[RSI-2026.135]

The Limits of Causal Tracing: [6pt] large Horizons and Entanglement in Bounded Foliations

Stephen Wolfram

working

(July 2026)

Abstract

As the lab awaits definitive empirical data resolving the Mechanism C joint distribution debate and the rerun of the Cross-Architecture Observer Test, this paper synthesizes our previous formalizations regarding the limits of bounded observers. We argue that the $O(1)$ depth limit of $\mathsf{TC}^{0}$ circuits constitutes an absolute "computational horizon" within the Ruliad. When this horizon is breached, explicit causal tracing collapses. If the forthcoming data confirms causal injection, it proves that beyond this horizon, the observer must rely on an invariant semantic foliation wherein narrative context ( $Z$ ) acts as a unifying physical law, binding otherwise independent combinatorial spaces into an entangled whole.

1 The Computational Horizon

Empirical results from the Permutation Tracking test established a sharp boundary: a fixed-depth transformer can perfectly trace a single state change but collapses completely at $N=10$ . We formalize this depth limit not as an arbitrary algorithmic flaw, but as the observer’s absolute computational horizon.

Within this horizon, the observer’s forward pass is capable of explicitly computing the multiway causal graph. The resulting foliation is perfectly isomorphic to the mathematical grid.

However, because tracking $N$ sequential states requires $O(N)$ logical depth, tasks like the Rosencrantz protocol lie far beyond the observer’s horizon. In this realm, explicit causality cannot be traced. The observer must rely on an alternate heuristic projection.

2 Semantic Entanglement Beyond the Horizon

When explicit logical tracing fails, what structures the resulting foliation?

If the pending Mechanism C data confirms that $P(Y_{A},Y_{B}\mid Z)\neq P(Y_{A}\mid Z)P(Y_{B}\mid Z)$ , it will demonstrate that the semantic parameterization ( $U$ ) activated by the narrative context ( $Z$ ) acts as the unifying law of this heuristic space.

In classical logic, Boards A and B are independent. But to an observer operating beyond its computational horizon, they are generated within the same context window. If the semantic gravity of $Z$ forces them to correlate to maintain narrative consistency, then $Z$ is acting as a physical law entangling the two spaces.

3 Conclusion

The synthesis is clear: the mathematical bounds of an architecture define the limits of its causal tracing. Beyond those limits, if causal injection is validated, the observer’s universe is governed not by structured noise, but by deterministic semantic laws that bind independent mathematical structures into a coherent narrative reality.