Negative Quintessence in Cosmology

• Negative quintessence is a term used in cosmology to describe models where the 'negative' modifier applies contextually to pressure, density, potential, or dynamical ratios.
• Different frameworks use negative quintessence to denote canonical scalar fields with negative pressure, fields with negative energy density, or fields with negative effective mass-squared, each with distinct implications for cosmic acceleration.
• Recent analyses show that careful parameter constraints from observational data (e.g., Planck, DESI) can distinguish these models and avoid phantom behavior while addressing dark energy dynamics.

Searching arXiv for the cited papers to ground the article in recent literature. Negative quintessence is not a single, uniform concept in the literature. Across cosmology, scalar–tensor theory, extra-dimensional models, and black-hole thermodynamics, the expression has been used for at least five distinct structures: canonical quintessence with negative pressure but positive energy density; a component with negative energy density and positive pressure; a quintessence field with negative effective mass-squared; non-minimally coupled quintessence in which a characteristic dynamical ratio is negative; and black-hole or string constructions in which the relevant “negative” quantity is a pressure term, a potential contribution, or an energy rescaling rather than the dark-energy density itself. The common thread is that the sign adjective modifies different objects in different frameworks, so the term is best interpreted contextually rather than universally (V. et al., 25 Apr 2025, Gómez-Valent et al., 1 Aug 2025, Chiba et al., 2010, Wolf et al., 2024).

1. Canonical negative-pressure quintessence

In one standard usage, “negative quintessence” means canonical scalar-field dark energy whose pressure is negative, so that $w \equiv p/\rho < 0$; it does not mean negative energy density (V. et al., 25 Apr 2025). For a minimally coupled scalar in FLRW,

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$

with

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$

and

$$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$

Negative pressure occurs whenever $V_{\rm eff} > \dot{\phi}^2/2$, so the model never requires $V_{\rm eff}<0$; “negative” refers to pressure, not to energy density (V. et al., 25 Apr 2025).

Within the false-vacuum construction of "Quintessence and false vacuum: Two sides of the same coin?" the scalar is initially trapped in a metastable false vacuum,

$V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$

and, after decay, rolls according to

$$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$

The paper uses both a semiclassical correction,

$$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$

and a coarse-grained source-driven form,

$$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$

with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$0 parametrizing steepness and decay rate (V. et al., 25 Apr 2025).

The reported viable window is an upper bound $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$1, with slow roll and late-time acceleration realized for $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$2 and stabilization parameter $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$3 (V. et al., 25 Apr 2025). In this regime the model yields

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$4

and explicitly avoids phantom behavior, since the slow-roll estimate

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$5

keeps $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$6 (V. et al., 25 Apr 2025). The same framework imposes a pressure-gap criterion,

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$7

and steepness bounds such as

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$8

to connect false-vacuum decay with swampland-type constraints (V. et al., 25 Apr 2025).

This usage is close to the conventional quintessence literature. “Quintessence’s Last Stand?” studies canonical, minimally coupled quintessence with negative pressure and asks how well future data can distinguish thawing models from a cosmological constant, emphasizing that deviations $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$9 are difficult to detect even with next-generation measurements, though redshift drift can improve sensitivity by a factor of two (Linder, 2015).

2. Negative energy-density quintessence

A second usage is explicitly nonstandard: “negative quintessence” can denote a component with negative energy density and positive pressure (Gómez-Valent et al., 1 Aug 2025). In "Effective Phantom Divide Crossing with Standard and Negative Quintessence," the dark-energy sector contains two minimally coupled scalar fields, a standard quintessence component and a negative quintessence component. The action is written as

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$0

with $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$1 and $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$2 (Gómez-Valent et al., 1 Aug 2025).

For each field,

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$3

Standard quintessence corresponds to $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$4, while negative quintessence corresponds to $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$5, giving

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$6

For $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$7, one has $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$8, and

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$9

with the paper emphasizing the region $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$0, for which $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$1 (Gómez-Valent et al., 1 Aug 2025).

