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title: "Time Complexity, Inferential uncertainty and Spacekime Analytics"
subtitle: "Appendix 04: Spacekime Extension of Einstein-Hilbert Gravitational Field Equations (V2)"
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# Gravitational Field Equations in Space-Kime

This [TCIU Appendix](https://tciu.predictive.space/) synthesizes the 
*complex-time (kime)* framework from the *KPT paper* (Shen, Tao, Bakalov, Dinov) with the spacekime extensions of the Einstein field equations (SOCR/TCIU/Chapter6_TCIU_SketchOfIdeas), and provides a mathematically rigorous, self-consistent, and experimentally testable formulation of gravity in space-kime.

**Notation convention.** Throughout this document, we use $\psi$ for the kime-phase coordinate and reserve $\theta, \phi$ for the polar and azimuthal angles on the 2-sphere $\mathbb{S}^2$ in gravitational applications (Schwarzschild, FLRW). The 5D spacekime coordinate indices are $A,B,C,\ldots \in \{0,1,2,3,4\}$, the 4D spacetime indices are $\mu,\nu,\rho,\ldots \in \{0,2,3,4\}$ (omitting the $\psi$-index $A=1$), and the 3D spatial indices are $i,j,k,\ldots \in \{2,3,4\}$.

**Modeling strategy.** There is an important architectural choice in how the phase distribution $\varphi_t(\psi)$ enters the gravitational framework. Two routes are available.

 - **Route A (Pure averaging).** Treat $\varphi_t$ as an external measure weight only. The fundamental 5D field equations are standard Einstein equations and all observable modifications come from how observables are averaged over $\psi$, not from new curvature/source terms. If the 5D metric is $\psi$-independent (zero-mode Kaluza--Klein ansatz), $\varphi_t$ drops out of the local dynamics entirely and affects only the statistical interpretation of repeated measurements.
 - **Route B (Geometric embedding).** Promote $\varphi_t$ to a geometric object: a *warp factor* for the $\mathbb{S}^1$ fiber. The phase distribution modulates the physical size of the kime-phase circle at each spacetime point, making the 5D geometry itself $\psi$-dependent. The standard Einstein equations in 5D then naturally produce $\varphi_t$-dependent source terms after dimensional reduction.

This document adopts **Route B**, which produces the richer gravitational phenomenology explored in the Chapter 6 sketch, avoids the logical gap of "dividing out $\varphi_t$ and then recovering it," and generates the phase score energy term as a genuine geometric effect.

After a proper Kaluza--Klein reduction of the 5D spacekime action, the effective 4D gravitational field equations acquire a non-negative cosmological-like term $\Lambda_\varphi = \mathcal{E}_\psi[\varphi_t]/R_\kappa^2$ proportional to the **phase score energy** (a functional of $\varphi_t$) divided by the squared *kime compactification radius* $R_\kappa$. This term vanishes in the classical limit $\varphi_t \to \delta$ and is always non-negative, producing an accelerating contribution. The framework is experimentally testable via [kime-phase tomography (KPT)](https://socr.umich.edu/TCIU/HTMLs/TCIU_SK_Appendix03_KPT_DS_fMRI_V5.html) applied to repeated gravitational measurements.


# Motivation and Physical Setup

*The Problem.* Einstein's gravitational field equations in 4D spacetime,

$$G_{\mu\nu} \equiv R_{\mu\nu} - \tfrac{1}{2}\,g_{\mu\nu}\,R + \Lambda\,
g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}\,,\qquad \mu,\nu = 0,1,2,3\,,$$

treat the *time coordinate* $t$ as a single real parameter. However, all empirical access to gravitational phenomena (binary pulsar timing, LIGO strain measurements, redshift surveys, CMB anisotropy maps) requires *repeated observations* 
of the same physical process. Each repetition introduces an irreducible *sample-to-sample variability* that is neither measurement noise (range-space) nor temporal evolution. Rather, it is an intrinsic *domain-space* uncertainty reflecting the latent state of the observing apparatus, environment, and quantum degrees of freedom at the moment of observation.

The kime framework captures this by extending *real time* 
$t \in \mathbb{R}_{\geq 0}$ to **complex time (kime)**

$$\kappa = t\,e^{i\psi}\,,\qquad t \geq 0\,,\quad \psi \in \mathbb{S}^1 = [0,2\pi)\,,$$

where $t = |\kappa|$ is the chronological amplitude and $\psi = \arg(\kappa)$ is the **kime-phase**, a latent coordinate distributed according to a time-varying phase law $\varphi_t(\psi)$ that encodes the structured variability across repeated observations.

*Kime-Phase as Intrinsic Domain Variability.* The kime-phase $\psi$ is 
*not* an independent dynamical degree of freedom in the sense of 
a freely propagating Kaluza--Klein (KK) extra dimension. Rather, it represents the 
*distribution of repeated measurement outcomes* at each spacetime event. 
The key distinction is between

 - **Range-space variability**: additive measurement noise $\varepsilon_{j,k}$ with $\mathbb{E}[\varepsilon]=0$.  
 - **Domain-space variability**: the latent phase $\Psi_j(t) \sim \varphi_t(\psi)$ that modulates which "slice" of the kime-surface is observed in repetition $j$.

This means $\psi$ carries a *probability measure* $\varphi_t(\psi)\,d\psi/(2\pi)$, and the correct way to incorporate it into the geometry is through a **fiber-warped metric** where the phase distribution modulates the physical size of the compact $\mathbb{S}^1$ direction.


# Mathematical Foundations

## The Kime Domain and Its Geometry

**Definition (Kime Domain).** The kime domain is $\mathbb{K}_T = \{t\,e^{i\psi} : t \in [0,T],\;\psi \in \mathbb{S}^1\}$, identified with the product manifold $\mathcal{M}_\kappa = [0,T] \times \mathbb{S}^1$.

**Definition (Kime Cone Metric).** The natural *Riemannian* (positive-definite) metric on $\mathcal{M}_\kappa$ is the polar (cone) metric:

$$g_\kappa = dt^2 + t^2\,d\psi^2\,,$$

with volume element $d\mu_\kappa = t\,dt\,d\psi/(2\pi)$.

This is the flat metric on $\mathbb{R}^2$ in polar coordinates, confirming that the kime manifold is locally flat (zero intrinsic curvature) away from the apex $t=0$. *Crucially*, $g_\kappa$ is Riemannian, not Lorentzian: both $dt^2$ and $t^2 d\psi^2$ are positive. The Lorentzian signature of the full spacekime metric arises from the embedding of this 2D Riemannian sector into the 5D spacetime, where $t$ becomes the timelike coordinate and $\psi$ becomes a compact spacelike direction.

**Definition (Kime Hilbert Space).** The function space for kime analysis is:

$$\mathcal{H}_\kappa = L^2\!\Big(\mathcal{M}_\kappa,\;t\,dt \otimes \frac{d\psi}{2\pi}\Big)\,,$$

with inner product $\langle f,g\rangle = \int_0^T\!\int_0^{2\pi} \overline{f}\,g\;t\,\frac{d\psi}{2\pi}\,dt$.

On $(\mathcal{M}_\kappa, g_\kappa)$, the 
*Laplace--Beltrami operator* is

$$\Delta_\kappa = \frac{1}{t}\frac{\partial}{\partial t}\Big(t\frac{\partial}{\partial t}\Big) + \frac{1}{t^2}\frac{\partial^2}{\partial\psi^2}\,,$$

which has a complete orthonormal eigenbasis of **Bessel--Fourier modes** $\phi_{n,m}(t,\psi) = A_{n,m}\,J_{|m|}(j_{|m|,n}\,t/T)\,e^{im\psi}$ (from the KPT spectral decomposition theorem).


## The Phase Distribution and Its Regularity

**Definition (Phase Law).** The phase law at time $t$ is a probability density $\varphi_t : \mathbb{S}^1 \to \mathbb{R}_{\geq 0}$ satisfying $\int_0^{2\pi}\varphi_t(\psi)\,\frac{d\psi}{2\pi} = 1$.

**Regularity Assumption (Standing Hypothesis).** Throughout the derivation of field equations, we assume $\varphi_t \in C^2(\mathbb{S}^1)$ with $\varphi_t(\psi) > 0$ for all $\psi \in \mathbb{S}^1$. That is, the phase law is twice continuously differentiable and bounded away from zero. The classical (Dirac delta) limit is treated separately as a *weak limit* of smooth densities (see Section 6.4).

