! TERA Charakter Latex

\section{v3t3m9 State Logic: Identity, Entity, and Admissibility}

Within TERA, v3t3m9 can be formalized as a state logic in which identity provides reference, entity provides instantiation, dependency provides coupling, and gating decides admissible transition in space.

\subsection{Identity Core}

Let
\[
a \in \mathbb{R}^3
\]
be an anchor, and let an identity core be defined as
\[
\mathcal{I} = (k, \ell, a, \Xi),
\]
where
\begin{itemize}
    \item $k$ is a unique identity key,
    \item $\ell$ is a human-readable label,
    \item $a$ is the spatial anchor,
    \item $\Xi$ is the set of relatively stable invariants.
\end{itemize}

For any spatial position
\[
x \in \mathbb{R}^3,
\]
the anchored relative position is
\[
r_a(x) = x - a.
\]

Thus, identity does not merely name a state; it provides the referential origin relative to which that state becomes readable.

\subsection{State Entity}

A concrete TERA entity is defined as
\[
\mathcal{E} =
(\mathcal{I}, x, \tau, \omega, I, E, P, M, U, L, \mathcal{D}),
\]
where
\begin{itemize}
    \item $\mathcal{I}$ is the identity core,
    \item $x \in \mathbb{R}^3$ is spatial position,
    \item $\tau \in \mathbb{R}_{\ge 0}$ is local time,
    \item $\omega \in \mathbb{R}_{\ge 0}$ is temporal frequency / resonance,
    \item $I \ge 0$ is information density,
    \item $E \ge 0$ is energy density,
    \item $P \in \mathbb{R}^n$ is the perception state vector,
    \item $M \in \mathbb{R}^n$ is the model state vector,
    \item $U \ge 0$ is the surrounding condition field (Umraum),
    \item $L \ge 0$ is the recurrence field (Umlauf),
    \item $\mathcal{D}$ is the set of dependencies.
\end{itemize}

The v3t3m9 body vector of the entity is
\[
b(\mathcal{E}) = (x_1, x_2, x_3, \tau, \omega).
\]

\subsection{Drift and Drift Energy}

The local drift vector is defined as
\[
C = M - P.
\]

The corresponding drift energy is
\[
V = \|C\|^2.
\]

Thus, drift measures the model--perception mismatch, while drift energy measures its magnitude.

\subsection{Coupling Dependency}

A coupling dependency may be written as
\[
d_j = (t_j, w_j, c_j, \kappa_j, \rho_j, m_j),
\]
where
\begin{itemize}
    \item $t_j$ is the target identity,
    \item $w_j$ is the dependency weight,
    \item $c_j$ is coupling strength,
    \item $\kappa_j$ is contextual fit,
    \item $\rho_j$ is recurrence strength,
    \item $m_j \in \{\text{by\_value}, \text{by\_ref}\}$ is the pass mode.
\end{itemize}

Its local pressure contribution is
\[
p(d_j) = w_j c_j \kappa_j \rho_j \chi(m_j),
\]
where
\[
\chi(m_j) =
\begin{cases}
1.2, & \text{if } m_j = \text{by\_ref},\\
1.0, & \text{if } m_j = \text{by\_value}.
\end{cases}
\]

The total dependency load of the entity is
\[
P_{\mathrm{dep}} = \sum_{d_j \in \mathcal{D}} p(d_j).
\]

\subsection{Contextual Coherence}

The local contextual coherence is defined as
\[
\kappa_{\mathrm{ctx}} = U \cdot L.
\]

This expresses the combined effect of surrounding condition and recurrence.

\subsection{Coupling Potential}

The local TERA coupling potential is defined as
\[
K = I E \omega \kappa_{\mathrm{ctx}} - \lambda V - P_{\mathrm{dep}},
\]
where $\lambda > 0$ is a drift penalty parameter.

This means that admissibility increases with information, energy, temporal resonance, and contextual coherence, but decreases with drift and dependency load.

\subsection{Admissibility Density}

The local admissibility density is defined as
\[
\mu = \sigma\big(K U L - \eta V\big),
\]
where
\[
\sigma(z) = \frac{1}{1+e^{-z}}
\]
is the logistic normalization function, and $\eta > 0$ weights drift energy.

Thus:
\begin{itemize}
    \item high $\mu$ indicates local admissibility,
    \item low $\mu$ indicates local instability or non-admissibility.
\end{itemize}

\subsection{Gate Decision}

The gate decision is defined as
\[
u = G(\mu, K, V),
\]
with
\[
u \in \{\mathrm{ACCEPT}, \mathrm{WAIT}, \mathrm{DEFER}, \mathrm{REJECT}\}.
\]

A simple threshold form may be written as
\[
u =
\begin{cases}
\mathrm{ACCEPT}, & \mu \ge 0.85,\; K>0,\; V<0.25,\\[4pt]
\mathrm{WAIT}, & \mu \ge 0.55,\; K>0,\\[4pt]
\mathrm{DEFER}, & \mu \ge 0.35,\\[4pt]
\mathrm{REJECT}, & \text{otherwise.}
\end{cases}
\]

This means that the gate does not decide from information alone, but from admissibility under coupling, contextual coherence, recurrence, and drift.

\subsection{World as Addressable State Space}

Let the TERA world be defined as
\[
\mathcal{W} = \{\mathcal{E}_1, \mathcal{E}_2, \dots, \mathcal{E}_m\},
\]
an addressable set of entities indexed by their identity keys.

A state snapshot of an entity $\mathcal{E}$ can then be written as
\[
S(\mathcal{E}) =
\big(
\mathcal{I},
a,
x,
r_a(x),
b(\mathcal{E}),
C,
V,
\kappa_{\mathrm{ctx}},
K,
\mu,
u,
P_{\mathrm{dep}}
\big).
\]

\subsection{One-Sentence Reading}

v3t3m9 is the hard-coded TERA state logic in which identity provides reference, entity provides instantiation, dependency provides coupling, and gating decides admissible transition within a spatiotemporal body of drift, coherence, and recurrence.


\[
\mathcal{E} =
(\mathcal{I}, x, \tau, \omega, I, E, P, M, U, L, \mathcal{D})
\]

\[
r_a(x)=x-a
\]

\[
C=M-P
\]

\[
V=\|C\|^2
\]

\[
\kappa_{\mathrm{ctx}}=U\cdot L
\]

\[
P_{\mathrm{dep}}=\sum_{d_j\in\mathcal{D}} p(d_j)
\]

\[
K=IE\omega\kappa_{\mathrm{ctx}}-\lambda V-P_{\mathrm{dep}}
\]

\[
\mu=\sigma(KUL-\eta V)
\]

\[
u=G(\mu,K,V)\in\{\mathrm{ACCEPT},\mathrm{WAIT},\mathrm{DEFER},\mathrm{REJECT}\}
\]