Homogeneous Phase-Flatness and Dark-Energy-Like Background Response in the Covariant Expansion-Phase Framework

In standard cosmology, late-time acceleration is modeled by either:

- a cosmological constant $\Lambda$TeX\Lambda, interpreted as vacuum energy [citation], or - a dark-energy sector, often realized as a scalar field (quintessence, $k$TeXk-essence, etc.) [citation].

In both cases one effectively adds a negative-pressure contribution to the right-hand side of Einstein's equations. Late-time acceleration was established by Type Ia supernova luminosity-distance measurements [citation] and is now constrained by CMB, supernova, and BAO analyses. Pantheon+ remains consistent with a late-time background close to $\Lambda$TeX\LambdaCDM when combined with external datasets [citation], and the final Planck PR4 release provides the updated CMB baseline for the corresponding minimal picture [citation]. DESI DR2 BAO analyses report low-redshift dark-energy-evolution preference in some joint BAO--CMB--SN fits under specified data-combination and parametrization choices [citation]. The inferred significance remains sensitive to the supernova compilation and to the adopted parametrization or reconstruction strategy [citation]. Recent studies of thawing quintessence and simple quintessence therefore find, at most, prior-sensitive or model-dependent support rather than an unambiguous non-phantom detection [citation].

All observational overlay remarks are read only relative to a declared late-time fitting convention: joint late-time BAO--CMB--SN background comparisons on an adopted supernova compilation and an adopted parametrization or reconstruction strategy. These fit-facing remarks are not part of the exact homogeneous background identities introduced below.

The CHC construction uses a global phase field $\HH$TeX\HH whose geometry and curvature govern the relevant sector across scales; in the cosmological sector this field encodes the expansion-response branch where applicable. On the homogeneous FRW branch, late-time acceleration arises when the mode $\HH(t)$TeX\HH(t) approaches a flat region of the effective potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH). In that regime,

\rho_{\HH}\approx U_{\rm flat},
\qquad
p_{\HH}\approx -U_{\rm flat},
\qquad
w_{\rm eff}\to -1,

so that the background expansion is equivalent to GR with

\Lambda_{\rm eff}=\frac{U_{\rm flat}}{M_{\rm Pl}^2},

with the constant read as phase-geometric rather than fluid-additive.

At the level of homogeneous FRW dynamics, a phase plateau with constant $U_{\rm flat}$TeXU_{\rm flat} is mathematically equivalent, on the declared branch, to an effective cosmological-constant background. The same phase field is required only to remain on an admitted homogeneous late-time plateau carrying non-phantom dynamics and explicit failure conditions.

Only homogeneous background dynamics are used. Distance mapping, clock inference, perturbation inference, and full halo--background synthesis are excluded from the construction. In particular, the late-time limit $\Xi\to 0$TeX\Xi\to 0 refers only to the homogeneous cosmological gradient hierarchy and is not itself an observational zero statement.

CHC-min action for cosmology

In the full CHC framework, the phase field $\HH$TeX\HH is coupled to constraint fields and invariants that enforce coherence across scales. The large-scale cosmological branch uses the CHC-min sector: a single-field effective theory obtained by integrating out auxiliary fields in a regime where spatial gradients are small and the dynamics is dominated by the homogeneous mode of $\HH$TeX\HH.

We use the metric signature $(-,+,+,+)$TeX(-,+,+,+) and define

X_{\HH}:=g^{\mu\nu}\nabla_\mu\HH\,\nabla_\nu\HH .

The CHC-min action (in the Jordan frame where matter is minimally coupled to the metric) can be written as

S_{\rm CHC}
=
\int d^4x\,\sqrt{-g}\Bigg[
\frac{M_{\rm Pl}^2}{2}R
-\frac{M_{\rm eff}^2}{2}X_{\HH}
-U_{\rm eff}(\HH)
\Bigg]
+S_m[g,\Psi],

where:

- $M_{\rm Pl}$TeXM_{\rm Pl} is the reduced Planck mass, - $M_{\rm eff}$TeXM_{\rm eff} is the effective phase rigidity scale at cosmological energies, - $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) is the effective potential of the phase, including integrated-out CHC structure and possible couplings to background matter, - $S_m$TeXS_m is the standard matter action for fields $\Psi$TeX\Psi.

Interpretation of the FRW variables..

The action reference introduces no independent dark-energy fluid variable beyond $\HH$TeX\HH. When ``fluid-like'' variables $(\rho_{\HH},p_{\HH})$TeX(\rho_{\HH},p_{\HH}) are introduced, they are the FRW decomposition of the stress--energy of $\HH$TeX\HH into effective background variables, not an ontic claim that a separate material component exists.

Connection to the CHC root branch..

The CHC-min action in Eq. reference is used here as the homogeneous effective reduction of the admitted constrained CHC branch. Any higher-sector or auxiliary objects in the broader framework are not imported as additional dynamical fields in this paper. The following schematic expression is a non-load-bearing motivation for the effective parameters in Eq. reference; none of its terms is used below as an independent source, perturbation, distance, clock, detector, or compact-object inference law. Schematically, a broader constrained sector may contain terms of the form

S_{\rm full} =
\int d^4x\sqrt{-g}\Big[
\frac{M_{\rm Pl}^2}{2}R
+ \alpha_1\,C_{\mu\nu}\nabla^\mu\nabla^\nu \HH
+ \alpha_2\,(\nabla \HH)^4/\Lambda_\Xi^4
+ \cdots
\Big].

