Cosmic Energy Inversion Theory (CEIT‑v5): A Self‑Consistent Geometric‑Field Framework for Gravity, Cosmology, and Entropy

1. Introduction

The standard $\mathsf{\Lambda }$CDM cosmological model has served for over two decades as the reference paradigm, successfully accounting for the large-scale structure of the Universe, cosmic microwave background (CMB) anisotropies, and the expansion history inferred from Type Ia supernovae [1],[2]. However, a growing body of high-precision observational evidence now challenges its foundational assumptions at a level beyond statistical fluctuations. The most prominent discrepancy, the Hubble tension, refers to the significant difference between the CMB-inferred value of the Hubble constant from the early universe ($H_0=67.4\pm 0.5\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$) from the Planck Collaboration [2] and the distance-ladder measurement from the late universe ($H_0=73.04\pm 1.04\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$) from the SH0ES program [3]. This discrepancy, commonly reported at the $5\sigma$ level [4], indicates that either the $\mathsf{\Lambda }$CDM model is incapable of simultaneously explaining early- and late-universe data, or that new physics beyond this model is at work.

Concurrently with the Hubble tension, four decades of direct searches for dark matter particles, despite unprecedented sensitivities, have failed to produce any credible candidate. The LUX-ZEPLIN (LZ) experiment currently sets the strongest limit, excluding spin-independent WIMP-nucleon cross sections above $9.2\times {10}^{-48}{\mathrm{cm}}^2$ for a $36\mathrm{GeV}/c^2$ WIMP at $90\mathrm{\%}$ confidence [5]. This long-standing observational void has severely questioned the hypothesis of an unknown fundamental particle. Most recently, the James Webb Space Telescope (JWST) has revealed massive, morphologically mature galaxy candidates at redshifts $z>10$, within the first $500\mathrm{Myr}$ after the Big Bang, whose stellar masses and formation timescales appear incompatible with the hierarchical assembly predicted by $\mathsf{\Lambda }$CDM by more than two orders of magnitude [6],[7]. Although this tension remains under active debate as stellar-mass estimates are revised, there is growing evidence for the need for new physics in the early universe [8]. These compounding tensions motivate a systematic exploration of alternatives that replace hypothetical dark-sector entities with intrinsic space-time dynamics.

The range of theoretical efforts to replace dark matter and dark energy encompasses a wide spectrum of approaches. Modified Newtonian Dynamics (MOND) successfully fits galactic rotation curves without dark matter [9], but fails on cluster scales and lacks a consistent relativistic cosmological completion [10]. Emergent gravity approaches offer conceptual insight [11], but remain largely phenomenological and untested at CMB scales. Einstein–Cartan theory introduces space-time torsion coupled to fermionic spin [12], yet the resulting effects are negligible at galactic densities [13]. What the observational data demand is a single covariant framework in which torsion is sourced by a field that is pervasive enough to operate from sub-galactic to cosmological scales.

The Cosmic Energy Inversion Theory (CEIT) has been introduced as an alternative framework in which torsion is sourced by gradients of a scalar energy field [14]. The fifth version (CEIT-v5), presented in this paper, builds upon the same foundational principles as earlier versions, but achieves a higher level of mathematical self-consistency and conceptual clarity through structural revisions of the equations. In this version, emphasis is placed on the principle that space-time curvature is the direct manifestation of the presence of energy-field gradients around massive objects. Matter (through its density) causes a local reduction of the field and hence a gradient. In the absence of matter, the field gradient vanishes and curvature also vanishes. Thus, general relativity does not exist as an independent theory within this framework; rather, it is a phenomenological description of the curvature arising from matter (via field gradients). Gravitational waves are interpreted as propagating disturbances in the energy field, travelling at the speed of light.

Earlier versions of this theory provided initial numerical estimates of parameters using computational tools, which remain valuable as reference points for future calibration. However, the aim of the present version is to provide a framework that can be directly tested against independent observational data, not merely rely on numerical fits. Observational validation, which requires high costs and extensive observational collaborations, is considered the main step in validating this framework. Accordingly, this paper does not provide definitive numerical values for parameters and postpones quantitative calibration with independent data (e.g., SPARC, Gaia, Planck, DESI, and LIGO/Virgo) to future work. This approach ensures methodological transparency and avoids premature claims of quantitative agreement with data.

2. Methodology

The Cosmic Energy Inversion Theory, fifth version (CEIT-v5), is formulated as a geometric-field framework in which the independent concepts of dark matter and dark energy are replaced by a single, fundamental scalar field $\mathsf{\Phi }$ that permeates all of spacetime. The theory is built upon six axioms and a single action principle in Einstein–Cartan–Ehresmann geometry. All equations presented in this section are derived from that action; no additional postulates are introduced for different physical scales.

2.1. Foundational Axioms

The six axioms below constitute the conceptual foundation of the theory. They are unchanged from earlier versions and remain the sole conceptual input; all subsequent equations are derived from them and from the action principle of Section 2.3.

Axiom I (Fundamental Space-time Energy Field):

A real, positive-definite scalar field $\mathsf{\Phi }\left(x^{\mu }\right)$ with dimensions of energy (mass-dimension 1 in natural units $\hslash =c=1$) exists, assigning a background energy density to every point of spacetime. The metric $g_{\mu \nu }$ is an independent degree of freedom; the relationship between curvature and field gradients is derived from the field equations, not assumed.

