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\begin{document}

\title{MD.TopoDevPOC: Topologically Unique Multi-Directional Developing Point-of-Control Patterns in Pre-Market K-Lines}

\author{\IEEEauthorblockN{ConQ Research Team}\\
\IEEEauthorblockA{\textit{Continual Quasars}\\
\today}
}

\maketitle

\begin{abstract}
This paper characterizes the complete space of developing Point of Control (POC) line patterns on pre-market candlestick sequences when no directional monotonicity is imposed. Allowing each inter-candle relation to be strict-up ($<$), strict-down ($>$), or equality ($=$), we prove that for $n$ candles there are exactly $3^{\,n-1}$ distinct patterns. The derivation uses elementary combinatorics and is validated by a state-transition matrix on three states. A ternary vector encoding with values $+1$, $0$, $-1$ is introduced, together with bidirectional conversion formulas to the symbolic sequence. Two natural aggregated metrics are defined: the migration count $M$ (total number of strict moves, the $L_1$ norm of the vector) and the end-to-end (E2E) net balance $S$ (the signed sum of the vector). Exact combinatorial counts for given $M$ are provided, and the joint distribution of $M$ and $S$ is outlined. Ten illustrative random multi-directional patterns are displayed in both symbolic and vector formats, with a right-to-left temporal chart. All results are purely combinatorial and require no simulation.
\end{abstract}

\begin{IEEEkeywords}
developing POC, multi-directional patterns, combinatorial enumeration, ternary encoding, end-to-end balance, pre-market analysis
\end{IEEEkeywords}

\section{Introduction}

The developing Point of Control (POC) during the pre-market session can exhibit rich behavior, moving up, down, or remaining flat between consecutive three-minute candles. While constrained (monotonic) patterns have been studied in a companion work, the unrestricted case where the POC may reverse direction multiple times within a single session captures the full complexity of pre-market price discovery. The central question is: Given $n$ consecutive pre-market candles, each associated with a developing POC value, how many distinct sequences of direction relations (strict-up, strict-down, flat) can be formed? Furthermore, what compact metrics describe the overall movement?

This paper provides a comprehensive combinatorial treatment of multi-directional developing POC patterns. The main contributions are:
\begin{enumerate}
    \item A closed-form enumeration: for $n$ candles, exactly $3^{\,n-1}$ distinct patterns exist.
    \item A ternary vector representation $(+1,0,-1)$ that encodes each transition, with explicit bidirectional conversion to the symbolic form.
    \item Definition and analysis of two end-to-end (E2E) metrics: the migration count $M$ (total strict moves) and the net balance $S$ (signed sum of moves). Exact formulas for the number of patterns with a given $M$ are derived.
    \item A state-transition matrix formulation on three states that independently confirms the total count.
    \item Ten illustrative patterns with a visual chart following the correct right-to-left temporal convention.
\end{enumerate}

No empirical data or simulations are used; the development is entirely theoretical.

\section{Problem Definition and Notation}

Let $n \in \mathbb{N}$ be the number of candles, with $n=39$ for a standard US pre-market session. Index candles from the most recent $C_0$ back to $C_{-(n-1)}$. The corresponding developing POC values form a real sequence
\begin{equation}
    \mathbf{p} = (p_0, p_1, \dots, p_{n-1}) \in \mathbb{R}^n,
\end{equation}
where $p_k$ is the POC of candle $C_{-k}$. For each adjacent pair $(p_k, p_{k+1})$ we observe one of three possible relations:
\begin{definition}[Multi-Directional Relation]
The relation $R_k$ between $p_k$ and $p_{k+1}$ belongs to the set $\{\, <,\; >,\; = \,\}$,
where $<$ means $p_k < p_{k+1}$ (up move), $>$ means $p_k > p_{k+1}$ (down move), and $=$ means $p_k = p_{k+1}$ (flat).
\end{definition}
A pattern is uniquely identified by the $(n-1)$-tuple of relations
\begin{equation}
    \mathbf{R} = (R_0, R_1, \dots, R_{n-2}), \quad R_k \in \{<, >, =\}.
\end{equation}
Two price sequences that yield the same $\mathbf{R}$ are considered equivalent. No further identification (e.g., by scaling or translation) is performed.