In the declared long-wavelength homogeneous sector, all such schematic auxiliary structure is represented only through the effective branch parameters $M_{\rm eff}$TeXM_{\rm eff}, $\Lambda_\Xi$TeX\Lambda_\Xi, and $U_{\rm eff}$TeXU_{\rm eff} in Eq. reference. The reduction used below is therefore Eq. reference itself; no tensor-constraint dynamics, multi-scale coherence law, source-side closure, distance/clock inference, perturbation reconstruction, or compact-object inference law is imported in this DE paper.

Equation reference is algebraically equivalent, on the declared homogeneous FRW branch, to a minimally coupled quintessence-type scalar with potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH), but in CHC, $\HH$TeX\HH is not introduced as a new dynamical fluid. The only imported statement used below is that the homogeneous background branch is read on one declared potential family and one admitted FRW window. Only the homogeneous late-time branch is analyzed below.

FRW reduction

Consider a spatially flat FRW metric

ds^2 = -dt^2 + a(t)^2 d\bm{x}^2,

and assume that on sufficiently large scales the phase field is spatially homogeneous:

\HH = \HH(t).

The energy--momentum tensor of the phase sector derived from reference is

T^{(\HH)}_{\mu\nu}
=
M_{\rm eff}^2\partial_\mu\HH\,\partial_\nu\HH
-g_{\mu\nu}\left(\frac{M_{\rm eff}^2}{2}X_{\HH}+U_{\rm eff}(\HH)\right).

With signature $(-,+,+,+)$TeX(-,+,+,+), the homogeneous specialization gives

X_{\HH}=-\dot{\HH}^{2},

so the admitted branch yields the effective FRW energy density and pressure

\rho_{\HH}
=
\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 + U_{\rm eff}(\HH),
[4pt]
p_{\HH}
=
\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 - U_{\rm eff}(\HH).

On the admitted homogeneous branch with $M_{\rm eff}^2>0$TeXM_{\rm eff}^2>0, one has

\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^{2}\ge 0,

so the branch remains non-phantom whenever $\rho_{\HH}>0$TeX\rho_{\HH}>0.

In CHC these are effective FRW background variables: they encode how the homogeneous phase sector sources gravity, not a separate fundamental fluid component.

The Friedmann equations become

3M_{\rm Pl}^2 H^2
=
\rho_m + \rho_r + \rho_{\HH},

[4pt]
-2M_{\rm Pl}^2 \dot{H}
=
\rho_m + \frac{4}{3}\rho_r + \rho_{\HH} + p_{\HH},

where $\rho_m$TeX\rho_m and $\rho_r$TeX\rho_r are the usual matter and radiation densities.

Varying reference with respect to $\HH$TeX\HH gives the covariant field equation

M_{\rm eff}^2\Box\HH-\partial_{\HH}U_{\rm eff}(\HH)=0.

On the homogeneous FRW branch, where $\Box\HH=-(\ddot{\HH}+3H\dot{\HH})$TeX\Box\HH=-(\ddot{\HH}+3H\dot{\HH}), this becomes

M_{\rm eff}^2
\left(\ddot{\HH} + 3H\dot{\HH}\right)
+
\partial_{\HH}U_{\rm eff}(\HH) = 0.

Equations reference, reference, reference, and reference are exact homogeneous background identities on the admitted FRW branch. Every later benchmark or fit-facing overlay statement is read only relative to the declared benchmark dark-energy/background family and the declared late-time fitting convention and does not modify those identities.

The restricted branch considered below consists of solutions in which $\HH$TeX\HH approaches a plateau of $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) at late times.

Declared potential-family admissibility on the homogeneous branch

The homogeneous late-time branch is read on one declared family of effective potentials and one admitted FRW window. The relevant requirement is not a completed multi-sector synthesis but the existence of an admissible plateau within one declared family.

Accordingly, the relevant parameter space is a common CHC set,

\Theta_{\rm CHC}=\{M_{\rm eff},\Lambda_\Xi,\theta_U\},

where $\theta_U$TeX\theta_U denotes the parameters specifying the chosen family of effective potentials. A full background posterior over $\Theta_{\rm CHC}$TeX\Theta_{\rm CHC} is excluded; only the existence of an admitted plateau family is used.

This family-level admissibility statement is made explicit in reference. Shared potential-family parametrization

To make the homogeneous plateau requirement explicit, fix a potential family

U_{\rm eff}(\HH)=\mu^4\,F\!\left(x;\alpha_i\right),
\qquad
x\equiv\frac{\HH-\HH_*}{\Delta},

where $\mu$TeX\mu sets the vertical scale, $\Delta$TeX\Delta the field width, and $\alpha_i$TeX\alpha_i a finite set of dimensionless shape parameters. For a plateau point $x_{\rm flat}$TeXx_{\rm flat}, define

U_{\rm flat}=\mu^4 F(x_{\rm flat}),
[3pt]
m_{\rm flat}^2:=\frac{\mu^4}{\Delta^2}F''(x_{\rm flat}).

A useful dimensionless plateau organizer is

\varepsilon_{\rm flat}^{(F)}
\equiv
\frac{\Delta^2 m_{\rm flat}^2}{U_{\rm flat}}
=
\frac{F''(x_{\rm flat})}{F(x_{\rm flat})}.

Once $(F,\mu,\Delta,\alpha_i)$TeX(F,\mu,\Delta,\alpha_i) is fixed, the plateau height $U_{\rm flat}$TeXU_{\rm flat} and the local plateau curvature $m_{\rm flat}$TeXm_{\rm flat} are not independent tunings. This non-independence statement is asserted only within a declared homogeneous potential family. Equations reference--reference therefore define the declared benchmark dark-energy/background family for all downstream plateau values, benchmark points, and family-level quantitative overlays. It is not a completed cosmological posterior and does not by itself establish a unique background fit.