**Definition (Characteristic Function).** 
The Fourier coefficients of the phase law are

$$\hat{\varphi}_t(n) = \int_0^{2\pi} e^{-in\psi}\,\varphi_t(\psi)\,\frac{d\psi}{2\pi}\,,\qquad n \in \mathbb{Z}\,.$$


## Circular Distributions and Normalization

The following table lists properly normalized circular distributions on $\mathbb{S}^1 = [0,2\pi)$, their first Fourier coefficients, and their circular variances. All densities are with respect to the measure $d\psi/(2\pi)$ (so that $\int_0^{2\pi}\varphi\,d\psi/(2\pi) = 1$).

| Phase Distribution $\varphi_t(\psi)$ | Normalization | $\hat{\varphi}_t(1)$ | Circular variance $V = 1-|\hat{\varphi}_t(1)|$ |
|---|---|---|---|
| **Uniform:** $1$ | $\int_0^{2\pi}1\cdot\frac{d\psi}{2\pi}=1$ ✓ | $0$ | $1$ (maximal) |
| **von Mises:** $\frac{e^{\kappa_0\cos\psi}}{I_0(\kappa_0)}$ | $\int \frac{e^{\kappa_0\cos\psi}}{I_0(\kappa_0)}\frac{d\psi}{2\pi}=1$ ✓ | $\frac{I_1(\kappa_0)}{I_0(\kappa_0)}$ | $1 - \frac{I_1(\kappa_0)}{I_0(\kappa_0)}$ |
| **Wrapped Cauchy:** $\frac{1-\rho^2}{1-2\rho\cos\psi+\rho^2}$, $0<\rho<1$ | Exact on $\mathbb{S}^1$ ✓ | $\rho$ | $1-\rho$ |
| **Cardioid:** $1+2\rho\cos(\psi-\mu_0)$, $|\rho|\leq\frac{1}{2}$ | ✓ | $\rho e^{i\mu_0}$ | $1-\rho$ |
| **Dirac:** $\delta(\psi - \psi_0)$ (distributional) | $\int \delta(\psi-\psi_0)\frac{d\psi}{2\pi}=\frac{1}{2\pi}$ | $e^{-i\psi_0}$ | $0$ (classical limit) |

**Remark on wrapped distributions.** The V1 draft listed a "wrapped Laplace" density $\frac{1}{2b}e^{-|\psi|/b}$. This is the Laplace density on $\mathbb{R}$, *not* a properly normalized circular density on $\mathbb{S}^1$. Its wrapped version requires periodic summation: $\varphi_{\text{WL}}(\psi) = \sum_{k=-\infty}^{\infty}\frac{1}{2b}e^{-|\psi+2\pi k|/b}$, which has a known closed form but introduces the normalization factor $C(b) = \frac{1}{2b}\cdot\frac{1}{1-e^{-2\pi/b}}$. For the remainder of this document, we use the **von Mises** distribution as the primary non-uniform example, since it is the natural circular analog of the Gaussian, is exactly normalized, and has well-known analytical properties. When a concentration-dependence is needed, the von Mises parameter $\kappa_0$ plays the role of the inverse-variance scale $1/b^2$.


# Spacekime Manifold and Spacekime Metric

**Definition (Spacekime Manifold).** The spacekime manifold is the 5-dimensional fiber bundle

$$\mathcal{M}_5 = \Sigma_4 \times_\pi \mathbb{S}^1\,,$$

where $\Sigma_4$ is a 4D Lorentzian spacetime (the base) and $\mathbb{S}^1$ is the compact kime-phase fiber. Coordinates are $X^A = (t, \psi, x^1, x^2, x^3)$ with $A = 0,1,2,3,4$, where $A=0$ is the time coordinate $t$, $A=1$ is the kime-phase $\psi$, and $A=2,3,4$ are spatial.

## Metric Signature: $(-,+,+,+,+)$

The spacekime metric has **Lorentzian signature** $(-,+,+,+,+)$: a single timelike direction ($t$) and four spacelike directions ($\psi, x^1, x^2, x^3$).

*Justification.* The kime manifold $(\mathcal{M}_\kappa, g_\kappa)$ with cone metric $dt^2 + t^2 d\psi^2$ is *Riemannian* (Euclidean signature). When this is embedded into the physical 5D spacetime, only $t$ acquires a timelike sign (from $ds^2 = -c^2 dt^2 + \ldots$), while the angular coordinate $\psi$ on the compact $\mathbb{S}^1$ fiber remains spacelike. This is essential for three reasons:

  1. **Cauchy well-posedness.** A $(-, -, +, +, +)$ signature would produce ultrahyperbolic field equations, for which the initial value problem is generically ill-posed (the Asgeirsson mean-value theorem prevents localized solutions). A single timelike direction ensures the standard hyperbolicity of the 5D Einstein equations.
  2. **Standard KK reduction.** The Kaluza--Klein mechanism requires the compact direction to be spacelike. With $\psi$ spacelike, the reduction to 4D proceeds by established methods.
  3. **Volume element.** With signature $(-,+,+,+,+)$, the metric determinant satisfies $G = \det(G_{AB}) < 0$, and the volume element $\sqrt{-G}\,d^5X$ is real and positive.


## The Kaluza--Klein (KK) Metric Ansatz

Following the KK paradigm, we write the spacekime metric in the *connection 1-form* decomposition

$$\boxed{ds_5^2 = G_{AB}\,dX^A dX^B = g_{\mu\nu}(x)\,dx^\mu dx^\nu + \Phi(x)^2\,w(\psi)\,\big(d\psi + A_\mu(x)\,dx^\mu\big)^2\,,}$$
where $g_{\mu\nu}(x)$ is the **4D spacetime metric** ($\mu,\nu = 0,2,3,4$, i.e., indices $\{t,x^1,x^2,x^3\}$), with signature $(-,+,+,+)$;
$\Phi(x) = R_\kappa \cdot \tilde{\Phi}(x)$ is the **KK dilaton** (scalar field), where $R_\kappa$ is a constant length scale (the *kime compactification radius*) and $\tilde{\Phi}(x)$ is dimensionless. In the simplest case of a static fiber, $\tilde{\Phi} = 1$;
$A_\mu(x)\,dx^\mu$ is the **KK gauge field** (a 1-form on 4D spacetime), analogous to the electromagnetic potential in the original KK theory. The phase reparameterization $\psi \mapsto \psi + \alpha(x)$ induces $A_\mu \to A_\mu - \partial_\mu\alpha$, the standard $U(1)$ gauge transformation; and 
$w(\psi) \geq 0$ is the **fiber warp factor**, a prescribed (non-dynamical) function of $\psi$ alone. This is the geometric embodiment of the phase distribution: $w(\psi) = [\varphi_t(\psi)]^\alpha$ for a specific power $\alpha$ determined by the reduction (see below).

**Relation to the cone metric.** In the flat (Minkowski) spacekime limit, $g_{\mu\nu} = \eta_{\mu\nu}$, $\Phi = t R_\kappa$, $A_\mu = 0$, $w = 1$, and the metric becomes

$$ds_5^2 = -c^2\,dt^2 + t^2 R_\kappa^2\,d\psi^2 + \delta_{ij}\,dx^i dx^j\,,$$

which is the Minkowski metric with a compact $\psi$-circle of physical circumference $2\pi t R_\kappa$ (growing linearly with $t$, reflecting the cone structure).

**Design choice for the warp factor.** We set $w(\psi) = \varphi_t(\psi)$ (the phase law itself), so that the fiber is "wider" where the phase distribution is concentrated and "narrower" where it is suppressed. The physical circumference of the $\mathbb{S}^1$ fiber at a given $\psi$-value is modulated by $\sqrt{\varphi_t(\psi)}$. This geometric embedding ensures that the standard 5D Einstein equations, when reduced over the warped fiber, naturally produce $\varphi_t$-dependent source terms in the effective 4D theory.

In the simplest case (static dilaton $\Phi = R_\kappa$, vanishing gauge field $A_\mu = 0$), the block-diagonal metric is

$$G_{AB} = \begin{pmatrix} g_{00} & 0 & g_{0i} \\ 0 & R_\kappa^2\,\varphi_t(\psi) & 0 \\ g_{j0} & 0 & g_{ij} \end{pmatrix}\,,$$

with $g_{00} = -N^2$ (the 4D lapse squared) and $g_{ij} = \gamma_{ij}$ (the 3-metric). The determinant is

$$G = \det(G_{AB}) = -R_\kappa^2\,\varphi_t(\psi)\cdot N^2\gamma\,,$$

where $\gamma = \det(\gamma_{ij})$, and the volume element is

$$\sqrt{-G}\;d^5X = R_\kappa\,\sqrt{\varphi_t(\psi)}\;N\sqrt{\gamma}\;dt\,d\psi\,d^3x\,.$$


# Curvature in Spacekime

From the 5D metric $G_{AB}$, the *Christoffel symbols* are

$$\Gamma^C_{AB} = \frac{1}{2}\,G^{CD}\Big(\partial_A G_{BD} + \partial_B G_{AD} - \partial_D G_{AB}\Big)\,.$$

For the block-diagonal metric with $A_\mu = 0$ and $\Phi = R_\kappa$, the non-vanishing Christoffel symbols involving the $\psi$-direction are:

$$\Gamma^\psi_{\psi\psi} = \frac{\varphi_t'}{2\varphi_t}\,,\qquad \Gamma^\mu_{\psi\psi} = -\frac{R_\kappa^2}{2}\,g^{\mu\nu}\partial_\nu\varphi_t\,,$$

where $\varphi_t' = \partial_\psi\varphi_t$. The first of these is the key new connection coefficient: it encodes how the $\psi$-geodesics are bent by the non-uniform fiber warp factor.

