---
title: "Time Complexity, Inferential uncertainty and Spacekime Analytics"
subtitle: "Appendix 03: Kime-Phase Tomography (KPT)"
author: "SOCR Team"
date: "`r format(Sys.time(),'%m/%d/%Y')`"
output:
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    highlight: tango
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header-includes:
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---

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


# Introduction

Consider *complex-time (kime)*, $\kappa = t \cdot e^{i\theta}$, as a polar-coordinate
parameterization of the the 2D domain, $\mathbb{R}^2$, of the 2D manifold. Both
of the domain parameters *time* (*t*) and the *phase* ($\theta$) are *random variables*,
not constants of controllable parameters. For instance, $t \sim Uniform[0, \infty)$
is a *time sampling distribution*, whereas $\theta \sim \Phi(t)$ is some 
*circular (periodic) phase sampling distribution*, such as Laplace, supported on
the unitary circle $S^1$, i.e., $\theta \in [-\pi, \pi)$. In this
[TCIU Appendix](https://tciu.predictive.space/) we explore the idea of *kime-phase tomography (KPT)* to recover the unobservable kime-phase, represent longitudinal processes, measured as 
repeated-measurement time-series, as 2D manifold kime-surfaces parameterized over 
$\kappa\in\mathbb{C}$.

As a motivation, consider the specific example of having repeated time-series measurements
of a controlled longitudinal process, $\{s_1(t), s_2(t), \cdots, s_N(t)\}$. For a 
fixed repeated-sample index, $j$, the observed intensities, $s_j(t)$ will reflect
the kime-surface amplitudes at radial location $t$. If $t \sim Uniform$ distribution, 
and $\theta \sim \Phi (t)$ is a circular (periodic) phase distribution supported on
the unitary circle $S^1$, kime-phase tomography maps the longitudinal process,
tracked as the entire collection of repeated time-series to a manifold, 
$Map: Process \{s_j(t)\} \to \mathcal{M}$. 

# Mapping Time-series $\longrightarrow$ Kime-surfaces

First we will explore the fundamentals of mathematically representing a 2D manifold
parameterized by *two distributions* — *time* \(t\) (radial) and *phase* \(\theta\) (angular).
Let's consider the repeated time-series measurements \(\{s_j(t)\}_{j=1}^N\). The goal
is to map the underlying process to a surface embedded in \(\mathbb{R}^3\). 
The complex-time (*kime*) parameterization \(\kappa = t e^{i\theta}\) defines the 
*kime domain*, where \(t \sim \text{Uniform}[0, \infty)\) and \(\theta \sim \Phi(t)\) 
(a circular Laplace distribution on \(S^1 \equiv [-\pi, \pi)\)). The observed intensity 
\(s_j(t)\) will impact the manifold amplitude at \(\kappa\). 

## Parameter Space and Distributions

*Kime Domain*: \(\mathcal{K} = \mathbb{C} \cong \mathbb{R}^2\), with kime points 
\(\kappa = t e^{i\theta}\).

*Time Distribution*: \(t \sim \mathcal{U}[0, \infty)\), with density \(p_t(t') = \lim_{T \to \infty} \frac{\mathbf{1}_{[0,T]}(t')}{T}\).

*Phase Distribution*: \(\theta \sim \Phi(t)\), a circular Laplace distribution 
conditioned on \(t\)
$$p_{\theta|t}(\theta' \mid t) = \frac{\beta(t)}{2 \sinh \beta(t)} e^{\beta(t) \cos(\theta' - \mu(t))}, \quad \theta' \in [-\pi, \pi),$$
where \(\mu(t) \in [-\pi, \pi)\) is the (Laplase distribution) *location parameter* and 
\(\beta(t) > 0\) is the *scale parameter.* This is a normalized periodic density 
(*wrapped Laplace*).

The observed data contains the entire time-series collection with all \(N\) realizations of
the longitudinal process, observed at discrete times. For each \(j \in \{1, \dots, N\}\),
the time points are \(\mathbf{t}_j = \{t_{j,k}\}_{k=1}^{M_j} \subset [0, \infty)\),
ant the corresponding process intensities are \(s_j(t_{j,k}) \in \mathbb{R}\).
Thus, the complete dataset is 
$\mathcal{D} = \{(j, t_{j,k}, s_j(t_{j,k})) \mid 1 \leq j \leq N, 1 \leq k \leq M_j\}.$

The time-series to kime-surfaces mapping \(\mathcal{M}: \mathcal{D} \to \mathbb{R}^3\)
may be defined in two steps. First, define the  *kime sampling*; for each 
$(j, t_{j,k}) \in \mathcal{D}\), sample \(\theta_{j,k} \sim \Phi(t_{j,k}).$ 
Second, define the *embedding* for each point
$$\mathbf{p}_{j,k} = \left( t_{j,k} \cos \theta_{j,k},\ t_{j,k} \sin \theta_{j,k},\ s_j(t_{j,k}) \right) \in \mathcal{M}\subset \mathbb{R}^3.$$

The *discrete kime-surface* is the set
$$\mathcal{M}_{\text{discrete}} = \left\{ \mathbf{p}_{j,k} \mid (j, t_{j,k}) \in \mathcal{D} \right\}.$$

Whereas the more general *continuous kime-surface* requires kernel regression to 
estimate the intensity function $f: \mathbb{R}^2 \to \mathbb{R}$
$$f(x, y) = \frac{\sum_{(j,k) \in \mathcal{D}} s_j(t_{j,k}) \cdot K\left( \frac{\| (x,y) - (x_{j,k}, y_{j,k}) \|}{h} \right)}{\sum_{(j,k) \in \mathcal{D}} K\left( \frac{\| (x,y) - (x_{j,k}, y_{j,k}) \|}{h} \right)},$$
where $(x_{j,k}, y_{j,k}) = (t_{j,k} \cos \theta_{j,k}, t_{j,k} \sin \theta_{j,k})$,
$K(u) = e^{-u^2}$ (Gaussian kernel), $h > 0$ is the bandwidth.
The *continuous kime-surface* may be displayed as
$$\mathcal{M}_{\text{cont}} = \left\{ (x, y, f(x,y)) \mid (x,y) \in \mathbb{R}^2 \right\}.$$

## Foundations

First we will explore the parameterization of $\mathcal{M}_{\text{cont}}$. The surface
\(\mathcal{M}_{\text{cont}}\) is parameterized by the radial and angular distributions. 
Let \(\Omega = [0, \infty) \times [-\pi, \pi)\) with the product measure 
$d\mu = p_t(t) p_{\theta|t}(\theta \mid t)  dt  d\theta$. The expected intensity
at \(\kappa = t e^{i\theta}\) is $z(t, \theta) = \mathbb{E}_j[s_j(t)],$
assuming \(s_j(t)\) are i.i.d. realizations of a stochastic process \(s(t)\). 
Then, the surface is
$$\mathcal{M}_{\text{cont}} = \left\{ (t \cos\theta, t \sin\theta, z(t, \theta)) \mid (t, \theta) \in \Omega \right\}.$$

### Interpreting $s_j(t,\theta)$ versus $s_j(t)$

*Latent phase path* may be defined in different ways, but here is one example.
For each replicate $j$, introduce a *latent* random trajectory
$$\Theta_j:(0,T]\longrightarrow S^{1},\qquad
\Theta_j(t)\underset{\text{over t}}{\overset{\text{i.i.d.}}{\sim}}\Phi_t,$$
Then, the *conditional process* is an *intrinsic* intensity surface
represented as a random field
$$S_j(t,\theta)=\bigl[\text{mechanistic model}\bigr]+\varepsilon_j(t,\theta),$$
where the $\varepsilon_j$’s are small-scale fluctuations.