Effective equation of state

Define the effective equation-of-state parameter of the homogeneous phase sector by

w_{\rm eff}
\equiv
\frac{p_{\HH}}{\rho_{\HH}}
=
\frac{\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 - U_{\rm eff}(\HH)}
{\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 + U_{\rm eff}(\HH)}.

Then:

w_{\rm eff} \approx -1
\quad\Longleftrightarrow\quad
\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 \ll U_{\rm eff}(\HH).

A phase-flatness regime is defined by:

\dot{\HH}\to 0,\qquad
\nabla\HH\to 0,\qquad
U_{\rm eff}(\HH)\to U_{\rm flat}\neq 0,

where $U_{\rm flat}$TeXU_{\rm flat} is the plateau value reached in the late-time Universe.

In this regime:

\rho_{\HH} \approx U_{\rm flat},\qquad
p_{\HH}\approx -U_{\rm flat},
\qquad
w_{\rm eff} \approx -1.

The phase sector therefore reproduces the same homogeneous FRW background equations as a cosmological-constant term with

\Lambda_{\rm eff} \equiv \frac{U_{\rm flat}}{M_{\rm Pl}^2}.

Meaning of the phase-flat limit..

The limit reference is a statement about the homogeneous dynamical response of $\HH$TeX\HH on cosmological scales: the background relaxes such that kinetic and gradient contributions become negligible compared to a finite plateau value.

Plateau conditions

A plateau region is characterized by the standard conditions:

U_{\rm eff}'(\HH_{\rm flat})\approx 0,
\qquad
U_{\rm eff}''(\HH_{\rm flat})>0,

with $U_{\rm eff}(\HH_{\rm flat})=U_{\rm flat}>0$TeXU_{\rm eff}(\HH_{\rm flat})=U_{\rm flat}>0.

Near the plateau, write

U_{\rm eff}(\HH)
=
U_{\rm flat}
+\frac12 m_{\rm flat}^2 (\HH-\HH_{\rm flat})^2
+\cdots,

with curvature $m_{\rm flat}^2\equiv U_{\rm eff}''(\HH_{\rm flat})$TeXm_{\rm flat}^2\equiv U_{\rm eff}''(\HH_{\rm flat}).

The CHC phase-gradient hierarchy variable used on this branch is

\Xi(t)
\equiv
\frac{M_{\rm eff}^2\,\dot{\HH}^2}{\Lambda_\Xi^4}

for the homogeneous cosmological specialization of the CHC gradient measure associated with the FRW mode $\HH(t)$TeX\HH(t). In this notation, $\Xi(t)$TeX\Xi(t) (and $\Xi(z)$TeX\Xi(z)) denotes only the homogeneous cosmological specialization of the CHC gradient measure. Local and propagation-side specializations are excluded. The scale $\Lambda_\Xi$TeX\Lambda_\Xi controls phase gradients in CHC.

Along the declared convex plateau-approach branch, FRW damping together with $U'_{\rm eff}(\HH)\to 0$TeXU'_{\rm eff}(\HH)\to 0 yields

\Xi(t)\to 0,
\qquad
U_{\rm eff}(\HH)\to U_{\rm flat}.

This is a branch-level consequence of the homogeneous reduction and the convex plateau conditions, not a standalone observational statement and not a stronger global basin-of-attraction claim.

Within the convex slow-roll-like approach branch, the homogeneous mode is driven toward the phase-flat plateau, so that $\Xi(t)\to 0$TeX\Xi(t)\to 0 and $1+w_{\rm eff}$TeX1+w_{\rm eff} becomes small at late times. No stronger global basin-of-attraction statement is assumed. In particular, $\Xi(t)\to 0$TeX\Xi(t)\to 0 is a statement about the cosmological homogeneous mode approaching a slow convex-plateau regime.

Slow-roll-like regime

Near the plateau and at late times, the damping term $3H\dot{\HH}$TeX3H\dot{\HH} in reference dominates over $\ddot{\HH}$TeX\ddot{\HH}, provided

\left|\ddot{\HH}\right|
\ll
3H\left|\dot{\HH}\right|.

In this slow-roll-like regime, the field equation reduces to

3H M_{\rm eff}^2 \dot{\HH}
\approx
-\,U_{\rm eff}'(\HH),

which shows explicitly that $\HH$TeX\HH is driven towards a point where $U_{\rm eff}'(\HH)\approx 0$TeXU_{\rm eff}'(\HH)\approx 0, i.e.\ the plateau.

Combining reference with reference and reference gives the deviation of $w_{\rm eff}$TeXw_{\rm eff} from $-1$TeX-1:

1+w_{\rm eff}
\approx
\frac{M_{\rm eff}^2 \dot{\HH}^2}{U_{\rm flat}}
\approx
\frac{1}{9H^2 M_{\rm eff}^2 U_{\rm flat}}
\big(U_{\rm eff}'(\HH)\big)^2.

As $\HH$TeX\HH approaches the plateau, $U_{\rm eff}'\to 0$TeXU_{\rm eff}'\to 0 and hence $w_{\rm eff}\to -1$TeXw_{\rm eff}\to -1.

Acceleration condition and transition redshift

Cosmic acceleration requires

\frac{\ddot{a}}{a}
=
-\frac{1}{6M_{\rm Pl}^2}
\left(\rho_{\rm tot}+3p_{\rm tot}\right) > 0.

