PRH | Aux | 4.11 • The One-Bit Barrier: "Each to All" and the Decisive Information

Many deep problems have the same logical shape: each individual instance can be resolved by a finite computation, yet the global statement "for all instances" behaves as if it sits at a boundary where information thins out. In the blur viewpoint, this is not a paradox: the passage from a finite model (where one can read individual outcomes) to an idealized infinite model (where one reads only compressed, stable aggregates) is a change of universe - a change of lens - and lenses may delete information. We isolate a simple meta-obstacle: a universal yes/no statement demands extraction of a single global bit. If the finite-to-infinite ladder fails to preserve at least one bit of task-relevant information in a blur-stable manner, then no method confined to the given universe can output a conclusive answer, even if it outputs increasingly strong statistical information. Collatz provides a sharp illustration: existing results certify nontrivial " $>0$ bits" of drift for almost all orbits, but the conjecture itself asks for a worst-case bit. The Collatz "one-ring" program makes this precise by packaging all finite-to-infinite obstructions into one master constant; proving a uniform gap $