundance relations (Conditions 1 and 2), compute a transfer function for the curvature perturbation through the break, and show that the associated induced gravitational wave background has two peaks detectable by LISA, DECIGO, ET, and LIGO A+. The central quantitative claim is that a suppression of the curvature amplitude between the two re-entries, approximated as P_R ∝ k^{-4}, can realize the required ratio β2/β1 ≈ 6×10^{-17}.","tokens_in":14279,"tokens_out":14215,"duration_ms":130630,"significance":"If the central claim were quantitatively established, the paper would present an appealing unified scenario connecting inflation, reheating, and dark matter through a single mechanism, with a falsifiable double-peak gravitational wave signature. The kinematic relations in Sec. II A, the mass bounds from Conditions 1 and 2, and the qualitative idea of dual horizon re-entries are clear and internally consistent. The paper also correctly emphasizes that the abundance ratio is exponentially sensitive to the curvature amplitude and collapse threshold. However, the quantitative demonstration of the suppression factor is not yet convincing: the transfer-function calculation is not reproducible as presented, the analytic finite-width formula has internal inconsistencies, and the parameter example used for the gravitational wave plots is not calibrated to the Press-Schechter abundance. These issues directly affect the load-bearing claim that PBH2 can be all of the cold dark matter while PBH1 reheats the universe.","major_comments":[{"comment":"The finite-width ratio β2/β1 in Eq. (20) does not follow from the preceding equations. Using the paper's own lognormal approximation and the transfer-function scaling P_R(k;t_f) = P_R^peak(k)(k/k0)^{-4}, the peak of the transferred spectrum is at k⋆2 = e^{-4Δ²}k⋆, but the value at the peak is A_R e^{-8Δ²}(k0/k⋆2)^4/(√(2π)Δ), not A_R(k0/k⋆2)^4/(√(2π)Δ) as stated after Eq. (19). Direct integration with the density-contrast factor (k/k⋆2)^4 gives σ2² = (16/81) A_R e^{16Δ²}(k0/k⋆)^4, whereas Eq. (19) contains an extra factor e^{8Δ²}. Neither this corrected expression nor the expression in the paper reduces to the prefactor (k0/k⋆)^2 and the exponent 4Δ² + (81/32)e^{-16Δ²}δ_th² A_R^{-1}[e^{8Δ²} - (k⋆/k0)^4] displayed in Eq. (20). Therefore the numerical threshold k⋆ > 8.24 k0, which is used to claim that Condition 2 is satisfied, is not supported by the displayed derivation.","section":"Sec. II.B, Eq. (20)"},{"comment":"The transfer function T(k) shown in Fig. 4 is the only quantitative input that suppresses β2, but the numerical setup is not documented. The caption specifies only 'smooth transitions' of w from −0.999 to 1/3 and back, with c_s² evolving from 1 to 1/3 and to 1, and durations of O(1) e-folds; no profile functions, initial conditions, discretization, or convergence checks are given. Moreover, the analytic replacement P_R(k;t_f) ≈ P_R^peak(k)(k/k0)^{-4} is applied for all k in the lognormal spectrum, including k < k0 where Fig. 4 shows the transfer function returning to unity. Because the Press-Schechter abundance is exponentially sensitive to the amplitude, even a small unsuppressed low-k tail can dominate β2; the paper does not quantify this contribution. The robustness of the suppression window to the transition profile and to the choice of k0 must be demonstrated, ideally with a reproducible numerical calculation or an analytic model with controlled errors.","section":"Sec. II.B, Fig. 4 and following text"},{"comment":"The parameter choices used for the gravitational wave predictions are not consistent with the Press-Schechter abundance formula used elsewhere. For A_R = 0.05, Δ = 0.5, and δ_th = 0.4, Eq. (18) gives σ1² = (16/81) × 0.05 × e² ≈ 0.073, so β1 ≈ γ erfc[δ_th/(√2 σ1)] ≈ 0.2 × erfc(1.05) ≈ 0.03, not the β1 = 10^{-2} adopted in Fig. 6. Conversely, the example in the text that satisfies Condition 2, A_R = 1 and k⋆ > 8.24 k0, has A1 ≈ A_R e^{8Δ²} ≈ 7.4, which is not a rare Gaussian fluctuation