In a matter + phase system (neglecting radiation at late times),

\rho_{\rm tot} = \rho_m + \rho_{\HH},

p_{\rm tot} = 0 + p_{\HH}.

The acceleration condition becomes

\rho_m + \rho_{\HH}(1+3w_{\rm eff}) < 0.

Near the plateau where $w_{\rm eff}\approx -1$TeXw_{\rm eff}\approx -1 and $\rho_{\HH}\approx U_{\rm flat}$TeX\rho_{\HH}\approx U_{\rm flat}, this reduces to

\rho_m \lesssim 2U_{\rm flat}.

With present-day matter density $\rho_{m0}$TeX\rho_{m0}, $\rho_m(z)=\rho_{m0}(1+z)^3$TeX\rho_m(z)=\rho_{m0}(1+z)^3, and a rough estimate of the transition redshift $z_\ast$TeXz_\ast (where $\ddot{a}=0$TeX\ddot{a}=0) is obtained from

\rho_{m0}(1+z_\ast)^3
\simeq
2U_{\rm flat}.

Thus

1+z_\ast
\simeq
\left(\frac{2U_{\rm flat}}{\rho_{m0}}\right)^{1/3}.

At the level of background expansion history, therefore, a phase-flat plateau with constant $U_{\rm flat}$TeXU_{\rm flat} produces the same $H(z)$TeXH(z) as a cosmological constant model with $\Omega_\Lambda=U_{\rm flat}/(3M_{\rm Pl}^2 H_0^2)$TeX\Omega_\Lambda=U_{\rm flat}/(3M_{\rm Pl}^2 H_0^2). The difference is conceptual and structural:

- $U_{\rm flat}$TeXU_{\rm flat} is not an independent vacuum term in the action but the plateau value of a phase potential on the homogeneous branch. - The same CHC field $\HH$TeX\HH remains responsible for the homogeneous phase-flatness branch, so the late-time background is carried by a geometric phase sector rather than by an added dark-energy fluid. - CHC imposes additional constraints (no phantom, restricted $w(z)$TeXw(z) forms) beyond generic quintessence models.

Behavior of the gradient measure $\Xi(z)$TeX\Xi(z)Xi(z)

Recall the gradient measure

\Xi(z)=\frac{M_{\rm eff}^2\,\dot{\HH}^2}{\Lambda_\Xi^4}.

Equation reference gives

\Xi(z)
\approx
\frac{1}{9H^2 M_{\rm eff}^2\Lambda_\Xi^4}
\big(U_{\rm eff}'(\HH)\big)^2.

Hence the adopted late-time statement is the weaker one: along a convex plateau-approach branch, $U_{\rm eff}'\to 0$TeXU_{\rm eff}'\to 0 and therefore $\Xi\to 0$TeX\Xi\to 0. A strict monotone decrease of $\Xi$TeX\Xi is not assumed globally.

Differentiating reference gives

\dot{\Xi}
=
\frac{2U'_{\rm eff}U''_{\rm eff}\dot{\HH}}{9H^2 M_{\rm eff}^2\Lambda_\Xi^4}
-\frac{2(U'_{\rm eff})^2\dot{H}}{9H^3 M_{\rm eff}^2\Lambda_\Xi^4}.

Using reference, this can be written as

\frac{\dot{\Xi}}{\Xi}
\approx
-\frac{2}{H}\left(\frac{U''_{\rm eff}}{3M_{\rm eff}^2}+\dot H\right).

Since $\dot H<0$TeX\dot H<0 in an expanding matter-plus-DE background, the sign is not fixed by convexity alone. A sufficient local condition for strict monotone decrease is

\frac{U''_{\rm eff}}{3M_{\rm eff}^2}>|\dot H|.

No global validity of reference is required; it is enough that along the convex late-time approach branch $\Xi\to 0$TeX\Xi\to 0 and hence $1+w_{\rm eff}$TeX1+w_{\rm eff} becomes small.

Relation to $\Lambda$TeX\LambdaLambda and scalar-field dark energy

The CHC phase-flatness branch differs from generic scalar-field dark energy in the following sense:

- Same background equations as scalar-field dark energy. The FRW-level equations reference--reference are algebraically identical, on the declared homogeneous branch, to those of a minimally coupled quintessence-type scalar with potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH). Any difference arises from how $U_{\rm eff}$TeXU_{\rm eff} is constrained by the CHC standing assumptions and by the admissible family-level plateau conditions on the homogeneous branch. - Effective cosmological constant from a plateau. When $\HH$TeX\HH freezes on a plateau, the phase sector reproduces the same homogeneous FRW background equations as a cosmological-constant term with $\Lambda_{\rm eff}=U_{\rm flat}/M_{\rm Pl}^2$TeX\Lambda_{\rm eff}=U_{\rm flat}/M_{\rm Pl}^2. At the level of background expansion, this is indistinguishable from $\Lambda$TeX\LambdaCDM with $\Lambda=\Lambda_{\rm eff}$TeX\Lambda=\Lambda_{\rm eff} on the declared homogeneous branch. - Phenomenological plateau scale. The plateau height $U_{\rm flat}$TeXU_{\rm flat} is treated as an observationally fixed background parameter, in the same way that $\Omega_\Lambda$TeX\Omega_\Lambda is fixed in $\Lambda$TeX\LambdaCDM. Whether $U_{\rm flat}$TeXU_{\rm flat} can be predicted from more fundamental CHC scales (e.g.\ the phase gradient scale $\Lambda_\Xi$TeX\Lambda_\Xi or other higher-level CHC parameters) is left open. - Structural restrictions and differences. CHC enforces: itemize[leftmargin=0.8cm] - No phantom behavior: $w_{\rm eff}\ge -1$TeXw_{\rm eff}\ge -1 in the CHC-min branch because the admitted branch obeys $\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0$TeX\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0 together with $\rho_{\HH}>0$TeX\rho_{\HH}>0. - A single phase field for the homogeneous background branch: late-time acceleration is carried by the phase-flatness regime of $\HH$TeX\HH rather than by an added dark-energy fluid. - A restricted class of $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) consistent with CHC standing assumptions (L4, L7, L9), narrowing the space of allowed $w(z)$TeXw(z).