Each of the *observed time series*, recorded time-courses, represents only *slices*
along the latent phase:
$$s_j(t):=S_j\!\bigl(t,\Theta_j(t)\bigr),\qquad t=t_{j,k}\ \text{in practice}.$$

In this framework,
$$\underbrace{s_j(t)}_{\text{measured}} \;=\;
\underbrace{S_j(t,\theta)}_{\text{latent surface}}
\Big|_{\theta=\Theta_j(t)}.$$
Hence $s_j(t,\theta)$ and $s_j(t)$ are not two different process
definitions. Rather, $s_j(t)$ is the *push‑forward* of $s_j(t,\theta)$ by the
random map $\theta=\Theta_j(t)$.

The table below decomposes the sources of stochasticity in this representation.

| Symbol              | Randomness from                              |
| ------------------- | ---------------------------------------------- |
| $\Theta_j(t)$       | *phase variability* (circular law $\Phi_t$) |
| $S_j(t,\theta)$     | *surface variability* (between replicates)  |
| $\varepsilon_{j,k}$ | *measurement noise* (optional)              |

Since $\Theta_j(t)$ and $S_j(t,\theta)$ are *independent* by construction,
the *moment relationships*

$$\mathbb E\bigl[s_j(t)\bigr]
  =\int_{-\pi}^{\pi}\mathbb E\bigl[S_j(t,\theta)\bigr]\,
     \varphi_t(\theta)\,\frac{d\theta}{2\pi} =:Z(t)$$
the quantity embedded as the kime-surface height function.  If
$\Theta_j(t)$ has temporal dependence, e.g., a wrapperd
[Ornstein-Uhlenbeck process (OU process)](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process),
where the state variable is defined on a compact space, like the unit circle $S^1$
or an interval with periodic boundary. The same formula holds with $\varphi_t$ 
replaced by the marginal law at time $t$.

Let
$$\Omega:=(0,\infty)\times S^{1},\qquad d\mu(t,\theta)=p_t(t)\,\varphi_t(\theta)\,
dt\,d\theta/2\pi$$
and define the *conditional mean surface*
$$z(t,\theta):=\mathbb E\bigl[S_1(t,\theta)\bigr].$$
The kime‑surface is the graph
$$\mathcal M_{\rm cont}=\Bigl\{\bigl(t\cos\theta,\;t\sin\theta,\;z(t,\theta)\bigr)\;:\;
(t,\theta)\in\Omega\Bigr\}.$$

The actually observed time‑series $s_j(t)$ are *1D stochastic slices* of this surface
along the random phase path $\theta=\Theta_j(t)$.
This interpretation is consistent, as long as the hierarchy
$S_j(t,\theta)\;\to\;s_j(t)=S_j\bigl(t,\Theta_j(t)\bigr)$ is explicit.  
All inference in KPT is aimed at *inverting* this hierarchy.
That is using the collection $\{s_j(t_k)\}$ to recover both
$z(t,\theta)$ (*kimesurface geometry*) and $\varphi_t(\theta)$ (*phase distirbution law*).
In a statistical-physics sense, $z(t,\theta)$ is the *conditional mean*, whereas 
$Z(t)=\mathbb E[s_j(t)]$ is the *marginal mean* after integrating over the 
latent kime-phase. 


### Theorem 1 (Consistency of Kernel Estimator)

*Theorem 1* (Consistency of Kernel Estimator): Let \(f(x,y)\) be the kernel estimator of 
\(z(t, \theta)\) at \((x,y) = (t \cos\theta, t \sin\theta)\). Assume that 

 1. *The process \(s_j(t)\) is ergodic with \(\mathbb{E}[|s_j(t)|^2] < \infty\) for all \(t\).*

 2. *Time points are i.i.d. from \(\mathcal{U}[0,T]\), and \(T \to \infty\).*

 3. *Bandwidth \(h = h(M)\) satisfies \(\lim_{M \to \infty} h = 0\) and \(\lim_{M \to \infty} M h^2 = \infty\) (where \(M = |\mathcal{D}|\)).*

Then, for all \((t, \theta) \in \Omega\), 
$f(x, y) \xrightarrow{a.s.} z(t, \theta) \quad \text{as} \quad M \to \infty.$

*Proof*: Define the regression function \(m(\mathbf{u}) = \mathbb{E}[s \mid \mathbf{u}]\)
for \(\mathbf{u} = (x,y)\). By the Strong Law of Large Numbers
$$\frac{\sum_{i=1}^M s_i K\left( \frac{\|\mathbf{u} - \mathbf{u}_i\|}{h} \right)}{\sum_{i=1}^M K\left( \frac{\|\mathbf{u} - \mathbf{u}_i\|}{h} \right)} \xrightarrow{a.s.} \frac{\mathbb{E}[s \cdot K(\|\mathbf{u} - \mathbf{u}'\| / h)]}{\mathbb{E}[K(\|\mathbf{u} - \mathbf{u}'\| / h)]}.$$

As \(h \to 0\), the right-hand side converges to \(\mathbb{E}[s \mid \mathbf{u}] = z(t, \theta)\) (by properties of Gaussian kernels). The conditions on \(h\) ensure consistency.

*Example* (Circular Laplace Distribution): The wrapped Laplace density is
$$p_{\theta|t}(\theta' \mid t) = \sum_{k=-\infty}^{\infty} \frac{1}{2b(t)} e^{-\frac{|\theta' - \mu(t) + 2\pi k|}{b(t)}}.$$
A simple sampling algorithm requires drawing \(\eta \sim \text{Laplace}(\mu(t), b(t))\)
(linear Laplace) and computing \(\theta_{j,k} = \eta \mod 2\pi\) (wrapped to \([-\pi, \pi)\)).
the *discrete kimesurface* (\(\mathcal{M}_{\text{discrete}}\)) can be
rendered as a 3D scatter plot of \(\mathbf{p}_{j,k}\), where points spread radially;
for fixed \(t\), they lie on a circle of radius \(t\) with heights \(s_j(t)\).
While the *continuous kimesurface* (\(\mathcal{M}_{\text{cont}}\))
can be displayed by sampling \(\{\theta_{j,k}\}\) for all \((j,k) \in \mathcal{D}\),
computing \((x_{j,k}, y_{j,k})\), and evaluating \(f(x,y)\) on a grid (e.g., Cartesian or polar).
It will show as a smooth surface, radial if \(z(t,\theta)\) is independent of \(\theta\),
e.g., if \(s_j(t)\) has no phase dependence.

Observe the intrinsic phase dependences. When \(s_j(t)\) depends on \(\theta\),
e.g., phase-sensitive processes, \(z(t,\theta)\) varies angularly, breaking radial symmetry.
It's more efficient to utilize a polar grid for efficient computation and visualization.
Also, the *bandwidth* hyperparameter \(h\) can be optimized via cross-validation.
Indeed, we can replace Laplace with any circular distribution, such as von Mises.


# Overview of Kime-Phase Tomography (KPT)

Kime-phase tomography is inspired by *quantum mechanics (QM)* and requires
translation of concepts from *quantum tomography* where state reconstruction is via
observable measurements, operator commutators, and distribution theory 
into the context of kime manifolds. The goal is to 
*reconstruct the unobservable phase distribution* $\Phi(t)$ from 
*repeated time-series measurements* $\{s_j(t)\}$.

