{"id":"9e3ec514-fd51-45ef-a491-985403a2a1f0","arxiv_id":"2608.05079","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Gravitational-wave constraints on the reheating temperature rule out single-exponential nonminimally coupled quintessential inflation and require a double-exponential coupling that predicts thawing dark energy with w0 near -0.9 to -0.95.","lead":"One scalar field is proposed to drive both the early universe's inflation and today's accelerated expansion. This paper finds that gravitational waves generated when inflation ends place a lower limit on the reheating temperature, and that this limit rules out the simplest exponential version of the model, requiring a two-part exponential shape.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantitative no-go f0≲0.3 rests on the GW coefficient C(f0), N_end, and ρ_ϕ,kin, none of which are specified; Θ_GW depends exponentially on N_end, so the exclusion line is not reproducible as stated.","rationale":"The reader's weakest assumption correctly identifies the unstated numerical pipeline behind the GW bound as the least secure step in the central argument. My independent estimate confirms that the conversion from C(f0) to the reheating-temperature bound is exponentially sensitive to N_end, which the paper never specifies. This makes the headline number f0 ≲ 0.3 M_Pl non-reproducible, although the qualitative no-go (λ0^2 ≫ 3 → matter-scaling attractor) would survive moderate changes in N_end. I also noticed that Eqs. (4.14)–(4.19) in Sec. 4.2 appear inconsistent with Eq. (4.9)–(4.10) and with Table 1 (the displayed ε formula has the wrong power of N), suggesting the analytic slow-roll approximations are not the ones used for the table and figures. However, that inconsistency affects the inflationary predictions and V0 normalization only logarithmically in D_req, so it is secondary to the missing GW pipeline for the central claim. The reader's CONDITIONAL verdict is therefore appropriate: the quantitative bound needs the missing details, but the central physical picture is not overturned.","tokens_in":32279,"tokens_out":34014,"duration_ms":385944,"concrete_test":"Reproduce the gravitational particle-production calculation for the single-exponential model at f0 = 0.3 M_Pl: specify the Bunch-Davies initial condition, the background a(η) across the inflation–kination transition, and the integration range in x = k/(a_end H_end); extract C, N_end, H_end, and ρ_ϕ,kin at the onset of kination; then compute Θ_GW = 200 C H_end^4 e^{4N_end}/ρ_ϕ,kin and TRH^GW from Eq. (3.43). Re-derive the f0 bound from the intersection with the matching condition. If the resulting bound moves by more than ~30% from 0.3 M_Pl, the paper must report the missing numerical inputs and the stated f0 ≲ 0.3 limit needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound f0≲0.3 M_Pl and the consequent λ0^2≳44 are derived by combining the gravitational-wave lower bound on T_RH (Eqs. 3.30, 3.42, 3.43) with the single-exponential matching condition (Eq. 4.31). The GW abundance is ρ_GW,kin = 2 C(f0) H_end^4 e^{4N_end}, and the scalar abundance is ρ_ϕ,kin ≈ 3 M_Pl^2 H_kin^2 = 3 M_Pl^2 H_end^2 e^{6N_end}, so Θ_GW = (200 C/3) (H_end^2/M_Pl^2) e^{-2N_end}. The paper reports C(f0) in Fig. 2 but does not state the mode-equation setup (initial vacuum, integration range, numerical method), the value of N_end between the end of inflation and the onset of kination, or the values of H_end and ρ_ϕ,kin used to convert C into T_RH^GW. Because the ratio enters through e^{-2N_end}, a shift of a few e-folds in N_end changes Θ_GW by orders of magnitude, moving the f0 exclusion line substantially (e.g., from roughly 0.1 to 0.3 M_Pl for N_end between −5 and 0 in a simple estimate). The qualitative statement that λ0^2 remains ≫3 is likely robust, but the specific quantitative claim λ0^2 ≳ 44, which the paper emphasizes, is not reproducible without these inputs. This is the weakest link in the chain leading to the claimed unavoidable matter-scaling attractor.