itemize

CHC standing assumptions L4 and L7 in the DE limit

Two further CHC standing assumptions constrain the homogeneous branch:

- L4 (Recovery principle). When $\nabla\HH\to 0$TeX\nabla\HH\to 0 and $U_{\rm eff}$TeXU_{\rm eff} is locally convex, the metric response reduces to GR with an effective cosmological constant: equation G_munu 1M_ Pl^2 T^(m)_munu -\Lambda_ eff g_munu, \Lambda_ eff = U_ flatM_ Pl^2. equation Thus, in the deep plateau limit, CHC explicitly reproduces GR+$\Lambda$TeX\Lambda as a special case, with $\Lambda$TeX\Lambda emergent from phase geometry. - L7 (Information--energy correspondence). Phase fluctuations generate thermodynamic curvature rather than contributing directly to vacuum energy. In the plateau regime, where $\Xi\to 0$TeX\Xi\to 0 and the phase is nearly homogeneous, the residual information content associated with $\HH$TeX\HH is encoded in a constant geometric term $U_{\rm flat}$TeXU_{\rm flat} rather than in a fluctuating vacuum-energy density. This supports the interpretation of $U_{\rm flat}$TeXU_{\rm flat} as a geometric phase constant entering the effective stress tensor. No microscopic cancellation of quantum-field vacuum contributions is derived, and no solution of the cosmological-constant problem is claimed.

Although these assumptions do not change the FRW equations themselves, they constrain the allowed interpretation and the space of viable $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH), and they ensure that the DE plateau is consistent with CHC's broader information-geometric picture.

Phase-flatness as a geometric phenomenon

The fluid-like variables $\rho_{\HH}$TeX\rho_{\HH} and $p_{\HH}$TeXp_{\HH} are not independent material components within CHC. They encode how the geometry responds to the phase field.

In particular, in the plateau regime

\rho_{\HH} \approx U_{\rm flat},

but $U_{\rm flat}$TeXU_{\rm flat} is not interpreted as vacuum energy of quantum fields in the usual sense. Rather, it is a geometric constant associated with a coherent phase configuration of the CHC field.

Late-time acceleration on this branch is generated by the homogeneous cosmic phase field $\HH$TeX\HH entering a flat plateau of the effective potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) at finite height $U_{\rm flat}$TeXU_{\rm flat}. At the level of the FRW equations this is equivalent to an effective cosmological-constant contribution, but the constant is read as a phase-geometric plateau rather than as an independently postulated vacuum term. The statement is restricted to the effective homogeneous stress tensor and does not constitute a microscopic derivation of vacuum-energy cancellation.

The CHC phase-flatness picture sits conceptually close to several well-studied frameworks---quintessence, $k$TeXk-essence, and unified dark-sector models such as scalar unified DM/DE and Chaplygin-type fluids [citation]. The relevant similarities and differences are the following.

Scalar-field quintessence vs.\ CHC-min

At the level of the declared homogeneous FRW branch, the CHC-min action reference is algebraically identical to a minimally coupled quintessence-type scalar with potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH). In both cases one has

\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi),\qquad
p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi),

with $w_\phi \ge -1$TeXw_\phi \ge -1 and late-time acceleration driven by a region where $V$TeXV is sufficiently flat.

The differences are structural rather than algebraic:

- In generic quintessence the field $\phi$TeX\phi is introduced as an additional low-energy component. The potential $V(\phi)$TeXV(\phi) is typically chosen phenomenologically. - In CHC-min, the field $\HH$TeX\HH is the homogeneous phase field of the root CHC construction and the potential $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) is read only on one declared family and one admitted FRW window. - The CHC standing assumptions (L4, L7, L9) further restrict the shape and interpretation of $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH), tying it to geometric and information-theoretic principles.

CHC-min is therefore a highly constrained subset of scalar-field dark-energy models with a standard-sign kinetic term, with fewer genuinely independent phenomenological freedoms than a generic quintessence potential. This distinction matters in the current observational landscape, where recent DESI-motivated analyses find only scant or prior-sensitive support for thawing and simple quintessence and thereby sharpen the importance of explicit structural restrictions [citation].

Unified dark-sector models as a boundary, not an import

There exists a broad class of unified dark-sector models in which a single component is asked to mimic both DM and DE:

- barotropic fluids with exotic equations of state (such as generalized Chaplygin gas) [citation], - nonstandard-kinetic scalar fields ($k$TeXk-essence, DBI-type actions) [citation], - scalar unified DM/DE models where the same field provides dust-like behavior at high redshift and DE-like behavior at late times.