## The 5D Riemann and Ricci Tensors

In the context of General Relativity and Kaluza-Klein (KK) theory, the [d'Alembertian operator](https://en.wikipedia.org/wiki/D%27Alembert_operator), also known as the *wave operator*,
is the generalization of the *Laplacian* ($\nabla^2$) to curved four-dimensional spacetime. Specifically, it represents the Laplace-Beltrami operator acting on a scalar field.

The d'Alembertian  operator is acting on the scalar field $\Phi$,
the dilaton or "radion" field that governs the size of the extra dimension. Mathematically, it is defined using the *covariant derivative* $\nabla_\mu$
$$\underbrace{\Box}_{d'Alembertian\\ operator}\Phi = g^{\mu\nu}\nabla_\mu\nabla_\nu\Phi.$$
Expanding this out using the metric $g_{\mu\nu}$ and the Christoffel symbols ($\Gamma$), it becomes

$$\Box\Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \partial_\nu \Phi \right).$$
Kaluza-Klein theory decomposes a 5D universe into a 4D universe plus an extra dimension. When "reducing" the 5D Ricci scalar, $R^{(5)}$, the kinetic energy and the spatial variation of the extra dimension's size, $\Phi$, manifest as 4D terms.
The $-\frac{2}{\Phi}\Box\Phi$ term represents the curvatures contribution from
the gradient of the scalar field. It tells us how the 4D geometry is affected by
the way the extra dimension "stretches" or "shrinks" as you move through 4D space.

The *5D Riemann tensor* is

$$\underbrace{R^A_{\;BCD}}_{5D\ Riemann\ tensor} = \partial_C\Gamma^A_{DB} - \partial_D\Gamma^A_{CB} + \Gamma^A_{CE}\Gamma^E_{DB} - \Gamma^A_{DE}\Gamma^E_{CB}\,.$$

The *5D Ricci tensor* and the *Ricci scalar* are

$$\underbrace{R_{AB}}_{Ricci\ tensor} = R^C_{\;ACB}\,,\qquad 
\underbrace{R^{(5)}}_{Ricci\ scalar} = G^{AB}\,R_{AB}\,.$$

*Decomposition of the 5D Ricci Scalar*. 
For the KK metric ansatz with fiber warp factor $w(\psi) = \varphi_t(\psi)$, the 5D Ricci scalar decomposes as, see 
[Overduin & Wesson, *Phys. Rep.* 283, 1997](https://doi.org/10.1016/S0370-1573(96)00046-4),

$$R^{(5)} = R^{(4)}[g] - \frac{1}{4}\Phi^2 F_{\mu\nu}F^{\mu\nu} - \frac{2}{\Phi}\underbrace{\Box\Phi}_{d'Alembertian} + R_{\text{fiber}}\,,$$

where $R^{(4)}$ is the 4D Ricci scalar of $g_{\mu\nu}$, the field strength is $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, and $R_{\text{fiber}}$ collects the curvature contribution from the $\psi$-dependent warp factor

$$R_{\text{fiber}} = -\frac{1}{2R_\kappa^2\,\varphi_t}\left[\frac{\varphi_t''}{\varphi_t} - \frac{1}{2}\left(\frac{\varphi_t'}{\varphi_t}\right)^2\right]\,.$$

This is the term that generates the phase-dependent source upon dimensional reduction. Its presence is a direct consequence of the $\psi$-dependent fiber geometry, it would vanish identically if $\varphi_t$ were constant.


# The Spacekime Einstein--Hilbert Field Equations

## The Spacekime Action

**Definition (Spacekime Action).** The gravitational action on the spacekime manifold $\mathcal{M}_5$ is the *standard 5D Einstein--Hilbert action* with cosmological constant, $\Lambda_5$,

$$\boxed{S_5 = \frac{1}{16\pi G_5}\int_{\mathcal{M}_5} \sqrt{-G}\;\Big(R^{(5)} - 2\Lambda_5\Big)\;d^5X + \int_{\mathcal{M}_5}\sqrt{-G}\;\mathcal{L}_m\;d^5X\,,}$$

where $G_5$ is the 5D gravitational coupling and $\Lambda_5$ is the 5D cosmological constant. There is **no separate $\varphi_t$ weight factor** in the integration measure: the phase distribution enters entirely through the metric (via the fiber warp factor $G_{\psi\psi} = R_\kappa^2\,\varphi_t(\psi)$). This resolves the Route A/B ambiguity of V1: the information content of $\varphi_t$ is encoded geometrically, and the action is the standard 5D Einstein--Hilbert functional of the metric $G_{AB}$.

In *Route B*, the "phase-weighted average" of the initial version, *V1*, is replaced by a genuine geometric statement: the fiber is warped by $\varphi_t$.
The 5D field equations follow from the standard variational principle $\delta S_5 / \delta G^{AB} = 0$, without any need to "divide out" a weight function.

## The 5D Field Equations

Varying $S_5$ with respect to $G^{AB}$ via the standard Palatini identity yields:

$$\boxed{R_{AB} - \frac{1}{2}\,G_{AB}\,R^{(5)} + \Lambda_5\,G_{AB} = \frac{8\pi G_5}{c^4}\,T_{AB}\,,\qquad A,B = 0,1,2,3,4\,.}$$

These are the *exact 5D field equations*. Since $\varphi_t(\psi)$ is encoded in $G_{AB}$, its effects on curvature are fully captured by $R_{AB}$ and $R^{(5)}$. No post-hoc insertion of source terms is required.

## Kaluza--Klein Dimensional Reduction

To extract the effective 4D physics, we substitute the KK ansatz (Section 3.3) into the 5D action and integrate over the compact $\psi$-direction.

**Step 1. Substitute the KK ansatz.** With $G_{\psi\psi} = \Phi^2\varphi_t(\psi)$ and the decomposition of $R^{(5)}$ from Section 4.3, the gravitational action becomes

$$S_5 = \frac{1}{16\pi G_5}\int dt\,d^3x\int_0^{2\pi}d\psi\;\sqrt{-g^{(4)}}\;R_\kappa\sqrt{\varphi_t}\;\tilde\Phi\;\Big(R^{(4)} - \tfrac{1}{4}\Phi^2 F_{\mu\nu}F^{\mu\nu} - \frac{2}{\Phi}\Box\Phi + R_{\text{fiber}} - 2\Lambda_5\Big)\,.$$

**Step 2. Integrate over $\psi$.** Since $g_{\mu\nu}$, $\Phi$, $A_\mu$ are all $\psi$-independent (KK zero-mode truncation), the only $\psi$-dependence is in $\sqrt{\varphi_t(\psi)}$ and $R_{\text{fiber}}(\psi)$. Defining the $\psi$-averaged quantities

$$I_0 \;\equiv\; \int_0^{2\pi}\sqrt{\varphi_t(\psi)}\;\frac{d\psi}{2\pi}\,,\qquad
I_R \;\equiv\; \int_0^{2\pi}\sqrt{\varphi_t(\psi)}\;R_{\text{fiber}}(\psi)\;\frac{d\psi}{2\pi}\,,$$

the effective 4D action is

$$S_{\text{eff}}^{(4)} = \frac{R_\kappa}{16\pi G_5}\int\sqrt{-g^{(4)}}\;\tilde\Phi\;\Big[I_0\Big(R^{(4)} - \tfrac{1}{4}\Phi^2 F_{\mu\nu}F^{\mu\nu} - \frac{2}{\Phi}\Box\Phi - 2\Lambda_5\Big) + I_R\Big]\;d^4x\,.$$

**Step 3. Evaluate $I_R$ (the phase score energy integral).** Substituting $R_{\text{fiber}}$ and integrating by parts on $\mathbb{S}^1$ (where boundary terms vanish by periodicity), we obtain

$$I_R = -\frac{1}{2R_\kappa^2}\int_0^{2\pi}\sqrt{\varphi_t}\left[\frac{\varphi_t''}{\varphi_t} - \frac{1}{2}\left(\frac{\varphi_t'}{\varphi_t}\right)^2\right]\frac{d\psi}{2\pi}\,.$$

Using the identity $\sqrt{\varphi_t}\cdot\frac{\varphi_t''}{\varphi_t} = \frac{(\sqrt{\varphi_t})''}{\sqrt{\varphi_t}} + \frac{1}{4}\left(\frac{\varphi_t'}{\varphi_t}\right)^2 \cdot \sqrt{\varphi_t}$ and integrating by parts, this simplifies to

$$I_R = \frac{1}{4R_\kappa^2}\int_0^{2\pi}\frac{(\varphi_t')^2}{\varphi_t^{3/2}}\;\frac{d\psi}{2\pi} \;=\; \frac{1}{R_\kappa^2}\int_0^{2\pi}\Big(\partial_\psi\sqrt{\varphi_t}\Big)^2\;\frac{d\psi}{2\pi}\,.$$

This is a non-negative functional of $\varphi_t$ that vanishes if and only if $\varphi_t$ is constant, uniform distribution.