## Operator algebra and commutators

On the Hilbert space
$\mathcal K \;=\;L^{2}\!\bigl((0,\infty)\times S^{1},dt\,d\theta/2\pi\bigr)$,
we can define operators acting on state functions as follows

| symbol              | action on $\psi(t,\theta)$                                  | domain (core)        |
| ------------------- | ----------------------------------------------------------- | -------------------- |
| $\widehat T$        | $(\widehat T\psi)(t,\theta)=t\,\psi(t,\theta)$              | $C_{c}^{\infty}$     |
| $\widehat\Theta$    | $(\widehat\Theta\psi)(t,\theta)=\theta\,\psi(t,\theta)$     | $C_{c}^{\infty}$     |
| $\widehat P_{\phi}$ | $(\widehat P_{\phi}\psi)(t,\theta)=e^{i\phi\widehat\Theta \psi(t,\theta)}$ | everywhere (unitary) |
| $\widehat P_{\phi}$ (alternative)| $(\widehat P_{\phi}\psi)(t,\theta)=e^{i\phi}\psi(t,\theta)$ | everywhere (unitary) |

Note that since both $\widehat T$ and $\widehat\Theta$ are *multiplication* operators,
they  commute, i.e., their commutator is trivial, $[\widehat T,\widehat\Theta]=0$.
*Non‑trivial commutators* arise only if at least *one* operator involves a
*derivative*, i.e., a generator of a one‑parameter unitary group.  Two classical
choices to obtain non-commuting operators are listed below.

| Canonical pair                                                                    | Commutator                                                    | Remarks                      |
| --------------------------------------------------------------------------------- | ------------------------------------------------------------- | ---------------------------- |
| $\bigl(\widehat\Theta,\ \widehat L_\theta:=-i\hbar_\kappa\,\partial_\theta\bigr)$ | $[\widehat\Theta,\widehat L_\theta]=i\hbar_\kappa\,\mathbf 1$ | angular "position/momentum" |
| $\bigl(\widehat T,\ \widehat\Omega:=-i\hbar_\kappa\,\partial_t\bigr)$             | $[\widehat T,\widehat\Omega]=i\hbar_\kappa\,\mathbf 1$        | radial analogue             |

### Kime cross‑non‑commutativity

In functional analysis and quantum mechanics, *Canonical Commutation Relation (CCR)*
operators are related to the algebraic structure that describes the commutation 
relations between position and momentum operators in quantum mechanics.
The CCR algebra over a symplectic vector space is a $C^*$-algebra generated by 
elements satisfying the 
[Weyl form of the canonical commutation relations](https://en.wikipedia.org/wiki/Canonical_commutation_relation). 
These relations imply that the elements are unitary and have specific commutation properties.

[Schwartz operators](https://doi.org/10.1142/S0129055X16300016) represent a non-commutative
analog of Schwartz functions. The space of Schwartz operators a Fréchet space
equipped with a set of seminorms. 
The [Schwartz core, or Schwartz kernel theorem](https://en.wikipedia.org/wiki/Schwartz_kernel_theorem)
states that *continuous linear operators between Schwartz spaces* 
(or between spaces of test functions and distributions) *can be represented by a kernel*
that is a distribution on the product space. 
The *Schwartz kernel theorem provides a framework for describing continuous linear operators using distributions*. 

For example, to introduce *kime cross‑non‑commutativity* between $t$ and $\theta$ may 
require [Moyal‑type plane](https://doi.org/10.1016/j.geomphys.2011.04.021),
$$[\widehat T,\widehat\Theta]=i\ell^{2}\mathbf 1
\qquad(\ell>0\;\text{a deformation length scale}),$$
in which case, $\widehat T,\widehat\Theta$ can't both be
multiplication operators, i.e., at least one operator must be represented by a
derivative plus a linear term so that the CCR holds.  

*One consistent realization* of *kime cross‑non‑commutativity* involves the
following pair of operators
$$\widehat T = t,\qquad \widehat\Theta = \theta + i\ell^{2}\partial_t,$$
defined on the Schwartz core.  All other commutators, e.g.,
$[\widehat T,\widehat P_\phi]$, are then automatically determined.


In practice, we select a desired canonical commutation relations and then pick
an operator representation that satisfies these relations.  

$$\boxed{
\begin{align}
{\text{For instance, }}
[\widehat T,\widehat P_\phi]=i\hbar_\kappa\widehat P_\phi\partial_\theta
{\text{ is not consistent with }}
\widehat P_\phi=e^{i\phi}\mathbf 1. \\
{\text{ To ensure that this commutator }}
[\widehat T,\widehat P_\phi]=i\hbar_\kappa\widehat P_\phi\partial_\theta
{\text{ requires defining }} \\
\widehat P_\phi:=e^{i\phi\widehat\Theta} {\text{and letting }}
\widehat\Theta = \theta + i\ell^{2}\partial_t
{\text{ carry a derivative component.}}
\end{align}
}$$


## Kime-Space Operators

*Definition 1* (Kime-Space): The kime domain $\mathcal{K} \cong \mathbb{R}^2$ is 
parameterized by $\kappa = te^{i\theta}$, with $t \sim \mathcal{U}[0, \infty)$, 
radial longitudinal coordinate, and $\theta \sim \Phi(t)$, angular coordinate, 
$\theta \in S^1 \equiv [-\pi, \pi)$.

*Definition 2*: (Operator Algebra): Define non-commutative operators on $\mathcal{K}$:

 - *Radial Operator* (time multiplication): $\hat{T} \psi(t, \theta) = t \cdot \psi(t, \theta)$  

 - *Angular Operator* (phase multiplication): $\hat{\Theta} \psi(t, \theta) = \theta \cdot \psi(t, \theta)$  

 - *Phase Shift Operator*: $\hat{P}_\phi \psi(t, \theta) = e^{i\phi} \psi(t, \theta)$  

The *commutator* $[\hat{T}, \hat{\Theta}]$ encodes uncertainty  
$$[\hat{T}, \hat{\Theta}] \psi = (t\theta - \theta t)\psi = 0, \quad \text{(commutative baseline)}.$$
To introduce quantum-like non-commutativity, define a *kime-deformation* 
$$[\hat{T}, \hat{P}_\phi] = i\hbar_\kappa \hat{P}_\phi \frac{\partial}{\partial \theta}, \quad \hbar_\kappa \text{ (kime "Planck constant")}.$$

<span style="border: 1px solid black; padding: 5px;">
This commutator may need to be replaced to reflect the specific 
canonical operator pair used, e.g.,
$[\widehat\Theta,\,-i\hbar_\kappa\partial_\theta]=i\hbar_\kappa$.
</span>


### Theorem 2: (Kime-Uncertainty Principle)

*Theorem 2*: (Kime-Uncertainty Principle): For any state $\psi(t, \theta)$ with $\|\psi\|=1$,  
$$\Delta t \cdot \Delta \theta \geq \frac{\hbar_\kappa}{2} \left| \left\langle \frac{\partial \psi}{\partial \theta} \right\rangle \right| .$$
*Proof*: Apply Cauchy-Schwarz to 
$(\Delta T)^2 (\Delta \Theta)^2 \geq \frac{1}{4} |\langle [\hat{T}, \hat{\Theta}] \rangle|^2$
with deformed commutator.

## Observables and Measurements

*Definition 5* (Kime-Observables): An observable is a self-adjoint operator $\hat{O}$ on
$\mathcal{H}_\kappa = L^2(\mathbb{R}^+ \times S^1, dt\, d\theta)$. The expectation 
for a state $\rho$, density operator, is
$\langle \hat{O} \rangle_\rho = \text{tr}(\rho \hat{O}).$

*Axiom 1*: (Measurement Data): The time-series $s_j(t)$ corresponds to measuring 
the "intensity observable" $\hat{S}$
$s_j(t) = \langle \hat{S} \rangle_{\rho_j(t)} + \epsilon_j(t), \quad \epsilon_j \sim \text{noise},$
where $\rho_j(t)$ is the state at $(t, \theta_j(t))$.

## Test Functions and Distribution Action

*Definition 6*: (Kime-Test Functions): Let $\mathcal{D}(\mathcal{K})$ be the space 
of smooth, compactly supported test functions $\varphi(t, \theta)$. The *distribution action*
of $\Phi(t)$ is  
$\langle \Phi, \varphi \rangle_t = \int_{S^1} \varphi(t, \theta)  d\Phi(t)(\theta).$

*Definition 7*: (Kime-Transform): The *Kime-Fourier Transform (KFT)* for $\Phi(t)$ is  
$\mathcal{F}_\kappa[\Phi](n, t) = \int_{S^1} e^{-in\theta}  d\Phi(t)(\theta), \quad n \in \mathbb{Z}$
recovering Fourier coefficients of $\Phi(t)$ at each $t$.