The analysis does not construct such a unified fit. Its object is narrower:

- Background-only branch. Only the homogeneous FRW reduction and the phase-flat plateau are developed here. - Non-phantom action package in CHC-min. In the CHC-min DE branch the action and FRW reduction imply $\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0$TeX\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0, excluding stable phantom behavior at the level of the effective action. Many unified models rely on nonstandard kinetic terms to realize more flexible $w(z)$TeXw(z), including $w<-1$TeXw<-1 regimes. - Boundary, not uptake. Unified DM/DE models provide a comparison class for the excluded claim class: no source-side weak-field response law, no joint halo-plus-background posterior, and no unified dark-sector fit are constructed here.

Thus CHC-DE is neither a new fluid model nor a generic scalar unifier. It is a phase-geometric homogeneous-sector construction in which late-time acceleration is carried by a positive locally stable phase-flat near-plateau region of the same global phase field, on one declared cosmological expansion-response branch.

Admissibility restrictions from the declared potential family

At homogeneous FRW level, the CHC-min branch is algebraically close to minimally coupled quintessence. The main restriction is a reduction of effective freedom: the same phase field $\HH$TeX\HH and the same effective potential family $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) must realize an admitted positive locally convex plateau or near-plateau within one declared family.

No full numerical background fit is constructed. The shared-family statement concerns admissibility within a declared parameter family, not a completed BAO-plus-CMB-plus-SN posterior. Relative to models in which the dark-energy potential is treated as freely adjustable, the CHC phase-flatness branch requires late-time acceleration to remain within one declared family and one admitted homogeneous window.

The requirement is admissibility within a common parameter set, not a completed late-time likelihood analysis. Let $U_{\rm flat}$TeXU_{\rm flat} denote the plateau height responsible for late-time acceleration and let $m_{\rm flat}$TeXm_{\rm flat} denote the local plateau curvature at a declared plateau point. A useful dimensionless organizer is

\varepsilon_{\rm flat}^{(F)}
\equiv
\frac{\Delta^2 m_{\rm flat}^2}{U_{\rm flat}}
=
\frac{F''(x_{\rm flat})}{F(x_{\rm flat})}.

The homogeneous CHC branch requires that one declared potential family admit a positive locally convex plateau or near-plateau with finite plateau organizer $\varepsilon_{\rm flat}^{(F)}$TeX\varepsilon_{\rm flat}^{(F)}.

Within the family reference, the admissibility statement is sharper:

\varepsilon_{\rm flat}^{(F)}
=
\frac{F''(x_{\rm flat})}{F(x_{\rm flat})}.

Hence the plateau height and the local plateau curvature are tied to the same $(\mu,\Delta,\alpha_i)$TeX(\mu,\Delta,\alpha_i). The branch is admissible only if there exists at least one point

\theta_*\in\Theta_{\rm CHC}

for which the following conditions hold simultaneously:

- $U_{\rm flat}>0$TeXU_{\rm flat}>0 at a plateau point with $U'_{\rm eff}(\HH_{\rm flat})\approx 0$TeXU'_{\rm eff}(\HH_{\rm flat})\approx 0 and $U''_{\rm eff}(\HH_{\rm flat})>0$TeXU''_{\rm eff}(\HH_{\rm flat})>0; - the same family admits a finite positive local plateau curvature $m_{\rm flat}^2$TeXm_{\rm flat}^2 at that point; - the plateau organizer $\varepsilon_{\rm flat}^{(F)}$TeX\varepsilon_{\rm flat}^{(F)} remains finite on the chosen family.

This condition identifies a non-empty admitted plateau set within one declared potential family. Because the benchmark point satisfies the stated inequalities explicitly, it supplies an existence witness for at least one admitted branch point within that family. It does not calibrate $U_{\rm flat}$TeXU_{\rm flat} or $m_{\rm flat}$TeXm_{\rm flat} to a cosmological posterior.

One-family existence demonstration

A compact existence demonstration shows that the admissible region need not be empty. Consider the explicit family

F_{\rm ex}(x;\alpha)=1+\alpha x^2 e^{-x^2},
\qquad
\alpha>0.

This family has a finite asymptotic plateau, $F_{\rm ex}(x)\to 1$TeXF_{\rm ex}(x)\to 1 as $|x|\to\infty$TeX|x|\to\infty, and a curved local region near $x=0$TeXx=0. Its first two derivatives are

F_{\rm ex}'(x;\alpha)
=
2\alpha x(1-x^2)e^{-x^2},
[3pt]
F_{\rm ex}''(x;\alpha)
=
2\alpha(2x^4-5x^2+1)e^{-x^2}.

Choose

x_{\rm flat}=4,\qquad \alpha=\frac12,\qquad \Delta=\mu.

Then

U_{\rm flat}
=
\mu^4 F_{\rm ex}(4;1/2)
=
\mu^4\!\left(1+8e^{-16}\right)
\approx \mu^4,
[3pt]
m_{\rm flat}^2
=
\frac{\mu^4}{\Delta^2}F_{\rm ex}''(4;1/2)
=
433\,e^{-16}\mu^2
\approx 4.9\times 10^{-5}\mu^2.

Hence

\varepsilon_{\rm flat}^{(F_{\rm ex})}
=
\frac{433\,e^{-16}}{1+8e^{-16}}
\approx 4.9\times 10^{-5},

while the plateau slope is already numerically small,

F_{\rm ex}'(4;1/2)=-60\,e^{-16}\approx -6.8\times 10^{-6}.

The admitted plateau set is therefore non-empty within at least one declared family. The benchmark supplies an explicit existence witness for such a set. It is not a cosmological likelihood fit and it does not calibrate a posterior.

Structural conditions and current-data status

The branch is subject to the following conditions.