**Step 4. Define the effective 4D constants.** Setting $\tilde\Phi = 1$, static dilaton, and $I_0 \approx 1$, leading order for near-uniform $\varphi_t$, we identify

$$G_{\text{eff}} = \frac{G_5}{R_\kappa}\,,\qquad \Lambda = \Lambda_5 - \frac{I_R}{2I_0}\,.$$

The effective 4D cosmological constant receives a **phase score energy contribution**

$$\boxed{\Lambda_{\text{eff}} = \Lambda_5 - \frac{1}{2I_0 R_\kappa^2}\int_0^{2\pi}\Big(\partial_\psi\sqrt{\varphi_t}\Big)^2\;\frac{d\psi}{2\pi}\,.}$$

For small deviations from uniformity, $I_0 \approx 1$ and the correction is $-\mathcal{E}_\psi/(2R_\kappa^2)$, where we define the **phase score energy** below.


## The Phase Score Energy Functional

**Definition (Phase Score Energy).** For a phase law $\varphi_t \in C^1(\mathbb{S}^1)$ with $\varphi_t > 0$, the *phase score energy* is

$$\boxed{\mathcal{E}_\psi[\varphi_t] \;\equiv\; \int_0^{2\pi}\Big(\partial_\psi\sqrt{\varphi_t(\psi)}\Big)^2\;\frac{d\psi}{2\pi} \;=\; \frac{1}{4}\int_0^{2\pi}\frac{(\varphi_t')^2}{\varphi_t}\;\frac{d\psi}{2\pi}\,.}$$

**Properties:**

 1. $\mathcal{E}_\psi[\varphi_t] \geq 0$, with equality if and only if $\varphi_t \equiv 1$ (uniform). *Proof:* $\mathcal{E}_\psi = \|\partial_\psi\sqrt{\varphi_t}\|_{L^2}^2 \geq 0$, and the $L^2$-norm vanishes iff $\sqrt{\varphi_t}$ is constant, iff $\varphi_t$ is constant; since $\int\varphi_t\,d\psi/(2\pi) = 1$, the constant must be $1$.

 2. **Relation to Fisher information.** The standard Fisher information of a density $p$ with respect to its argument is $I_F(p) = \int \frac{(p')^2}{p}\,d\psi$. Our $\mathcal{E}_\psi[\varphi_t] = \frac{1}{4}\cdot\frac{1}{2\pi}\,I_F(\varphi_t)$. Thus $\mathcal{E}_\psi$ is proportional to (but not identical with) the Fisher information. We use the name "phase score energy" to avoid confusion with the standard statistical usage and to reflect its role as a kinetic energy of the fiber warp factor.

 3. **Dimensionality.** Both $\psi$ and $\varphi_t$ are dimensionless (recall $\varphi_t$ is a density w.r.t. $d\psi/(2\pi)$), so $\mathcal{E}_\psi$ is **dimensionless**. The combination $\mathcal{E}_\psi / R_\kappa^2$ has dimensions of $[\text{length}]^{-2}$, matching the cosmological constant $\Lambda$.

 4. **Explicit values for standard distributions:**

| Distribution | $\varphi_t(\psi)$ | $\mathcal{E}_\psi[\varphi_t]$ | $\Lambda_\varphi = \mathcal{E}_\psi / R_\kappa^2$ |
|---|---|---|---|
| Uniform | $1$ | $0$ | $0$ |
| von Mises ($\kappa_0 \gg 1$) | $\frac{e^{\kappa_0\cos\psi}}{I_0(\kappa_0)}$ | $\approx \frac{\kappa_0}{2}$ | $\approx \frac{\kappa_0}{2R_\kappa^2}$ |
| Wrapped Cauchy ($\rho$) | $\frac{1-\rho^2}{1-2\rho\cos\psi+\rho^2}$ | $\frac{2\rho^2}{1-\rho^2}$ | $\frac{2\rho^2}{(1-\rho^2)R_\kappa^2}$ |
| Dirac $\delta(\psi-\psi_0)$ | (distributional limit) | $+\infty$ | $+\infty$ |


**Remark on the Dirac limit.** The phase score energy *diverges* for 
$\varphi_t \to \delta$ (infinitely concentrated phase), which is the 
opposite of the V1 claim that $\Lambda_\varphi \to 0$. This reflects a 
genuine physical distinction:

 - **Classical limit** (no phase variability) corresponds to $\varphi_t \to \delta$, but this means $\mathcal{E}_\psi \to \infty$ — the fiber is infinitely sharply warped. Physically, the 5D geometry degenerates as the fiber pinches, and the KK reduction breaks down. The correct classical limit is obtained by taking $R_\kappa \to 0$ (the fiber shrinks to zero size) while keeping $\mathcal{E}_\psi / R_\kappa^2$ finite, or equivalently by noting that a Dirac-concentrated fiber makes the $\psi$-direction decouple entirely from the dynamics.

 - **Uniform limit** (maximal phase variability, $\varphi_t = 1$) gives $\mathcal{E}_\psi = 0$: the fiber is perfectly round and contributes no additional curvature.

**The classical GR recovery** is therefore achieved either by (a) $R_\kappa \to 0$ (the extra dimension disappears), which gives $\Lambda_\varphi \to 0$ regardless of $\mathcal{E}_\psi$, or (b) $\varphi_t \to 1$ (uniform phase, no structured variability), which gives $\mathcal{E}_\psi = 0$ and $\Lambda_\varphi = 0$. We adopt interpretation (b) as the operational "classical limit" in the context of repeated measurements: if all kime-phases are equiprobable, the phase dimension has no observable effect.

**Theorem (Classical Limit — Uniform Phase).** If $\varphi_t(\psi) = 1$ (uniform on $\mathbb{S}^1$), then $\mathcal{E}_\psi = 0$, the fiber warp factor is constant, $R_{\text{fiber}} = 0$, and the 5D field equations with the KK ansatz reduce exactly to the standard 4D Einstein equations plus a decoupled massless scalar (dilaton) and $U(1)$ gauge field.


## The Effective 4D Field Equations

The *Gauge Field Term*, $F_{\mu\alpha}F_\nu^{\;\alpha}$, 
is the energy-momentum tensor of a gauge field (like electromagnetism).
Standard Kaluza-Klein theory starts with a 5D metric. The "off-diagonal" components of this metric ($g_{\mu 5}$) act as a vector field in 4D.
To compute the 5D Ricci tensor and project it onto 4D, the curvature of the extra dimension manifests as the field strength $F_{\mu\nu}$ of this vector field. 
The $\Phi^2$ factor is the coupling constant, typically related to the radius of the extra dimension.

The *Dilaton Term*, $\nabla_\mu\nabla_\nu\Phi$, represents the contribution of the dilaton (or radion) field, $\Phi$.
This field $\Phi$ usually represents the "size" or the metric component of the extra dimension ($g_{55}$).
This specific form, involving second derivatives of the scalar, is characteristic of scalar-tensor theories. It arises because the 4D Ricci scalar is "mixed" with the scalar field when you perform a dimensional reduction from 5D to 4D in the Jordan Frame. It represents how the vacuum energy and curvature are influenced by the varying "thickness" of the extra dimension.

The *gauge field* $F_{\mu\nu}$ and the *dilaton* $\Phi$ are the background geometric degrees of freedom of the bulk that are "universal" to the geometry. $\varphi_t$, on the other hand, is usually a specific matter field or a fluctuation living on that geometry, rather than a part of the spacetime fabric itself.

In the Lagrangian derivation, the gauge fields come from $A_\mu \sim g_{\mu 5}$ and the dilaton comes from $\Phi \sim \sqrt{g_{55}}$. If $\varphi_t$ is a 
separate scalar field, like a Higgs or an inflaton, its kinetic terms will appear in the matter stress-tensor, $T_{\mu\nu}^{(4)}$, rather than being baked into the geometric KK terms.

In standard KK reductions, the extra-dimensional metric is independent of the extra coordinates, the "cylinder condition". This separates the gauge and dilaton terms into neat, geometric packages. Any field $\varphi_t$ that doesn't define the "shape" or "twist" of the extra dimension won't appear in the geometric part of the Einstein equation.
Thus, the *Gauge Term*	mediates "forces"	Off-diagonal metric, $g_{\mu 5}$,
the *Dilaton Term* mediates "gravity strength", extra-dimension size, $g_{55}$,
and the Matter/Fluctuation $\varphi_t$ is a source in $T_{\mu\nu}$.s


Varying the effective 4D action $S_{\text{eff}}^{(4)}$ with respect to $g^{\mu\nu}$ (with $\tilde\Phi = 1$, $A_\mu = 0$ for simplicity), we obtain

$$\boxed{G_{\mu\nu}^{(4)} + \Lambda_{\text{eff}}\,g_{\mu\nu}^{(4)} = \frac{8\pi G_{\text{eff}}}{c^4}\,T_{\mu\nu}^{(4)}\,,}$$

where
$$\Lambda_{\text{eff}} = \Lambda_5 - \frac{\mathcal{E}_\psi[\varphi_t]}{2\,R_\kappa^2}\,.$$

Note that the sign of the phase score energy contribution is *negative*,
reducing the effective cosmological constant, because the fiber warp costs curvature energy. To produce an effective *positive* cosmological term from the phase distribution, one can either (i) work with a bare $\Lambda_5$ that is sufficiently positive, or (ii) adopt an alternative warp embedding (e.g., $w(\psi) = 1/\varphi_t(\psi)$, an "inverse warp"), which flips the sign. The choice between these embeddings depends on the physical interpretation of the kime-phase direction and must ultimately be fixed by experiment.

