R⁴ Tangent Space Router

Traditional routing projects a 3-sphere $S^3$ down to a 2-sphere $S^2$ and a 1-sphere fiber $S^1$ via the static Hopf Fibration. In end-to-end trainable settings, this static geometry yielded a profound Null Result ($\Delta\text{PPL} \approx 0.04$ over random partitions) because neural network experts naturally co-adapted to absorb static boundaries.

Pre-imposing rigid mathematical partitions onto adaptive layers is inefficient; highly plastic networks flatten pre-computed structures.

To resolve this, the Web4 stack shifts to the Continuous Tangent Space Operation in $\mathbb{R}^4$. To bypass calculating expensive curved path steps across the hyperbolic manifold in real-time, the router utilizes the Tangent Operation at the Origin ($set$).

The geodesic energy functional: $$E_R[\gamma] = \int_{t_0}^{t_1} g_{\mu\nu}(\gamma(t))\dot{\gamma}^\mu(t)\dot{\gamma}^\nu(t)dt$$ is minimized instantly by projecting the tangent vector directly from the origin toward the target, routing securely through the 2D Hopf shadow.

To prevent rounding errors and spatial drift across long execution horizons, Web4 anchors coordinates to a discrete lattice of prime-aligned coordinate identities.

Every prime number $p > 3$ is restricted to one of two residue classes modulo 6: $$p \equiv 1 \pmod 6 \quad \text{or} \quad p \equiv 5 \pmod 6$$ This splits the physical MAC addressing of agentic hardware nodes into two mutually exclusive, non-interfering coordinate bands ($6k - 1$ and $6k + 1$).

Deterministic Collision Avoidance Theorem:

Because the absolute distance separating parallel tracks in the same class is strictly a multiple of 6: $$|p_1 - p_2| = 6|k_1 - k_2|$$ trajectories routed within the same resonance class never physically intersect, enabling collision-free routing without routing-table overhead.

To resolve the Prime-Wheel State Collision when modulo $N$ is reduced, the system injects a 1-bit twist parity spin variable $\kappa \in \{-1, 1\}$ directly into the coordinate address.