! TERA Header

## Main Formula

TERA evaluates whether an informational state is stable enough for commitment by combining:

1. **anchored readability**
2. **geodesic logical cost**
3. **gate-based decision logic**

### 1. Perspective Anchor

Let

\[
a \in \mathbb{R}^n
\]

be the anchor (reference state), and let

\[
x \in \mathbb{R}^n
\]

be the current state.

The anchored reading is

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

This is the first condition of legibility:  
without an anchor, there is only position;  
with an anchor, there is direction, distance, and relational meaning.

---

### 2. Main TERA Functional

Let

- \(M\) be a state space,
- \(x_0\) the current anchored state,
- \(x_g\) the goal state,
- \(\gamma : [0,1] \to M\) a candidate trajectory.

Define the TERA cost functional as

\[
J[\gamma \mid a]
=
\int_0^1
\left(
\|\dot\gamma(t)\|_{G(\gamma(t))}^2
+
\alpha A(\gamma(t))
+
\beta R(\gamma(t))
-
\mu K_a(\gamma(t),x_g)
\right)\,dt
\]

where

- \(G(x)\) is the local geometry / movement cost,
- \(A(x)\) is artifact load,
- \(R(x)\) is residual noise,
- \(K_a(x,x_g)\) is goal coherence measured relative to the anchor \(a\),
- \(\alpha,\beta,\mu \ge 0\) are weighting parameters.

### Interpretation

TERA minimizes:

- traversal cost,
- artifact exposure,
- residual noise,

while maximizing:

- anchor-aligned goal coherence.

This means TERA does not ask for the naive shortest path.  
It asks for the most coherent low-distortion trajectory relative to a readable reference.

---

### 3. Informational Tension Score

For a current informational state \(b_n\), define a tension score

\[
s_n
=
\lambda_1 I(b_n)
+
\lambda_2 \Delta I_n
+
\lambda_3 J^\*_n
\]

with

\[
J^\*_n = \min_{\gamma} J[\gamma \mid a]
\]

where

- \(I(b_n)\) is informational stress / uncertainty,
- \(\Delta I_n\) is informational drift,
- \(J^\*_n\) is the minimal anchored logical path cost.

This gives one combined quantity describing whether the current state is readable, stable, and decision-worthy.

---

### 4. Gate Logic

The TERA gate is defined by thresholds:

\[
\mathrm{Gate}(b_n)=
\begin{cases}
\mathrm{ACCEPT}, & s_n \le \tau_{\mathrm{acc}} \;\land\; \mathrm{Rob}_\Gamma(b_n)\ge \eta \\
\mathrm{WAIT}, & \tau_{\mathrm{acc}} < s_n < \tau_{\mathrm{rej}} \\
\mathrm{REJECT}, & s_n \ge \tau_{\mathrm{rej}} \;\lor\; \mathrm{Rob}_\Gamma(b_n)<\eta
\end{cases}
\]

where

- \(\tau_{\mathrm{acc}}\) is the acceptance threshold,
- \(\tau_{\mathrm{rej}}\) is the rejection threshold,
- \(\mathrm{Rob}_\Gamma(b_n)\) is robustness under admissible context variation \(\Gamma\),
- \(\eta\) is the minimum admissible robustness level.

---

### 5. One-Line Reading

TERA decides by asking whether an anchored informational state remains coherent enough, under drift, artifact load, residual noise, and context variation, to justify commitment.

---

### 6. Compact Summary

\[
\boxed{
\text{Anchor} \to \text{Readable State} \to \text{Logical Cost} \to \text{Tension} \to \text{Gate}
}
\]

or more explicitly:

\[
\boxed{
r_a(x)=x-a
\;\to\;
J[\gamma \mid a]
\;\to\;
s_n
\;\to\;

## Main Formula

TERA evaluates whether a current state is stable enough for commitment by combining:

1. anchored readability
2. logical path cost
3. informational drift
4. gate-based decision logic

Let \(a\) be the anchor, \(x_n\) the current state, and \(x_g\) the goal state.

### 1. Anchored Reading

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

### 2. Minimal Logical Cost

\[
J_n^\* = \min_{\gamma : x_n \to x_g} J[\gamma \mid a]
\]

### 3. Normalized Tension Score

\[
\hat s_n
=
\lambda_1 \hat I(x_n)
+
\lambda_2 \widehat{\Delta I}_n
+
\lambda_3 \hat J_n^\*
\]

where all terms are normalized to a comparable scale.

### 4. Gate

\[
\mathrm{Gate}(x_n)=
\begin{cases}
\mathrm{ACCEPT}, & \hat s_n \le \tau_{\mathrm{acc}} \land \mathrm{Rob}_\Gamma(x_n)\ge \eta \\
\mathrm{WAIT}, & \tau_{\mathrm{acc}} < \hat s_n < \tau_{\mathrm{rej}} \\
\mathrm{REJECT}, & \hat s_n \ge \tau_{\mathrm{rej}} \lor \mathrm{Rob}_\Gamma(x_n)<\eta
\end{cases}
\]
\{\mathrm{ACCEPT},\mathrm{WAIT},\mathrm{REJECT}\}
}
\]

This is the minimal TERA chain:
from reference,
to structure,
to stability,
to decision.



\boxed{
\hat s_n = \lambda_1 \hat I_n + \lambda_2 \widehat{\Delta I}_n + \lambda_3 \hat J_n^\*
\quad\Longrightarrow\quad
\mathrm{Gate}(x_n)\in\{\mathrm{ACCEPT},\mathrm{WAIT},\mathrm{REJECT}\}
}