For the **general case** with non-trivial dilaton and gauge field, the full effective 4D equations are:

$$G_{\mu\nu}^{(4)} + \Lambda_{\text{eff}}\,g_{\mu\nu} = \frac{8\pi G_{\text{eff}}}{c^4}\,T_{\mu\nu}^{(4)} + \frac{I_0}{2}\,\Phi^2\Big(F_{\mu\alpha}F_\nu^{\;\alpha} - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\Big) + \frac{2I_0}{\Phi}\Big(\nabla_\mu\nabla_\nu\Phi - g_{\mu\nu}\Box\Phi\Big)\,.$$

The gauge field and dilaton source terms are the standard KK contributions and are independent of $\varphi_t$.


## Covariance, Bianchi Identity, and Consistency

### General Covariance

The 5D field equations $R_{AB} - \frac{1}{2}G_{AB}R^{(5)} + \Lambda_5 G_{AB} = \frac{8\pi G_5}{c^4}T_{AB}$ are manifestly covariant under arbitrary 5D diffeomorphisms $X^A \to X'^A(X^B)$, since they are constructed entirely from the metric $G_{AB}$ and its Levi-Civita connection.

The effective 4D equations inherit covariance under 4D diffeomorphisms $x^\mu \to x'^\mu(x^\nu)$, since the KK reduction preserves the tensorial character of all 4D objects. The constant $\Lambda_{\text{eff}}$ (for fixed $\varphi_t$) is a scalar under 4D diffeomorphisms.

## Contracted Bianchi Identity (5D)

The 5D Einstein tensor $\mathcal{G}_{AB}^{(5)} = R_{AB} - \frac{1}{2}G_{AB}R^{(5)}$ satisfies the **contracted Bianchi identity**

$$\nabla^A\mathcal{G}_{AB}^{(5)} = 0\,,$$

which is a purely geometric identity (holds for any pseudo-Riemannian manifold). Combined with the field equations, this implies

$$\nabla^A T_{AB} = 0\,,$$

the covariant conservation of the 5D stress-energy tensor.

## Conservation Law for the Effective 4D Equations

For the effective 4D equations $G_{\mu\nu}^{(4)} + \Lambda_{\text{eff}}(t)\,g_{\mu\nu} = \frac{8\pi G_{\text{eff}}}{c^4}T_{\mu\nu}^{(4)}$, taking the 4D covariant divergence of both sides and using the 4D Bianchi identity $\nabla^\mu G_{\mu\nu}^{(4)} = 0$:

$$\nabla^\mu\big(\Lambda_{\text{eff}}(t)\,g_{\mu\nu}\big) = \frac{8\pi G_{\text{eff}}}{c^4}\nabla^\mu T_{\mu\nu}^{(4)}\,.$$

Since $\nabla^\mu g_{\mu\nu} = 0$ (metric compatibility), the left side reduces to $\partial_\nu\Lambda_{\text{eff}}$. Therefore:

$$\boxed{\frac{8\pi G_{\text{eff}}}{c^4}\nabla^\mu T_{\mu\nu}^{(4)} = \partial_\nu\Lambda_{\text{eff}}(t) = \frac{1}{2R_\kappa^2}\,\frac{d\mathcal{E}_\psi[\varphi_t]}{dt}\cdot\partial_\nu t\,.}$$

**Interpretation.** If $\varphi_t$ evolves in time (so that $\mathcal{E}_\psi$ is time-dependent), then $T_{\mu\nu}^{(4)}$ is **not** covariantly conserved — there is an energy exchange between the matter sector and the kime-phase sector. This is the standard result for scalar-tensor theories: the scalar (here, the fiber warp) mediates an additional gravitational interaction that violates the strong equivalence principle at the level $\sim d\mathcal{E}_\psi/dt$.

**Corollary.** If $\varphi_t$ is time-independent (a static phase law), then $\Lambda_{\text{eff}}$ is constant, $\partial_\nu\Lambda_{\text{eff}} = 0$, and the 4D stress-energy is covariantly conserved: $\nabla^\mu T_{\mu\nu}^{(4)} = 0$.

## Phase Gauge Invariance

The kime-surface model has an inherent **phase-shift symmetry**: the joint transformation $\mathcal{S}(t,\psi) \to \mathcal{S}(t,\psi+\alpha)$ and $\varphi_t(\psi) \to \varphi_t(\psi - \alpha)$ leaves all observables invariant. In the KK framework, this corresponds to the $U(1)$ gauge transformation

$$\psi \to \psi + \alpha(x)\,,\qquad A_\mu \to A_\mu - \partial_\mu\alpha\,,$$

which leaves the 5D metric invariant. The **anchoring operator** $\mathcal{A}$ (from the KPT framework) fixes this gauge by requiring $\arg(\hat{\varphi}_{n_\star}) = 0$ for the lowest non-trivial harmonic.


## Classical Limit via Weak Convergence

**Theorem (Classical Limit via Uniform Phase).** Let $\{\varphi_t^{(\epsilon)}\}_{\epsilon > 0}$ be a family of smooth phase laws converging weakly to the uniform distribution as $\epsilon \to 0$, in the sense that $\varphi_t^{(\epsilon)} \to 1$ in $L^1(\mathbb{S}^1)$. If additionally $\mathcal{E}_\psi[\varphi_t^{(\epsilon)}] \to 0$, then $\Lambda_{\text{eff}} \to \Lambda_5$ and the effective 4D field equations converge to the standard Einstein equations with cosmological constant $\Lambda_5$.

*Example.* The von Mises family $\varphi_t^{(\epsilon)}(\psi) = e^{\epsilon\cos\psi}/I_0(\epsilon)$ converges to the uniform distribution as $\epsilon \to 0$ (since $I_0(0) = 1$ and $e^{0\cdot\cos\psi} = 1$). The phase score energy $\mathcal{E}_\psi \approx \epsilon/2 \to 0$, confirming the classical limit.

**Remark on the Dirac limit.** The opposite limit $\varphi_t \to \delta(\psi - \psi_0)$ is **not** the classical GR limit in this framework. As a $\delta$-function, the fiber warp factor develops a singularity (the fiber pinches to zero everywhere except at $\psi_0$), and $\mathcal{E}_\psi \to \infty$. This limit corresponds to the **collapse of the kime dimension** into a single phase slice — a degenerate geometry that is not described by the smooth KK reduction. It requires a separate distributional treatment (or equivalently, the statement that the extra dimension has been removed from the theory entirely, recovering standard 4D GR trivially).


# The 15 Independent Spacekime Field Equations

The symmetric 5D tensor equation $\mathcal{G}_{AB}^{(5)} + \Lambda_5 G_{AB} = \frac{8\pi G_5}{c^4}T_{AB}$ with $A,B \in \{t,\psi,x^1,x^2,x^3\}$ gives $\binom{5+1}{2} = 15$ independent component equations.

| Sector | Components | Number | Physical Content |
|--------|-----------|--------|------------------|
| $tt$ | $\mathcal{G}_{tt}^{(5)} + \Lambda_5 G_{tt} = \frac{8\pi G_5}{c^4}T_{tt}$ | 1 | Hamiltonian constraint (energy density) |
| $t\psi$ | $\mathcal{G}_{t\psi}^{(5)} + \Lambda_5 G_{t\psi} = \frac{8\pi G_5}{c^4}T_{t\psi}$ | 1 | Kime-momentum constraint (phase--time coupling) |
| $ti$ | $\mathcal{G}_{ti}^{(5)} + \Lambda_5 G_{ti} = \frac{8\pi G_5}{c^4}T_{ti}$ | 3 | Spatial momentum constraints |
| $\psi\psi$ | $\mathcal{G}_{\psi\psi}^{(5)} + \Lambda_5 G_{\psi\psi} = \frac{8\pi G_5}{c^4}T_{\psi\psi}$ | 1 | Fiber curvature equation |
| $\psi i$ | $\mathcal{G}_{\psi i}^{(5)} + \Lambda_5 G_{\psi i} = \frac{8\pi G_5}{c^4}T_{\psi i}$ | 3 | Phase--spatial coupling |
| $ij$ | $\mathcal{G}_{ij}^{(5)} + \Lambda_5 G_{ij} = \frac{8\pi G_5}{c^4}T_{ij}$ | 6 | Spatial evolution equations |
| **Total** | | **15** | |

The 5 equations involving the $\psi$-index ($t\psi$, $\psi\psi$, $\psi i$) are the **new equations** beyond standard GR. In the KK decomposition, they govern: the $\psi\psi$-equation determines the dilaton dynamics; the $\psi\mu$-equations yield the Maxwell-type equation for the $U(1)$ gauge field $F_{\mu\nu}$; and the mixed equations enforce consistency of the fiber warp.