and lies outside the regime where the Press-Schechter formula is reliable. The paper therefore does not present a single parameter point where the required β2/β1 ≈ 6×10^{-17}, the consistency of β1 with the density-variance calculation, and the plotted gravitational wave amplitudes are all mutually consistent.","section":"Sec. II.B, Eqs. (18)-(20) and Fig. 6"},{"comment":"The statement that A(t_re,2) ≲ 10^{-2} 'provided that A(t_re,1) ≃ 10^{-1}' is necessary but not sufficient for Condition 2. For A(t_re,1) = 0.1 and δ_th = 0.4, the ratio in Eq. (15) is β2/β1 ≈ (A2/0.1) exp[-20.3(A2^{-1} - 10)], and the required value 6×10^{-17} is reached only in a narrow interval of A2. The paper does not translate the numerical transfer function into a predicted value of A(t_re,2) with the accuracy needed for an exponentially sensitive quantity. This gap, together with the issues in Eqs. (19)-(20), means the central assertion that the mechanism can realize the dark-matter abundance is not quantitatively established.","section":"Sec. II.B, text after Eq. (15)"}],"minor_comments":[{"comment":"The sentence containing 'we assume radiation domination during both formation stages we assume wA = wB = 1/3' has a duplicated 'we assume' and should be edited.","section":"Sec. II.B"},{"comment":"The caption should specify the transition profile functions, the duration of each transition, and how k0 is determined from T(k0) = 1; currently the description is too vague to reproduce the curve.","section":"Sec. II.B, Fig. 4"},{"comment":"The text says the approximate equality sign in Eq. (13) corresponds to f_PBH = 1, but the equation is written as an inequality; please clarify whether the equality is meant as the maximum allowed value or as a separate benchmark.","section":"Sec. II.B, Eq. (13)"},{"comment":"The abstract says PBHs form at 'nearly the same comoving scales', but the finite-width calculation places the second peak at k⋆2 = e^{-4Δ²}k⋆, which for Δ = 0.5 is about 0.37 k⋆; please reconcile the wording with the shifted peak.","section":"Abstract and Sec. II.B"},{"comment":"The peak frequency f⋆ uses k⋆, but if PBH2 forms predominantly from modes near k⋆2, the peak of the induced GW spectrum should be evaluated at the corresponding frequency; this point should at least be discussed.","section":"Sec. III, Eq. (27)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a genuinely new idea: the same comoving scale, re-entering the horizon twice in a two-stage inflation scenario, can produce two PBH populations—an ultra-light one that evaporates and reheats the universe, and a heavy one that serves as dark matter. The analytic skeleton in Sec. II is clean: the mass relation and the two necessary conditions (1) and (2) follow from simple horizon and energy-density arguments, and the two-peak GW spectrum is a concrete, falsifiable target. I have not seen this specific combination in the earlier interrupted-inflation literature cited here.\n\nThe soft spot is the suppression mechanism. The entire viability of the CDM family rests on the transfer function T(k) that damps the curvature perturbation between the first and second re-entry. This T is computed numerically for one undocumented smooth transition profile and then approximated by a k^-4 scaling. From that approximation comes Eq. (20) and the threshold k* > 8.24 k0. The derivation of Eq. (20) is not shown in enough detail to reproduce it, and the result is exponentially sensitive: a 10% change in the transferred amplitude or in delta_th changes beta2/beta1 by factors of e^4 or more. The paper claims the threshold is satisfied for a specific example, but it does not demonstrate a parameter set where beta2/beta1 actually lands at the CDM value rather than far below it. The Fig. 6 example uses AR=0.05 with beta1=1e-2, but the Press-Schechter estimate from Eq. (18) gives beta1 closer to 2e-2 for that AR and delta_th=0.4, so the curves are not calibrated to a consistent abundance calculation. These are fixable with a reproducible transfer-function computation and a consistent parameter scan.