Geometric & Mathematical Foundations

Rather than processing symbols or boolean values, Spatioz processes decision logic within a continuous, curved differential manifold. This section defines the formal mathematical foundation of our geometric decision framework.

1. Relational Riemannian Graph Space

Let the decision space be modeled as a Riemannian manifold $(M, g)$, where $M$ represents the state coordinates and $g$ is the metric tensor defining the "distance" or cost of transition between states.

Unlike Euclidean spaces where distance is $d(x,y) = \sqrt{\sum (x_i-yi)^2}$, the metric tensor $g{ij}$ varies depending on the logical states of sensors and anchors:

$$ds^2 = \sum{i,j} g{ij}(x) dx^i dx^j$$

Here, $g_{ij}(x)$ represents the logical tension or reluctance between different components. If a robotic actuator is near a joint limit or facing high resistance, the metric tensor distorts, making that direction "further" in the decision space.

2. Relational Valence & Tension Field

Every edge in the decision graph represents a relational valence $v \in [-1, 1]$, representing attraction ($+1$) or repulsion ($-1$). The total energy (action) $E$ of the system is modeled as:

$$E = \frac{1}{2} \int_M g^{ij} \frac{\partial \Phi}{\partial x^i} \frac{\partial \Phi}{\partial x^j} dV$$

where $\Phi$ is the decision potential field. The system constantly seeks to minimize $E$:

$$\delta E = 0 \implies \Delta_g \Phi = 0$$

where $\Delta_g$ is the Laplace-Beltrami operator on the manifold, which resolves to local homeostasis across all graph segments.

graph LR
    A[Node A] -- v = +0.8 --> B[Node B]
    B -- v = -0.5 --> C[Node C]
    A -- v = -0.2 --> C
    style B fill:#1a1a2e,stroke:#007bff,stroke-width:2px

3. Relative Tension Distribution (RTD)

To normalize and balance competing intents (e.g. alignment vs speed) without conditional branches, Spatioz uses Root Mean Square (RMS) normalization over the active activations:

$$\text{RTD}(x_i) = \frac{xi^p}{\sum{j} x_j^p}$$

where $p \geq 1$ determines the selection pressure (higher $p$ acts like a soft maximum, while $p=1$ distributes tension linearly).