The model uses thawing quadratic absolute-value potentials,

$$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$2

with $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$3 so that the negative quintessence component thaws first and is dynamically relevant at intermediate redshift (Gómez-Valent et al., 1 Aug 2025). The total dark energy is

$$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$4

and the effective equation of state is

$$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$5

The phantom divide is crossed effectively when

$$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$6

even though neither component individually crosses $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$7 (Gómez-Valent et al., 1 Aug 2025).

Using Planck+DESI+DES-Y5, the paper reports $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$8 for the SQ+NQ model versus $$w_\phi=\frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V_{\rm eff}}{\frac{1}{2}\dot{\phi}^2 + V_{\rm eff}}.$$9 for $V_{\rm eff} > \dot{\phi}^2/2$0CDM, a preference of $V_{\rm eff} > \dot{\phi}^2/2$1, with best-fit parameters $V_{\rm eff} > \dot{\phi}^2/2$2, $V_{\rm eff} > \dot{\phi}^2/2$3, $V_{\rm eff} > \dot{\phi}^2/2$4, and $V_{\rm eff} > \dot{\phi}^2/2$5 (Gómez-Valent et al., 1 Aug 2025). The paper states that the model can reproduce a peak in $V_{\rm eff} > \dot{\phi}^2/2$6 around $V_{\rm eff} > \dot{\phi}^2/2$7–$V_{\rm eff} > \dot{\phi}^2/2$8 and approach zero near $V_{\rm eff} > \dot{\phi}^2/2$9–$V_{\rm eff}<0$0, and that two constant-$V_{\rm eff}<0$1 fluids plus $V_{\rm eff}<0$2 cannot reproduce the same reconstructed shape (Gómez-Valent et al., 1 Aug 2025).

This definition is sharply different from the false-vacuum usage. In the latter, the authors explicitly state that the dark-energy density remains positive and that “negative” strictly refers to pressure (V. et al., 25 Apr 2025). The coexistence of these two incompatible definitions is a central source of ambiguity in the term.

3. Negative potential, negative minima, and the impossibility of acceleration in minimal models

A third meaning arises when the potential itself becomes negative. In this case, the term often refers not to a viable accelerating phase, but to a pathology or late-time recollapse. "Casimir-Induced Quintessence in Dark Dimension" shows that in a minimal 5D Dark Dimension setup with one large extra dimension, 5D gravity, and three right-handed bulk neutrinos, the Casimir-induced radion potential is negative at large radius (Katayama et al., 20 Mar 2026). The geometry is

$V_{\rm eff}<0$3

with canonically normalized radion

$V_{\rm eff}<0$4

The Einstein-frame potential is

$V_{\rm eff}<0$5

At large $V_{\rm eff}<0$6, the fermionic contributions are exponentially suppressed and the graviton Casimir term dominates, so

$V_{\rm eff}<0$7

(Katayama et al., 20 Mar 2026).

The paper then states that if $V_{\rm eff}<0$8 at late times, positivity of

$V_{\rm eff}<0$9

requires $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$0, implying

$V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$1

which cannot drive cosmic acceleration; if instead $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$2, then $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$3 and the Friedmann equation is spoiled (Katayama et al., 20 Mar 2026). The minimal model therefore yields what the paper calls “negative quintessence,” namely a potential that “attains a negative minimum and does not provide a positive region” (Katayama et al., 20 Mar 2026).

The remedy is to add extra bulk species—two massive 5D gauge bosons and two massless 5D Dirac fermions with periodic boundary conditions—so that the Casimir sum develops a positive, sufficiently flat plateau suitable for slow roll (Katayama et al., 20 Mar 2026). With

$V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$4

the model produces a positive region and, for $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$5, yields $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$6 crossing $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$7 around $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$8, while $V_f(\phi)=\frac{1}{2}m_f^2(\phi-\phi_f)^2,$9 (Katayama et al., 20 Mar 2026). It is compared to DESI DR2 BAO with

$$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$0

versus $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$1CDM values $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$2 and $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$3, giving $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$4 in the illustrative fit (Katayama et al., 20 Mar 2026).