\n\nThe paper is honest about what is new: it cites the existing interrupted-inflation papers, and the novelty is the dual formation and its reheating-plus-CDM outcome. There is no circularity in the definitions of mass or abundance.\n\nOverall, the idea deserves the attention of the PBH community, and the GW prediction is worth taking seriously. But the central quantitative claim needs the numerical support to be made transparent before the scenario can be called a viable model. A serious referee could push for that revision. I would send it to peer review rather than desk reject.","headline":"A novel dual-PBH scenario with a clean analytic skeleton, but the load-bearing suppression of the heavy PBH abundance rests on an undocumented numerical transfer function and a parameter example that is not internally consistent; it deserves peer review with a demand for reproducible numerics.","tokens_in":14806,"tokens_out":5144,"would_cite":false,"duration_ms":47227,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single inflationary scale can forge two black-hole families: ultra-light holes that reheat the universe and heavy ones that survive as dark matter.","keywords":["primordial black holes","two-stage inflation","reheating","dark matter","induced gravitational waves","quantum evaporation","curvature perturbation","break stage"],"falsifier":"To settle the central claim, compute the collapse threshold and smoothing window for the break and post-inflation equations of state rather than assuming $\\delta_{\\mathrm{th}}=0.4$ and a sharp lognormal peak, and check whether $\\beta_2/\\beta_1$ can stay below $6\\times10^{-17}$ while $\\beta_1$ satisfies Condition 1; observationally, a null detection of either predicted gravitational-wave peak by LISA or DECIGO in the $10^{-1}$ to $1$ Hz window would rule out the advertised parameter space.","tokens_in":13512,"feed_emoji":"🕳️","tokens_out":8546,"duration_ms":80826,"temperature":0.7,"pith_summary":"This paper argues that one particular fluctuation scale from inflation can produce two separate families of primordial black holes. The scale leaves the horizon during a first stage of inflation, falls back inside during a temporary break between two inflationary stages, and then leaves and re-enters again after inflation ends. The first re-entry makes ultra-light black holes ($M_1 \\lesssim 5\\times 10^8$ g) whose quantum evaporation can reheat the universe before Big Bang nucleosynthesis; the second re-entry makes heavier black holes ($M_2 \\gtrsim 10^{17}$ g) that can serve as all of the cold dark matter. If the scenario is right, reheating, dark matter, and a distinctive two-peak gravitational-wave background all trace back to a single mechanism, with peaks in the band of planned space-based gravitational-wave observatories.","feed_headline":"One fluctuation scale can forge two black-hole families","feed_subtitle":"Ultralight primordial holes from inflation's pause reheat the universe; their heavy siblings stay as dark matter.","key_machinery":"The load-bearing object is the double horizon crossing of one comoving mode $k_\\star$ in a two-stage inflationary model with a break. The mass scale at each crossing is set by $M_{\\mathrm{PBH}} = 4\\pi\\gamma M_{\\mathrm{pl}}^2/H(t_{\\mathrm{re}})$, so two different Hubble rates yield two different black-hole masses. The abundance of each family is set through the standard collapse-abundance formula $\\beta \\propto \\mathrm{erfc}[\\delta_{\\mathrm{th}}/(\\sqrt{2}\\sigma)]$, where $\\sigma^2$ is the smoothed density-contrast variance; the mechanism that makes the ratio small enough is the curvature transfer function $T(k) \\equiv P_{\\mathcal R}(k,t_f)/P_{\\mathcal R}(k,t_{\\mathrm{re},1})$, obtained by solving the linear curvature perturbation equation $\\mathcal R''_k + 2(z'/z)\\mathcal R_k' + c_s^2 k^2 \\mathcal R_k = 0$, which the authors find behaves as $k^{-4}$ in the