Axiom II (Spatial Inversion – Energy Hierarchy):

Formation of gravitationally bound material structures causes a local reduction of $\mathsf{\Phi }$. Thus, $\mathsf{\Phi }$ reaches a local minimum at the centres of massive objects and increases monotonically with distance from the mass, approaching the cosmic background value ${\mathsf{\Phi }}_{\mathrm{\infty }}$. This establishes a nested hierarchy:

$${\mathsf{\Phi }}_{\mathrm{core}}<{\mathsf{\Phi }}_{\mathrm{edge}}<{\mathsf{\Phi }}_{\mathrm{IGM}}.$$

Axiom III (Separation of Density and Gradient Roles):

The field $\mathsf{\Phi }$ plays two independent roles:

(a) the density $\mathsf{\Phi }$ acts as an energy-absorption agent for particles and radiation, coupling with torsion;

(b) the gradient $\nabla \mathsf{\Phi }$ acts as a source of geometric pressure that enters the modified Poisson equation and produces additional gravitational force, equivalent to dark matter.

Axiom IV (Orbital Stability Condition):

A test particle in a stable circular orbit remains in that orbit only when the net gradient of $\mathsf{\Phi }$ vanishes in the commoving frame:

$$\nabla {\mathsf{\Phi }}_{\mathrm{total}}=0.$$

Axiom V (Positive Feedback and Critical Threshold):

The rate of energy absorption by the field is directly proportional to the field itself. Hence, as $\mathsf{\Phi }$ increases, the absorption rate also increases. This positive feedback drives the field toward a critical value ${\mathsf{\Phi }}_c$, near which absorption grows superlinearly. This linear proportionality holds only in the intermediate regime $\mathsf{\Phi }\ll {\mathsf{\Phi }}_c$ (such as the current epoch); the complete nonlinear form, which exhibits different behavior in the two extreme limits $\mathsf{\Phi }\to 0$ (near black-hole singularities) and $\mathsf{\Phi }\to {\mathsf{\Phi }}_c$ (at the quantum bounce), is given in Equation (21).

Axiom VI (Quantum Bounce and Cyclic Reset):

When $\mathsf{\Phi }$ reaches ${\mathsf{\Phi }}_c$, a non-classical phase transition (quantum bounce) occurs, in which $\mathsf{\Phi }$ jumps to a very large initial value ${\mathsf{\Phi }}_0\gg {\mathsf{\Phi }}_c$, and the field’s entropy drastically decreases. This event closes the cosmic cycle and gives birth to a new universe. In the absence of ordinary matter, spacetime curvature is not an independent entity; its value is completely determined by the gradients of $\mathsf{\Phi }$.

2.2. Geometric Foundations and Torsion

The geometric arena is a four-dimensional manifold equipped with an Einstein–Cartan–Ehresmann connection, in which torsion is present as an independent geometric degree of freedom alongside Riemannian curvature. The torsionful connection is written as

$${\mathsf{\Gamma }}_{\mu \nu }^{\alpha }={\stackrel{\mathrm{‾}}{\mathsf{\Gamma }}}_{\mu \nu }^{\alpha }+K_{\mu \nu }^{\alpha },$$

Where ${\stackrel{\mathrm{‾}}{\mathsf{\Gamma }}}_{\mu \nu }^{\alpha }$ is the usual Christoffel symbol (constructed solely from the metric $g_{\mu \nu }$) and $K_{\mu \nu }^{\alpha }$ is the contortion tensor. The torsion tensor is defined as the antisymmetric part of the connection:

$$T_{\mu \nu }^{\alpha }\equiv {\mathsf{\Gamma }}_{\mu \nu }^{\alpha }-{\mathsf{\Gamma }}_{\nu \mu }^{\alpha }.$$

In CEIT, torsion is sourced not by fermionic spin, but by the gradient of the fundamental energy field $\mathsf{\Phi }$. This constitutive relation is chosen to be the simplest covariant expression that vanishes when the field is uniform, respects local Poincaré invariance, and automatically satisfies the Bianchi identities. Specifically,

$$T_{\mu \nu }^{\alpha }={\displaystyle \frac{\kappa }{E_H}}\left({\delta }_{\mu }^{\alpha }{\partial }_{\nu }\mathsf{\Phi }-{\delta }_{\nu }^{\alpha }{\partial }_{\mu }\mathsf{\Phi }\right),$$

(1)

and, correspondingly,

$$K_{\mu \nu \alpha }={\displaystyle \frac{\kappa }{E_H}}\left(g_{\mu \alpha }{\partial }_{\nu }\mathsf{\Phi }-g_{\nu \alpha }{\partial }_{\mu }\mathsf{\Phi }\right),$$

(2)

Where $\kappa$ is a dimensionless coupling constant to be calibrated, and $E_H\equiv 246\mathrm{GeV}$ is the fixed electroweak scale. This structure ensures that torsion vanishes in regions where the field is uniform, recovering the symmetric connection of general relativity in that limit.

