ZERO AND INFINITY ON DISCRETE TERMS

Abstract. The divergences and paradoxes of modern physics sit at zero or at infinity. Among them are the

ultraviolet catastrophe, the self-energy divergences, the singularity theorems, renormalization, instantaneous

collapse in measurement and Zeno’s paradoxes. This paper states, as one position argued across its companion

records, that zero and infinity are boundary markers of a bounded quantized system, names for edges and not

places where anything stands. Three results carry the case. A completed infinity cannot be performed: finite-time

singularity formation is a supertask, excluded for any finite-energy system by the Margolus–Levitin bound, with

reachable depth logarithmic in every resource. A completed limit cannot be certified: no finite set of finite-

precision measurements distinguishes it from a rational witness constructed at every precision. The one

empirical test of an actual infinity, classical equipartition in a finite cavity, ended when Planck replaced it with a

count. For zero the position is symmetric: what acts is never zero, every realized quantity has a least count, and

zero is reached by notation and never by process. In place of the two ends stand a floor, the least count, and a

ceiling, the finite supply of operations. Within the integers halving terminates, octave equivalence reduces the

doubling ladder to its root, and the chords of the doubling map envelope a cardioid with its cusp at −1/3 and

scale 2/3. Analysis as mathematics is unchanged. The position is refuted by a physically realized completed

infinite process, or by a realized exact zero that acts.