The following result illustrates the tomographic kime-phase reconstruction theorem.

### Theorem 3: (Reconstruction of $\Phi(t)$)

Let $t>0$ be fixed and assume that the 
kime-surface *height observable* admits the spectral representation

$$\widehat{S}(t)=\sum_{n\in\mathbb Z} f_n(t)\,e^{in\widehat{\Theta}},
\qquad
f_{-n}(t)=\overline{f_n(t)},
\tag{B.1}$$
where $\widehat{\Theta}$ is a *(self‑adjoint) phase operator* whose
spectrum is the unit circle and whose functional calculus satisfies
$\exp\{in\widehat{\Theta}\}\,|{\theta}\rangle = e^{in\theta}|{\theta}\rangle$.
For independent realizations
$\{s_j(t_k),\theta_{j,k}\}_{\overbrace{1\le j\le M}^{repeats},\;\overbrace{1\le k\le K}^{time}}$,
define the empirical *mixed moment*

$$\widehat m_{n}(t):=\frac{1}{M}\sum_{j=1}^{M}s_j(t)\,e^{-in\theta_{j}},
\qquad
n\in\mathbb Z.
\tag{B.2}$$

*Theorem 3 (Information‑completeness, empirical form)*: If $\mathbb E|s_1(t)|^{2}<\infty$
and the kime-phase distribution derivative 
$\varphi_t(\theta)=\dfrac{d\Phi_t}{d\theta}\in L^{2}(S^{1})$, then for each $n$
$$\boxed{\;
   \widehat m_{n}(t)\;\xrightarrow[M\to\infty]{\text{a.s.}}\;
   f_n(t)\,\mathcal F[\varphi_t](n)
 },
 \tag{B.3} $$
where the Fourier transform is
$\mathcal F[\varphi_t](n)=\int_{-\pi}^{\pi}e^{-in\theta}\varphi_t(\theta)\,\dfrac{d\theta}{2\pi}$.

*Proof*: Let;s start with the *KKM decomposition*. Equation (B.1) implies
$$s_j(t)=\langle\psi_j|\widehat{S}(t)|\psi_j\rangle
          =\sum_{n}f_n(t)\,e^{in\theta_j}, \tag{B.4}   $$
where $\theta_j$ is the phase outcome (eigenvalue of $\widehat{\Theta}$) 
obtained in the $j$-th experiment.

Since the $\theta_j$ are i.i.d. with density $\varphi_t$, the expectaiton is
$$\mathbb E\bigl[s_j(t)\,e^{-in\theta_j}\bigr]
   =\sum_{k}f_k(t)\,\underbrace{\mathbb E\bigl[e^{i(k-n)\theta_j}\bigr]}_{\mathcal F[\varphi_t](n-k)}
   =f_n(t)\,\mathcal F[\varphi_t](n).
   \tag{B.5}$$

The summands in (B.2) are i.i.d. with finite first moment, so by the strong law
of large numbers
$$\widehat m_{n}(t)\;\xrightarrow{\text{a.s.}}\;
     \mathbb E\bigl[s_1(t)\,e^{-in\theta_1}\bigr],$$
and equation (B.5) gives the desired limit (B.3). $\square$

Note that directly explicating the operator domains and using explicit
Fourier coefficient $\mathcal F[\varphi_t](n)$, the more *formal convergence* 
stated in equation (B.3) can be *informally* stated as
$$\lim_{M\to\infty}\frac1M\sum_{j=1}^{M}s_j(t)\,e^{-in\theta_j}
    =f_n(t)\,\mathcal F[\Phi](n,t).$$


The *Information‑completeness (Theorem 3)* directly connects to the 
*Identifiability (Theorem 1)*, as knowing the entire sequence
$\{\widehat m_n(t)\}$ gives the full set of Fourier coefficients of
$\varphi_t$, and hence completely described the kime-phase distribution $\Phi_t$ itself.
Also, equation (B.3) reflects a finite‑$t$ instance of the
*Kime-Uncertainty Principle (Theorem 2)*, as uniform‑in‑$t$ consistency follows 
from standard triangular‑array sLLN arguments for a dense grid $\{t_k\}$.


*Corollary 3*: If $f_n(t) \neq 0$, $\Phi(t)$ is reconstructed via
$$\mathcal{F}_\kappa[\Phi](n,t) = \lim_{M \to \infty} \frac{1}{M f_n(t)} \sum_j s_j(t) e^{-in\theta_j}.$$

## Kime-Phase Tomography Algorithm

*Step 1*: Empirical Estimation of Moments. For discrete time bins $\{t_k\}$, 
compute the empirical moments 
$\hat{m}_n(t_k) = \frac{1}{M} \sum_{j=1}^M s_j(t_k) e^{-in\theta_j}$ and 
estimate $f_n(t_k)$ via basis regression (e.g., $f_n(t) = t \alpha_n$).

*Step 2*: Inverse Kime-Transform. Solve for $\Phi(t)$ at each $t_k$  
$\Phi(t_k) = \arg \min_\Phi \sum_n \left| \hat{m}_n(t_k) - f_n(t_k) \mathcal{F}_\kappa[\Phi](n,t_k) \right|^2.$

*Theorem 4*: (Consistency): When $\theta_j$ are i.i.d. $\sim \Phi(t)$, and $f_n(t) \neq 0$,
$\hat{\Phi}(t) \xrightarrow{M \to \infty} \Phi(t)$ in distribution. 

*Proof*: Follows directly from the Law of Large Numbers due to the continuity of 
$\mathcal{F}_\kappa^{-1}$.


*Example*: (Laplace Phase Reconstruction): Let's assume 
$\Phi(t) = \text{Laplace}(\mu(t), b(t))$ on $S^1$ with density 
$$p(\theta|t) = \frac{\beta(t)}{2\sinh \beta(t)} e^{\beta(t) \cos(\theta - \mu(t))}, \quad \beta(t) = 1/b(t).$$

Then the *KFT coefficients* are   
$$\mathcal{F}_\kappa[\Phi](n,t) = I_n(\beta(t)) / I_0(\beta(t)) \cdot e^{in\mu(t)},$$
where $I_n$ is the modified Bessel function.

The *KPT reconstruction* involves estimation of
$\hat{m}_n(t_k)$ from $\{s_j(t_k)\}$, solving for $\mu(t_k), \beta(t_k)$ via  
$$|\hat{m}_1(t_k)| = \left| \frac{I_1(\beta(t_k))}{I_0(\beta(t_k))} \right|, \quad \arg(\hat{m}_1(t_k)) = \mu(t_k),$$  
and interpolation of $\mu(t), \beta(t)$ across $t$.

After the kime-phase distribution $\Phi(t)$ is reconstructed, the *kime-manifold embedding*
gives the kime-surface $\mathcal{M}$ 
$$\mathcal{M} = \left\{ \left( t\cos\theta,  t\sin\theta,  \mathbb{E}_j[s_j(t)] \right) \mid t \geq 0,  \theta \sim \Phi(t) \right\}.$$

The physical interpretations of kime tomography reflects a quantum analogy
where $\Phi(t)$ is the *state* reconstructed from *measurements* $s_j(t)$ via 
operator expectations. Analogously to QM, the significance of the kime-operator
commutator is realized as non-commutativity probes sensitivity to phase perturbations,
enabling resolution in tomography. 
The kime-space information completeness is guarantees as
$\{e^{in\hat{\Theta}}\}_{n \in \mathbb{Z}}$ spans $\mathcal{H}_\kappa$, and hence,
$\Phi(t)$ is uniquely reconstructible (cf. Pauli's theorem in QM).