2.3. Complete Action

The dynamics of the system are derived from a well-defined action principle. The complete CEIT action is

$$S=\int d^{4}x\sqrt{-g}[{\displaystyle \frac{R\left(\mathsf{\Gamma }\right)}{16\pi G}}+{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\mathsf{\Phi }{\partial }_{\nu }\mathsf{\Phi }-V(\mathsf{\Phi })+{\displaystyle \frac{\xi }{E_H}}\mathsf{\Phi }{T}_{\mu \nu }^{\alpha }{T}_{\alpha }^{\mu \nu }+{\displaystyle \frac{{\eta }_m}{E_H}}\mathcal{G}(\mathsf{\Phi }\left)T^{\left(\mathrm{matter}\right)}\right]+S_{\mathrm{matter}},$$

(3)

$$\mathcal{G}\left(\mathsf{\Phi }\right)\equiv {\displaystyle \frac{\mathsf{\Phi }}{1-(\mathsf{\Phi }/{\mathsf{\Phi }}_c{)}^{n_{\mathsf{\Phi }}}}},n_{\mathsf{\Phi }}>1$$

(3a)

The linear matter–field coupling $\mathsf{\Phi }{T}^{\left(\mathrm{matter}\right)}$ in previous versions is only the first-order approximation of the full function $\mathcal{G}\left(\mathsf{\Phi }\right)$ in the regime $\mathsf{\Phi }\ll {\mathsf{\Phi }}_c$. This nonlinear generalization ensures that the energy exchange between matter and the field exhibits significant nonlinear behavior only in the vicinity of the two limiting events of the theory — black-hole singularities ($\mathsf{\Phi }\to 0$) and the quantum bounce ($\mathsf{\Phi }\to {\mathsf{\Phi }}_c$) — while in the intermediate epoch (such as today) it reduces to the earlier linear form.

Where:

• $R\left(\mathsf{\Gamma }\right)$ is the curvature scalar of the torsionful connection, decomposing into the Riemannian curvature (from the Christoffel symbol) and second-order contortion terms.
• ${\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\mathsf{\Phi }{\partial }_{\nu }\mathsf{\Phi }$ is the kinetic term of the scalar field.
• $V\left(\mathsf{\Phi }\right)$ is the self-interaction potential, which determines the intrinsic energy of the field and, through its curvature, the screening length.
• ${\displaystyle \frac{\xi }{E_H}}\mathsf{\Phi }{T}_{\mu \nu }^{\alpha }{T}_{\alpha }^{\mu \nu }$ is the torsion–field coupling term, responsible for the energy absorption mechanism from radiation and particles.
• ${\displaystyle \frac{{\eta }_m}{E_H}}\mathsf{\Phi }{T}^{\left(\mathrm{matter}\right)}$ is the explicit matter–field coupling term, where $T^{\left(\mathrm{matter}\right)}=g^{\mu \nu }{T}_{\mu \nu }^{\left(\mathrm{matter}\right)}$ is the trace of the matter energy–momentum tensor.

$S_{\mathrm{matter}}$ is the matter action, minimally coupled to the metric.

2.4. Field Equations

Variation of the action with respect to the metric $g^{\mu \nu }$ yields the modified Einstein equations. There is no separate cosmological constant term:

$$G_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\left(\mathrm{matter}\right)}+T_{\mu \nu }^{\left(\mathsf{\Phi }\right)}\right),$$

(4)

Where $G_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R$ is the conventional Einstein tensor constructed solely from $g_{\mu \nu }$. The energy–momentum tensor of the $\mathsf{\Phi }$ field is

$$T_{\mu \nu }^{\left(\mathsf{\Phi }\right)}={\displaystyle \frac{\beta }{E_H^2}}\left({\partial }_{\mu }\mathsf{\Phi }{\partial }_{\nu }\mathsf{\Phi }-g_{\mu \nu }\left({\displaystyle \frac{1}{2}}(\partial \mathsf{\Phi }{)}^2-V(\mathsf{\Phi })\right)\right),\beta \equiv 1+{\displaystyle \frac{4\xi {\kappa }^2}{E_H^2}}.$$

(4a)

In the limit $\xi \to 0$, $\beta \to 1$, recovering the standard form of a minimal scalar field.

Variation with respect to $\mathsf{\Phi }$ gives the nonlinear wave equation:

$$◻\mathsf{\Phi }+V^{\prime }\left(\mathsf{\Phi }\right)-{\displaystyle \frac{4\xi {\kappa }^2}{E_H^3}}(\partial \mathsf{\Phi }{)}^2+{\displaystyle \frac{{\eta }_m}{E_H}}{\mathcal{G}}^{\prime }(\mathsf{\Phi })T^{\left(\mathrm{matter}\right)}=0.$$

(5)

The term $-{\displaystyle \frac{4\xi {\kappa }^2}{E_H^3}}(\partial \mathsf{\Phi }{)}^2$ acts as a “cosmic frictional force” arising from the torsion–field coupling; the term ${\displaystyle \frac{{\eta }_m}{E_H}}{T}^{\left(\mathrm{matter}\right)}$ is the matter source, which in the non-relativistic limit reduces to a screened Poisson equation.

$${\mathcal{G}}^{\prime }\left(\Phi \right)={\displaystyle \frac{1+(n_{\mathsf{\Phi }}-1)(\mathsf{\Phi }/{\mathsf{\Phi }}_c{)}^{n_{\mathsf{\Phi }}}}{(1-(\mathsf{\Phi }/{\mathsf{\Phi }}_c{)}^{n_{\mathsf{\Phi }}}{)}^2}}$$

(5a)

For $\mathsf{\Phi }\ll {\mathsf{\Phi }}_c$, ${\mathcal{G}}^{\prime }\left(\mathsf{\Phi }\right)\approx 1$, and the equation reduces to the previous linear form. Therefore, the screened Poisson equation (7) and all derived galactic/cosmological results (Equations (7)–(20)) remain unchanged, as they all operate in the regime $\mathsf{\Phi }\ll {\mathsf{\Phi }}_c$.

