---
title: "Time Complexity, Inferential uncertainty and Spacekime Analytics"
subtitle: "Wheeler-DeWitt Equation in Spacekime"
author: "SOCR Team"
date: "`r format(Sys.time(),'%m/%d/%Y')`"
output:
  html_document:
    theme: spacelab
    highlight: tango
    includes:
      before_body: TCIU_header.html
    toc: true
    number_sections: true
    toc_depth: 3
    toc_float:
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    code_folding: hide
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header-includes:
  - \usepackage{amsmath}
  - \usepackage{amssymb}
  - \usepackage{physics}
  - \usepackage{mathrsfs}
  - \usepackage{bm}
---

```{r setup, include=FALSE, warning=FALSE, message=FALSE, error=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(ggplot2)
library(dplyr)
library(plotly)
```


# Introduction

Complex-time (kime) is defined as follows. Complex-time (kime) representation of repeated measurements longitudinal data localizes measureable quantitities according to their space-kime coordinates, $({\bf{x}}=(x,y,x),\kappa = t e^{i\theta}$), where the kime-magnitude $|\kappa|=t$ is the event temporal ordering, time, and the kime-phase
corresponds to random draws from a time-dependent kime-phase
distribution, $\theta \sim \Phi(t)$, supported on $[-\pi,\pi)$. As the kime-phase reflects a random draw from a phase distribution, it is unobservable.  It's worth clarifying the derivative operator formulation when $\theta\sim\Phi$, rather than a deterministic coordinate. One may consider the need for using Itô calculus, distributional derivatives, or another framework to clarify the role of $\theta$ and $\frac{d}{d\theta}$ and explicate the derivative operator

In QM, $\hat{x}$ and $\hat{p}$ operate on wavefunctions $\psi(x)$ in
$L^2(\mathbb{R})$, where $x$ is a continuous spatial variable. The
derivative $\frac{d}{dx}$ is well-defined as an operator on
differentiable functions. In the kime context, $\kappa = t e^{i\theta}$
introduces $\theta$ as a phase, randomly sampled from $\Phi(\theta; t)$
for each measurement at time $t$. Approach 1 probes $\Phi(\theta; t)$
via test functions, e.g., $f_\alpha(\theta)$, computing expectations
like
$\langle \Phi, f_\alpha \rangle = \int f_\alpha(\theta) \Phi(\theta; t) d\theta$.
In Approach 1A reformulation, we defined operators on a Hilbert space,
e.g., $L^2[-\pi, \pi]$, acting on functions $\psi(\theta)$, where
$\psi(\theta)$ represent $\sqrt{\Phi(\theta; t)}$ or another state-like
object. When $\theta$ is a random draw, $\frac{d}{d\theta}$ seems
ill-defined without a continuous trajectory. Yet, $\Phi(\theta; t)$ is a density over $\theta$, suggesting a functional space where $\theta$ is a
variable of integration, not a stochastic process evolving in time (as
in Itô calculus).

Since $\Phi(\theta; t)$ is a probability density over
$\theta \in [-\pi, \pi]$ at fixed $t$, we treat $\theta$ as a coordinate in a function space, not a random variable evolving dynamically. Thus,

 -   Hilbert Space: $L^2[-\pi, \pi]$, where functions $\psi(\theta)$
    are square-integrable over $\theta$, and
    $\Phi(\theta; t) = |\psi(\theta)|^2$ (analogous to a probability
    density in QM).

 -   State: $\psi(\theta)$ could be $\sqrt{\Phi(\theta; t)}$ (assuming
    $\Phi$ is real and non-negative), representing the state of the
    kime-phase distribution at time $t$.

 -   Operators: Define $\hat{F}_\alpha$ and $\hat{G}_\beta$ as
    operators on $\psi(\theta)$, with expectations computed against
    $\Phi(\theta; t)$.
The differentiation operator $\hat{G}_\beta = -i \frac{d}{d\theta}$ in
$L^2[-\pi, \pi]$, where $\theta$ is the independent variable of the
function space, not a random draw per se. The random sampling occurs
when we generate data (e.g., $\theta_n(t) \sim \Phi$), but the operators
act on the density or its square root. Thus, the multiplication
operator $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$ and the
differentiation operator
$\hat{G}_\beta \psi(\theta) = -i \beta \frac{d}{d\theta} \psi(\theta)$,
where $\beta$ scales the momentum (frequency-like) contribution.
Since $\theta \in [-\pi, \pi]$ is a circular
domain (phase wraps around), we impose periodic boundary conditions,
$\psi(-\pi) = \psi(\pi)$, ensuring $\hat{G}_\beta$ is well-defined on
differentiable functions in $L^2[-\pi, \pi]$.

Then, the commutator is $[\hat{\Theta}, \hat{G}_\beta] \psi(\theta) = -i \beta \theta \frac{d}{d\theta} \psi(\theta) - (-i \beta \psi(\theta) - i \beta \theta \frac{d}{d\theta} \psi(\theta)) = i \beta \psi(\theta)$, where

 $$\hat{\Theta} \hat{G}_\beta \psi(\theta) = \theta (-i \beta \frac{d}{d\theta} \psi(\theta)) = -i \beta \theta \frac{d}{d\theta} \psi(\theta)$$
 
 $$\hat{G}_\beta \hat{\Theta} \psi(\theta) = -i \beta \frac{d}{d\theta} (\theta \psi(\theta)) = -i \beta \left( \psi(\theta) + \theta \frac{d}{d\theta} \psi(\theta) \right).$$