\section{Combinatorial Enumeration}

\begin{theorem}[Total Pattern Count]
For $n \ge 1$ candles, the number of distinct multi-directional developing POC patterns is
\begin{equation}
    N_n = 3^{\,n-1}.
\end{equation}
\end{theorem}

\begin{proof}
For each of the $n-1$ adjacent pairs, there are three independent choices ($<$, $>$, $=$). Hence $N_n = 3^{\,n-1}$.
\end{proof}

For $n=39$, this evaluates to $3^{38} \approx 4.36 \times 10^{18}$. The space is vast, accommodating all possible sequences of directional moves and flats.

\subsection{State-Transition Matrix Confirmation}

The same result follows from a 3-state Markovian view. Define states corresponding to the last observed relation: $U$ (strict-up, $<$), $D$ (strict-down, $>$), and $F$ (flat, $=$). There are no restrictions on transitions, so any state can follow any state. The transition (adjacency) matrix is the all-ones matrix
\begin{equation}
    \mathbf{A} = \begin{pmatrix}
        1 & 1 & 1 \\
        1 & 1 & 1 \\
        1 & 1 & 1
    \end{pmatrix} = \mathbf{1}\mathbf{1}^\top,
\end{equation}
where $\mathbf{1} = (1,1,1)^\top$. With $n$ candles we have $n-1$ transitions. Let the initial state vector $\mathbf{v}_0 = (1,1,1)^\top$ (all states permissible before the first comparison). The total number of patterns is
\begin{equation}
    N_n = \mathbf{1}^\top \mathbf{A}^{n-2} \mathbf{v}_0 .
\end{equation}
Since $\mathbf{A}^k = 3^{k-1} \mathbf{A}$ for $k \ge 1$, we obtain
\begin{align}
    N_n &= \mathbf{1}^\top \left(3^{n-3} \mathbf{A}\right) \mathbf{v}_0 \nonumber \\
        &= 3^{n-3} \, \mathbf{1}^\top \mathbf{A} \mathbf{1} \nonumber \\
        &= 3^{n-3} \cdot 9 = 3^{\,n-1}.
\end{align}
This cross-check confirms the elementary counting argument.

\section{Ternary Encoding and Conversion}

A compact, signed representation of a pattern is given by a ternary vector of length $n-1$.

\begin{definition}[Ternary Transition Vector]
For a pattern with relations $\mathbf{R} = (R_0, \dots, R_{n-2})$, define the vector $\mathbf{m} = (m_0, \dots, m_{n-2})$ by
\begin{equation}
    m_k = \begin{cases}
        +1, & \text{if } R_k = \; < \;\;(\text{up}),\\
        -1, & \text{if } R_k = \; > \;\;(\text{down}),\\
        0,  & \text{if } R_k = \; = \;\;(\text{flat}).
    \end{cases}
\end{equation}
\end{definition}

Thus $\mathbf{m} \in \{-1,0,+1\}^{n-1}$. The mapping is a bijection between the $3^{\,n-1}$ symbolic sequences and the ternary vectors.

\subsection{Conversion Formulas}

Let $\Sigma$ be a string of length $n-1$ over the alphabet $\{<, >, =\}$, ordered from $\Sigma_0$ (transition between $C_0$ and $C_{-1}$) to $\Sigma_{n-2}$.

\textbf{Symbolic $\to$ Ternary:}
\begin{equation}
    m_j = f(\Sigma_j), \quad j=0,\dots,n-2,
\end{equation}
with $f(<)=+1$, $f(>)=-1$, $f(=)=0$.

