A bitstring $B$ exists if some physical configuration encodes it. In a computable universe where $B$ does exist but only occurs once, and a perfect fidelity simulation of that universe is run within it on an unphysically large computer (except without the existence of the hardware running that simulation), $B$ then occurs twice, because it is specified in some way on the hardware of the computer on which that simulation is run. If $B^{\prime }$ refers to the simulated bitstring:

• What should distinguish $B$, as a bitstring existing in a real external world, and $B^{\prime }$?
• If our simulation was run on a hypercomputer such that it could infinitely recur in perfect fidelity, is $B^{\prime }$ 'more real' than $B^{\prime \prime \prime \prime \prime }$?

These are the kinds of questions I want to find answers to, because I want to be able to prove things about simulators. As a first step, let's carve up the simulator.

Given the probability space $(\Omega ,F,P)$ where $\Omega$ is the sample space, $F$ is the event space, and $P$ is the probability measure, mapping events in $F$ to the interval $[0,1]$, let $\omega$ refer to individual outcomes in $\Omega$, each of which describe a discrete simulacrum, and $C^k$ as individual discrete events in $F$. Classifying simulacra as programs, we denote the maximum Kolmogorov complexity $K$ with respect to a universal Turing machine $U$ for any given space $(\Omega ,F,P)$ as $v={max}_{\omega \in \Omega }{K}_U\left(\omega \right)$.

Proceeding, let ${\omega }^{\ast }\left(C^k\right)$ be the complete set of simulacra for some Cartesian object $C^k$, where individual simulacra are addressed by ${\omega }^n\left(C^k\right)$ as sets of actions indexed by $m$. Let $\Xi :{\Omega }^{\ast }\left(C^k\right)\times C^k\to {\omega }_m^n$ be a function that maps from choices for each $\omega$ to a world $w_m^k\in W^k$, where $w_m^k$$= refers to the $m^{\text{th}}$ world in the set of possible worlds for the $k^{\text{th}}$ object, and $k$ indexes the object $W$ describes possible worlds for, culminating in $C^k=({\omega }_1,...,{\omega }^n,\Xi )$.

By modeling the coupling of the probability space $(\Omega ,F,P)$ and its contained simulacra as a dynamical system, the following are considered to describe sampling tokens from a simulation state $S$ at time-step $t$ given the complete simulation history prior $S^{\ast }t=(S^0,...,S^t)$ as a trajectory through states, where states are given by the set of worlds for all objects $W^{\ast }$ realized in the set of actualized objects $A$ at time-step $t$, ${\bigcup }_{w_m^k\in A_t}{\Xi }^{-1}\left(w_m^k\right)$:

• The token selection function $\psi :S^{\ast t}\to \tau$, where $\tau$ is a distribution over all tokens in an alphabet $T$.
• The evolution operator $\varphi$ which evolves a trajectory $S^{\ast t}$ to $S^{\ast t+1}$ by appending the token sampled with $\psi$.

Assuming the model defined above, the simulation forward pass becomes a simple operation delineated as follows:
We begin at simulation state $S^0$, which denotes the empty or null state, whereby $A_0=\varnothing$, which is also maximally entropic.
 

$\psi \left(S^0\right)$ is applied for one time-step:

• $P$ selects $C^k$ from $F$ under $v$, aggregating the set of realized worlds in $A^1$ as $S^1$
• The token selection function is applied to the current state as $\varphi \left(S^{\ast 1}\right)$