Existence of a single recursion-theorem fixed point for all simulation horizons

Ascertain whether there exists a single Turing machine index n* (a fixed point guaranteed by Kleene’s recursion theorem) that simultaneously satisfies the self-simulation construction for every time horizon Δt, i.e., establish whether one can choose n* such that the equality T^{n*}(w_0) = g(Δt, w_0, n*) holds uniformly for all Δt in the self-simulation framework described in the paper.

Background

The self-simulation lemma uses Kleene’s recursion theorem to construct, for each fixed Δt, a program index n* that enables a universe to simulate its own future state at time Δt. The author notes that the recursion theorem often yields multiple fixed points (potentially infinitely many) and raises the question of whether this multiplicity can be leveraged to obtain a single n* that works for all Δt at once.

This issue is central to strengthening self-simulation: a uniform n* would obviate the need to re-derive or re-load a new program for each Δt, potentially simplifying the self-simulation procedure and deepening its theoretical implications. The author explicitly flags that it is not clear whether such an n* exists, and does not resolve it within the paper.

References

A subtlety is that the recursion theorem can in general be satisfied by more than one $n*$ --- by an infinite number in fact. It is not clear though that there is a way to exploit this flexibility so that there is at least one $n*$ that satisfies the recursion theorem for all $t$. So for simplicity, this possibility is not considered in this paper.

— Implications of computer science theory for the simulation hypothesis (2404.16050 - Wolpert, 9 Apr 2024) in Appendix C: Why Δt is not a physical variable