In this [KPT Overview section](TCIU_SK_Appendix03_KPT.html#3_Overview_of_Kime-Phase_Tomography_(KPT)), we presented the main ideas of kime-phase tomography
without some of the technical foundations that are included in this more detailed KPT 
formalism. The *Table* below showcases some of the extensions of the above
basics KPT overview, which are extended in the more 
[advanced KPT mathematical foundations](TCIU_SK_Appendix03_KPT.html#5_Generalized_KPT)
presented in the next section.

| Aspects                             |  [Basic KPT (Overview)](TCIU_SK_Appendix03_KPT.html#3_Overview_of_Kime-Phase_Tomography_(KPT))                           | [Advanced KPT](TCIU_SK_Appendix03_KPT.html#5_Generalized_KPT)                                                                                                              |
| ---------------------------------- | ---------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------ |
| *Mathematical setting*           | Uses intuitive statements; e.g., ignoring $\sigma$‑algebra, probability space, and measurability of random fields                           | Explicates the probability triple $(\Omega,\mathcal F,\mathbb P)$ and defines $t,\theta,s_j$ as measurable maps                    |
| *Distribution of $t$*            | Uses “Uniform $[0,\infty)$” without normalizing constant                                                                  | Corrects by using the *truncated uniform* on $[0,T]$ with $T\to\infty$, introduces an improper prior and works under conditional densities   |
| *Phase law $\Phi(t)$*            | Introduced as a “circular Laplace” without the wrapped‑Laplace pdf written later is not normalized in $\theta$                  | Provides a wrapped‑Laplace or von Mises pdf with explicit normalizer, dependent parameters $(\mu(t),\beta(t))$.                       |
| *Map $\{s_j(t)\}\to \mathcal M$* | Definition (6) of $\mathbf p_{j,k}$ needs to analyze the continuity, injectivity, and topological type of $\mathcal M$ | Offers a precise embedding $F: (t,\theta)\mapsto\mathbb R^3$, shows $F$ is $C^1$ a.s., and includes the kimesurface metric                        |
| *Tomography (inverse) problem*   | Offers an analogy with quantum tomography, but identifiability and completeness results are over-simplified                  | Formulates and proves an identifiability theorem (*Kime Pauli theorem*) and gives a constructive inversion algorithm with error bounds |
| *Statistical consistency*        | Theorem 1 on kernel regression cites classical results without explicit conditions for circular covariate                | Extends to mixed radial/circular covariate; specifies bandwidth sequence $(h_n)$ and moment assumptions                                |


# Mathematical Foundations of KPT 

Let's extend the [KPT Overview section overview](TCIU_SK_Appendix03_KPT.html#3_Overview_of_Kime-Phase_Tomography_(KPT))
above to include all
necessary the technical details. First, we will limit the kime-phase distribution 
to Laplace o von Mises, and later will generalize this to any symmetric 
(periodic) distribution supported on $S^1$.

## Probability model of complex‑time (*kime*)

Let $(\Omega,\mathcal F,\mathbb P)$ be a complete probability space and define the 
*radial time* as
$$t:\Omega\to\mathbb R_{\ge 0},\qquad 
t\sim\mathcal U_T:=\mathsf{Uniform}(0,T)\;(T<\infty),$$

with the *improper‑uniform limit* recovered by letting $T\to\infty$ in statements 
that remain finite. Conditional on $t$, let the *phase* be
$$\theta:\Omega\to(-\pi,\pi],\qquad
\theta\mid t\sim\Phi_{(\mu(t),\beta(t))},$$
where $\Phi$ is the *wrapped Laplace* (or von Mises) family with density

$$f_{\Phi}(\theta\mid t)=
\frac{\beta(t)}{2\sinh\!\bigl(\beta(t)\bigr)}
\exp\!\bigl\{\beta(t)\cos\bigl(\theta-\mu(t)\bigr)\bigr\}.
\tag{1}$$

The *complex‑time (kime) coordinate* is the measurable map

$$\kappa:\Omega\to\mathbb C,\qquad 
\kappa(\omega)=t(\omega)\,e^{i\theta(\omega)}.$$

When obseving repeatedly a longitudinal process, let
$\bigl\{s_j(t):t\ge 0,\;j=1,\dots,N\bigr\}$
be i.i.d. real‑valued stochastic processes on $(\Omega,\mathcal F,\mathbb P)$ 
with continuous sample paths and second moments.  Observations are made at design points

$$\{t_{j,k}\}_{k=1}^{M_j}\subset(0,T]
\quad\Longrightarrow\quad 
\mathcal D=\bigl\{(j,k):s_{j,k}=s_j(t_{j,k})\bigr\}.$$

Of course, additive noise models, e.g., $s$ plus measurement error, can also 
be incorporated, but are omitted here for clarity of presentation.

## Kime‑surface embedding

Let's define the *embedding map*

$$F:\;(0,T]\times(-\pi,\pi] \longrightarrow \mathbb R^3,\qquad
F(t,\theta)=
\bigl(t\cos\theta,\;t\sin\theta,\;Z(t,\theta)\bigr),
\tag{2}$$

where $Z(t,\theta)=\mathbb E\bigl[s_1(t)\bigr]$ is the *mean intensity*.
Assuming $Z\in C^1\bigl((0,T]\times(-\pi,\pi]\bigr)$, $F$ is a $C^1$ immersion 
and its image
$\mathcal M:=F\!\bigl((0,T]\times(-\pi,\pi]\bigr)\subset\mathbb R^3$
is a *2‑dimensional manifold with boundary*, e.g., a punctured cone when $Z$ is radial.

In the kime-coordinates, in coordinates $(t,\theta)$, the induced *metric* $g$ is

$$g=\begin{pmatrix}
1+Z_t^2 & Z_t Z_\theta\\[4pt]
Z_t Z_\theta & t^2+Z_\theta^2
\end{pmatrix},
\qquad
Z_t=\partial_t Z,\;Z_\theta=\partial_\theta Z.$$

And the *curvatures* follow from standard metric curvature formulas; see [Appendix A](TCIU_SK_Appendix03_KPT.html#6_Appendix_A:_Geometry_of_the_kime‑surface).



Consider the *forward measurement model*. For each datum $(j,k)$ draw an 
*independent phase sample*
$\vartheta_{j,k}\mid t_{j,k}\;\sim\;\Phi_{(\mu(t_{j,k}),\beta(t_{j,k}))},$
and form the Euclidean coordinate
$$\mathbf p_{j,k}=F\!\bigl(t_{j,k},\vartheta_{j,k}\bigr)
           =\bigl(x_{j,k},y_{j,k},s_{j,k}\bigr).
\tag{3}$$

Equation (3) realizes a *random point cloud* sampling of the kime‑surface.
When $M:=\sum_jM_j\to\infty$, the cloud becomes dense under mild conditions 
(law of large numbers on manifolds).

The corresponding *inverse problem* (*Kime‑phase tomography*) aims to
recover the unknown *conditional phase law* $\Phi_{(\mu(t),\beta(t))}$ from $\mathcal D$
and hence reconstruct the embedding, eq. (2).

The *harmonic expansion* assumption requires that for each fixed $t$ the process $s_j(t)$
admits a circular harmonic expansion

$$\mathbb E\bigl[s_j(t)\,|\,\theta\bigr]
      =\sum_{n\in\mathbb Z}f_n(t)\,e^{in\theta},
\qquad f_{-n}=\overline{f_n}.
\tag{4}$$

To state the *“Kime–Pauli” identifiability theorem*, let's define the 
*empirical moments*
$$\widehat m_n(t_k)=\frac1N\sum_{j=1}^{N}s_{j,k}\,e^{-in\vartheta_{j,k}}.
\tag{5}$$

Using eq. (4), $\mathbb E[s_j(t)e^{-in\theta}]=f_n(t)\,\varphi_n(t)$ 
where $\varphi_n(t)$ is the $n$-th circular Fourier coefficient of $f_\Phi(\theta\mid t)$.
We'll prove the following *Information Completeness Theorem* using the fact that
the moment sequence uniquely identifies $\{\varphi_n(t)\}$, $f_n(t)\neq 0$,
and Fourier inversion recovers $f_\Phi$.


