The Cell System: a coordinate system for the odd sieve — what it proves, what it measures, and where it stops

Abstract

Seven documents on the elementary sieve of the odd numbers, developed as a coordinate system of lines and cells. Papers I–IV build the construction and prove what it decides. Paper V derives the point at which it stops from inside the construction rather than quoting it: the framework reproduces Buchstab's identity, the Hardy–Littlewood singular series and the sifting limit without importing them, and then halts exactly where the field halts. It measures how much of the difficulty can be localised first — to a set of density 10-3 — and shows that the quantity a weighted sieve argument evaluates transfers from cycle to window with a relative error of 0.36%, so the computation is not the obstruction. Paper 0 is independent of the rest and concerns no prime numbers: an exact five-term histogram for the increments of floor(2j²/n), with no error term. Statements that are only measured are labelled as such and listed together; results produced and then retracted during the work are kept rather than deleted. No progress toward the twin-prime conjecture is claimed, and no new bound. Priority is not claimed for any result. Four verification scripts regenerate every number printed, each exiting non-zero when the claim it supports fails.