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

Eleven papers, one overview and nine verification scripts on the elementary sieve of the odd numbers, developed as a coordinate system of lines and cells. Papers 1 to 4 build the coordinates and the transport laws: an exact histogram for the increments of floor(2j²/n) with no error term, the line/cell coordinate system, the exact inheritance law on a full sieve cycle, and what survives when that law is evaluated on a short interval. Papers 5 to 8 are the construction and what it decides about twin pairs: the three-letter gap alphabet, the six exception positions of the twin criterion, clocks and the capacity of a single line, and the belts between consecutive prime squares. Papers 9 to 11 derive the point at which the construction stops, from inside it rather than by quoting it: the depth cut, the unique-owner theorem, the exact twin count T = C − R + S with no error term and the parity identity 2(R − C) = Σ(−1)Ω; then why no ordinary sieve crosses it; then four tests of the framework against problems whose answers are known independently; and finally what an external ingredient would have to supply, with every route already tried recorded and closed. The framework reproduces Buchstab's decomposition, the Hardy–Littlewood singular series and the exact factor-of-two loss I(s) = 2f1(s) without importing them, and then halts exactly where the field halts. (The sifting limits are quoted from the literature, not derived here.) Paper 1 is independent of the rest and concerns no prime numbers. 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. Release 2.0.0 is a structural release. The two largest papers of the previous seven-document set were each carrying four or five independent threads; they are now seven separate papers, and the whole set is numbered 1 to 11 in reading order, with a title on each saying what it contains. Companion papers are now cited as [P1] to [P11] so that a bracket can no longer mean both a paper of the set and an entry in a reference list; every paper's reference list is renumbered contiguously and every entry in it is cited in that paper's own body. Thirty-two stale cross-references left by the split were repaired, together with the damaged brackets that produced them. In Paper 11 the linear sieve functions are now given in their correct ranges with F continued above s = 3, and the level-of-distribution assumption on the semiprime rail is stated as an assumption rather than as available. Paper 6 is reorganised so that its numbered sections precede its appendices and the sharpened forms of its central corollary sit in an appendix. Because every filename and every citation form changed, this is a major version rather than a point release. Nine verification scripts regenerate every number printed, each exiting non-zero when the claim it supports fails.