*Theorem 1: (Information Completeness)*: Let's fix $t\in(0,T]$ and let 
$m_{n}(t):=\mathbb E\!\bigl[s_1(t)\,e^{-in\theta}\bigr], \qquad n\in\mathbb Z,$
where $s_1(t)$ admits the harmonic expansion
$\mathbb E\!\bigl[s_1(t)\mid \theta\bigr]=\sum_{n\in\mathbb Z}f_n(t)\,e^{in\theta}, \quad f_{-n}=\overline{f_n}.$
Assume that there exists at least one $n_\star\neq0$ with 
$f_{n_\star}(t)\neq0$ and the conditional phase density 
$f_\Phi(\,\cdot\mid t)\in L^{2}(S^{1})$.
Then the map $f_\Phi(\,\cdot\mid t)\;\longmapsto\; \bigl\{m_{n}(t)\bigr\}_{n\in\mathbb Z}$
is *injective*; i.e. the entire moment sequence determines $f_\Phi$ uniquely.

*Proof*: Let's consider the moment factorization using the law of total expectation
and eq. (4)
$$m_{n}(t)=\mathbb E_\theta\!\bigl[\,\mathbb E[s_1(t)\mid\theta]\,e^{-in\theta}\bigr]
           \;=\;\sum_{k\in\mathbb Z} f_k(t)
                 \underbrace{\mathbb E_\theta[e^{i(k-n)\theta}]}_{=\varphi_{n-k}(t)},$$
where $\varphi_{m}(t)=\int_{-\pi}^{\pi}e^{im\theta}\,f_\Phi(\theta\mid t)\,d\theta$ 
is the $m$-th Fourier coefficient of $f_\Phi$.

Setting $n=n_\star$ and using $f_{n_\star}(t)\neq0$ gives
$$\varphi_{0}(t)=\frac{m_{n_\star}(t)}{f_{n_\star}(t)}.
   \tag{1.1}$$

To recover of all $\varphi_{m}(t)$, fix any $m\neq0$ and choose $n=n_\star+m$.
Then
$$m_{n_\star+m}(t)
   =f_{n_\star}(t)\varphi_{m}(t)+\sum_{k\neq n_\star}f_k(t)\varphi_{n_\star+m-k}(t).$$

The term $\varphi_{0}(t)$ is shown in eq. (1.1) and we can proceed inductively on
$|m|$ starting with $|m|=1$. The linear system is upper‑triangular in $\varphi_{m}$
and hence solvable because $f_{n_\star}(t)\neq0$.  Thus, *every* Fourier coefficient
$\{\varphi_m(t)\}_{m\in\mathbb Z}$ is uniquely obtained from the moments $\{m_n(t)\}$.

The *uniqueness of the density* follows since $f_\Phi(\,\cdot\mid t)\in L^{2}(S^{1})$
and its Fourier series converges to the function in $L^{2}$‑norm.  Two $L^{2}$ functions
with identical Fourier coefficients coincide *almost everywhere.*  Therefore,
no two different circular densities can produce the same moment sequence, establishing injectivity.
$\square$

The following theorem guarantees the consistency of the estimator. It relies on
a plug‑in inversion $\widehat \varphi_n(t_k)=\widehat m_n(t_k)/f_n(t_k)$
followed by truncated Fourier synthesis, to shows the estimator consistency, 
$\widehat \Phi(t_k)$, and a kernel smoothing in $t$ to recover $\widehat\Phi(t)$ on $(0,T]$.

*Theorem 2 (LLN‑based consistency)*. For a fixed radial location $t_k$ and integer $n$, 
define
$$\widehat m_{n}(t_k)=\frac{1}{N}\sum_{j=1}^{N} X_{j}^{(n,k)}, \qquad  
X_{j}^{(n,k)} := s_{j,k}\,e^{-in\vartheta_{j,k}},$$
where $\vartheta_{j,k}\stackrel{\text{i.i.d.}}{\sim}\Phi(\,\cdot\mid t_k)$ and
$\{s_{j,k}\}_{j=1}^{N}$ are independent copies of $s_1(t_k)$ satisfying
$\sup_{j,k}\mathbb E|s_{j,k}|^{2}<\infty$. Then,
$$\widehat m_{n}(t_k)\xrightarrow[N\to\infty]{\text{a.s.}} 
      m_{n}(t_k)=\mathbb E[X_{1}^{(n,k)}].$$

*Proof*: For fixed $t_k$ the pairs $\{(s_{j,k},\vartheta_{j,k})\}_{j=1}^{N}$ are 
*independent and identically distributed* by hypothesis. Hence, so are the 
random variables $X_{j}^{(n,k)}$.

The finite variance follows by the Cauchy–Schwarz enequality,
$$\mathbb E\left |X_{j}^{(n,k)}\right |^{2}=\mathbb E|s_{j,k}|^{2}\le \sup_{j,k}\mathbb E|s_{j,k}|^{2}<\infty.$$

The *Kolmogorov Strong Law of Large Numbers* (sLLN) indicates that if 
$X_1,X_2,\dots$ are i.i.d. with $\mathbb E|X_1|<\infty$, then 
$\frac1N\sum_{j=1}^{N}X_j\to\mathbb E[X_1]$ almost surely.
In our case, $\mathbb E|X_{1}^{(n,k)}| \le (\mathbb E|s_{1,k}|^{2})^{1/2}<\infty$, 
so the sLLN applies and yields
$\widehat m_{n}(t_k)\;\xrightarrow{\text{a.s.}}\;\mathbb E[X_{1}^{(n,k)}]=m_{n}(t_k).$

Because the argument is carried out *separately for each fixed* $t_k$, we do not
require joint convergence along a growing grid of $k$.  If a finite set of $\{t_k\}$
is considered, take the intersection of the corresponding probability‑one events.  
When $k$ also grows, we can apply a uniform sLLN (Kolmogorov–Chentsov or Vapnik–Červonenkis)
once measurability and envelope conditions for $\{X_{j}^{(n,k)}\}$ are verified.
Hence, the estimator $\widehat m_{n}(t_k)$ is strongly consistent. $\square$

Because $\widehat f_{n}$ is obtained from a consistent regression or known a priori
and each empirical moment is strongly consistent, the plug‑in estimator
$$\widehat\varphi_{n}(t_k)=\widehat m_{n}(t_k)\bigl/\widehat f_{n}(t_k),$$
inherits strong consistency whenever $\widehat f_{n}(t_k)$ converges in probability 
to a non‑zero limit.
Therefore, a truncated Fourier inversion of the consistent sequence 
$\{\widehat\varphi_n(t_k)\}_{|n|\le n_{\max}}$ yields a *consistent reconstruction*
$\widehat\Phi(t_k)$ of the circular kime-phase density.


## Practical KPT Algorithm

| Step                          | Action                                                                                                                 |   |                           |
| ----------------------------- | ---------------------------------------------------------------------------------------------------------------------- | - | ------------------------- |
| *1. Phase sampling*         | For each $(j,k)$ simulate $\vartheta_{j,k}$ by inverse‑CDF or accept–reject on eq. (1)                                    |   |                           |
| *2. Moment computation*     | Compute $\widehat m_n(t_k)$ for $| n | \le n\_{\max}$ using eq. (5) |
| *3. Coefficient estimation* | If $f_n(t)$ is unknown, fit $f_n(\cdot)$ by non‑parametric regression on $(t_k,\widehat m_n(t_k))$                    |   |                           |
| *4. Inversion*              | Obtain $\widehat\Phi(t_k)$ by Fourier inversion of $\{\widehat\varphi_n(t_k)\}$                                       |   |                           |
| *5. Surface rendering*      | Build $\widehat Z(t,\theta)=\sum_n \widehat f_n(t)e^{in\theta}$ and embed via eq. (2) to visualize $\widehat{\mathcal M}$ |   |                           |

In practice, the bandwidths and truncation orders $(h,n_{\max})$ can be selected by 
*leave‑one‑trajectory‑out* cross‑validation.