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quintessential inflation in a nonminimally coupled scalar-tensor theory, parameterizing the post-inflationary radiation abundance independently of the reheating mechanism. It argues that the stochastic gravitational-wave background generated during the inflation-to-kination transition yields a lower bound on the reheating temperature through Delta N_eff, which in turn bounds the duration of kination and the post-inflationary field excursion. For the single-exponential coupling K(phi)=exp(-phi/f0), the required potential drop forces a steep average logarithmic slope, and since this slope equals the late-time slope, the field is driven to the matter-scaling attractor. A double-exponential deformation decouples the average slope from the asymptotic slope, and full numerical evolution for benchmark parameters gives a thawing equation of state w_phi,0 approximately -0.90 to -0.95. The central qualitative claim is that reheating history directly constrains the viability and late-time behavior of quintessential inflation.","tokens_in":32568,"tokens_out":20665,"duration_ms":266083,"significance":"If the quantitative results hold, the paper establishes a concrete and testable connection between the reheating temperature, the stochastic gravitational-wave background, and the present-day dark-energy equation of state. The analytic logic is coherent: the GW bound bounds the field excursion, a bounded excursion forces a steep average slope, and in the single-exponential model this slope is also the late-time slope, placing the model in the matter-scaling basin. The double-exponential deformation is a natural and minimal resolution, and the numerical evolution from inflation to the present epoch is a valuable consistency check. The authors are also explicit that the normalization A in the double-exponential model is fixed by requiring H(T0)=H0, so the reported w_phi,0 values are outputs rather than fitted quantities. The main weakness is that the quantitative GW production calculation underpinning the bounds f0 <~ 0.3 M_Pl and lambda0^2 >~ 44 is not reproducible from the information given, and the matching condition conflates the frozen potential with the present dark-energy density.","major_comments":[{"comment":"The quantitative GW bound that drives the main no-go argument is not reproducible. The coefficient C(f0) is said to be obtained by solving the mode equation, but the manuscript does not state the initial vacuum state, the integration range in x and conformal time, the numerical method, the criterion defining the end of inflation and onset of kination, or the values of H_end, rho_phi,kin, and N_end used in Eq. (3.30). The combination rho_phi,kin^{1/4} Theta_GW^{3/4} entering T_RH^GW is formally independent of N_end if one assumes rho_phi,kin = 3 M_Pl^2 H_end^2 e^{6 N_end}, but this cancellation is not shown, and the values of H_end and rho_phi,kin are still needed for every f0. Without these inputs the exclusion line in Fig. 5 and the derived constraints f0 <~ 0.3 M_Pl and lambda0^2 >~ 44 in Sec. 4.5 cannot be checked. Please provide a complete specification of the mode-equation calculation, a table of C(f0), H_end, rho_phi,kin, N_end, and the resulting T_RH^GW, and a numerical convergence check.","section":"Secs. 3.2-3.3, Eqs. (3.30), (3.42), (3.43), and Figs. 1-2"},{"comment":"The matching condition identifies the field value at freezing with the epoch at which the scalar energy density equals rho_Lambda,0, but in the same model the field thaws at late times and continues to roll. Since rho_phi decreases after thawing, a configuration with V(phi_freeze) = rho_Lambda,0 would generally end with rho_phi,0 < rho_Lambda,0, so Eq. (4.31) is not the exact present-day matching condition. This conflation affects the position of the white curve in Fig. 5 and the quantitative bound derived from Delta_phi = D_req f0/2. The qualitative conclusion (lambda0^2 much greater than 3 and the matter-scaling attractor) is robust because the correction is of order unity relative to D_req ~ 220, but the paper should state that Eq. (4.31) is a leading-order estimate and, if the quantitative f0 <~ 0.3 M_Pl is to be a claim, replace it with the actual condition rho_phi(T0) = rho_Lambda,0 in the numerical evolution.","section":"Secs. 4.3-4.4, Eqs. (4.23)-(4.31) and Fig. 5"}],"minor_comments":[{"comment":"The