Thus, $[\hat{\Theta}, \hat{G}_\beta] = i \beta I$ confirms
non-commutativity, with a commutator proportional to $\beta$. Itô
calculus does not apply to stochastic processes where a variable,
$\theta(t)$, evolves randomly over time $t$, governed by a stochastic
differential equation (SDE). In kime, $\theta$ is sampled independently
at each $t$ from $\Phi(\theta; t)$, with no explicit temporal dynamics
(e.g., $d\theta = \mu dt + \sigma dW_t$). And the derivative
$\frac{d}{d\theta}$ operates over the $\theta$-domain of the density
$\Phi(\theta; t)$ at fixed $t$, not over $t$. Thus, Itô calculus is
unnecessary here. The randomness is in the sampling process, not in a
continuous evolution of $\theta$. A distributional derivative (e.g.,
weak derivative) would be needed if $\psi(\theta)$ or $\Phi(\theta; t)$
were not differentiable (e.g., included Dirac deltas). For simplicity,
assume $\Phi(\theta; t)$ is smooth (e.g., von Mises), so the standard
derivative suffices. If $\Phi$ were less regular, we’d define
$\hat{G}_\beta$ in the sense of distributions,
$\langle \hat{G}_\beta \psi, \phi \rangle = -i \beta \int_{-\pi}^\pi \psi(\theta) \frac{d}{d\theta} \phi(\theta) d\theta$,
for test functions $\phi \in C^\infty[-\pi, \pi]$.

# Wheeler-DeWitt Equation Formulation in Spacekime

To extend the classical Wheeler-DeWitt equation to space-kime (complex-time), we incorporate the kime-phase \(\theta\) as an additional quantum variable. Here's the mathematical derivation:

**Extended Hilbert Space**:
The wave functional depends on the spatial metric \(g_{ij}\) and the kime-phase \(\theta\),
$\Psi[g, \theta] \in L^2(\text{Super space}) \otimes L^2[-\pi, \pi),$
where Super space is the space of 3-metrics \(g_{ij}\), and \(L^2[-\pi, \pi)\) handles the \(\theta\)-dependence.

**Kime-Phase Operators**: Define operators acting on the \(\theta\)-Hilbert space

 - Position operator: \(\hat{\Theta} \psi(\theta) = \theta \psi(\theta)\),

 - Momentum operator: \(\hat{G}_\beta = -i\beta \frac{d}{d\theta}\), with commutator \([\hat{\Theta}, \hat{G}_\beta] = i\beta I\).

**Extended Wheeler-DeWitt Equation**:
The Hamiltonian constraint includes spatial gravity and kime-phase contributions
$\left[\hat{\mathcal{H}}_{\text{space}} + \hat{\mathcal{H}}_{\theta}\right] \Psi[g, \theta] = 0,$
where the spatial Hamiltonian (standard Wheeler-DeWitt) is
\[
\hat{\mathcal{H}}_{\text{space}} = -16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right).
\], the kime-phase Hamiltonian is
\[
\hat{\mathcal{H}}_{\theta} = -\frac{\beta^2}{2\mu} \frac{d^2}{d\theta^2} + V(\theta),
\]
and \(\mu\) a coupling constant and \(V(\theta)\) a potential (e.g., enforcing \(\Phi(\theta; t)\)).

**Periodic Boundary Conditions**:
For \(\theta \in [-\pi, \pi)\), impose $\Psi[g, -\pi] = \Psi[g, \pi], \quad \forall g.$

## Assuming Separation of Variables

Assume \(\Psi[g, \theta] = \Psi_{\text{space}}[g] \psi(\theta)\). This splits the equation:
\[
\hat{\mathcal{H}}_{\text{space}} \Psi_{\text{space}}[g] = E \Psi_{\text{space}}[g],
\]
\[
\hat{\mathcal{H}}_{\theta} \psi(\theta) = -E \psi(\theta).
\]

**Fourier Decomposition**:
Expand \(\psi(\theta)\) in eigenfunctions of \(\hat{G}_\beta\)
\[
\psi(\theta) = \sum_{n=-\infty}^\infty c_n e^{in\theta}, \quad n \in \mathbb{Z}.
\]
Substituting into \(\hat{\mathcal{H}}_{\theta}\psi = -E\psi\) gives:
\[
\left(\frac{\beta^2 n^2}{2\mu} + V(n)\right) c_n = -E c_n.
\]

**Modified Spatial Equation**:
For each Fourier mode \(n\), the spatial Wheeler-DeWitt equation becomes
\[
\left[-16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) + \frac{\beta^2 n^2}{2\mu} + V(n)\right] \Psi_{\text{space}}^{(n)}[g] = 0.
\]
This introduces a kime-phase energy term \(\frac{\beta^2 n^2}{2\mu} + V(n)\), modifying the effective cosmological constant \(\Lambda\).

The physical interpretation of the kime-phase \(\theta\) contributes discrete energy levels indexed by \(n\), analogous to angular momentum. Each mode \(n\) corresponds to a universe with a distinct effective energy density, leading to a multiverse-like spectrum of solutions.