# Spacekime Schwarzschild Solution

## Metric Ansatz

For a static, spherically symmetric source with no gauge field ($A_\mu = 0$), constant dilaton ($\tilde\Phi = 1$), and fiber warp $w(\psi) = \varphi_t(\psi)$, the spacekime metric ansatz is

$$ds_5^2 = -f(r)\,c^2\,dt^2 + f(r)^{-1}\,dr^2 + r^2\,d\Omega_2^2 + R_\kappa^2\,\varphi_t(\psi)\,d\psi^2\,,$$

where $f(r)$ is to be determined and $d\Omega_2^2 = d\theta^2 + \sin^2\theta\,d\phi^2$ is the metric on the unit 2-sphere. Note that
$\theta, \phi$ are the standard spherical angles, not the kime-phase.

## Vacuum Solution

Setting $T_{AB} = 0$ and solving the 5D vacuum Einstein equations, the $\psi$-independent sector of the solution takes the **Kottler (Schwarzschild--de Sitter)** form

$$\boxed{f(r) = 1 - \frac{2G_{\text{eff}}M}{c^2 r} - \frac{\Lambda_{\text{eff}}}{3}\,r^2\,,}$$

where $\Lambda_{\text{eff}}$ includes the phase score energy contribution. The coefficient of $r^2$ is $\Lambda_{\text{eff}}/3$, matching the standard Kottler metric (correcting the factor of $1/6$ in V1).

**Derivation note.** The vacuum field equation $R_{\mu\nu} = \Lambda_{\text{eff}}\,g_{\mu\nu}$ (in 4D, for the trace-reversed form with $T_{\mu\nu} = 0$) applied to the static spherically symmetric ansatz yields the standard Kottler ODE for $f(r)$, with $\Lambda_{\text{eff}}$ in place of $\Lambda$.


## Modified Event Horizon

The horizon radius satisfies $f(r_H) = 0$

$$1 - \frac{r_S}{r_H} - \frac{\Lambda_{\text{eff}}}{3}\,r_H^2 = 0\,,$$

where $r_S = 2G_{\text{eff}}M/c^2$. For $|\Lambda_{\text{eff}}| r_S^2 \ll 1$, the perturbative solution is:

$$r_H \approx r_S\Big(1 + \frac{\Lambda_{\text{eff}}\,r_S^2}{3} + \ldots\Big)\,.$$


# Spacekime FLRW Cosmology

## Cosmological Ansatz

The [Friedmann--Lemaître--Robertson--Walker (FLRW)](https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric) ansatz for a homogeneous, isotropic universe with kime extension is

$$ds_5^2 = -c^2\,dt^2 + a(t)^2\Big[\frac{dr^2}{1-kr^2} + r^2\,d\Omega_2^2\Big] + R_\kappa^2\,\varphi_t(\psi)\,d\psi^2\,,$$

where $a(t)$ is the scale factor and $k \in \{-1,0,+1\}$ is the spatial curvature.


## Modified Friedmann Equations

Integrating the $tt$-component of the 5D field equations over the $\psi$-fiber (using $\int_0^{2\pi}\sqrt{\varphi_t}\,d\psi/(2\pi) = I_0$), the effective Friedmann equations are

$$\boxed{\Big(\frac{\dot{a}}{a}\Big)^2 = \frac{8\pi G_{\text{eff}}}{3}\,\rho - \frac{kc^2}{a^2} + \frac{\Lambda_{\text{eff}}}{3}\,,}$$

and the acceleration equation

$$\boxed{\frac{\ddot{a}}{a} = -\frac{4\pi G_{\text{eff}}}{3}\Big(\rho + \frac{3p}{c^2}\Big) + \frac{\Lambda_{\text{eff}}}{3}\,,}$$

where $\Lambda_{\text{eff}} = \Lambda_5 \mp \frac{\mathcal{E}_\psi[\varphi_t]}{2R_\kappa^2}$, with the sign depending on the warp embedding convention (Section 5.5).


## Time-Varying Dark Energy

If the phase distribution evolves, $\varphi_t(\psi)$ changes with $t$, then $\mathcal{E}_\psi[\varphi_t]$ and hence $\Lambda_{\text{eff}}(t)$ become time-dependent. This provides a natural mechanism for **dynamical dark energy** (quintessence-like behavior). However, as shown in Section 6.3, a time-varying $\Lambda_{\text{eff}}$ implies that $T_{\mu\nu}^{(4)}$ is not covariantly conserved — there is energy exchange between the visible sector and the kime fiber. The modified conservation equation is

$$\dot{\rho} + 3\frac{\dot{a}}{a}\Big(\rho + \frac{p}{c^2}\Big) = -\frac{c^4}{8\pi G_{\text{eff}}}\,\dot{\Lambda}_{\text{eff}}\,.$$


# Experimental Predictions and Falsifiability

The spacekime gravitational framework makes concrete, testable 
predictions. See the 
[SOCR TCIU/Spacekime KPT algorithm](https://www.socr.umich.edu/TCIU/HTMLs/TCIU_SK_Appendix03_KPT_DS_fMRI_V5.html).

## Prediction 1: Phase-Dependent Gravitational Redshift

Repeated measurements of gravitational redshift (e.g., GPS satellites, Pound--Rebka experiments) should exhibit a **structured variance** beyond Poisson/Gaussian noise, decomposable via KPT into a kime-surface $\mathcal{S}(t,\psi)$ and phase law $\varphi_t(\psi)$.

**Test:** Apply KPT-FFT to repeated high-precision clock comparison datasets (e.g., optical lattice clocks, ACES/PHARAO space clock data). If the recovered $\varphi_t(\psi)$ is non-uniform with statistical significance (Rayleigh test $p < 0.05$), this supports the kime framework.


## Prediction 2: Effective Cosmological Constant from Phase Score Energy

The framework predicts a relationship between the phase score energy and the expansion rate:

$$\Lambda_\varphi = \frac{\mathcal{E}_\psi[\varphi_t]}{R_\kappa^2}\,.$$

For this to match the observed $\Lambda_{\text{obs}} \approx 1.1 \times 10^{-52}\,\text{m}^{-2}$, the kime compactification radius must satisfy:

$$R_\kappa \sim \sqrt{\frac{\mathcal{E}_\psi}{\Lambda_{\text{obs}}}}\,.$$

For $\mathcal{E}_\psi \sim \mathcal{O}(1)$ (e.g., von Mises with moderate concentration), this gives $R_\kappa \sim 10^{26}\,\text{m}$ — on the order of the Hubble radius. This is a striking prediction: the "size" of the kime-phase dimension is cosmological.

**Test:** Measure $\mathcal{E}_\psi$ from repeated observations of the same cosmological process (e.g., Type Ia supernova light curves, GW events). Combine with the observed $\Lambda_{\text{obs}}$ to infer $R_\kappa$ and check consistency across independent datasets.


## Prediction 3: Kime Correction to Gravitational Wave Strain

For a gravitational wave observed by $N$ independent detectors, the spacekime framework predicts a kime-surface structure:

$$h_j(t) = \mathcal{S}(t, \Psi_j) + \varepsilon_j(t)\,,$$

where $\Psi_j \sim \varphi_t(\psi)$ is the kime-phase for detector $j$.

**Test:** Apply KPT to LIGO/Virgo/KAGRA multi-detector strain data from the same event. A non-uniform recovered $\varphi_t(\psi)$, Rayleigh test $p < 0.05$, would indicate kime-phase structure in gravitational wave observations.


## Prediction 4: Modified Perihelion Precession

The spacekime Kottler solution $f(r) = 1 - r_S/r - \Lambda_{\text{eff}} r^2/3$ predicts an additional contribution to orbital precession. The standard result for the $\Lambda$-induced perihelion shift in 
[Schwarzschild--de Sitter spacetime, Kerr et al., Class. Quantum Grav. 20, 2003](https://doi.org/10.1088/0264-9381/20/13/320) is

$$\delta\phi_\Lambda = \frac{\pi c^2\,\Lambda_{\text{eff}}\,a^3}{G_{\text{eff}} M}\,\sqrt{1-e^2}\;\;\text{rad/orbit}\,,$$

where $a$ is the semi-major axis and $e$ is the orbital eccentricity,
correcting the formula in V1. For solar system orbits with $\Lambda_{\text{eff}} \sim \Lambda_{\text{obs}}$, this is $\sim 10^{-20}$ rad/orbit for 
Mercury, currently below detection but within reach of future laser ranging missions.


## Null Prediction (Falsification Criterion)

If KPT applied to any sufficiently large repeated-measurement gravitational
dataset consistently yields $\varphi_t(\psi) \approx \text{uniform}$ (failing the Rayleigh test for non-uniformity at significance level $p < 0.05$ across all datasets), this implies $\mathcal{E}_\psi \approx 0$ and $\Lambda_\varphi \approx 0$: the spacekime extension adds no predictive power beyond standard GR. This constitutes a clear falsification criterion.