2.5. Vacuum Curvature Theorem

Taking the trace of (4) in a matter-free region ($T_{\mu \nu }^{\left(\mathrm{matter}\right)}=0$, hence $T^{\left(\mathrm{matter}\right)}=0$), and using $g^{\mu \nu }{G}_{\mu \nu }=-R$ in four dimensions, we obtain

$$R_{\mu \nu |\mathrm{vacuum}}={\displaystyle \frac{8\pi G\beta }{E_H^2}}\left({\partial }_{\mu }\mathsf{\Phi }{\partial }_{\nu }\mathsf{\Phi }-g_{\mu \nu }V\left(\mathsf{\Phi }\right)\right).$$

(6)

Vacuum Curvature Theorem: In the absence of ordinary matter, the Ricci tensor decomposes into a part proportional to the metric $(-g_{\mu \nu }V(\mathsf{\Phi }\left)\right)$, arising from the field’s potential, and a part directly sourced by the tensor product of the field gradient with itself $\left({\partial }_{\mu }\mathsf{\Phi }{\partial }_{\nu }\mathsf{\Phi }\right)$. If $\mathsf{\Phi }$ is uniform and the potential is at its minimum with $V\left({\mathsf{\Phi }}_{\mathrm{\infty }}\right)=0$, then $R_{\mu \nu }=0$ and space-time becomes flat (special relativity). This theorem expresses the central claim: space-time curvature in vacuum is the direct manifestation of energy-field gradients.

2.6. Weak-Field Limit and Screened Poisson Equation

In the quasi-static limit relevant for galactic dynamics, the wave Equation (5) reduces to a screened Poisson equation. Neglecting temporal derivatives and the nonlinear friction term, defining $\delta \mathsf{\Phi }={\mathsf{\Phi }}_{\mathrm{\infty }}-\mathsf{\Phi }$, expanding the potential around its minimum (${\mathsf{\Phi }}_{\mathrm{\infty }}$) with $V^{\prime }\left({\mathsf{\Phi }}_{\mathrm{\infty }}\right)=0$, and using ${\eta }_m\equiv 4\pi G{E}_H/c^4$, we get

$${\nabla }^2\left(\delta \mathsf{\Phi }\right)-{\displaystyle \frac{\delta \mathsf{\Phi }}{{\lambda }^2}}={\displaystyle \frac{4\pi G}{c^2}}{\rho }_m,$$

(7)

Where the screening length is derived self-consistently from the curvature of the potential at its minimum:

$${\lambda }^{-2}={\displaystyle \frac{V^{″}\left({\mathsf{\Phi }}_{\mathrm{\infty }}\right)}{c^2}}.$$

(7a)

The general solution is the Yukawa-type Green’s integral:

$$\delta \mathsf{\Phi }\left(\mathbf{r}\right)={\displaystyle \frac{G}{c^2}}\int d^3{r}^{\prime }{\displaystyle \frac{{\rho }_m\left({\mathbf{r}}^{\prime }\right)e^{-|\mathbf{r}-{\mathbf{r}}^{\prime }|/\lambda }}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}.$$

(7b)

The negative sign encodes the inversion principle: increasing matter density reduces $\mathsf{\Phi }$. Non-thermal contributions from magnetic fields and turbulence, which are physically present sources of stress–energy, are included in the source term ${\rho }_m$ where appropriate, without altering the structure of the equation.

2.7. Dark Matter as Geometric Pressure

Substituting (7b) into (4) and taking the (00) component in the weak-field limit yields the effective Poisson equation:

$${\nabla }^2\mathsf{\Psi }=4\pi G{\rho }_m+{\displaystyle \frac{c^2\beta }{2E_H^2}}(\nabla \mathsf{\Phi }{)}^2.$$

(8)

The second term acts as a geometric pressure. It can be written as an effective dark-matter density:

$${\rho }_{\mathrm{DM}}^{\left(\mathrm{eff}\right)}={\displaystyle \frac{c^2\beta }{16\pi G{E}_H^2}}(\nabla \mathsf{\Phi }{)}^2.$$

(8a)

The orbital velocity of test particles in a galaxy, from the orbital stability condition (Axiom IV), is

$$v^2\left(r\right)={\displaystyle \frac{G{M}_{\mathrm{vis}}\left(r\right)}{r}}+{\displaystyle \frac{c^2\beta }{E_H^2}}{\int }_0^r r^{\prime }{\left({\displaystyle \frac{d\mathsf{\Phi }}{d{r}^{\prime }}}\right)}^{2}d{r}^{\prime }.$$

(9)

The effective dark-matter density is always non-negative and grows where $\mathsf{\Phi }$ varies most rapidly. In the outer regions of galaxies, the geometric term dominates and compensates for the Keplerian decline, producing flat rotation curves.

2.8. Effective Potential and Acceleration (Solar System Scale)

For solar-system tests, the effective gravitational potential is defined as

$${\mathsf{\Psi }}_{\mathrm{eff}}=-{\displaystyle \frac{c^2\beta }{2E_H^2}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_{\mathrm{bg}}}}\right),$$

(10)

Where ${\mathsf{\Phi }}_{\mathrm{bg}}$ is the local background field. The gravitational acceleration is

$$a=-\nabla {\mathsf{\Psi }}_{\mathrm{eff}}={\displaystyle \frac{c^2\beta }{2E_H^2}}{\displaystyle \frac{\nabla \mathsf{\Phi }}{\mathsf{\Phi }}}.$$

(11)

The field gradient created by a single mass is obtained from (7b):

$$\nabla {\mathsf{\Phi }}_{\mathrm{object}}\left(r\right)={\displaystyle \frac{G{M}_{\mathrm{object}}}{c^2{r}^2}}{\mathsf{\Phi }}_{\mathrm{bg}}\left(R_{\mathrm{gal}}\right)\cdot F_{\mathrm{screen}}(r,\lambda ),$$

(12)

Where $F_{\mathrm{screen}}$ is a screening function that tends to 1 at distances much smaller than $\lambda$ and to 0 at distances much larger than $\lambda$. Its explicit form is derived from the action (3).