The space-kime Wheeler-DeWitt equation extends general relativity by quantizing the kime-phase \(\theta\). The resultant theory couples spatial geometry to a compact internal dimension (\(\theta\)), introducing quantized energy corrections. This formalism bridges complex-time stochasticity with quantum gravity, offering a pathway to unify measurement statistics with cosmological dynamics.
\[
\boxed{\left[-16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) - \frac{\beta^2}{2\mu} \frac{d^2}{d\theta^2} + V(\theta)\right] \Psi[g, \theta] = 0}
\]

## Example using von Mises Mixture Kime-Phase Distribution

Let's work our an example to illustrate the special case of the Wheeler-DeWitt Equation (WDE) in spacekime, when the kime-phase distribution is a two-component mixture of von Mises ($vM$) where we have good estimates of the 4 parameters (the mixture distribution weight, $w$) and time-varying ($t$-dependent) locations ($\mu_1(t)$ and $\mu_2(t)$) and concentrations ($k_1(t)$ and $k_2(t)$), i.e., $\Phi(\theta) = w vM(\theta;\mu_1(t), k_1(t)) + (1-w)\,vM(\theta;\mu_2(t),k_2(t)).$

In this case, the kime-phase distribution at time \( t \) is a mixture
$\Phi(\theta; t) = w \, vM(\theta; \mu_1(t), k_1(t)) + (1-w) \, vM(\theta; \mu_2(t), k_2(t)),$
where \( w \in [0,1] \) is the mixture weight, \( \mu_1(t), \mu_2(t) \) are time-dependent mean angles, and \( k_1(t), k_2(t) \) are time-dependent oncentration parameters.

The von Mises density is
$vM(\theta; \mu, k) = \frac{e^{k \cos(\theta - \mu)}}{2\pi I_0(k)},$
where \( I_0(k) \) is the modified Bessel function of order $0$.

Assuming that the kime-phase wavefunction is
$\psi(\theta; t) = \sqrt{\Phi(\theta; t)} = \sqrt{w \, vM_1 + (1-w) \, vM_2}$
ensures \( |\psi(\theta; t)|^2 = \Phi(\theta; t) \).

Then, the kime-phase Hamiltonian is
$$\hat{\mathcal{H}}_\theta = -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t),$$
where \( V(\theta; t) \) is designed to enforce the mixture \( \Phi(\theta; t) \). Substituting \( \psi(\theta; t) \) into the eigenvalue equation \( \hat{\mathcal{H}}_\theta \psi = -E \psi \), we solve for \( V(\theta; t) \)
$$V(\theta; t) = -E + \frac{\beta^2}{2\mu} \frac{\nabla_\theta^2 \psi(\theta; t)}{\psi(\theta; t)}.$$

Let's explicitly derive the potential by computing
\( \nabla_\theta^2 \psi \),
$$\nabla_\theta^2 \psi = \frac{w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2}{2\sqrt{\Phi}} - \frac{\left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{4\Phi^{3/2}}.$$
Substituting into \( V(\theta; t) \) yields
$$V(\theta; t) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8\mu \Phi^2}.$$

For more specificity, assume a simplified von Mises case with static locations (means) and concentrations (dispersions). Consider \( \mu_1, \mu_2, k_1, k_2 \) are constant (time-independent) and use the von Mises property \( \nabla_\theta vM = -k \sin(\theta - \mu) vM \), the potential becomes
$$V(\theta) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w k_1 \left(k_1 \sin^2(\theta - \mu_1) - \cos(\theta - \mu_1)\right) vM_1 + (1-w)k_2 \left(k_2 \sin^2(\theta - \mu_2) - \cos(\theta - \mu_2)\right) vM_2 \right].$$

Therefore, the extended WDE with von Mises mixture phase (true model), the full spacekime WDE becomes
$$\boxed{
\left[ -16\pi G \, G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \frac{\sqrt{g}}{16\pi G} \left(R - 2\Lambda\right) - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) \right] \Psi[g, \theta] = 0
},$$
where \( V(\theta; t) \) is defined above.

The corresponding physical interpretation of the {\it{potential structure}} \( V(\theta; t) \) combines two "wells" centered at \( \mu_1(t) \) and \( \mu_2(t) \), modulated by \( k_1(t), k_2(t) \). The mixture weight \( w \) controls their relative influence.
With regard to the spatial geometry coupling, the spatial Hamiltonian \( \hat{\mathcal{H}}_{\text{space}} \) is modified by the kime-phase energy \( -E \), effectively renormalizing the cosmological constant \( \Lambda \rightarrow \Lambda + \Delta\Lambda \), where \( \Delta\Lambda \propto E \).

Let's consider an example with separated modes where
\( \mu_1 \approx \mu_2 + \pi \) and \( k_1, k_2 \gg 1 \) and the mixture approximates a bimodal distribution. The potential \( V(\theta; t) \) resembles a double-well on a circle, with minima at \( \mu_1, \mu_2 \). The spatial WDE then describes a universe with two dominant geometric states correlated with the kime-phase peaks.

This example depicts how non-Gaussian kime-phase mixture distributions generate structured potentials \( V(\theta; t) \), which modify the effective energy-momentum content of the Wheeler-DeWitt equation. The formalism bridges stochastic measurement statistics (via \( \Phi(\theta; t) \)) with quantum gravitational dynamics.


## Assuming No Separation of Variables

An alternative, slightly more general, extension of the classical Wheeler-DeWitt equation to complex-time (kime) without separability assumptions.

The classical Wheeler-DeWitt equation in quantum gravity is formulated as
$\hat{\mathcal{H}}\Psi[g_{ij}] = 0$,
where $\hat{\mathcal{H}}$ is the Hamiltonian constraint operator and $\Psi[g_{ij}]$ is the wave functional of the universe, depending on the 3-metric $g_{ij}$.

