Can a computer compute any algorithm in this universe? Is that even true once we remember the universe has a finite memory and every irreversible bit flip turns into heat?
I take the abstract claim of universality as a guiding ideal, then ask what survives contact with Landauer and Bekenstein. Only computations that fit the energy and memory budget actually happen, so the winning style must squeeze meaning per bit and minimize erasure.
That pushed me to a simple lens. Computation rides on connectivity, not on coordinates. If I draw a circuit on a rubber sheet and stretch or crumple the sheet without cutting a wire, the program does not change. Only rewiring changes behavior. In other words, logic lives in topological classes of the wiring, while geometry is costume.
Once constraints are encoded as topological invariants, energy based model predictive flows on graphs beat stepwise generation.
Think a ball rolling on a landscape carved by the network itself. The search does not waste bits exploring directions the wiring already forbids. Guidance replaces guesswork, so you move less entropy to reach the same answer.
Thermodynamics also agrees. Coz we only pay when we erase information, so I organize computations to be mostly reversible, push entropy to the boundary, and erase only when learning truly demands forgetting. Throughput then scales with boundary more than with bulk, echoing holography.
With a finite cosmic memory, the smart move is to store structure rather than raw state, compressing meaning into invariants that a boundary can carry.
Message passing turns out to be gauge transport in plain clothes. Treat edges as connections and cycles as holonomy loops. The accumulated phase around a loop enforces global consistency without a global overseer. Small perturbations do not matter unless they jump the system to a different topological class, which is why robustness emerges from the wiring itself.
So can a computer compute any algorithm in this universe? In the abstract sense yes. In the physical sense it can compute the subclass that fits within time, energy, and memory, and it does best when it leans on topology. I now see useful computation as homotopy search - start in the right class, deform within it, and rewire only when you choose a different class.
That respects the information limit of the universe and trades costly bits for conserved structure, which is why I believe it is true.