2.9. Three-Component Field Decomposition

The total field is decomposed into three physically distinct components:

$$\mathsf{\Phi }(\mathrm{x},\mathrm{t})=\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\left(\mathrm{t}\right)+{\mathsf{\Phi }}_{\mathrm{host}}\left(\mathrm{x}\right)+\mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{local}}(\mathrm{x},\mathrm{t}).$$

(13)

Here $\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\left(\mathrm{t}\right)$ is the homogeneous background field governing cosmic expansion; ${\mathsf{\Phi }}_{\mathrm{host}}\left(\mathrm{x}\right)$ is the quasi-static profile of a galaxy or cluster obtained from (7b); and $\mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{local}}$ captures time-dependent perturbations from satellite galaxies, spiral arms, magnetic fields, or turbulence. This decomposition enables consistent treatment of phenomena across vastly different scales.

2.10. Orbital Stability, Satellite Perturbations, and the Galactic Warp

From (9), the local stability condition (Axiom IV) defines the excess centripetal demand beyond the Keplerian contribution:

$$\mathrm{F}\left(\mathrm{r}\right)\equiv {\displaystyle \frac{{\mathrm{v}}^2\left(\mathrm{r}\right)}{\mathrm{r}}}-{\displaystyle \frac{\mathrm{G}{\mathrm{M}}_{\mathrm{vis}}\left(\mathrm{r}\right)}{{\mathrm{r}}^2}}={\displaystyle \frac{{\mathrm{c}}^2\mathsf{\beta }}{{\mathrm{E}}_{\mathrm{H}}^2}}{\left({\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_{\mathrm{host}}}{\mathrm{d}\mathrm{r}}}\right)}^2\mathrm{r}.$$

(14)

This specifies the required field gradient at radius $\mathrm{r}$ to maintain a stable orbit.

For a two-body system (e.g., the Milky Way and a massive satellite like the Large Magallanes Cloud), the presence of the satellite perturbs the axisymmetric host field. The total perturbation $\mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{comp}}$ is the sum of host and satellite contributions from (7b). The asymmetric vertical gradient of $\mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{comp}}$ at the galactic disk produces a vertical force per unit mass:

$${\mathrm{F}}_{\mathrm{z}}^{\mathrm{warp}}(\mathrm{R},\mathsf{\varphi })={\displaystyle \frac{{\mathrm{c}}^2\mathsf{\beta }}{4\mathsf{\pi }\mathrm{G}{\mathrm{E}}_{\mathrm{H}}^2}}\mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{comp}}{\displaystyle \frac{\partial \mathsf{\delta }{\mathsf{\Phi }}_{\mathrm{comp}}}{\partial \mathrm{z}}}.$$

(15)

This force, varying with azimuthal angle $\mathsf{\varphi }$, induces a coherent vertical displacement of the outer disk — the Galactic warp. The predicted warp profile ${\mathrm{z}}_{\mathrm{w}}\left(\mathrm{R}\right)$ is directly comparable to astrometric data (e.g., Gaia DR3/DR4) once the underlying parameters are calibrated. The relaxation timescale ${\mathsf{\tau }}_{\mathrm{relax}}\sim \mathsf{\lambda }/\mathrm{c}$ is typically many orders of magnitude shorter than the satellite’s orbital period, implying that the warp phase remains locked to the satellite’s azimuth — a natural geometric resolution of the warp precession problem.

2.11. Background Cosmology and Dynamical Dark Energy

For the homogeneous background field $\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\left(t\right)$, the wave Equation (5) reduces to

$$\ddot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}+3H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}+V^{\prime }\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)-{\displaystyle \frac{4\xi {\kappa }^2}{E_H^3}}{\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2=0.$$

(16)

The effective energy density and pressure of the field, from (4a) in a Friedmann–Lemaître–Robertson–Walker metric, are

$${\rho }_{\mathsf{\Phi }}={\displaystyle \frac{1}{2}}{\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2+V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)+{\displaystyle \frac{3}{2}}H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}\stackrel{\mathrm{‾}}{\mathsf{\Phi }},$$

(17)

$$p_{\mathsf{\Phi }}={\displaystyle \frac{1}{2}}{\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2-V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)-{\displaystyle \frac{1}{2}}H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}\stackrel{\mathrm{‾}}{\mathsf{\Phi }}.$$

(18)

The dynamical equation of state is then

$$w_{\mathrm{DE}}(t)={\displaystyle \frac{p_{\mathsf{\Phi }}}{{\rho }_{\mathsf{\Phi }}}}={\displaystyle \frac{\frac{1}{2}{\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2-V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)-\frac{1}{2}H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}{\frac{1}{2}{\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2+V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)+\frac{3}{2}H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}.$$

(19)

In the current epoch, where ${\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}}^2\ll V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)$ and $H\dot{\stackrel{\mathrm{‾}}{\mathsf{\Phi }}}\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\ll V\left(\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\right)$, we have $w_{\mathrm{DE}}\to -1$, reproducing cosmological-constant behaviour.