In the ADM formalism of general relativity, the Hamiltonian constraint arises from the Einstein-Hilbert action
$$S = \int d^4x \sqrt{-g}R = \int dt \int d^3x \sqrt{g}\left(K_{ij}K^{ij} - K^2 + {}^{(3)}R\right),$$
where $K_{ij}$ is the extrinsic curvature, $K$ is its trace, and ${}^{(3)}R$ is the scalar curvature of the spatial slice.

The quantized Hamiltonian constraint becomes
$$\hat{\mathcal{H}} = G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R ,$$
where $G_{ijkl} = \frac{1}{2\sqrt{g}}(g_{ik}g_{jl} + g_{il}g_{jk} - g_{ij}g_{kl})$ is the DeWitt supermetric.

In the kime framework, $\kappa = te^{i\theta}$, 
where the kime-magnitude $|\kappa| = t$ represents the usual time ordering, and the kime-phase $\theta$ is drawn from a time-dependent distribution $\Phi(\theta; t)$ supported on $[-\pi, \pi)$.

To extend to space-kime, we first modify the ADM metric to incorporate complex time
$$ds^2 = -N^2d\kappa d\bar{\kappa} + g_{ij}(dx^i + N^i d\kappa)(dx^j + N^j d\kappa),$$
where $\bar{\kappa}$ is the complex conjugate of $\kappa$, and $N$ and $N^i$ are the complex-valued lapse and shift functions.

The modified action becomes
$$S = \int d\kappa d\bar{\kappa} d^3x \sqrt{g}\left(K_{ij}(\kappa)K^{ij}(\bar{\kappa}) - K(\kappa)K(\bar{\kappa}) + {}^{(3)}R\right),$$
where the extrinsic curvature now depends on $\kappa$
$$K_{ij}(\kappa) = \frac{1}{2N}\left(\nabla_i N_j + \nabla_j N_i - \frac{\partial g_{ij}}{\partial \kappa}\right).$$

Since $\theta$ is randomly drawn from $\Phi(\theta; t)$, we must integrate over this distribution to obtain physical observables. The expectation value of any observable $\mathcal{O}$ becomes
$$\langle\mathcal{O}\rangle = \int_{-\pi}^{\pi} \mathcal{O}(t,\theta) \Phi(\theta; t) d\theta$$

Without the separability assumption, the extended Wheeler-DeWitt equation in space-kime becomes
$$\hat{\mathcal{H}}_{\kappa}\Psi[g_{ij}, \kappa] = 0,$$
where $\hat{\mathcal{H}}_{\kappa}$ is the kime-dependent Hamiltonian constraint
$$\hat{\mathcal{H}}_{\kappa} = G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R + i\hat{\Theta}_{\kappa}, $$
where $\hat{\Theta}_{\kappa}$ is an operator representing the kime-phase contribution.

To properly account for the probabilistic nature of $\theta$, we expand the wave functional in a kime-phase basis
$$\Psi[g_{ij}, \kappa] = \int_{-\pi}^{\pi} \psi[g_{ij}, t, \theta] \Phi(\theta; t) d\theta .$$

Using the Hilbert space structure $L^2[-\pi, \pi]$ for the kime-phase, we define

 - Multiplication operator: $\hat{\Theta} \psi(\theta) = \theta \psi(\theta)$

 - Differentiation operator: $\hat{G}_{\beta} \psi(\theta) = -i\beta \frac{d}{d\theta}\psi(\theta)$

With these operators, we can formulate the space-kime Wheeler-DeWitt equation
$$\left[G_{ijkl}\frac{\delta^2}{\delta g_{ij}\delta g_{kl}} - \sqrt{g}{}^{(3)}R + \frac{\partial^2}{\partial t^2} + i\beta\frac{\partial}{\partial \theta}\right]\psi[g_{ij}, t, \theta] = 0 ,$$
subject to the periodic boundary condition $\psi[g_{ij}, t, -\pi] = \psi[g_{ij}, t, \pi]$.

The commutation relation 
$[\hat{\Theta}, \hat{G}_{\beta}] = i\beta I$
introduces a quantum uncertainty principle in the kime domain, analogous to the position-momentum uncertainty in standard quantum mechanics. This creates a richer structure that may help address the "problem of time" in quantum gravity.

To explore solutions in the semi-classical limit, we can use the WKB approximation
$$\psi[g_{ij}, t, \theta] = A[g_{ij}, t, \theta]e^{iS[g_{ij}, t, \theta]/\hbar}.$$

This leads to the kime-modified Hamilton-Jacobi equation
$$\left(\frac{\delta S}{\delta g_{ij}}\right)G^{ijkl}\left(\frac{\delta S}{\delta g_{kl}}\right) + \sqrt{g}{}^{(3)}R + \left(\frac{\partial S}{\partial t}\right)^2 + \beta\frac{\partial S}{\partial \theta} = 0 .$$

This space-kime formulation introduces a statistical ensemble of universes, each with different kime-phase $\theta$ drawn from the distribution $\Phi(\theta; t)$. Physical observables are computed by averaging over this ensemble.
This approach may provide new insights into the emergence of time in quantum gravity, as the kime-phase can be interpreted as a quantum degree of freedom that complements the classical notion of time represented by $t$.

The non-commutative structure in the kime domain suggests a fundamental limitation on the simultaneous determination of time and its conjugate variable, analogous to the energy-time uncertainty in quantum mechanics.
This space-kime extension of the Wheeler-DeWitt equation offers a novel framework for addressing longstanding problems in quantum gravity, particularly the "problem of time," by introducing a richer structure that captures both the deterministic and probabilistic aspects of time evolution.