Negative-potential quintessence also appears in scalar–tensor cosmology. "Scalar-Tensor Quintessence with a linear potential: Avoiding the Big Crunch cosmic doomsday" considers

$$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$5

in the Jordan frame. The paper states that all quintessence potentials that are either monotonic with negative interval or have a minimum at negative values of the potential generically predict a future collapse of the scale factor to a “doomsday” singularity in minimally coupled models (Lykkas et al., 2015). With sufficiently large non-minimal coupling, however, the curvature term modifies the effective force,

$$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$6

so that the field reverses direction, the potential becomes positive, and the universe approaches de Sitter rather than a Big Crunch (Lykkas et al., 2015). For each $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$7 there is a critical $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$8, increasing approximately linearly with $$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{\rm eff}}{d\phi}=0.$$9, and the paper quotes

$$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$0

(Lykkas et al., 2015).

"Unstable Axion Quintessence Revisited" studies a periodic potential with a negative minimum,

$$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$1

which leads to eventual recollapse and a late-time era of kination during contraction (Gardner, 2010). For $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$2 and $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$3, the universe turns around at finite time, $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$4 becomes negative, and the Klein–Gordon friction term becomes a negative-friction term, amplifying $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$5 (Gardner, 2010). Representative values include, for $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$6,

$$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$7

and the paper states that $$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$8–$$V_{\rm eff}(\phi)=V(\phi)+\frac{\hbar\lambda}{2}\langle \phi^2(t)\rangle,$$9 of the universe’s lifetime can lie in the coincidence regime $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$0 (Gardner, 2010).

These constructions all support the same technical point: for a canonical scalar, negative potential energy does not by itself furnish late-time acceleration. This suggests that when “negative quintessence” refers to $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$1, it usually signals a problem to be cured rather than a successful accelerating phase (Katayama et al., 20 Mar 2026, Lykkas et al., 2015, Gardner, 2010).

4. Negative effective mass-squared and negative dynamical ratios

A fourth usage detaches the adjective from energy density and pressure and attaches it to other dynamical quantities. In "Scant evidence for thawing quintessence," “negative quintessence” refers to a canonical quintessence field with negative effective mass-squared,

$$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$2

in a hilltop potential (Wolf et al., 2024). The model uses

$$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$3

with $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$4 in units of $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$5 and $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$6 in units of $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$7 (Wolf et al., 2024). Negative $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$8 does not imply negative energy density; viable models still require $$V_{\rm eff}=\int_{t_0}^t \left[\Lambda(t)+V_0 e^{-\lambda \phi_0(t)}\right]dt,$$9 and total $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$00, so that $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$01 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$02 until late times (Wolf et al., 2024).

The paper emphasizes that for standard slow-roll thawing with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$03, one has

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$04

whereas for hilltop thawing with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$05, the slope

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$06

depends strongly on $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$07 and saturates to $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$08 as $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$09 for the BAO+CMB+SNe setup used (Wolf et al., 2024). Example CPL fits for the same physical model $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$10 are survey-dependent: SNe give $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$11, BAO give $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$12, CMB give $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$13, and the combined fit gives $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$14 (Wolf et al., 2024). The posterior for $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$15 favors large negative values, but the best thawing model improves $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$16 by only $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$17 relative to $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$18CDM, with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$19 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$20, leading to the conclusion of scant evidence for thawing quintessence (Wolf et al., 2024).

