One observer, one computationally irreducible world, three views of the same limit: how much it can predict, how its grain changes what is knowable, and how its coarse-graining manufactures the arrow of time. Pick a rule and move the sliders.
The top image is the world running from a single black cell. Below, the observer never sees this whole picture. In demos 1 and 2 it watches only the center column over time. In demo 3 it can only see a coarse, blocky version. Everything it knows is squeezed through that narrow channel.
Shown: prediction error vs how many past symbols (k) the observer may use. Lower is better, 1 bit is a coin flip, 0 is perfect. How it is derived: on the first half of the center-cell stream the observer tallies how often each k-symbol pattern was followed by 0 or 1, then is graded on the held-out second half by the surprise of the true next symbol, averaged to bits/symbol. Black is your rule; faint lines are 250, 110, 30.
Shown: the best prediction error the observer can reach (lowest over all k) when it views the same world through a coarser channel. How it is derived: instead of one cell it reads the majority vote of a width-w strip over time, and we take the floor of that channel's error. Same world, different observer, so the floor moves with the channel. Try Rule 30 to see this panel come alive.
Shown: how disordered a coarse view looks as time runs, starting from an ordered state. Lower is ordered, higher is scrambled. How it is derived: each step, lump the row into blocks of b cells and take the Shannon entropy of the block-pattern frequencies (bits per cell). The micro-rule is deterministic, yet the coarse entropy still climbs, most for chaos. The arrow of time is in the grain.