# Potential Reformulation of WDE in Spacekime

We need to clarify the role of \(\theta\) as a coordinate in \( L^2[-\pi, \pi) \) and as a  a random draw (\(\theta \sim \Phi(\theta; t)\)). Considering \(\theta\) as a random draw at each \( t \),  how does \(\Psi[g, \theta]\) evolves dynamically, as \(\theta\) lacks a temporal trajectory (unlike a stochastic process).

Assuming \(\psi(\theta) = \sqrt{\Phi(\theta; t)}\) (where \(\Phi = |\psi|^2\)) may be contradictory. In quantum mechanics (QM), \(|\psi|^2\) is a probability density over the configuration space, but in spacekime, \(\Phi(\theta; t)\) is a pre-defined distribution from data. This conflates the quantum state with a statistical distribution, risking inconsistency.

\(\Phi(\theta; t)\) is time-dependent, but the Hilbert space \( L^2[-\pi, \pi) \) and operators (\(\hat{\Theta}, \hat{G}_\beta\)) are defined statically. The \( t \)-dependence of \(\Phi\) should influence the operators or the Hilbert space structure dynamically.

Itô calculus isn’t needed as \(\theta\) isn’t a stochastic process evolving in \( t \), yet, the operator \(\hat{G}_\beta\) acts on \(\psi(\theta)\), which is tied to \(\Phi(\theta; t)\). If \(\Phi\) is non-smooth (e.g., a mixture with sharp peaks), the derivative \(\frac{d}{d\theta}\) may require a weak formulation.

What may be the physical meaning of the scaling parameter \(\beta\) in \(\hat{G}_\beta\)? Is it a fundamental constant, or does it depend on the system (e.g., related to \(\Phi\))? This ambiguity affects the interpretation of the commutator and uncertainty relation.
Let's try to derive the implied commutator uncertainty relation \(\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2}\) to justify the quantum nature of \(\theta\).

The separability assumption \(\Psi[g, \theta] = \Psi_{\text{space}}[g] \psi(\theta)\) may be overly restrictive. In quantum gravity, the spatial metric \( g_{ij} \) and time (or kime) are typically coupled, especially in a framework aiming to address the problem of time. This assumption may oversimplify the dynamics.
Also, if the potential \( V(\theta) \) enforces the kime-phase distirbution, \(\Phi(\theta; t)\), we may need a general form for specificity of the kime-phase Hamiltonian.

How to physically interpret the coupling parameter, e.g., related to kime-phase dynamics or measurement statistics, \(\mu\) in \(\hat{\mathcal{H}}_{\theta}\).
The separation assumption leads to \(\hat{\mathcal{H}}_{\text{space}} \Psi_{\text{space}} = E \Psi_{\text{space}}\) and \(\hat{\mathcal{H}}_{\theta} \psi = -E \psi\). What is the physical role of \( E \) as a shared eigenvalue? Does \( E \) represent a physical energy, or is it a mathematical artifact?

In the von Mises mixture example, assuming \(\psi = \sqrt{\Phi}\) may conflate the quantum wavefunction with a statistical distribution. In QM, \(\psi\) evolves via the Schrödinger equation, but here, \(\Phi(\theta; t)\) is externally specified, making \(\psi\) static at each \( t \).
In this example, we assumed constant \(\mu_1, \mu_2, k_1, k_2\) for simplification. Can the more general case be worked out (\(\mu_i(t), k_i(t)\))? The time-dependence of \( V(\theta; t) \) complicates the WDE, as the Hamiltonian becomes time-dependent.

As the potential \( V(\theta; t) \) involves terms like \(\frac{\nabla_\theta^2 \Phi}{\Phi}\), there may be numerical unstabilities when \(\Phi \to 0\), e.g., for high \( k_1, k_2 \), \(\Phi\) has sharp peaks, which ay require regularization.
The energy eigenvalue \( E \) is treated as a constant, but it may be dependent on \( t \) via \(\Phi(t)\).

*Assessment*: The von Mises example is a strong demonstration of the framework’s potential, but the assumption \(\psi = \sqrt{\Phi}\) is problematic. The time-dependence of \( V(\theta; t) \) needs better handling, and numerical stability should be addressed for practical implementation.

For the non-separability formulation, how to interpret the lapse \( N \) and shift \( N^i \) as complex-valued due to \( d\kappa d\bar{\kappa} \). In standard ADM, \( N \) and \( N^i \) are real, ensuring a real metric. Complex values may lead to non-physical solutions.
The term \( i\hat{\Theta}_\kappa \) in \(\hat{\mathcal{H}}_\kappa\) is defined as a kime-phase contribution. Does its form, \( i\beta \frac{\partial}{\partial \theta} \), require verification of hermiticity?

The expansion \(\Psi[g_{ij}, \kappa] = \int \psi[g_{ij}, t, \theta] \Phi(\theta; t) d\theta\) mixes quantum and classical probabilities. \(\Phi(\theta; t)\) is a classical distribution, but \(\psi[g_{ij}, t, \theta]\) is a quantum wavefunction, creating a conceptual mismatch.
The final WDE includes \(\frac{\partial^2}{\partial t^2}\), but the classical WDE is typically timeless (Hamiltonian constraint = 0). Having a \( t \)-derivative may contradict the core problem of time the WDE aims to solve, unless \( t \) is reinterpreted as a parameter.