\textbf{Ternary $\to$ Symbolic:}
\begin{equation}
    \Sigma_j = g(m_j), \quad g(+1)=\;<,\; g(-1)=\;>,\; g(0)=\;=.
\end{equation}

The vector $\mathbf{m}$ is a row vector of size $1 \times (n-1)$.

\section{End-to-End Metrics: Migration Count and Net Balance}

Two aggregate properties naturally emerge from the ternary vector.

\begin{definition}[Migration Count $M$]
The number of strict directional moves, i.e., the count of non-zero entries in $\mathbf{m}$, is
\begin{equation}
    M(\mathbf{m}) = \sum_{k=0}^{n-2} |m_k| = \|\mathbf{m}\|_1.
\end{equation}
\end{definition}

\begin{definition}[End-to-End (E2E) Net Balance $S$]
The signed sum of the moves, capturing the net directional bias from the most recent candle to the oldest, is
\begin{equation}
    S(\mathbf{m}) = \sum_{k=0}^{n-2} m_k .
\end{equation}
\end{definition}

$M$ measures the total activity (number of POC level changes), while $S$ indicates whether the cumulative movement was net up ($S>0$), net down ($S<0$), or perfectly cancelled ($S=0$). The two are related by $M \equiv |S| \pmod{2}$ when only strict moves are present.

\subsection{Exact Counts by Migration Count}

For a fixed $M = k$ ($0 \le k \le n-1$), the number of patterns with exactly $k$ strict moves is
\begin{equation}
    C_{n,k} = \binom{n-1}{k} \, 2^{\,k}.
\end{equation}
\textit{Derivation:} Choose $k$ positions out of $n-1$ to be strict; at each chosen position assign either $+1$ or $-1$ ($2$ choices). The remaining $n-1-k$ positions are $0$. Summing over $k$,
\[
\sum_{k=0}^{n-1} \binom{n-1}{k} 2^{\,k} = (1+2)^{\,n-1} = 3^{\,n-1},
\]
which recovers the total.

If each transition is chosen uniformly at random among the three symbols, then $M$ follows a binomial distribution: $M \sim \mathrm{Binomial}\bigl(n-1,\; \frac{2}{3}\bigr)$.

\subsection{Joint Distribution of $M$ and $S$}

The number of patterns with $M = k$ and a given net balance $S = s$ can be expressed as
\[
N_{k,s} = \binom{n-1}{k} \cdot \binom{k}{\frac{k+s}{2}},
\]
provided $s \equiv k \pmod{2}$ and $|s| \le k$; otherwise $N_{k,s}=0$. Summation of $N_{k,s}$ over $s$ returns $\binom{n-1}{k}2^{k}$.

\section{Illustrative Examples}

Table~\ref{tab:examples} shows ten randomly generated multi-directional patterns for $n=6$ candles (five transitions). Each row gives a unique ID, the symbolic sequence, the $1 \times 5$ ternary vector $\mathbf{m}$, migration count $M$, and net balance $S$.

\begin{table}[h]
\centering
\caption{Ten Random Multi-Directional Developing POC Patterns ($n=6$)}
\label{tab:examples}
\begin{tabular}{c c c c c}
\toprule
\textbf{ID} & \textbf{Symbolic Sequence} & \textbf{Ternary Vector $\mathbf{m}$} & \textbf{$M$} & \textbf{$S$} \\
\midrule
1 & $> \; < \; = \; > \; >$ & $(-1,\, +1,\, 0,\, -1,\, -1)$ & 4 & -2 \\
2 & $< \; = \; < \; > \; =$ & $(+1,\, 0,\, +1,\, -1,\, 0)$ & 3 & +1 \\
3 & $= \; > \; < \; = \; <$ & $(0,\, -1,\, +1,\, 0,\, +1)$ & 3 & +1 \\
4 & $< \; < \; < \; = \; >$ & $(+1,\, +1,\, +1,\, 0,\, -1)$ & 4 & +2 \\
5 & $> \; = \; > \; < \; <$ & $(-1,\, 0,\, -1,\, +1,\, +1)$ & 4 & 0 \\
6 & $= \; = \; < \; > \; =$ & $(0,\, 0,\, +1,\, -1,\, 0)$ & 2 & 0 \\
7 & $< \; > \; = \; > \; <$ & $(+1,\, -1,\, 0,\, -1,\, +1)$ & 4 & 0 \\
8 & $> \; < \; > \; = \; =$ & $(-1,\, +1,\, -1,\, 0,\, 0)$ & 3 & -1 \\
9 & $= \; > \; = \; < \; >$ & $(0,\, -1,\, 0,\, +1,\, -1)$ & 3 & -1 \\
10& $< \; = \; = \; > \; <$ & $(+1,\, 0,\, 0,\, -1,\, +1)$ & 3 & +1 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Visual Chart}