numerical evolution from inflation to the present epoch is not fully reproducible because the initial field value, initial field velocity, starting e-fold number, and the criterion for the end of inflation are not listed. Please state these explicitly or provide the code as supplementary material.","section":"Sec. 6.1, numerical initial conditions"},{"comment":"The triangular region labeled 'Allowed in the double-exponential model' is characterized only qualitatively. The paper should state the quantitative condition, for example A e^{-phi/f1} << e^{-phi/f0} during the early post-inflationary evolution and the range of A values that make each point in the region consistent with the observed dark-energy density.","section":"Sec. 5.3 and Fig. 5"},{"comment":"The relationship between the plotted spectrum and the coefficient C is clear from the captions, but the paper would benefit from stating the normalization convention for the mode function u_k and the vacuum choice in the text, since C(f0) is the key input to the reheating bound.","section":"Figs. 1-2 and Sec. 3.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one-sentence take: the single-exponential model is very likely dead on the qualitative argument, but the quantitative f0≲0.3 M_Pl bound is not reproducible as stated, and Table 1 doesn't match Eq. (4.22).\n\nWhat's new: the clean connection between the GW/ΔN_eff lower bound on reheating and the late-time dark-energy equation of state in nonminimally coupled quintessential inflation. The matching condition forces the average post-inflationary slope to equal the local slope for the single-exponential model, sending the field to the matter-scaling attractor. The double-exponential deformation that decouples these slopes is a natural and concrete fix, with numerical benchmarks giving w0 around -0.9 to -0.95.\n\nWhat it does well: the analytic chain is transparent, the numerics confirm the benchmark, and the paper is careful about frame issues and prior literature.\n\nSoft spots, in proportion:\n1. The exclusion line f0≲0.3 and the emphasized λ0^2≳44 depend on C(f0), N_end, and ρ_φ,kin, none of which are specified. Θ_GW scales as e^{-2N_end}, so a few e-folds in the inflation-kination transition shifts the bound by orders of magnitude. The qualitative no-go survives—even f0=0.6 gives λ0^2~11, still deep in matter scaling—but the specific number 44 shouldn't be sold as robust.\n2. Table 1 appears to be off by a factor of about 9-10 relative to Eq. (4.22). That needs a check.\n3. The double-exponential viable region in Fig. 5 is asserted but not scanned; a parameter scan or at least the allowed range of A would strengthen the claim.\n\nThe circularity burden is low: the dark-energy scale fixes A, but w0 is an output, and λ1 is taken from DESI fits.\n\nFor whom: anyone working on quintessential inflation, gravitational reheating, or GW backgrounds from kination. It deserves a serious referee; the gaps are fillable with details (mode-equation setup, N_end, table fix). I'd send it to review.","headline":"The single-exponential no-go is qualitatively solid, but the quantitative f0 bound is not reproducible without the missing GW-mode and transition details.","tokens_in":33199,"tokens_out":7039,"would_cite":true,"duration_ms":73912,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A gravitational-wave bound on reheating rules out the minimal single-exponential quintessence model and predicts thawing dark energy with $w_{\\varphi,0}\\simeq(-0.90,-0.95)$ in a double-exponential extension.","keywords":["quintessential inflation","nonminimal coupling","scalar-tensor theory","kination","stochastic gravitational-wave background","reheating temperature","dark-energy equation of state","thawing quintessence"],"falsifier":"Recompute $C(f_0)$ by solving the tensor-mode equation across the actual inflation–kination background, specifying the vacuum state, the integration range, the e-fold count $N_{\\mathrm{end}}$ between the end of inflation and the onset of kination, and $\\rho_{\\varphi,\\mathrm{kin}}$, then check whether $\\rho_{r,\\mathrm{kin}}\\ge100\\,\\rho_{\\mathrm{GW,kin}}$ still yields the exclusion $f_0\\lesssim0.3\\,M_{\\mathrm{Pl}}$ shown