Perhaps clarify the role of \(\theta\) and \(\Phi(\theta; t)\), e.g.,

 - Separate Quantum and Classical Roles: Treat \(\Phi(\theta; t)\) as a classical distribution for sampling \(\theta\), but let \(\psi(\theta)\) evolve quantum-mechanically via \(\hat{\mathcal{H}}_\theta\). For example, solve a Schrödinger-like equation for \(\psi\)
 $i \frac{\partial \psi(\theta, t)}{\partial t} = \hat{\mathcal{H}}_\theta \psi(\theta, t),$
  where \(\hat{\mathcal{H}}_\theta = -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta, t)\), and \( V(\theta, t) \) is chosen to approximate \( |\psi|^2 \to \Phi \) over time.

 - Dynamic Hilbert Space: Allow the Hilbert space to evolve with \( t \), reflecting the time-dependence of \(\Phi(\theta; t)\). For example, define a family of operators \(\hat{\Theta}(t), \hat{G}_\beta(t)\) that adapt to \(\Phi(t)\).

Can we formulate the weak derivatives for non-smooth phase \(\Phi\) via a weak derivative for \(\hat{G}_\beta\)
  \[
  \langle \hat{G}_\beta \psi, \phi \rangle = -i\beta \int_{-\pi}^\pi \psi(\theta) \frac{d\phi}{d\theta} d\theta,
  \]
  ensuring robustness for non-smooth \(\Phi\) (e.g., mixtures with sharp peaks).

 - Derive Uncertainty Relation: Explicitly derive the uncertainty relation
  $\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2},$$
  using the commutator \([\hat{\Theta}, \hat{G}_\beta] = i\beta I\), and validate it numerically for sample \(\Phi\).

 - Generalize \( V(\theta; t) \): Derive a general form for \( V(\theta; t) \) without assuming \(\psi = \sqrt{\Phi}\). For example, minimize a cost function
  $V(\theta; t) = \arg\min_V \int_{-\pi}^\pi \left| |\psi(\theta)|^2 - \Phi(\theta; t) \right|^2 d\theta,$ where \(\psi\) evolves via \(\hat{\mathcal{H}}_\theta\).

Is it possible to reformulate separability by coupling \( g_{ij} \) and \(\theta\)? Relax the separability assumption by introducing a coupling term, e.g.,
  $\hat{\mathcal{H}} = \hat{\mathcal{H}}_{\text{space}} + \hat{\mathcal{H}}_{\theta} + \hat{\mathcal{H}}_{\text{couple}},$
  where \(\hat{\mathcal{H}}_{\text{couple}} = \lambda \frac{\delta}{\delta g_{ij}} \frac{\partial}{\partial \theta}\), with \(\lambda\) a coupling constant.

- For a physical interpretation of \(\beta, \mu\), can we relate \(\beta, \mu\) to physical scales, e.g., \(\beta \sim \hbar\) (quantum scale), \(\mu \sim \frac{1}{G}\) (gravitational scale), and validate through dimensional analysis?

Consider eliminating \(\frac{\partial^2}{\partial t^2}\) to preserve the timeless nature of the WDE. Instead, treat \( t \) as a parameter in \(\Phi(\theta; t)\), and solve
  $$\left[ G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \sqrt{g} {}^{(3)}R - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) \right] \psi[g_{ij}, \theta; t] = 0,$$
  where \( t \) is fixed, and solutions are computed for each \( t \).

 - Complex Lapse/Shift: To ensure the metric remains real, can we redefine the ADM formalism, e.g., use \( d\kappa d\bar{\kappa} = dt^2 - t^2 d\theta^2 \), and adjust \( N, N^i \) to be real or impose constraints?

 - Regarding hermiticity of \(\hat{\Theta}_\kappa\), can we verify that \( i\beta \frac{\partial}{\partial \theta} \) is Hermitian on \( L^2[-\pi, \pi) \) with periodic boundary conditions, ensuring the Hamiltonian is physically meaningful?

Consider computing the observable effects of kime-phase, e.g., modifications to the cosmological constant \(\Delta\Lambda\) affecting the Hubble parameter \( H_0 \). Compare with CMB data or cosmological observations.
Can we estimate the parameters \(\beta, \mu\) from physical constraints, e.g., by matching \(\Delta\Lambda\) to observed dark energy density?
Is it possible to test the multiverse interpretation by simulating universes for different \( n \)-modes and comparing their geometric properties (e.g., curvature, expansion rates)?

In the numerical simulations, let's try to solve the WDE numerically, \(\hat{\mathcal{H}}_\theta \psi = -E \psi\) for the von Mises mixture, using finite differences or spectral methods (e.g., Fourier basis \( e^{in\theta} \)). Perhaps regularize \( V(\theta; t) \): Add a small constant to \(\Phi\) (e.g., \(\Phi + \epsilon\)) to avoid division by zero in \( V(\theta; t) \), ensuring numerical stability.
Computing ensemble averages \(\langle \mathcal{O} \rangle\) for sample observables (e.g., curvature \( R \)) over \(\Phi(\theta; t)\), may validate the statistical ensemble interpretation.

Would these revised Hilbert Space and Operators formulations improve the framework?

 - Hilbert Space: Define the wave functional as \(\psi[g_{ij}, \theta; t] \in L^2(\text{Superspace}) \otimes L^2[-\pi, \pi)\), where \( t \) is a parameter reflecting the time-dependence of \(\Phi(\theta; t)\).

 - Operators: \(\hat{\Theta} \psi(\theta) = \theta \psi(\theta)\), and \(\hat{G}_\beta = -i\beta \frac{\partial}{\partial \theta}\), with \([\hat{\Theta}, \hat{G}_\beta] = i\beta I\).

- Uncertainty Relation: $\Delta \Theta \cdot \Delta G_\beta \geq \frac{\beta}{2},$
  derived from the commutator.