2.12. Field-Dependent Particle Stability and Early Structure Formation

Axioms III and V imply that the effective potential governing bound-state stability depends on the ambient field $\mathsf{\Phi }$, since $\mathsf{\Phi }$ mediates energy exchange with matter through the fifth term of the action (3). Generalizing the earlier phenomenological relation to the present field notation, the characteristic lifetime of a structure-forming species is

$${\mathsf{\tau }}_{\mathrm{i}}\left(\mathsf{\Phi }\right)={\mathsf{\tau }}_{\mathrm{i}}^{\left(0\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\mathsf{\beta }}_{\mathrm{struct}}{\displaystyle \frac{\mathsf{\Phi }-{\mathsf{\Phi }}_0}{{\mathsf{\Phi }}_0}}\right],$$

(20)

Where ${\mathsf{\Phi }}_0$ is the present-day background value and ${\mathsf{\beta }}_{\mathrm{struct}}$ is a dimensionless coupling to be calibrated against high-redshift data. At early times, the background field $\stackrel{\mathrm{‾}}{\mathsf{\Phi }}\left(\mathrm{z}\right)$ is larger than its present value, so (20) predicts a reduction in characteristic formation timescales — offering a natural internal mechanism for the rapid assembly of massive, morphologically mature galaxies observed by JWST at $\mathrm{z}>10$. This mechanism is structurally built into the theory; its quantitative predictions require calibration of ${\mathsf{\beta }}_{\mathrm{struct}}$ against galaxy mass functions and are deferred to future work.

2.13. Black Hole Entropy, Modified Evaporation, and the Cosmic Cycle

The rate of entropy absorption from Hawking radiation by the $\mathsf{\Phi }$ field, based on Axioms III and V, is

$${\displaystyle \frac{d{S}_{\mathrm{abs}}}{dt}}={\displaystyle \frac{\alpha }{E_H}}{\int }_{\mathcal{H}} \mathcal{G}\left(\mathsf{\Phi }\right(x\left)\right){\rho }_{\mathrm{rad}}\left(x\right)d^{3}x,$$

(21)

With this form, three distinct physical regimes emerge from a single formula without additional assumptions: (1) Near the event horizon, where the local field reaches its minimum ($\mathsf{\Phi }\to 0$), the entropy absorption also reaches its minimum ($\mathcal{G}\to 0$), and Hawking radiation effectively escapes without absorption — precisely the behavior expected at black-hole singularities. (2) In the intermediate epoch, away from both extremes (such as the current cosmic era), $\mathcal{G}\left(\mathsf{\Phi }\right)\approx \mathsf{\Phi }$, meaning no bounce effects are observed in the ordinary evolution of the universe. (3) As the background field approaches the critical threshold ($\mathsf{\Phi }\to {\mathsf{\Phi }}_c$), the absorption grows superlinearly and diverges; this divergence itself can serve as the physical driving mechanism for reaching the critical conditions necessary for the quantum bounce (Axiom VI), smoothly connecting the inherently nonlinear and discontinuous “birth moment” of the new universe to the preceding behavior.

The field also modifies the Hawking mass-loss rate directly:

$${\displaystyle \frac{dM}{dt}}=-{\displaystyle \frac{\hslash c^4}{15360\pi G^2{M}^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_{\mathrm{crit}}}}\right)-{\gamma }^{\prime }{\displaystyle \frac{(\nabla \mathsf{\Phi }{)}^2}{E_H{M}^{3/2}}},$$

(22)

Where ${\mathsf{\Phi }}_{\mathrm{crit}}$ is a critical threshold and ${\gamma }^{\prime }$ is a dimensionless constant. The first term recovers the standard Hawking rate, suppressed in low-$\mathsf{\Phi }$ environments; the second, gradient-driven term dominates where $\nabla \mathsf{\Phi }$ is large, such as in the intergalactic medium.

The visible entropy evolves as

$${\displaystyle \frac{d{S}_{\mathrm{vis}}}{dt}}={\displaystyle \frac{d{S}_{\mathrm{BH}}}{dt}}-{\displaystyle \frac{d{S}_{\mathrm{abs}}}{dt}},$$

(23)

and the second law is preserved by field-entropy production:

$${\displaystyle \frac{d{S}_{\mathrm{tot}}}{dt}}={\displaystyle \frac{d{S}_{\mathrm{vis}}}{dt}}+{\displaystyle \frac{d{S}_{\mathsf{\Phi }}}{dt}}\ge 0,{\displaystyle \frac{d{S}_{\mathsf{\Phi }}}{dt}}\equiv {\displaystyle \frac{d{S}_{\mathrm{abs}}}{dt}}+{\displaystyle \frac{d{S}_{\mathrm{irr}}}{dt}},$$

(24)

With $d{S}_{\mathrm{irr}}$ a dissipative term from positive feedback. The Page time is modified as

$$t_{\mathrm{Page}}=t_{\mathrm{Page}}^{\left(0\right)}\left[1+\eta {\left({\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_c}}\right)}^n\right],$$

(25)

Where $t_{\mathrm{Page}}^{\left(0\right)}=3G^2{M}^3/(\hslash c^4)$ is the standard Page time, and $\eta ,n$ are free parameters to be calibrated against gravitational-wave data.