suppression band. That $k^{-4}$ suppression, combined with the exponential sensitivity of $\\beta$ to $\\delta_{\\mathrm{th}}/\\sigma$, is what lets the heavy black-hole abundance fall below the required ceiling while remaining large enough to be all of the cold dark matter.","core_discovery":"The central discovery claim is the 'dual' formation: at a fixed comoving wavenumber $k_\\star$, the Hubble horizon is crossed twice, so the same primordial curvature perturbation seeds black holes at two epochs. During the non-inflationary break between the two inflationary stages, the mode re-enters at a high Hubble rate $H_{\\mathrm{re},1}$, forming PBH1 with mass $M_1 = 4\\pi\\gamma M_{\\mathrm{pl}}^2/H_{\\mathrm{re},1} \\lesssim 5\\times 10^8$ g; after inflation it re-enters at a lower $H_{\\mathrm{re},2}$, forming PBH2 with $M_2 \\gtrsim 10^{17}$ g. The paper derives two necessary conditions: PBH1 must dominate the energy density and then evaporate before Big Bang nucleosynthesis (Condition 1), and PBH2 must remain subdominant at matter-radiation equality unless it is the dark matter (Condition 2, $\\beta_2/\\beta_1 \\lesssim 6\\times 10^{-17}$). It then shows numerically that the curvature power spectrum at the end of inflation can be suppressed relative to the first re-entry, scaling as $P_{\\mathcal R} \\propto k^{-4}$ in the relevant band, which can supply the needed suppression of $\\beta_2$ while keeping PBH2 abundant enough to be all of the dark matter. The observable signature is a two-peak stochastic gravitational-wave background, one peak from the isocurvature perturbations that source gravitational waves during PBH1 evaporation and one from second-order curvature-induced gravitational waves at PBH2 formation.","pith_inferences":["If the $k^{-4}$ suppression is a general feature of a break stage, then measuring the two peak heights in the gravitational-wave background would directly constrain the break's duration $\\Delta N$, its equation-of-state parameter $w_A$, and the smoothness of the transitions, quantities that are otherwise hard to access.","The same double-re-entry logic could produce more than two black-hole populations if inflation contains several pauses; each additional re-entry would add a peak, so the bi-peak spectrum is the minimal case of a broader family.","The paper's illustrative example ($A_R=1$, $\\Delta=0.5$, $\\delta_{\\mathrm{th}}=0.4$) leads to the condition $k_\\star > 8.24 k_0$, a threshold that depends on the assumed collapse threshold; a dedicated numerical-relativity threshold calculation for the break equation of state would either support or close this window.","Because the numerical transfer function is computed with $w_A=1/3$ and smooth transitions while the threshold argument allows $w_B>w_A$, the most direct theoretical test is to evolve the transfer function across the actual equation-of-state history and recompute both abundances self-consistently."],"forward_implications":["The same $k_\\star$ predicts two gravitational-wave peaks, one near $10^{-1}$ Hz from the evaporation of PBH1 and one near $10^{-1}$ to $1$ Hz from curvature-induced gravitational waves at PBH2 formation, placing both in the LISA, Taiji, TianQin, and DECIGO band.","If PBH1 mass is near $5\\times10^8$ g and PBH2 near $10^{17}$ g, the scenario yields both a Big Bang nucleosynthesis-safe reheating and $f_{\\mathrm{PBH}}=1$ cold dark matter, with an abundance ratio $\\beta_2/\\beta_1$ as small as $10^{-17}$.","With a radiation-like post-inflation epoch ($w_B=1/3$), the mass of the dark-matter black holes is bounded by $M_2 \\lesssim 10^{19}$ g when $M_1<5\\times10^8$ g; with a kination epoch ($w_B=1$), the bound relaxes to $M_2\\lesssim10^{23}$ g.","Because the abundance ratio is exponentially sensitive to the collapse threshold and the amplitude ratio, the scenario makes sharp, testable predictions about the relative heights of the two gravitational-wave peaks rather than a loose order-of-magnitude range."],"supporting_citations":[{"why":"Supplies