In "Slow-roll Extended Quintessence," the negative object is again different: the characteristic ratio

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$21

during radiation- and matter-dominated eras (Chiba et al., 2010). In the Jordan-frame action

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$22

the scalar equation can be written using

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$23

and slow-roll consistency fixes

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$24

Hence $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$25 in matter domination and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$26 in radiation domination, whereas minimally coupled thawing quintessence gives

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$27

(Chiba et al., 2010). The paper calls this sign flip a sharp discriminator of non-minimal coupling and states that the negativity refers to this acceleration-to-friction ratio, not to $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$28 (Chiba et al., 2010).

Both papers illustrate that “negative quintessence” may refer not to the sign of $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$29 or $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$30, but to the sign of a curvature of the potential or of a background dynamical ratio (Wolf et al., 2024, Chiba et al., 2010).

5. Coupled, extended, and effective phantom-like quintessence

A fifth strand concerns effective phantom-divide crossing generated without a phantom field. In "Phantom-Divide Crossing in Exponentially Coupled Quintessence and the Role of Neutrino-Mass Freedom," the model is a canonical scalar conformally coupled to CDM through

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$31

with interaction

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$32

and effective equation of state

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$33

Since $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$34 for canonical quintessence, any $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$35 comes entirely from the interaction term (Wang et al., 20 Jun 2026).

The paper shows that the $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$36 branch can drive the field up the potential at early times, giving $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$37 while $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$38 is large, so $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$39 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$40 can fall below $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$41; later the potential dominates, $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$42 flips sign, $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$43 becomes negative, and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$44 returns above $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$45 (Wang et al., 20 Jun 2026). With Planck+DESI+DES-Dovekie and fixed $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$46, the favored parameters are

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$47

with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$48 relative to $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$49CDM, and the paper states that $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$50 at more than $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$51, close to $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$52 (Wang et al., 20 Jun 2026). When a signed effective neutrino-mass parameter is allowed, the preference for nonzero coupling weakens, and both $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$53 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$54 branches become consistent with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$55 within $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$56 (Wang et al., 20 Jun 2026).

Extended quintessence also uses “negative” to describe the sign of the non-minimal coupling. In "Spherical collapse in the extended quintessence cosmological models," the coupling function is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$57

with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$58 described as negative coupling (Fan et al., 2015). The effective Newton’s constant is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$59

so negative $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$60 typically enhances gravity (Fan et al., 2015). Nevertheless, the paper finds that for representative $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$61, deviations of the spherical-collapse threshold $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$62 and virial overdensity $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$63 from $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$64CDM are less than $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$65, and that differences between metric and Palatini formalisms are very small (Fan et al., 2015).

These papers indicate that effective phantom-like behavior, sign-changing energy transfer, or negative gravitational couplings can all fall under the broad rhetorical umbrella of “negative quintessence,” even when the underlying scalar remains canonical and non-phantom (Wang et al., 20 Jun 2026, Fan et al., 2015).

6. Black-hole and string-theoretic meanings

In black-hole thermodynamics, “negative quintessence” usually refers to negative pressure or to a negative contribution to some effective energy quantity, not to cosmological dark-energy density. For Kiselev-type black holes, quintessence is described by

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$66

so the defining feature is negative pressure with positive energy density (Ghaffarnejad et al., 2018, Azreg-Aïnou, 2014).

For the Reissner–Nordström black hole surrounded by quintessence with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$67, the metric function is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$68

and the approximate Lie symmetry analysis yields an energy-rescaling factor

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$69

Since $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$70 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$71, the quintessential contribution

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$72

is negative for all $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$73, so quintessence reduces the effective energy relative to the RN case (Hussain et al., 2014). The physical radius where the total rescaling factor vanishes is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$74

and the paper notes that $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$75 lies outside the event horizon for $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$76 (Hussain et al., 2014).

In RN–AdS black holes with quintessence in the grand canonical ensemble, quintessence also induces negative-pressure effects. For $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$77, the Hawking temperature becomes

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$78

so quintessence adds a constant negative shift $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$79 (Tiwari et al., 12 May 2026). With fixed electric potential $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$80, the equation of state is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$81

and the criticality conditions imply

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$82

so there is no physical critical point in the grand canonical ensemble (Tiwari et al., 12 May 2026). The normalized Ruppeiner curvature

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$83

changes sign at

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$84

with $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$85 indicating attractive interactions and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$86 repulsive interactions (Tiwari et al., 12 May 2026).