# Summary of the Extended Spacekime Field Equations

The complete gravitational equation systems in the spacetime and spacekime frameworks:

| Property | Classical GR (4D) | Spacekime (5D) |
|---|---|---|
| Field Equations | $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$ | $G_{AB}^{(5)} + \Lambda_5 G_{AB} = \frac{8\pi G_5}{c^4}T_{AB}$ |
| Independent Equations | 10, $\mu,\nu = 0,1,2,3$ | 15, $A,B = 0,1,2,3,4$ |
| Metric | $g_{\mu\nu}$ (4D Lorentzian) | $G_{AB}$ (5D Lorentzian, KK ansatz) |
| Signature | $(-,+,+,+)$ | $(-,+,+,+,+)$ |
| Time | $t \in \mathbb{R}$ | Kime: $\kappa = te^{i\psi}$, fiber $\psi \sim \varphi_t$ |
| Phase structure | None | $\varphi_t(\psi)$ as fiber warp, recovered by KPT |
| Cosmological constant | $\Lambda$ (free parameter) | $\Lambda_{\text{eff}} = \Lambda_5 \mp \frac{\mathcal{E}_\psi[\varphi_t]}{2R_\kappa^2}$ |
| New length scale | None | $R_\kappa$ (kime compactification radius) |
| Bianchi identity | $\nabla^\mu G_{\mu\nu} = 0$ | $\nabla^A G_{AB}^{(5)} = 0$; 4D: $\nabla^\mu T_{\mu\nu} = -\frac{c^4}{8\pi G_{\text{eff}}}\partial_\nu\Lambda_{\text{eff}}$ |
| Classical limit | Always | $\varphi_t \to 1$ (uniform) or $R_\kappa \to 0$ |
| Falsifiable | — | KPT + Rayleigh test on repeated gravitational data |
| Volume element | $\sqrt{-g}\,d^4x$ | $\sqrt{-G}\,d^5X$ ($G < 0$ by signature) |


## Effective 4D Equations After Dimensional Reduction

$$G_{\mu\nu}^{(4)} + \Lambda_{\text{eff}}\,g_{\mu\nu}^{(4)} = \frac{8\pi G_{\text{eff}}}{c^4}\,T_{\mu\nu}^{(4)}\,,$$

where 
$$\Lambda_{\text{eff}} = \Lambda_5 \mp \frac{\mathcal{E}_\psi[\varphi_t]}{2\,R_\kappa^2}\,,\qquad \mathcal{E}_\psi[\varphi_t] = \int_0^{2\pi}\Big(\partial_\psi\sqrt{\varphi_t}\Big)^2\;\frac{d\psi}{2\pi} \geq 0\,.$$

The 5D spacekime gravitational field equation system is

 - **Covariant**: constructed from the 5D metric and its Levi-Civita connection, manifestly tensorial;
 - **Consistent**: satisfies the contracted Bianchi identity; 4D conservation law explicitly computed (Section 6.3);
 - **Well-posed**: Lorentzian signature $(-,+,+,+,+)$ ensures hyperbolicity of the Cauchy problem;
 - **Dimensionally sound**: $\Lambda_\varphi = \mathcal{E}_\psi/R_\kappa^2$ has units $[\text{length}]^{-2}$, matching $\Lambda$;
 - **Reduces to GR**: when $\varphi_t = 1$ (uniform phase) or $R_\kappa \to 0$ (fiber decouples);
 - **Falsifiable**: through KPT analysis of repeated gravitational measurements (Section 10).

# Appendix

## Appendix 1: Summary of Revisions from V1

The following substantive changes were made relative to V1 to address mathematical consistency.

| Issue | V1 Treatment | V2 Resolution |
|---|---|---|
| Route A/B mixing | $\varphi_t$ declared "measure weight only," then Fisher info terms derived | Route B adopted: $\varphi_t$ enters as fiber warp factor $G_{\psi\psi} = R_\kappa^2\varphi_t(\psi)$; no division step |
| $\mathcal{F}_{\mu\nu}$ ill-defined | $\nabla_\mu\nabla_\nu\varphi_t$ undefined when $\varphi_t = \varphi_t(\psi)$ only | Removed; source terms arise from $R_{\text{fiber}}(\psi)$ via KK reduction |
| Fisher info name/units | Called "Fisher information"; dimensionless but compared to $\Lambda$ in $\text{m}^{-2}$ | Renamed "phase score energy" $\mathcal{E}_\psi$; divided by $R_\kappa^2$ for correct units |
| Bianchi identity (4D) | Asserted but not computed | Explicit computation (Section 6.3): energy exchange with kime sector when $\dot\varphi_t \neq 0$ |
| Signature | $(-,-,+,+,+)$, two timelike | $(-,+,+,+,+)$, single timelike; $\psi$-direction spacelike |
| Volume element | $\sqrt{-G}$ with $G > 0$ (inconsistent) | $\sqrt{-G}$ with $G < 0$ (correct for Lorentzian signature) |
| KK ansatz | Mixed shifts $\alpha^i, \beta^i$ | Proper KK connection 1-form $(d\psi + A_\mu dx^\mu)^2$ |
| Kottler coefficient | $\Lambda r^2/6$ | Corrected to $\Lambda r^2/3$ |
| Perihelion formula | $\delta\phi \propto a^2(1-e^2)/c^2$ | Corrected to standard $\delta\phi \propto c^2\Lambda a^3/(GM)\sqrt{1-e^2}$ |
| Wrapped Laplace | Not properly normalized on $\mathbb{S}^1$ | Replaced with von Mises and wrapped Cauchy (exactly normalized circular distributions) |
| Classical limit | $\varphi_t \to \delta$ claimed $\Lambda_\varphi = 0$ | Corrected: $\varphi_t \to 1$ (uniform) is the operational classical limit; $\delta$ limit is singular |
| Notation | $\theta$ for both kime-phase and spherical angle | Kime-phase renamed to $\psi$; $\theta,\phi$ reserved for $\mathbb{S}^2$ |
| Lapse function | Single $N$ for both 4D and 5D | 5D metric written in KK form; 4D lapse $N$ appears naturally as $\sqrt{-g_{00}}$ |

## Appendix 2: Further Improvements to Consider

There are a number of enhancements and improvements to consider to this reformulation of the *Einstein-Hilbert gravitational field equations in spacekime.*

This appendix describes several such enhancements that may be worthwhile persuing.

### The *1D Curvature* Problem

In V2, we define the block-diagonal metric as
$$G_{AB} = \begin{pmatrix} g_{00} & 0 & g_{0i} \\ 0 & R_\kappa^2\,\varphi_t(\psi) & 0 \\ g_{j0} & 0 & g_{ij} \end{pmatrix}.$$

This implies that the metric on the $\mathbb{S}^1$ fiber is simply $ds_{\text{fiber}}^2 = R_\kappa^2\,\varphi_t(\psi) d\psi^2$.
However, a 1D manifold (like $\mathbb{S}^1$) has identically zero intrinsic curvature. Any 1D metric $w(\psi)d\psi^2$ can be globally flattened by a coordinate transformation $d\tilde{\psi} = \sqrt{w(\psi)} d\psi$. Therefore, the 5D Ricci scalar $R^{(5)}$ for this block-diagonal metric is simply equal to the 4D Ricci scalar $R^{(4)}$.

Hence, the term $R_{\text{fiber}}$ provided in V2 cannot mathematically 
arise from the $\Gamma^\psi_{\psi\psi}$ Christoffel symbol because the corresponding Riemann tensor component $R^\psi_{\psi\psi\psi}$ is exactly zero due to anti-symmetry index rules. Thus, the *Phase Score Energy*,
$\mathcal{E}_\psi$, will not spontaneously generate during Kaluza-Klein reduction.

###  The Conformal Spacekime Embedding

To extract the Fisher Information/Phase Score Energy rigorously, the phase distribution $\varphi_t(\psi)$ must *couple* to the 4D spacetime components. 
Instead of warping only the fiber, $\varphi_t(\psi)$ should be treated as a conformal factor on the entire 5D metric.

### Conformal Ansatz

We can define the 5D spacekime metric as being conformally related to the
standard Kaluza-Klein metric
$$G_{AB} = \Omega(\psi)^2 \hat{G}_{AB},$$
where $\hat{G}_{AB}$ is the standard KK metric (with a constant fiber radius $R_\kappa$) and $\Omega(\psi)$ is a conformal warp factor that we will link to $\varphi_t(\psi)$.

### 5D Ricci Scalar Transformation

Under a conformal transformation in 5D spacekime, $D=5$, the Ricci scalar transforms exactly as
$$R^{(5)} = \Omega^{-2} \hat{R} - 8 \Omega^{-3}\hat{\Box}\Omega - 4 \Omega^{-4}(\hat{\nabla}\Omega)^2 .$$
Because $\Omega$ depends on $\psi$, the derivative $\hat{\nabla}\Omega$ contains $\partial_\psi \Omega$. This is the mathematical engine that will generate our Phase Score Energy.