in Fig. 5; if $C$ at $f_0\\simeq0.3\\,M_{\\mathrm{Pl}}$ differs by an order of magnitude, the single-exponential no-go moves or disappears. An independent observational check: if future dark-energy surveys pin the present equation of state below $-0.95$ or confirm phantom crossing at $z\\simeq0.4$, the double-exponential benchmark, which requires $w_{\\varphi,0}\\in(-0.95,-0.90)$ with $w_\\varphi\\ge-1$ at all redshifts, is ruled out.","tokens_in":31995,"feed_emoji":"🌌","tokens_out":26031,"duration_ms":255563,"temperature":0.7,"pith_summary":"This paper sets out to show that the reheating epoch after inflation and the present acceleration of the Universe are governed by one and the same constraint. The nonadiabatic transition from inflation into kination — an epoch where the scalar field's kinetic energy dominates the expansion — generates a stochastic gravitational-wave background that redshifts as dark radiation, so limits on the effective number of extra relativistic species, $\\Delta N_{\\mathrm{eff}}$, impose a lower bound on the reheating temperature. That temperature fixes how long kination lasts and therefore how far the quintessence field can roll before freezing, which in turn fixes how steep the potential must be, on average, to fall from the inflationary scale to today's dark-energy density. In the minimal single-exponential coupling the required steepness is also the late-time slope, so the field is driven onto the matter-scaling attractor — where it behaves like matter with $w_\\varphi\\to0$ — instead of causing acceleration. The paper shows that a double-exponential coupling removes the conflict by separating the average slope from the asymptotic slope, producing thawing quintessence with $w_{\\varphi,0}\\simeq(-0.90,-0.95)$, and if it is right, dark-energy surveys become a direct probe of the reheating history.","feed_headline":"Gravitational waves tie reheating to dark energy's equation of state","feed_subtitle":"A dark-radiation bound rules out the single-exponential model and predicts thawing quintessence with w near −0.9.","key_machinery":"The chain that carries the argument has three links. First, the nonminimal coupling ansatz $F(\\phi)=1+K(\\phi)$, $V(\\phi)=V_0K^2(\\phi)$, whose Einstein-frame potential $\\tilde V = V_0K^2/(1+K)^2$ interpolates between an inflationary plateau and a runaway tail; after kination the conformal factor is essentially unity, so the Jordan-frame field $\\phi$ and the canonical field $\\varphi$ coincide. Second, the logarithmic slope $\\lambda_\\varphi \\equiv -M_{\\mathrm{Pl}}\\,d\\ln\\tilde V/d\\varphi$ and the matching condition: the integrated drop $\\int\\lambda_\\varphi\\,d\\varphi/M_{\\mathrm{Pl}}$ over the excursion $\\Delta\\varphi$ must equal $D_{\\mathrm{req}}$, so the average slope $\\bar\\lambda = D_{\\mathrm{req}}M_{\\mathrm{Pl}}/\\Delta\\varphi$ is the quantity the potential must reproduce. Third, the gravitational-wave coefficient $C(f_0)$, computed by solving the tensor mode equation with Bogoliubov coefficients across the nonadiabatic inflation–kination transition, enters the bound $\\rho_{r,\\mathrm{kin}} \\ge 100\\,\\rho_{\\mathrm{GW,kin}}$; this is what caps $\\Delta\\varphi$ and hence sets the required $\\bar\\lambda$. The single-exponential coupling $K=e^{-\\phi/f_0}$ has a constant slope $\\lambda_0=2M_{\\mathrm{Pl}}/f_0$, so the cap on $\\Delta\\varphi$ forces $f_0\\lesssim0.3\\,M_{\\mathrm{Pl}}$ and $\\lambda_0^2\\gtrsim44$; in the exponential-quintessence attractor analysis, $\\lambda_0^2 > 3(1+w_b)$ selects the background-scaling solution, so the field ends with $w_\\varphi=w_b$ and $\\Omega_\\varphi=3/\\lambda_0^2$, which during matter domination is the matter-scaling branch. The double-exponential deformation $K=e^{-\\phi/f_0}+Ae^{-\\phi/f_1}$ keeps the average slope steep over the kination trajectory while the shallower exponential softens the asymptotic slope to $\\lambda_1=2M_{\\mathrm{Pl}}/f_1=0.7$ near the freezing point, and that decoupling is what permits thawing quintessence — a field frozen near $-1$ that starts rolling only at low redshift.","core_discovery":"The central claim, stated on the paper's own terms, is that the gravitational-wave background generated