The cosmic cycle closes through the integrated entropy reduction:

$$\mathsf{\Delta }{S}_{\mathrm{vis}}^{\left(\mathrm{cycle}\right)}=-{\int }_{t_{\mathrm{start}}}^{t_{\mathrm{bounce}}} {\displaystyle \frac{d{S}_{\mathrm{abs}}}{dt}}dt=-\mathsf{\Delta }{S}_{\mathsf{\Phi }},$$

(26)

and at the bounce (Axiom VI):

$$\mathsf{\Phi }\left(t_{\mathrm{bounce}}^{+}\right)={\mathsf{\Phi }}_0\gg {\mathsf{\Phi }}_c,S_{\mathsf{\Phi }}\left(t_{\mathrm{bounce}}^{+}\right)\ll S_{\mathsf{\Phi }}\left(t_{\mathrm{bounce}}^{-}\right).$$

(27)

In the limit $\mathsf{\Phi }\to 0$, the entropy reduces to the Bekenstein–Hawking area law:

$$S_{\mathrm{BH}}\stackrel{\mathsf{\Phi }\to 0}{\to }{\displaystyle \frac{k_B{c}^{3}A}{4G\hslash }}.$$

(28)

2.14. Cluster-Scale Relaxation and the Bullet Cluster Analogue

Merging galaxy clusters provide another independent test. During a cluster merger, the collisional intracluster gas is decelerated by ram pressure, while the effectively collision less galaxy population — and the $\mathsf{\Phi }$-field profile that tracks it through (7b) — continue largely undisturbed. The field relaxes toward its new equilibrium configuration on a timescale

$${\tau }_{\mathrm{relax}}\sim {\displaystyle \frac{\lambda }{c}},$$

(29)

Generalized to the merger geometry by the field’s interaction with the intracluster plasma. This relaxation induces a spatial offset $\mathsf{\Delta }r$ between the peak of the effective lensing potential (tracing ${\mathsf{\Phi }}_{\mathrm{host}}$) and the X-ray gas centroid, analogous to the Bullet Cluster phenomenon. Once the parameters $\kappa ,\xi ,\lambda$ are calibrated, $\mathsf{\Delta }r$ is a fixed, falsifiable prediction of the theory without additional cluster-specific parameters.

2.15. Limiting Cases

The framework consistently recovers known physics in all appropriate limits:

• ${\partial }_{\mathsf{\mu }}\mathsf{\Phi }=0$: Torsion vanishes; $\mathsf{\beta }\to 1$; ${\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\left(\mathsf{\Phi }\right)}$ reduces to a minimal scalar field. If $\mathrm{V}\left({\mathsf{\Phi }}_{\mathrm{\infty }}\right)=0$, (4) reduces to the vacuum Einstein equations.
• $\mathsf{\Phi }\ll {\mathsf{\Phi }}_{\mathrm{c}}$: Positive-feedback terms become negligible; (9) reduces to the Keplerian relation.
• Φ→0 (near black-hole singularity/horizon): G(Φ)→0; both the entropy absorption rate (Equation (21)) and the matter source term in the field Equation (5) reach their minimum, and Hawking radiation escapes without significant absorption.
• Φ→Φ_c (near the quantum bounce): G(Φ)→∞; the entropy absorption and matter–field coupling diverge superlinearly, consistent with Axiom V and leading to the boundary condition of Axiom VI.
• $\mathsf{\Phi }\to {\mathsf{\Phi }}_{\mathrm{\infty }}$: (7) reduces to the Laplace equation ${\nabla }^2\left(\mathsf{\delta }\mathsf{\Phi }\right)=0$.
• $\mathsf{\Phi }\to 0$(Near horizon): (28) reduces to the Bekenstein–Hawking area law.
• $\mathsf{\xi }\to 0$: $\mathsf{\beta }\to 1$; (6) reduces to ${\mathrm{R}}_{\mathsf{\mu }\mathsf{\nu }}{|}_{\mathrm{vacuum}}=(8\mathsf{\pi }\mathrm{G}/{\mathrm{E}}_{\mathrm{H}}^2)({\partial }_{\mathsf{\mu }}\mathsf{\Phi }{\partial }_{\mathsf{\nu }}\mathsf{\Phi }-{\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}\mathrm{V}(\mathsf{\Phi }\left)\right)$.

$\mathsf{\kappa }\to 0$: All torsion-sourced effects (dark matter, warp, modified evaporation) vanish; the theory reduces to a minimally coupled scalar-field cosmology with a matter source term.

2.16. Fixed Constants and Free Parameters

The theory contains a single fixed reference scale and a set of free parameters, each with a clear physical role and calibration source. The fixed scale is ${\mathrm{E}}_{\mathrm{H}}\equiv 246\mathrm{GeV}$. The free parameters are listed in Table 1.

Parameter	Physical role	Calibration source
$\mathsf{\kappa }$	Torsion–field coupling strength	Galactic rotation curves (SPARC)
$\mathsf{\xi }$	Torsion–field self-interaction (enters $\mathsf{\beta }$)	CMB, BAO, large-scale structure
${\mathsf{\Phi }}_{\mathrm{c}}$	Critical bounce density	Large-scale structure / entropy sector
$\mathrm{V}\left(\mathsf{\Phi }\right)$ coefficients	Self-interaction potential	Supernovae (Pantheon+), BAO (DESI)
$\mathsf{\lambda }$ / ${\mathrm{V}}^{″}\left({\mathsf{\Phi }}_{\mathrm{\infty }}\right)$	Screening length	Gravitational lensing, PPN/solar-system data
$\mathsf{\alpha }$	Entropy coupling constant	Gravitational-wave data (LIGO/Virgo)
$\mathsf{\eta },\mathrm{n}$	Positive-feedback Page-time parameters	Gravitational-wave / black-hole data
${\mathsf{\beta }}_{\mathrm{struct}}$$n_{\mathsf{\Phi }}$	Early-structure-formation coupling Nonlinear exponent of the coupling function $\mathcal{G}\left(\mathsf{\Phi }\right)$ near ${\mathsf{\Phi }}_c$ and $\mathsf{\Phi }\to 0$	JWST high-$\mathrm{z}$ mass functions Entropy sector / Page time / gravitational-wave data (along with $\alpha ,\eta ,n$)

Determination of the numerical values of these parameters using independent observational data and statistical methods (e.g., MCMC) is deferred to future work.