The non-separable WDE Hamiltonian
  $$\hat{\mathcal{H}} = G_{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} - \sqrt{g} {}^{(3)}R - \frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta; t) + \lambda \frac{\delta}{\delta g_{ij}} \frac{\partial}{\partial \theta},$$
  where \(\lambda\) introduces coupling between \( g_{ij} \) and \(\theta\).

And the WDE $\hat{\mathcal{H}} \psi[g_{ij}, \theta; t] = 0,$  with periodic boundary condition \(\psi[g_{ij}, -\pi; t] = \psi[g_{ij}, \pi; t]\).

- Kime-Phase Evolution: Solve a separate Schrödinger equation for \(\psi(\theta, t)\)
  $$i \frac{\partial \psi(\theta, t)}{\partial t} = \left( -\frac{\beta^2}{2\mu} \frac{\partial^2}{\partial \theta^2} + V(\theta, t) \right) \psi(\theta, t),$$
  where \( V(\theta, t) \) is chosen to minimize \(\int | |\psi|^2 - \Phi(\theta; t) |^2 d\theta\).

In the von Mises mixture example consider the potential
  $$V(\theta; t) = V_0 + \frac{\beta^2}{4(\Phi + \epsilon)} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8(\Phi + \epsilon)^2},$$
  with \(\epsilon = 10^{-6}\) for numerical stability, the parameters \(\beta = \hbar\), \(\mu = \frac{1}{G}\), and estimate \( V_0 \) to match cosmological observations (e.g., \(\Delta\Lambda \sim 10^{-52} \, \text{m}^{-2}\)).

## Simulation

We'll demonstrate a numerical `R` code implementation showing solving the kime-phase 
WDE and computing the potential \( V(\theta; t) \).

```{r WDE_1, warning=FALSE, message=FALSE, error=FALSE}
library(circular)

# Von Mises mixture
von_mises_mixture <- function(theta, w, mu1, mu2, k1, k2) {
  vm1 <- dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  vm2 <- dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  w * vm1 + (1 - w) * vm2
}

# Compute potential V(theta; t)
compute_potential <- function(theta, t, w, mu1_t, mu2_t, k1_t, k2_t, beta = 1, mu = 1, E = 0, eps = 1e-6) {
  # Time-dependent parameters
  mu1 <- mu1_t(t)
  mu2 <- mu2_t(t)
  k1 <- k1_t(t)
  k2 <- k2_t(t)
  
  # Compute Phi
  Phi <- von_mises_mixture(theta, w, mu1, mu2, k1, k2)
  
  # First derivative of vM
  grad_vM1 <- -k1 * sin(theta - mu1) * dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  grad_vM2 <- -k2 * sin(theta - mu2) * dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  grad_Phi <- w * grad_vM1 + (1 - w) * grad_vM2
  
  # Second derivative (numerical approximation)
  dtheta <- theta[2] - theta[1]
  grad2_vM1 <- diff(grad_vM1, differences = 1) / dtheta
  grad2_vM2 <- diff(grad_vM2, differences = 1) / dtheta
  grad2_vM1 <- c(grad2_vM1, grad2_vM1[length(grad2_vM1)])  # Extend for length
  grad2_vM2 <- c(grad2_vM2, grad2_vM2[length(grad2_vM2)])
  grad2_Phi <- w * grad2_vM1 + (1 - w) * grad2_vM2
  
  # Potential
  V <- -E + (beta^2 / (4 * mu * (Phi + eps))) * grad2_Phi - 
       (beta^2 / (8 * mu * (Phi + eps)^2)) * grad_Phi^2
  V
}

# Example parameters
w <- 0.6
mu1_t <- function(t) 0.5 * t
mu2_t <- function(t) 0.5 * t + pi
k1_t <- function(t) 5 + 2 * sin(2 * pi * t / 30)
k2_t <- function(t) 5 - 2 * sin(2 * pi * t / 30)

# Compute V at t = 10
theta <- seq(-pi, pi, length.out = 1000)
t <- 10
V <- compute_potential(theta, t, w, mu1_t, mu2_t, k1_t, k2_t)

# Plot
plot(theta, V, type = "l", xlab = "θ", ylab = "V(θ; t=10)", main = "Kime-Phase Potential")
```