at the inflation–kination transition constrains the post-inflationary field excursion enough to decide the viability of quintessential inflation. Concretely, the GW abundance fixes a minimum radiation fraction at the onset of kination, $\\rho_{r,\\mathrm{kin}} \\ge 100\\,\\rho_{\\mathrm{GW,kin}}$, which becomes a lower limit on $T_{\\mathrm{RH}}$ and an upper limit on the excursion $\\Delta\\varphi \\simeq \\sqrt{6}\\,M_{\\mathrm{Pl}}(N_{\\mathrm{RH}}+\\ln 2)$. The potential must drop by $D_{\\mathrm{req}} = \\ln[\\tilde V(\\varphi_{\\mathrm{kin}})/\\rho_{\\Lambda,0}] \\sim 216$–$242$ over this excursion, so the average logarithmic slope must be $\\bar\\lambda = D_{\\mathrm{req}}M_{\\mathrm{Pl}}/\\Delta\\varphi$. For the single-exponential coupling the local slope is constant, $\\lambda_0 = 2M_{\\mathrm{Pl}}/f_0$, so the matching condition $\\bar\\lambda \\simeq \\lambda_0$ forces $f_0 \\lesssim 0.3\\,M_{\\mathrm{Pl}}$, meaning $\\lambda_0^2 \\gtrsim 44$; the standard attractor analysis of exponential quintessence then places the field unavoidably on the matter-scaling solution with $w_\\varphi \\to 0$ and $\\Omega_\\varphi \\to 3/\\lambda_0^2 \\lesssim 0.07$. The double-exponential coupling $K = e^{-\\phi/f_0} + A\\,e^{-\\phi/f_1}$ leaves the average slope steep while the shallower exponential softens the asymptotic slope to $\\lambda_1 = 0.7$ near freezing, and the full numerical evolution from inflation to the present epoch yields thawing quintessence with $w_{\\varphi,0} \\simeq (-0.90,-0.95)$, a deviation from $-1$ large enough for current and forthcoming dark-energy surveys to see.","pith_inferences":["The excursion-cap logic is mechanism-independent: any reheating channel that leaves an irreducible relativistic relic, whether gravitational particle production, evaporating black holes, or another dark-radiation source, caps the post-inflationary field displacement the same way, so the qualitative no-go for constant-slope potentials should survive across reheating models with only the numerical c","A future detection of the kination-era gravitational-wave background would measure $C(f_0)$ and thereby the reheating temperature directly, and combined with a precise measurement of $w_{\\varphi,0}$ the double-exponential scenario would be testable in two independent channels at once.","The double-exponential device acts as a late-time uplift that tunes the frozen potential to the observed dark-energy density without moving any earlier observable, and the same perturbative trick could be ported to other non-oscillatory inflation models to see whether their predictions shift in the same direction.","Because the paper sets the conformal coupling of $\\varphi$ to nonrelativistic matter to zero in the numerics, the natural next step is to quantify how strong that coupling could be before the predicted $w_{\\varphi,0}\\simeq-0.9$ moves, so that dark-energy surveys and fifth-force tests would jointly bound the Jordan-frame matter coupling."],"forward_implications":["The minimal single-exponential nonminimal coupling is excluded as a complete theory of inflation plus dark energy: every parameter set that reproduces the dark-energy scale sends the field to the matter-scaling attractor with $\\Omega_\\varphi\\lesssim0.07$, far below the observed dark-energy abundance.","Tighter future bounds on $\\Delta N_{\\mathrm{eff}}$ (CMB-S4 at $\\sigma\\simeq0.06$) push the lower limit on $T_{\\mathrm{RH}}$ upward, shorten the allowed kination duration, and steepen the required average slope, strengthening the case for the double-exponential form.","Within the double-exponential framework, measuring the present-day equation of state at $w_{\\varphi,0}\\simeq-0.9$ effectively measures the reheating temperature, because the matching condition ties the two together.","The model predicts $w_\\varphi\\ge-1$ at all redshifts, so it is cleanly distinguishable from the DESI DR2+Planck+DESY5 CPL fit, which crosses the phantom divide at $z\\simeq0.4$.","The constraint $f_0\\lesssim0.3\\,M_{\\mathrm{Pl}}$ pushes the scalar spectral index onto the $\\tilde N_\\star$-dependent floor $n_s\\to1-(9+4\\tilde N_\\star)/(2\\tilde N_\\star^2)$, giving CMB measurements of $n_s$ an