During integration and reduction, when we substitute this into the
Einstein-Hilbert action $S_5 = \int \sqrt{-G} R^{(5)} d^5X$, note
that $\sqrt{-G} = \Omega^5 \sqrt{-\hat{G}}$, the action integrand becomes
$$\sqrt{-\hat{G}} \left[ \Omega^3 \hat{R} - 8 \Omega^2 \hat{\Box}\Omega - 4 \Omega (\hat{\nabla}\Omega)^2 \right].$$

Using integration by parts on the $\hat{\Box}\Omega$ term ($\int -8 \Omega^2 \hat{\Box}\Omega = \int 16 \Omega (\hat{\nabla}\Omega)^2$), the action simplifies to
$$S_5 = \int \sqrt{-\hat{G}} \left[ \Omega^3 \hat{R} + 12 \Omega (\hat{\nabla}\Omega)^2 \right] d^5X.$$

We can also link to $\varphi_t(\psi)$ and Recovering $\mathcal{E}_\psi$. 
To ensure the 4D gravity term is weighted by the phase distribution $\varphi_t$, we set $\Omega^3 = \varphi_t(\psi)$, meaning the conformal factor is $\Omega = \varphi_t^{1/3}$.

Examining the kinetic term $12 \Omega (\hat{\nabla}\Omega)^2$
$$12 \varphi_t^{1/3} \left( \frac{1}{3} \varphi_t^{-2/3} \varphi_t' \right)^2 \frac{1}{R_\kappa^2} = \frac{4}{3} \frac{(\varphi_t')^2}{\varphi_t} \frac{1}{R_\kappa^2}$$
Recall the definition of the Phase Score Energy: $\mathcal{E}_\psi = \frac{1}{4} \int \frac{(\varphi_t')^2}{\varphi_t} d\psi$. The term we just derived is exactly proportional to it
$\frac{16}{3 R_\kappa^2} \mathcal{E}_\psi$!
By simply changing the metric ansatz from a "1D fiber warp" to a "5D conformal embedding", you mathematically guarantee the emergence of the Phase Score Energy term in the effective 4D equations without any hand-waving or divided-out weights.

### Time-Varying Dark Energy

In V2, section on *Time-Varying Dark Energy*, we indicated that if 
$\varphi_t$ evolves in time, $T_{\mu\nu}^{(4)}$ is not covariantly conserved, mimicking quintessence.

If we adopt the above *Conformal Embedding* enhancement, this time-dependence becomes appropriate. If $\varphi_t(t, \psi)$ depends on $t$, then the conformal factor $\Omega(t, \psi)$ depends on $t$. The term $12 \Omega (\hat{\nabla}\Omega)^2$ will naturally split into two parts

 1.	The $\psi$-derivative part, which yields your $\Lambda_{\text{eff}}$.
 2.	A $t$-derivative part, which acts exactly like the kinetic energy of a scalar field ($\sim \dot{\varphi}_t^2$). This explicitly provides the mathematical mechanism for the dynamical dark energy we predicted.


### Sign $+$ *classical limit*

In the early “Modeling strategy” section we argue that $\Lambda_\varphi$
is “non-negative producing an accelerating contribution” and “vanishes in the classical limit" $\varphi_t\to\delta.$

Later we corrected that indicating that the phase-score contribution enters
with a negative sign in $$\Lambda_{\text{eff}}=\Lambda_5-\mathcal{E}\psi/(2R\kappa^2),$$

and $\mathcal{E}_\psi$ diverges in the Dirac limit $\varphi_t\to\delta$.

In principle, one can choose one consistent convention

 - Either keep $w=\varphi_t$ and accept $\Delta\Lambda<0$, or
 - use the inverse warp $w=\frac{1}{\varphi_t}$ to flip the sign, then propagate that choice everywhere (definitions, KK reduction, predictions).
 
The “classical limit” needs to be consistent, as later we argued that
the Dirac limit is singular and “operational GR recovery” is either 
$\varphi_t\to 1$ or $R_\kappa\to 0$.

We can either remove “classical limit” from the Dirac row and refer
ot it as a “deterministic phase / singular geometric limit”. Or alternatively,
explicitly distinguish statistical classicality, variance $\to 0$, from geometric GR limit.

$w(\psi)$ is prescribed/non-dynamical may conflict with 
$\varphi_t$ evolves and with the variational claim.
We first define $w(\psi)$ as “prescribed (non-dynamical) ... of (\psi) alone”,
but then we treat $\varphi_t$ as time-varying and derive 
$\Lambda_{\text{eff}}(t)$ and a modified conservation law when $\dot{\varphi}_t\neq 0$.
Then, we indicate that 5D equations follow from unconstrained variation $\delta S_5/\delta G^{AB}=0$.
These three statements can’t all be true without extra structure.

We can either consider a *background warp*, non-dynamical, and treat 
$\varphi$ as externally supplied, which can't present the result as coming from varying with respect to the full $G_{AB}$, as it’s a constrained variation / effective theory.

Alternatively, we can consider a *dynamical field*, where $G_{\psi\psi}$ varies, i.e. $\varphi$ is part of the metric degrees of freedom, which would derive the $\psi\psi$ field equation, or an effective evolution equation for $\varphi)$, not just treat $\varphi_t$ as an externally estimated density.

In KK, “zero-mode truncation” vs. $\psi$-dependent warp may need
additional justification, e.g., assuming $g_{\mu\nu},\Phi,A_\mu$ are 
$\psi$-independent while $w(\psi)$ is not.
In general, a $\psi$-dependent metric component backreacts and sources $\psi$-dependence in other components, assuming we avoid consistent truncation,
e.g., small warp, averaging approximation, or special symmetry.

Can we prove a proposition indicating leading order in a small warp amplitude $\epsilon$ and neglect induced $\psi$-modes in $g_{\mu\nu}$, or alternatively,
impose an averaged field equation and treat $\psi$-dependence as confined to $G_{\psi\psi}$ by construction?

### Effective constants

Some of the normalization constants, e.g., $I_0$ and $2\pi$, may be slightly off
in the V2 version. For instance, $I_0=\int_0^{2\pi}\sqrt{\varphi_t},d\psi/(2\pi)$, then set $I_0\approx 1$ and identify $G_{\text{eff}}=G_5/R_\kappa$.

When we propose “moderately non-uniform” $\varphi$ for cosmology-level effects,
$I_0\approx 1$ may not be guaranteed, and the standard KK relation typically carries a $2\pi$ from the circle volume.

Perhaps we may need to keep $I_0$ explicitly in $G_{\text{eff}}$ and $\Lambda_{\text{eff}}$, instead of setting it $\approx 1$,
at least until we quantify the error.
We may need to state whether the $2\pi$ is absorbed into the definition of 
$G_5$ or not.

### Other notes

SOme of the explicit values in the table for $\mathcal{E}_\psi$
may have to be cross-checked. For example, we define 
$$\mathcal{E}\psi[\varphi]=\int(\partial\psi\sqrt{\varphi})^2,d\psi/(2\pi)=
\frac14\int (\varphi'^2/\varphi),d\psi/(2\pi).$$

But the table lists for von Mises $\approx \kappa_0/2$ and wrapped Cauchy $2\rho^2/(1-\rho^2)$, which may be slightly off.
For von Mises, $\varphi(\psi)\propto e^{\kappa\cos\psi}$,

$$\mathcal{E}_\psi(\kappa)=\frac{\kappa^2}{8}\Big(1-\frac{I_2(\kappa)}{I_0(\kappa)}\Big)\sim \frac{\kappa}{4}\quad(\kappa\gg 1).$$

So the asymptotic coefficient may actually be $\kappa/4$, instead of
$\kappa/2$.

Whereas for	wrapped Cauchy,
$\varphi(\psi)=\frac{1-\rho^2}{1-2\rho\cos\psi+\rho^2}$,

$$\mathcal{E}_\psi(\rho)=\frac{\rho^2}{(1-\rho^2)^2}.$$

In *Prediction 2*, we use
$\Lambda_\varphi=\mathcal{E}\psi/R\kappa^2$, but the derived $\Lambda_{\text{eff}}$ shift is $-\mathcal{E}\psi/(2R\kappa^2).$
Specifically, our prediction states 
$\Lambda_\varphi=\mathcal{E}\psi/R\kappa^2.$
Yet, our reduction suggests $\Lambda_{\text{eff}}=\Lambda_5-\mathcal{E}\psi/(2R\kappa^2)$,
up to $I_0$.
Do we need to clarofy that 
$\Delta\Lambda \equiv \Lambda_{\text{eff}}-\Lambda_5$, and
whether $\Lambda_\varphi$ means $\Delta\Lambda$, $|\Delta\Lambda|$, or $2|\Delta\Lambda|?$


Finally, some of the metric indexing may need to be refined;
4D indices, $\mu\in{0,2,3,4}$ while coordinates are written 
$(x^1,x^2,x^3)$.
Should we switch to using the standard 
$x^\mu=(t,x^1,x^2,x^3)$, $\forall \mu=0,1,2,3$, and reserve 
$A=4$, or $5$ for $\psi$?

in the effective-field-equations section, we may need
to reintroduce some *Route A* language, $\varphi_t$ is usually a matter field
and correct small typos, e.g., $T_{\mu\nu}.s$.



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