To obtain *higher‑order manifolds* requires extending $t\in\mathbb R^d$ with $d>1$
to a $d+1$‑dimensional kime‑surface in $\mathbb R^{d+2}$. 
Additive observation noise $s_{j,k}=S_{j,k}+\varepsilon_{j,k}$ may affect
the moment variance but not identifiability under zero‑mean assumption.
In eq. (1), we can also use *alternative phase laws* such as wrapped Cauchy or non‑parametric 
circular densities. Theorems 1 and 2 hold with $f_\Phi\in L^2(S^1)$.
Finally, a *geometric inference* based on the curvature of $\mathcal M$ 
encodes dynamical features of the underlying process
and bootstrap on $\widehat g$ provides quantification of uncertainty.

# Generalized KPT

Next we will expand the details in previous section starting with 
fully explicating the construction of the probability space,
$(\Omega,\mathcal F,\mathbb P)$, that underlies *kime‑phase tomography* (KPT).
Then will well generalize the phase law beyond the wrapped‑Laplace or vonMises
distributions.

The probability space $(\Omega,\mathcal F,\mathbb P)$ involves the following.

| Component                     | Symbol                                  | Measurable structure                                                                                | Description                                                                          |
| ----------------------------- | --------------------------------------- | --------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------ |
| *Radial time*               | $\Omega_t=(0,\infty)$                   | Borel $\sigma$‑algebra $\mathcal B((0,\infty))$                                                            | One outcome $t$ is drawn for each realization                                       |
| *Phase*                     | $\Omega_\theta=S^{1}=(-\pi,\pi]$        | Borel $\sigma$‑algebra $\mathcal B(S^{1})$                                                                 | Conditional on $t$, an angular outcome $\theta$ is drawn from some circular density |
| *Longitudinal paths*        | $\Omega_s=C([0,T];\mathbb R)^{N}$       | Cylindrical $\sigma$‑algebra generated by point evaluations $ \pi_{j,\tau}\bigl(\mathbf s\bigr)=s_j(\tau)$ | A vector of $N$ continuous‑time sample paths $ \mathbf s=(s_1,\dots,s_N)$           |
| *Indexing noise (optional)* | $\Omega_\varepsilon=\mathbb R^{\infty}$ | Product Borel $\sigma$‑algebra                                                                             | Measurement errors $\{\varepsilon_{j,k}\}$ if needed                                |

The *master sample space* is the Cartesian product

$$\boxed{\;\Omega=\Omega_t\times\Omega_\theta\times\Omega_s\times\Omega_\varepsilon\;}$$

with $\Omega_\varepsilon$ omitted if no measurement noise is modeled.

The $\sigma$‑*algebra* is

$$\boxed{\;
  \mathcal F=\mathcal B((0,\infty))\otimes
             \mathcal B(S^{1})\otimes
             \sigma\!\bigl\{\pi_{j,\tau}\bigr\}_{1\le j\le N,\;0\le\tau\le T}
             \otimes
             \mathcal B(\mathbb R^{\infty})
  \;}$$

and the corresponding *probability measure* is

$$
\boxed{\;
  \mathbb P=\mathbb P_t\otimes\mathbb P_{\theta\mid t}\otimes
            \bigl(\mathbb P_{s}\bigr)^{\otimes N}\otimes\mathbb P_\varepsilon
  \;}
$$
where *$\mathbb P_t$* is a (truncated) uniform or any proper law on $(0,\infty)$, 
*$\mathbb P_{\theta\mid t}$* is a *family* of circular distributions, 
$\{\Phi_t\}_{t>0}$, *$\mathbb P_s$* is the law of one real‑valued, second‑moment
continuous process $s(t)$; i.i.d. copies indexed by $j$, and 
*$\mathbb P_\varepsilon$* is a product measure for i.i.d. zero‑mean noise 
$\varepsilon_{j,k}$, which is set to a point mass at $0$ if the noise is ignored.

Each coordinate map, e.g. $t(\omega)$, $\theta(\omega)$,
$s_j(\cdot)(\omega)$, is measurable by construction, so all derived
maps—such as the complex‑time coordinate
$\kappa(\omega)=t(\omega)\,e^{i\theta(\omega)}$ are measurable as well.

We will explore the minimal assumptions to ensure the kime-phase law in full generalizable.
For every $t>0$, let $\Phi_t$ be a probability measure on $S^{1}$, subject
to the following 3 assumptions:

 - *(A1) Symmetry*: $\Phi_t(A)=\Phi_t(-A)$ for every Borel set $A\subseteq S^{1}$, which can be dropped this if we permit asymmetric phases,
 - *(A2) Absolute continuity*: $\Phi_t$ admits a density $\varphi_t$ w.r.t. the uniform measure $d\theta/2\pi$, and 
 - *(A3) Square‑integrability*: $\varphi_t\in L^{2}(S^{1})$, which is necessary for the Fourier arguments in *Theorem 1*.

*Examples* of distributions that satisfy these assumptions include *Wrapped Laplace*, 
*von Mises*, *wrapped Cauchy*, *wrapped Gaussian*, *mixtures*, and non‑parametric
kernel densities all satisfy $\{(A1), (A2), (A3)\}$.

In the general case, a more unified notation can be used
$$\theta\mid t\;\sim\;\Phi_t,\qquad
d\Phi_t(\theta)=\varphi_t(\theta)\,\frac{d\theta}{2\pi},\qquad
 \varphi_t\in L^{2}(S^{1})\; \forall t.$$

In practice, a parametric family may support reduction of complexity or dimensionality,
in which case, the paretric distribution may explicate its parameters
$\Phi_{(\eta(t))}$ with parameter curve $\eta(t)$ taking values in
some finite‑dimensional manifold. For instance, the wrapped‑Laplace and von Mises
cases above correspond to the special choice
$\eta(t)=(\mu(t),\beta(t))$ and density shown in eq. (1).

Because $\varphi_t\in L^{2}(S^{1})$, it has Fourier coefficients
$$\varphi_{t,n}:=\int_{-\pi}^{\pi}e^{-in\theta}\,\varphi_t(\theta)\,\frac{d\theta}{2\pi},
\qquad n\in\mathbb Z,$$
and the series $\theta\mapsto\sum_{n}\varphi_{t,n}e^{in\theta}$ converges to
$\varphi_t$ in $L^{2}$.  This is exactly the requirement in Theorem 1.
Hence, the identifiability proof above is unchanged.

The table below illustrated that the earlier proofs remain valid as the corresponding
argument directly generalize.

| Section                                     | Dependence on a specific $\Phi$                                                 | Need for Revision | How to phrase generally                                                            |
| ------------------------------------------- | ------------------------------------------------------------------------------- | ------------------ | ---------------------------------------------------------------------------------- |
| *Embedding* $F(t,\theta)$                 | None (only needs angles)                                                        | *No*             | Keep $F(t,\theta)=(t\cos\theta,\,t\sin\theta,\,Z(t,\theta))$                      |
| *Curvatures* (see [Appendix A](TCIU_SK_Appendix03_KPT.html#6_Appendix_A:_Geometry_of_the_kime‑surface))                 | Only uses partials of $Z$                                                       | *No*             | Unchanged                                                                        |
| *Theorem 1 (information completeness)*    | Uses *Fourier* of phase density; needs (A3)                                     | *No*             | Replace “wrapped Laplace density” by “any $L^{2}$ circular density $\varphi_t$”   |
| *Theorem 2 (LLN consistency)*             | No assumption on $\Phi_t$ beyond finite variance of $e^{-in\theta}$ (automatic) | *No*             | Proof is unchanged                                                                   |
| *Algorithm step 1 (moment calculation)*   | Works for *any* phase law because only $\theta$ appears inside $e^{-in\theta}$  | *No*            |                                                                                    |
| *Algorithm step 4 (parametric inversion)* | A model (e.g. von Mises) is optional                                    | Optional           | Offer both parametric (fewer moments) and non‑parametric (Fourier series) options |

Hence only the *model statement* in the set‑up section may need revisions.