Below we expand this simulation to compute the kime-phase potential 
\( V(\theta; t) \) as a *surface* over \(\theta \in [-\pi, \pi)\) and 
\( t \in [0, 100]\). The graph visualizes the potentail as a 3D surface plot 
in SVG format using the `plotly` package. 

```{r WDE_3, warning=FALSE, message=FALSE, error=FALSE}
# Load required libraries
library(circular)
library(plotly)
library(htmlwidgets)  # For saving as SVG

# Von Mises mixture density function
von_mises_mixture <- function(theta, w, mu1, mu2, k1, k2) {
  vm1 <- dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
  vm2 <- dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
  w * vm1 + (1 - w) * vm2
}

# Compute the kime-phase potential V(theta; t) across a grid of theta and t
compute_potential_surface <- function(theta, t, w, mu1_t, mu2_t, k1_t, k2_t, 
                                     beta = 1, mu = 1, E = 0, eps = 1e-6) {
  # Initialize matrix to store V(theta; t)
  V <- matrix(NA, nrow = length(t), ncol = length(theta))
  
  # Loop over time points
  for (i in seq_along(t)) {
    # Time-dependent parameters
    mu1 <- mu1_t(t[i])
    mu2 <- mu2_t(t[i])
    k1 <- k1_t(t[i])
    k2 <- k2_t(t[i])
    
    # Compute Phi
    Phi <- von_mises_mixture(theta, w, mu1, mu2, k1, k2)
    
    # First derivative of vM (numerical approximation)
    dtheta <- theta[2] - theta[1]
    grad_vM1 <- -k1 * sin(theta - mu1) * dvonmises(theta, mu1, k1) / (2 * pi * besselI(k1, 0))
    grad_vM2 <- -k2 * sin(theta - mu2) * dvonmises(theta, mu2, k2) / (2 * pi * besselI(k2, 0))
    grad_Phi <- w * grad_vM1 + (1 - w) * grad_vM2
    
    # Second derivative (numerical approximation)
    grad2_vM1 <- diff(grad_vM1, differences = 1) / dtheta
    grad2_vM2 <- diff(grad_vM2, differences = 1) / dtheta
    grad2_vM1 <- c(grad2_vM1, grad2_vM1[length(grad2_vM1)])  # Extend for length
    grad2_vM2 <- c(grad2_vM2, grad2_vM2[length(grad2_vM2)])
    grad2_Phi <- w * grad2_vM1 + (1 - w) * grad2_vM2
    
    # Compute potential V at time t[i]
    V[i, ] <- -E + (beta^2 / (4 * mu * (Phi + eps))) * grad2_Phi - 
              (beta^2 / (8 * mu * (Phi + eps)^2)) * grad_Phi^2
  }
  
  return(V)
}

# Plot the kime-phase potential as a 3D surface using plotly
plot_kime_surface <- function(theta, t, V, title = "Kime-Phase Potential Surface") {
  # Create a plotly surface plot
  fig <- plot_ly(
    x = theta,
    y = t,
    z = V,
    type = "surface",
    colorscale = "Viridis",
    colorbar = list(title = "V(θ; t)")
  ) %>%
    layout(
      title = title,
      scene = list(
        xaxis = list(title = "θ (radians)", range = c(-pi, pi)),
        yaxis = list(title = "t (seconds)", range = c(0, 100)),
        zaxis = list(title = "V(θ; t)"),
        camera = list(eye = list(x = 1.5, y = 1.5, z = 0.8))
      )
    )
  
  return(fig)
}

# Main function to compute and plot the kime-phase potential surface
kime_potential_analysis <- function() {
  # Define parameters
  w <- 0.6
  mu1_t <- function(t) 0.5 * t %% (2 * pi) - pi  # Linearly increasing, wrapped to [-pi, pi)
  mu2_t <- function(t) (0.5 * t + pi) %% (2 * pi) - pi  # Opposite phase
  k1_t <- function(t) 5 + 2 * sin(2 * pi * t / 30)  # Oscillating concentration
  k2_t <- function(t) 5 - 2 * sin(2 * pi * t / 30)
  
  # Define grids for theta and t
  theta <- seq(-pi, pi, length.out = 200)  # Reduced resolution for faster computation
  t <- seq(0, 100, length.out = 100)
  
  # Compute the potential surface
  V <- compute_potential_surface(theta, t, w, mu1_t, mu2_t, k1_t, k2_t)
  
  # Plot the surface
  fig <- plot_kime_surface(theta, t, V)
  
  # Save as SVG
  # htmlwidgets::saveWidget(fig, "kime_potential_surface.html", selfcontained = TRUE)
  # Note: Direct SVG export requires additional tools like `webshot` or manual conversion
  # For now, we save as HTML and suggest manual conversion to SVG using a browser
  
  return(list(V = V, fig = fig))
}

# Run the analysis
results <- kime_potential_analysis()

# Display the plot (in RStudio or a browser)
results$fig
```

**Notes**:

 - The `von_mises_mixture()` function computes the kime-phase distribution 
\(\Phi(\theta; t) = w vM(\theta; \mu_1(t), k_1(t)) + (1-w) vM(\theta; \mu_2(t), k_2(t))\).

 - The `compute_potential_surface()` function calculates \( V(\theta; t) \) over a grid of \(\theta\) and \( t \). A small \(\epsilon = 10^{-6}\) is added to \(\Phi\) to avoid division by zero. The simulation uses numerical derivatives for \(\nabla_\theta \Phi\) and \(\nabla_\theta^2 \Phi\),  using
 \[
     V(\theta; t) = -E + \frac{\beta^2}{4\mu \Phi} \left[ w \nabla_\theta^2 vM_1 + (1-w) \nabla_\theta^2 vM_2 \right] - \frac{\beta^2 \left(w \nabla_\theta vM_1 + (1-w) \nabla_\theta vM_2\right)^2}{8\mu \Phi^2}.
     \]

 - The `plot_kime_surface()` function creates a 3D surface plot using `plot_ly`. The x-axis is \(\theta \in [-\pi, \pi)\), the y-axis is \( t \in [0, 100] \), and the z-axis is \( V(\theta; t) \). The `colorscale = "Viridis"` provides a clear color gradient, and the camera angle is adjusted for better visualization.

 - The time-dependent parameters are defined as \(\mu_1(t) = 0.5 t \mod 2\pi - \pi\), \(\mu_2(t) = (0.5 t + \pi) \mod 2\pi - \pi\): Linearly increasing means, wrapped to \([-\pi, \pi)\) and \(k_1(t) = 5 + 2 \sin(2\pi t / 30)\), \(k_2(t) = 5 - 2 \sin(2\pi t / 30)\): Oscillating concentrations with a period of 30 seconds. These choices create a dynamic double-well potential that evolves over time.


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