- Non-phantom branch condition. On the admitted homogeneous branch with $M_{\rm eff}^2>0$TeXM_{\rm eff}^2>0, one has $\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0$TeX\rho_{\HH}+p_{\HH}=M_{\rm eff}^2\dot{\HH}^2\ge 0, so stable phantom behavior is excluded. If robust background inference requires persistent $w(z)<-1$TeXw(z)<-1 or a stable phantom crossing as a necessary feature, the branch is excluded. - Structural plateau condition. If no single declared potential family admits a non-empty positive locally convex plateau or near-plateau set on the homogeneous branch, the shared-family admissibility reading fails on the stated branch. - $\Lambda$TeX\Lambda-equivalent limit. If the background posterior collapses onto a purely $\Lambda$TeX\Lambda-equivalent limit with no resolved need for dynamical evolution, the branch may remain algebraically admissible but carries no observational necessity claim beyond an alternative phase-geometric reading of the same FRW background.

The current-data remarks in this subsection are read only relative to the declared late-time fitting convention introduced above and are not promoted here to exact background identities or to a completed posterior over $\Theta_{\rm CHC}$TeX\Theta_{\rm CHC}.

On the declared same-stack public route, the representative homogeneous non-phantom plateau witness has only the status DE-BACKGROUND-NONEXCLUSION: it is not excluded by the selected background envelope under the stated fitting convention. Broader BAO--CMB--SN inference remains sensitive to the supernova compilation, parametrization, reconstruction strategy, and imposed phantom/non-phantom domain. Pantheon+ and Planck PR4/NPIPE supply a near-$\Lambda$TeX\LambdaCDM baseline, while DESI DR2 BAO analyses report evolving-dark-energy preference in some joint fits; that preference is not used here as a CHC-DE selection claim. Recent thawing and simple-quintessence studies in this data environment find either scant evidence or strong prior dependence for non-phantom explanations [citation].

The admitted non-phantom plateau branch is therefore retained only as a declared homogeneous-background nonexclusion interpretation, not as a data-selected endpoint and not as a preference statement over $\Lambda$TeX\LambdaCDM.

A bounded public background-stress route evaluates the homogeneous phase-flatness reading on one selected DESI DR2 BAO--Pantheon+ SN--Planck PR4/NPIPE--ACT DR6 background stack [citation]. The route compares a representative homogeneous non-phantom plateau witness against the selected public-chain posterior envelope and assigns

\texttt{DE-BACKGROUND-NONEXCLUSION}

when the witness lies inside the pointwise 95% envelope on the declared observables. For the selected route the representative plateau witness is not excluded. This status is a same-stack nonexclusion witness for the background envelope only. It is not a data-selected CHC-DE fit, not a preference statement over $\Lambda$TeX\LambdaCDM, not a full CHC-DE family posterior, and not a sealed reconstruction stack. Distance mapping, perturbation-side inference, clock inference, and halo--background synthesis remain outside the DE lane.

The homogeneous FRW reduction of the CHC-min action admits a late-time convex phase-flatness branch in which $w_{\mathrm{eff}}\approx -1$TeXw_{\mathrm{eff}}\approx -1 on the admitted homogeneous-FRW, background-only non-phantom domain. In that regime,

\rho_{\HH}\approx U_{\mathrm{flat}},
\qquad
p_{\HH}\approx -U_{\mathrm{flat}},
\qquad
\Lambda_{\mathrm{eff}}=\frac{U_{\mathrm{flat}}}{M_{\mathrm{Pl}}^2},

so the background expansion reproduces the same homogeneous FRW equations as an effective cosmological-constant contribution, with the constant read as phase-geometric. Within the homogeneous reduction, $U_{\rm flat}$TeXU_{\rm flat} enters as a geometric phase constant in the effective homogeneous stress tensor; no microscopic cancellation of quantum-field vacuum contributions is derived.

The main structural result is a family-level admissibility statement within a single declared potential family. Once a family $U_{\rm eff}(\HH)=\mu^4F((\HH-\HH_*)/\Delta;\alpha_i)$TeXU_{\rm eff}(\HH)=\mu^4F((\HH-\HH_*)/\Delta;\alpha_i) is fixed, the plateau height $U_{\rm flat}$TeXU_{\rm flat} and the local plateau curvature $m_{\rm flat}$TeXm_{\rm flat} are not independent tunings. The explicit benchmark in reference--reference shows that the admitted plateau set need not be empty. This family-level non-independence is not a completed cosmological posterior statement and does not calibrate $U_{\rm flat}$TeXU_{\rm flat} or $m_{\rm flat}$TeXm_{\rm flat} to a background fit.

The branch is excluded only if, within the declared late-time BAO--CMB--SN fitting convention, robust background inference on the adopted domain requires persistent phantom behavior, or if no admitted potential family supports a positive locally convex plateau or near-plateau on the homogeneous branch. If the background posterior collapses onto a purely $\Lambda$TeX\Lambda-equivalent limit, the branch remains an alternative phase-geometric reading rather than an observationally required extension. Within the declared literature-facing background route, Pantheon+, Planck PR4/NPIPE, ACT DR6, and DESI DR2 do not by themselves exclude the representative non-phantom homogeneous plateau reading under the stated convention; they do not select, require, or calibrate the full CHC-DE family [citation].

Only homogeneous background dynamics are used. Distance mapping, clock inference, perturbation inference, and full halo--background synthesis are excluded.