*Phase law (general form)*: Conditional on $t>0$, draw a phase
$$
 \boxed{\;
   \theta\mid t\;\sim\;\Phi_t,\qquad
   \Phi_t\hbox{ supported on }S^{1},\;
   d\Phi_t(\theta)=\varphi_t(\theta)\,\frac{d\theta}{2\pi},\;
   \varphi_t\in L^{2}(S^{1}),
 \;}$$
where $\varphi_t$ is *any symmetric (or asymmetric, if desired) circular density* 
satisfying $\int_{-\pi}^{\pi}\varphi_t(\theta)\tfrac{d\theta}{2\pi}=1$.
The wrapped‑Laplace and von Mises distributions represent examples,
but these cases are *not limiting* the KPT theory.

When implementing hte most general KPT algorithm, we need to consider

 1. *Simulation / sampling*: We can plug in *any* circular RNG (e.g. `rwrappedcauchy`, `rvonmises`, or a kernel density sampler) for $\theta_{j,k}$.
 2. *Moment estimates*: The empirical moments $\widehat m_n(t_k)=\frac1N\sum_j s_{j,k}e^{-in\vartheta_{j,k}}$ remain the sufficient statistics for KPT, regardless of the form of $\Phi_t$.
 3. *Reconstruction*: For *non‑parametric* distribution: truncate the Fourier series of $\varphi_t$ at $|n|\le n_{\max}$ and estimate those coefficients directly without any distributional assumptions. While for *parametric* distributions, if any domain-specific prior knowledge suggests that $\Phi_t$ is, say, von Mises, we can plug the empirical moments into the closed‑form relationships $ \varphi_{t,1}=I_1(\kappa_t)/I_0(\kappa_t)\,e^{i\mu_t}$, etc.

This suggests that *KPT is agnostic* to the specific circular distribution chosen for the phase.
All theoretical guarantees require only that $\varphi_t\in L^{2}(S^{1})$, so its Fourier
series exists.

# Applications

The [TCIU Appendix](https://tciu.predictive.space/) includes implementation of 
a simulated fMRI time-series and the corresponding the 
[Kime-Phase Tomography (KPT) fMRI Time-Series Simulation](https://www.socr.umich.edu/TCIU/HTMLs/TCIU_SK_Appendix03_KPT_sim.html) 
showing the mapping of repeated fMRI time-series measurements to 2D kime-surfaces. 
This simulation specifically tests the KPT algorithm on synthetic fMRI data with 
ON (*stimulus*) and OFF (*rest*) conditions.


# Appendix A: Geometry of the kime‑surface

To study the geometry of the kime‑surface, let's start the smooth kimesurface embedding
$$F:(0,T]\times(-\pi,\pi]\;\longrightarrow\;\mathbb R^{3},
\qquad F(t,\theta)=\bigl(x(t,\theta),y(t,\theta),Z(t,\theta)\bigr)
           =(t\cos\theta,\;t\sin\theta,\;Z(t,\theta)),$$
with $Z\in C^{2}((0,T]\times(-\pi,\pi])$ and 
express the partial derivatives as $Z_t=\partial_tZ,\;Z_\theta=\partial_\theta Z,$ etc.

The *first fundamental form* is

$$\begin{aligned}
E&=\langle X_t,X_t\rangle =1+Z_t^{2},\\[2pt]
F&=\langle X_t,X_\theta\rangle =Z_tZ_\theta,\\[2pt]
G&=\langle X_\theta,X_\theta\rangle =t^{2}+Z_\theta^{2}.
\end{aligned}$$
and the *Normal vector* is
$$W=X_t\times X_\theta
=(\sin\theta\,Z_\theta-t\cos\theta\,Z_t,\;
  -\cos\theta\,Z_\theta-t\sin\theta\,Z_t,\;
  t),\qquad
\|W\|^{2}=t^{2}\bigl(1+Z_t^{2}\bigr)+Z_\theta^{2}.$$

Let's define the squared normal magnitude, 

$$\Delta :=\boxed{\;\|W\|^{2}=EG-F^{2} \;} \tag{A.1}$$

Then the *second fundamental form* is

$$\begin{aligned}
e_{\text{num}} &= t\,Z_{tt},\\
f_{\text{num}} &= t\,Z_{t\theta}-Z_\theta,\\
g_{\text{num}} &= t\bigl(tZ_t+Z_{\theta\theta}\bigr).
\end{aligned}
\qquad
e=\frac{e_{\text{num}}}{\|W\|},\;
f=\frac{f_{\text{num}}}{\|W\|},\;
g=\frac{g_{\text{num}}}{\|W\|}.$$

Using $K=(eg-f^{2})/(EG-F^{2})$ and equation (A.1), the *Gaussian curvature* is

$$\boxed{\;
\underbrace{K}_{Gaussian}=\frac{e_{\text{num}}g_{\text{num}}-f_{\text{num}}^{2}}
          {\Delta^{2}}
  \;=\;
  \frac{\displaystyle
        t^{2}Z_{tt}\bigl(tZ_t+Z_{\theta\theta}\bigr)\;-\;
        \bigl(tZ_{t\theta}-Z_\theta\bigr)^{2}}
       {\bigl[t^{2}\bigl(1+Z_t^{2}\bigr)+Z_\theta^{2}\bigr]^{2}}
} \tag{A.2}$$

Note that the overall sign of the numerator is *positive*, not negative. 

Also, the *mean curvature*

$$\boxed{\;
\underbrace{H}_{mean}=\frac{1}{2}\frac{Eg+Ge-2Ff}{EG-F^{2}}
  =\frac{(1+Z_t^{2})\,g_{\text{num}}+(t^{2}+Z_\theta^{2})\,e_{\text{num}}
          -2Z_tZ_\theta\,f_{\text{num}}}
         {2\,\Delta^{3/2}}
} \tag{A.3}$$

may be expanded as

$$H=\frac{(1+Z_t^{2})\,t\bigl(tZ_t+Z_{\theta\theta}\bigr)
        +(t^{2}+Z_\theta^{2})\,tZ_{tt}
        -2Z_tZ_\theta\bigl(tZ_{t\theta}-Z_\theta\bigr)}
       {2\,\bigl[t^{2}(1+Z_t^{2})+Z_\theta^{2}\bigr]^{3/2}}.$$

Finally, the *principal curvatures* are

$$\underbrace{k_{1,2}}_{principal}=H\;\pm\;\sqrt{H^{2}-K}.$$

## Shape operator

With respect to the *kime coordinate* basis $(X_t,X_\theta)$, the *shape operator* is

$$\boxed{\;
S=I^{-1}II
   =\frac{1}{\Delta}
     \begin{pmatrix}
       G & -F\\ -F & E
     \end{pmatrix}
     \begin{pmatrix}
       e_{\text{num}} & f_{\text{num}}\\
       f_{\text{num}} & g_{\text{num}}
     \end{pmatrix}
     \frac{1}{\|W\|}
}\tag{A.4} $$

Jointly, equations (A.2) - (A.4) give a complete and full account
of the *intrinsic* and *extrinsic geometry* of kime‑surfaces.
All classical surface *invariants* (*Christoffel symbols*, *geodesics*,
*Laplace–Beltrami operator*, etc.) can now be written explicitly from 
the scalar *coefficients* $E,F,G$ and their $t,\theta$ derivatives.


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