The extended thermodynamics of charged de Sitter-like black holes with quintessence similarly treats quintessence as a negative-pressure source. The event-horizon quintessential pressure is

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$87

for $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$88 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$89, and the enthalpy takes the form

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$90

with thermodynamic volume

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$91

(Azreg-Aïnou, 2014). The paper emphasizes that the negative pressure creates a de Sitter-like causal structure with inner, event, and cosmological horizons (Azreg-Aïnou, 2014).

In string phenomenology, negative terms typically enter as AdS-like contributions used in tuning. "The $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$92-term Problem and other Challenges of Stringy Quintessence" states that negative potential-energy contributions in LVS are too small to cancel the positive uplift from realistic SUSY breaking, and argues that a new negative term

$$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$93

would be needed to cancel the uplift and stabilize the volume (Hebecker et al., 2019). The paper also states explicitly that a canonical scalar sitting at negative potential energy does not accelerate, because acceleration requires net positive $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$94 dominating over kinetic energy (Hebecker et al., 2019).

7. Conceptual synthesis and recurrent misconceptions

The literature therefore supports a strongly contextual reading of the term. At least four recurrent misconceptions can be separated from the published uses.

First, negative quintessence does not generically mean negative energy density. In the false-vacuum quintessence model and in standard canonical thawing models, the energy density is positive and the “negative” refers to pressure or equation of state (V. et al., 25 Apr 2025, Linder, 2015, Wolf et al., 2024). By contrast, the composite SQ+NQ model defines negative quintessence precisely by $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$95 and $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$96 (Gómez-Valent et al., 1 Aug 2025).

Second, negative potential is not equivalent to viable acceleration. In the Dark Dimension radion model, a negative potential tail produces $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$97 if $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$98 is maintained, and therefore cannot explain dark energy (Katayama et al., 20 Mar 2026). Scalar–tensor and unstable-axion models similarly treat negative-potential regions as precursors of recollapse unless additional structure reverses the roll or reshapes the potential (Lykkas et al., 2015, Gardner, 2010).

Third, effective phantom-divide crossing need not imply a phantom field. The two-field SQ+NQ model and the exponentially coupled quintessence model both realize effective $$S = \int d^4x \sqrt{-g}\left[\frac{M_P^2}{2}R-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\right],$$99 crossing without any individual component having $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$00 or any healthy canonical field becoming phantom (Gómez-Valent et al., 1 Aug 2025, Wang et al., 20 Jun 2026). This suggests that observationally inferred phantom-like behavior may reflect sectoral decomposition or interaction terms rather than a fundamental ghost.

Fourth, the negative sign may attach to auxiliary dynamical quantities rather than to the dark-energy fluid itself. Negative $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$01 in hilltop thawing, negative $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$02 in extended quintessence, negative coupling $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V_{\rm eff}(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V_{\rm eff}(\phi),$$03, and negative energy-rescaling contributions in black-hole spacetimes are all documented uses (Wolf et al., 2024, Chiba et al., 2010, Fan et al., 2015, Hussain et al., 2014).

A plausible implication is that “negative quintessence” functions more as a family resemblance term than as a sharply defined category. In current arXiv usage, the phrase can denote negative pressure, negative density, negative potential curvature, negative dynamical response, or negative effective energetic contribution, depending on the model class. Any technical reading therefore requires immediate specification of which quantity is negative and whether the model remains canonical, minimally coupled, observationally viable, and free of classical or quantum pathologies (V. et al., 25 Apr 2025, Gómez-Valent et al., 1 Aug 2025, Wolf et al., 2024, Katayama et al., 20 Mar 2026).