Bright and lonely universe

Welcome to lonely universe , an infinite realm, each occupied by a single particle!

In this universe, space and time are constructed as follows:

• Space is divided into countably infinite small box-like regions between which the lonely particle may jump back and forth without any limitations. Let $x(\cdot)$ denote the position of the particle. Then it is possible for $x(t)\in B_{i}$ in one moment and then the particle may suddenly appear elsewhere $x(t')\in B_{j}$ in the next.
• Time is curiously strewn together and composed of two resolutions : For observers of lonely universe, one may only observe the state of the universe every Planck-second $\mathbb{T}=\bigcup_{n\in\mathbb{Z}}\big[nt_{P},(n+1)t_{P}\big)=\mathbb{R},$ where $t_{P}$ is the Planck-time. But the lonely particle itself experiences a series of ‘eternities’: it experiences each time interval $\big[nt_{P},(n+1)t_{P}\big)$ as an infinite series of moments $\big[nt_{P},(n+1)t_{P}\big)=\bigcup_{j\geq 0}\big[nt_{P}+t_{j},(n+1)t_{P}+t_{j+1}\big)$ , where $t_{j}=\sum_{0\leq k<j}2^{-(k+1)}t_{P}.$ For the internal life consciousness of the particle, it perceives each of these finer time intervals as lasting equally long.

Now, one might think that an observer of the universe will only see one particle each time it looks. But that’s not how lonely universe works. The particle radiates each Planck-second a total of $\mathcal{E}>0$ units of energy. But the location of this radiation depends on how the particle whizzes around. Each time the particle visits a box, it leaves behind a trace. These traces remain linked to each other and the lonely particle divides up the energy amongst them proportional to their frequency of being visited. This means concretely that, for any time point $t\in\big[nt_{P}+t_{j},nt_{P}+t_{j+1}\big),$ the energy in Box $B_{i}$ is given by $E_{i}(t) = \dfrac{\Big|\{k\mid 0\leq k\leq j,~x(nt_{P}+t_{k})\in B_{i}\}\Big|}{j+1}\cdot\mathcal{E}.$ These trace values of energy in the boxes clearly fluctuate in the course of the moments. What the observer finally sees in box $B_{i}$ at time $(n+1)t_{P}$ is then something between the asymptotic infimum and asymptotic supremum of the trace energy: $\liminf_{[nt_{P},(n+1)t_{P})\ni t\rightarrow (n+1)t_{P}}E_{i}(t) \leq E_{\text{obs},i}\big((n+1)t_{P}\big) \leq \limsup_{[nt_{P},(n+1)t_{P})\ni t\rightarrow (n+1)t_{P}}E_{i}(t).$ So, for example, if $x(t)\in B_{173}$ at times $t\in\big[nt_{P}+t_{j},nt_{P}+t_{j+1}\big)$ for $j=2,5,8,11,14,\ldots,$ then $E_{\text{obs},173}\big((n+1)t_{P}\big)\geq\frac{\mathcal{E}}{3}$ . Or if $x(t)\in B_{44}$ only at times $t\in\big[nt_{P}+t_{j},nt_{P}+t_{j+1}\big)$ for $j=3\cdot 10,\, 3\cdot 100,\, 3\cdot 1000,\, \ldots,$ then $E_{\text{obs},44}\big((n+1)t_{P}\big)=0$ .

Say that the particle lights up the region $B_{i}$ for the observer exactly in case $E_{\text{obs},i}\big((n+1)t_{P}\big).$ If multiple regions are lit up, the observer may be fooled into thinking that there are particles in all these regions, despite there only being one particle in this universe.

Question: Is it possible for the particle to move in such a fashion as to light up all regions ? That is, can the lonely particle fool an observer into thinking that the universe is full?

The lonely universe and the lonely particle