the two-stage inflationary background with a temporary halt, the setting in which a mode can exit, re-enter, and exit the Hubble horizon again.","marker":"[19]"},{"why":"Provides the earlier analysis of primordial black hole formation during a break in inflation that the dual-scenario mass estimates and horizon re-entry logic build on.","marker":"[25]"},{"why":"Supplies the collapse-abundance formula $\\beta=\\gamma\\,\\mathrm{erfc}(\\delta_{\\mathrm{th}}/\\sqrt{2}\\sigma)$ that converts variance into the abundance ratio used in Conditions 1 and 2.","marker":"[49]"},{"why":"Supplies equation-of-state dependent collapse threshold values, used as the mechanism for suppressing PBH2 via $w_B>w_A$ and for the fiducial $\\delta_{\\mathrm{th}}=0.4$.","marker":"[50]"},{"why":"Establishes the isocurvature-induced gravitational-wave signal from an early PBH-dominated era, the basis for the evaporation peak from PBH1.","marker":"[14]"},{"why":"Provides the second-order gravitational-wave amplitude for a lognormal curvature spectrum, used for the PBH2-formation peak.","marker":"[64]"},{"why":"Relates the UV cutoff of the PBH isocurvature spectrum to the peak gravitational-wave frequency today, used to place the peaks in the LISA and DECIGO bands.","marker":"[53]"},{"why":"Sets the observational asteroid-mass window in which primordial black holes can constitute all of the cold dark matter, the target range for PBH2.","marker":"[7]"}],"fun_headline_variants":["Two black-hole families from one cosmic scale","One scale seeds dual black-hole populations","Black-hole twins: one reheats, one dark matter","Inflation's pause forges two black-hole families","Single fluctuation yields two black-hole fates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the claim that the same fluctuation that made the ultra-light black holes is strongly damped by the time it re-enters the horizon a second time, by an amount large enough to keep the heavy black-hole abundance below its ceiling; small changes in the break equation of state, the smoothness of the transitions, the smoothing window, or the collapse threshold can move the abundance ratio by many orders of magnitude.","fun_headline_variants_meta":{"raw":{"variants":["Two black-hole families from one cosmic scale","One scale seeds dual black-hole populations","Black-hole twins: one reheats, one dark matter","Inflation's pause forges two black-hole families","Single fluctuation yields two black-hole fates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000887,"raw_usage":{"total_tokens":3877,"prompt_tokens":1043,"completion_tokens":2834,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":2764}},"tokens_in":659,"tokens_out":2834,"duration_ms":18145,"temperature":1.0,"reasoning_tokens":2764,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:37:29.742615+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To settle the central claim, compute the collapse threshold and smoothing window for the break and post-inflation equations of state rather than assuming $\\delta_{\\mathrm{th}}=0.4$ and a sharp lognormal peak, and check whether $\\beta_2/\\beta_1$ can stay below $6\\times10^{-17}$ while $\\beta_1$ satisfies Condition 1; observationally, a null detection of either predicted gravitational-wave peak by LISA or DECIGO in the $10^{-1}$ to $1$ Hz window would rule out the advertised parameter space.","supporting_citations":[{"cited_title":"Multiple Inflationary Stages with Varying Equation of State","cited_arxiv_id":"1207.3638","evidence_quote":"Provides the earlier analysis of primordial black hole formation during a break in inflation that the dual-scenario mass estimates and horizon re-entry logic build on."},{"cited_title":"See a proof in Appendix A","cited_arxiv_id":null,"evidence_quote":"Supplies equation-of-state dependent collapse threshold values, used as the mechanism for suppressing PBH2 via $w_B>w_A$ and for the fiducial $\\delta_{\\mathrm{th}}=0.4$."}],"review_version":1}