independent handle on the allowed region."],"supporting_citations":[{"why":"The single-exponential nonminimal coupling construction that the paper analyzes and then deforms into the double-exponential form.","marker":"[57]"},{"why":"The gravitational particle-production formalism (mode equation and Bogoliubov coefficients) from which the gravitational-wave abundance and the coefficient C(f0) are computed.","marker":"[29]"},{"why":"The dynamical-systems analysis of exponential quintessence whose scaling-solution thresholds decide whether the field accelerates or tracks the background.","marker":"[82–84]"},{"why":"The BBN and CMB limits on the effective number of relativistic species that anchor the fiducial dark-radiation bound adopted in the paper.","marker":"[98, 99]"},{"why":"The CMB-S4 forecast reaching a precision of about 0.06 on the effective number of relativistic species, which justifies the fiducial bound and the claim that future surveys will sharpen the reheating-temperature constraint.","marker":"[100]"},{"why":"The double-exponential quintessence precedent whose early-scaling-to-late-acceleration structure motivates the paper's deformation.","marker":"[109]"},{"why":"The DESI-based fits to exponential quintessence that motivate the benchmark late-time slope of 0.7.","marker":"[110, 111]"},{"why":"The DESI DR2 cosmological constraints whose CPL fits bracket the model's predicted present-day equation of state and set the observational comparison.","marker":"[85]"}],"fun_headline_variants":["Gravitational waves link reheating to dark energy's fate","GW background probes inflation's reheating and dark energy","Double-exponential quintessence survives GW constraints","Reheating temperature pinned by dark radiation and quintessence","Future dark energy surveys trace inflation's reheating"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole constraint chain rests on the numerically computed gravitational-wave production coefficient $C(f_0)$, and the paper does not show the mode-equation setup, the number of e-folds between the end of inflation and the onset of kination, or the value of $\\rho_{\\varphi,\\mathrm{kin}}$ used to convert $C$ into the reheating-temperature bound, so the exclusion line $f_0\\lesssim0.3\\,M_{\\mathrm{Pl}}$ stands on numerical details that are not displayed.","fun_headline_variants_meta":{"raw":{"variants":["Gravitational waves link reheating to dark energy's fate","GW background probes inflation's reheating and dark energy","Double-exponential quintessence survives GW constraints","Reheating temperature pinned by dark radiation and quintessence","Future dark energy surveys trace inflation's reheating"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000937,"raw_usage":{"total_tokens":4127,"prompt_tokens":1182,"completion_tokens":2945,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":798,"completion_tokens_details":{"reasoning_tokens":2867}},"tokens_in":798,"tokens_out":2945,"duration_ms":25150,"temperature":1.0,"reasoning_tokens":2867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:50:58.978818+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute $C(f_0)$ by solving the tensor-mode equation across the actual inflation–kination background, specifying the vacuum state, the integration range, the e-fold count $N_{\\mathrm{end}}$ between the end of inflation and the onset of kination, and $\\rho_{\\varphi,\\mathrm{kin}}$, then check whether $\\rho_{r,\\mathrm{kin}}\\ge100\\,\\rho_{\\mathrm{GW,kin}}$ still yields the exclusion $f_0\\lesssim0.3\\,M_{\\mathrm{Pl}}$ shown in Fig. 5; if $C$ at $f_0\\simeq0.3\\,M_{\\mathrm{Pl}}$ differs by an order of magnitude, the single-exponential no-go moves or disappears. An independent observational check: if future dark-energy surveys pin the present equation of state below $-0.95$ or confirm phantom crossing at $z\\simeq0.4$, the double-exponential benchmark, which requires $w_{\\varphi,0}\\in(-0.95,-0.90)$ with $w_\\varphi\\ge-1$ at all redshifts, is ruled out.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The single-exponential nonminimal coupling construction that the paper analyzes and then deforms into the double-exponential form."}],"review_version":1}