*Minimal action and homogeneous reduction

We use the metric signature $(-,+,+,+)$TeX(-,+,+,+) and define

X_{\HH}:=g^{\mu\nu}\nabla_\mu\HH\,\nabla_\nu\HH .

The background branch uses the CHC-min action

S_{\rm CHC}
=
\int d^4x\,\sqrt{-g}\Bigg[
\frac{M_{\rm Pl}^2}{2}R
-\frac{M_{\rm eff}^2}{2}X_{\HH}
-U_{\rm eff}(\HH)
\Bigg]
+S_m[g,\Psi],

together with the spatially homogeneous specialization $\HH=\HH(t)$TeX\HH=\HH(t) on a flat FRW metric. With signature $(-,+,+,+)$TeX(-,+,+,+), the homogeneous specialization gives $X_{\HH}=-\dot{\HH}^{2}$TeXX_{\HH}=-\dot{\HH}^{2} on that branch. The resulting effective background variables are

\rho_{\HH}
=
\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 + U_{\rm eff}(\HH),
\qquad
p_{\HH}
=
\frac{M_{\rm eff}^2}{2}\dot{\HH}^2 - U_{\rm eff}(\HH),

and

w_{\rm eff}=\frac{p_{\HH}}{\rho_{\HH}}.

No independent dark-energy fluid variable is introduced beyond the homogeneous mode of $\HH$TeX\HH.

*Declared potential-family admissibility and common parameter set

The background branch is read only on one declared potential family and one admitted homogeneous FRW window. The common parameter set is

\Theta_{\rm CHC}=\{M_{\rm eff},\Lambda_\Xi,\theta_U\},

with $\theta_U$TeX\theta_U denoting the parameters of the chosen family $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH). Within the chosen family reference, the local plateau curvature is

m_{\rm flat}^2=\frac{\mu^4}{\Delta^2}F''(x_{\rm flat}),

and the plateau organizer is

\varepsilon_{\rm flat}^{(F)}=
\frac{F''(x_{\rm flat})}{F(x_{\rm flat})}.

These quantities enter only the family-level homogeneous plateau admissibility statement. *Scope

Included statements:

- the homogeneous FRW reduction of the CHC-min action; - the admitted non-phantom plateau branch with $w_{\rm eff}\simeq -1$TeXw_{\rm eff}\simeq -1; - the shared-family plateau organizer and the resulting non-independent tuning of $U_{\rm flat}$TeXU_{\rm flat} and $m_{\rm flat}$TeXm_{\rm flat}; - an explicit one-family existence demonstration showing that the admitted plateau set need not be empty on one declared family; - the observational and structural conditions used to assess the background branch.

Excluded statements:

- a cosmological posterior fit; - a full halo--background synthesis over $\Theta_{\rm CHC}$TeX\Theta_{\rm CHC}; - perturbation-side inference, ISW extraction, or late-time cross-correlation fitting; - clock or distance-mapping layers beyond the FRW background.

The CHC standing assumptions used above are summarized below in the form adapted to the cosmological branch.

*L4: Recovery principle

In regimes where phase gradients are negligible and the effective potential is locally convex,

\nabla\HH \to 0,
\qquad
U_{\rm eff}''(\HH)>0,

the CHC field equations reduce to Einstein's equations with an effective cosmological constant:

G_{\mu\nu}
=
\frac{1}{M_{\rm Pl}^2}\,T^{(m)}_{\mu\nu}
-\Lambda_{\rm eff} g_{\mu\nu}
+\mathcal{O}\!\left(|\nabla\HH|_{\rm bg}^{2}\right), 

\quad \Lambda_{\rm eff} = \frac{U_{\rm eff}(\HH)}{M_{\rm Pl}^2}.

In particular, on a phase-flat plateau with $U_{\rm eff}\to U_{\rm flat}$TeXU_{\rm eff}\to U_{\rm flat}, one recovers GR+$\Lambda$TeX\Lambda with $\Lambda=\Lambda_{\rm eff}$TeX\Lambda=\Lambda_{\rm eff} up to corrections organized by the background-adapted phase-gradient norm, equivalently by $\Xi$TeX\Xi on the homogeneous FRW branch.

*L7: Information--energy correspondence

Phase fluctuations in CHC carry information-theoretic content that is encoded as thermodynamic curvature rather than as a naive sum of zero-point energies. In homogeneous cosmology this is expressed by the statement that the phase-sector contribution to the energy-- momentum tensor has the form

T^{(\HH)}_{\mu\nu}
=
M_{\rm eff}^2\partial_\mu\HH\,\partial_\nu\HH
-g_{\mu\nu}\left(\frac{M_{\rm eff}^2}{2}X_{\HH}+U_{\rm eff}(\HH)\right),

with $U_{\rm eff}(\HH)$TeXU_{\rm eff}(\HH) representing a geometric phase constant on the plateau, rather than a direct sum of vacuum-mode energies. This statement concerns the effective homogeneous stress tensor only and does not derive a microscopic cancellation of quantum-field vacuum contributions.

*L9: Phase-gradient hierarchy

The phase-gradient hierarchy assumption states that, at late times and on sufficiently large scales, the homogeneous FRW branch used here is organized by the dimensionless gradient measure

\Xi
=
\frac{M_{\rm eff}^2\dot{\HH}^2}{\Lambda_\Xi^4}.

This is the homogeneous cosmological specialization of the more general CHC gradient hierarchy. In the slow-roll regime near a convex plateau, the late-time statement is that $\Xi\to 0$TeX\Xi\to 0 along the approach branch. A strict local monotone decrease follows only under the sufficient condition reference.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.