Figure~\ref{fig:multipattern} illustrates pattern ID~7 ($< > = > <$) with the conventional right-to-left temporal orientation. The POC values are set arbitrarily to reflect the signed moves: starting at $y=3$, then $+1\,(4)$, $-1\,(3)$, $0\,(3)$, $-1\,(2)$, $+1\,(3)$. The annotations between points display the relation symbols. The migration count $M=4$ and net balance $S=0$.

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.6]
    \draw[->] (0,0) -- (6,0) node[right] {$t$ (candle index)};
    \draw[->] (0,0) -- (0,5.5) node[above] {POC};
    \foreach \x in {0,1,2,3,4,5} {
        \pgfmathtruncatemacro{\idx}{-\x}
        \draw (5-\x,0.1) -- (5-\x,-0.1) node[below] {\small $C_{\idx}$};
    }
    % Pattern ID 7: < > = > <   (m = +1, -1, 0, -1, +1)
    % Points: (5,3), (4,4), (3,3), (2,3), (1,2), (0,3)
    \draw[thick, blue] (5,3) -- (4,4) -- (3,3) -- (2,3) -- (1,2) -- (0,3);
    \filldraw (5,3) circle (2pt) (4,4) circle (2pt) (3,3) circle (2pt) (2,3) circle (2pt) (1,2) circle (2pt) (0,3) circle (2pt);
    \node[anchor=south] at (4.5,3.5) {$<$};
    \node[anchor=south] at (3.5,3.5) {$>$};
    \node[anchor=west]  at (2.5,3)   {$=$};
    \node[anchor=south] at (1.5,2.5) {$>$};
    \node[anchor=south] at (0.5,2.5) {$<$};
\end{tikzpicture}
\caption{Visualization of multi-directional pattern ID~7: $< > = > <$ ($M=4$, $S=0$). Right-to-left orientation places current candle $C_0$ on the right.}
\label{fig:multipattern}
\end{figure}

\section{Conclusion}

We have fully enumerated the set of all multi-directional developing POC line patterns on a pre-market session of $n$ candles. The total number is $3^{\,n-1}$, arising from the independence of the three possible relations ($<$, $>$, $=$) at each of the $n-1$ transitions. A ternary encoding with $\{+1,0,-1\}$ provides a compact signed representation. The migration count $M$ and end-to-end net balance $S$ offer interpretable summaries of pattern activity and directional bias. Exact combinatorial formulas give the number of patterns possessing any prescribed $M$, and jointly with $S$. The state-transition matrix formulation independently verifies the count. These results form a theoretical foundation for pattern classification, statistical testing, and machine-readable feature construction in pre-market analysis.

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\bibitem{marketprofile}
J.~F.~Dalton, \emph{Mind Over Markets: Power Trading with Market Generated Information}. Marketplace Books, 2013.
\bibitem{combinatorics}
R.~P.~Stanley, \emph{Enumerative Combinatorics}, vol.~1, 2nd~ed. Cambridge University Press, 2011.
\end{thebibliography}

\end{document}