2.17. Falsifiable Predictions

The framework yields a set of structurally distinct, testable predictions, summarized in Table 2.

Prediction	Physical origin	Facility	Timeline
Precise shape of galactic rotation curves	Geometric pressure (Equation (9))	Gaia DR4, SPARC	Ongoing
Galactic warp, asymmetric drift, vertex deviation	Satellite-induced perturbations (Equations (14)–(15))	Gaia DR4, Cepheids	Ongoing
Environment-dependent black-hole evaporation	Gradient term in (22)	Future BH population studies	Long-term
Modified Page time (Equation 25)	Positive-feedback entropy exchange	LIGO/Virgo/LISA	2030s
Bullet-Cluster-type lensing/X-ray offset	Cluster-scale relaxation (Equation (29))	Weak-lensing + X-ray surveys	Ongoing
$w_{\mathrm{DE}}\left(t\right)$	Background field evolution (Equations (16)–(19))	DESI, Euclid, Roman	2025–2030
High-redshift fine-structure variation	Coupling of $\mathsf{\Phi }$ to matter sector	JWST/NIRSpec	2025–2030

3. Discussion and Conclusions

The Cosmic Energy Inversion Theory in its fifth version (CEIT-v5) has been presented as a geometric-field framework in which the independent concepts of dark matter and dark energy are replaced by a fundamental space-time energy field ($\mathsf{\Phi }$). This theory, based on six axioms and a single action in Einstein–Cartan–Ehresmann geometry, coherently explains key cosmological and astrophysical phenomena. Dark matter emerges as geometric pressure ${\rho }_{\mathrm{DM}}^{\left(\mathrm{eff}\right)}={\displaystyle \frac{c^2\beta }{16\pi G{E}_H^2}}(\nabla \mathsf{\Phi }{)}^2$, arising directly from the field gradient and explaining flat galactic rotation curves without unknown particles. Dark energy is modeled as the temporal evolution of the field with density ${\rho }_{\mathsf{\Phi }}={\displaystyle \frac{1}{2}}{\dot{\mathsf{\Phi }}}^2+V\left(\mathsf{\Phi }\right)+{\displaystyle \frac{3}{2}}H\dot{\mathsf{\Phi }}\mathsf{\Phi }$ and a dynamical equation of state $w_{\mathrm{DE}}\left(t\right)$, which in the current epoch approaches $-1$ and reproduces the cosmological-constant behaviour. Black-hole entropy is redefined as a flux exchange between Hawking radiation and the $\mathsf{\Phi }$ field, and the Page time is modified with positive feedback. The cosmic cycle is closed through the reduction of visible entropy over a period and the reset of field entropy at the ${\mathsf{\Phi }}_c$ threshold, giving birth to a new universe.

The vacuum curvature theorem (Equation 6) clearly expresses the central claim of the theory: in the absence of ordinary matter, the Ricci tensor decomposes into a part proportional to the metric (arising from the field’s potential, playing the role of a dynamical cosmological constant) and a part sourced directly by the field gradient. This means that space-time curvature in vacuum is not an independent entity; it is the direct manifestation of the presence of energy-field gradients. If the field is uniform and the potential is at its minimum, curvature vanishes and space-time becomes flat (special relativity). General relativity does not exist as an independent theory in this framework; rather, it is a phenomenological description of the curvature arising from matter (via field gradients) that reduces to flat space-time in the limit of field uniformity and appropriate potential normalization. Gravitational waves are interpreted as propagating disturbances in the energy field, travelling at the speed of light and producing time-varying curvature.

Earlier versions of this theory provided initial numerical estimates of parameters using computational tools, which remain valuable as reference points for future calibration. However, the aim of the present version is to provide a framework that can be directly tested against independent observational data. Accordingly, quantitative calibration of parameters using observational data (e.g., SPARC, Gaia, Planck, DESI, and LIGO/Virgo) is deferred to future work, and this paper does not provide definitive numerical values.

The future outlook of the theory encompasses three main axes: (a) calibration of parameters using new observational data from next-generation telescopes; (b) development of three-dimensional N-body numerical simulations within the CEIT-v5 framework for precise prediction of the large-scale structure; and (c) numerical investigation of the quantum bounce dynamics. Among the testable predictions of the theory are the precise shape of galactic rotation curves (Equation (9)), the Galactic warp and asymmetric drift (Equations (14) and (15)), environment-dependent black-hole evaporation (Equation (22)), the modified Page time (Equation (25)), and the scale dependence of $w_{\mathrm{DE}}\left(t\right)$ (Equations (16)–(19)).

In summary, CEIT-v5, by providing a unified and principled description of many observed phenomena, and by offering a mathematical foundation for precise calibration, presents a self-consistent and testable framework that may serve as an alternative to the standard $\mathsf{\Lambda }$CDM model in explaining currently unresolved phenomena. The final validation of this framework requires independent experimental tests and calibration with observational data, which depends on collaboration of the scientific community and investment in next-generation observational projects.