The Cell System
A coordinate system for the odd sieve — what it proves, what it measures, and where it stops
Mohamed Osman · ORCID 0009-0004-5912-999X · independent researcher
Twelve papers, one overview and thirty-seven verification scripts. Papers 1 to 4 build the coordinates
and the transport laws; papers 5 to 8 are the construction and what it decides about twin pairs;
papers 9 to 11 derive the point at which it stops from inside the construction rather than
quoting it, and measure how much of the difficulty the framework can localise before failing to
cross it.
Every headline result of papers 1 to 11 already exists in the literature, and each paper names
the source. For the overlap weight of paper 12 no source has been found, and no novelty is claimed for it. No progress toward the twin-prime conjecture is claimed.
Priority is not claimed for any result.
What the collection offers instead: a construction that produces Buchstab's decomposition, the
Hardy–Littlewood singular series and the exact factor-of-two loss without importing them, and
then stops exactly where the field stops; negative results and withdrawn claims kept rather than deleted; and a
script in code/ that regenerates every number printed.
Repository ·
DOI 10.5281/zenodo.22661626 ·
Prime Number Studies (the author's other work)
Papers
- The Cell System: an Overview
Every result of the set stated once, with its status attached: proved or measured
2026-10-16 · PDF - Paper 1. An Exact Histogram for a Quadratic Staircase
The increments of floor(2j^2/n), and the local maxima of 2j^2 mod n
2026-10-16 · PDF - Paper 2. Cells and Lines
A coordinate system for the odd sieve
2026-10-16 · PDF - Paper 3. The Inheritance Law on the Cycle
Exact transport of the divisor census, and of weights, over a sieve cycle
2026-10-16 · PDF - Paper 4. From Cycle to Window
What survives when an exact periodic law is evaluated on a short interval
2026-10-16 · PDF - Paper 5. The Gap Alphabet
Which gaps can occur between consecutive odd composites, and where the ladder of proofs stops
2026-10-16 · PDF - Paper 6. The Twin Criterion: Six Exception Positions
A single open cell in a sector is a twin pair unless it sits at one of six named places
2026-10-16 · PDF - Paper 7. Clocks and Inheritance
Primality as a zero-test, and the capacity of a single line across sectors
2026-10-16 · PDF - Paper 8. Belts and Short Windows
What a line can do between its own square and the next, and the anatomy of the window between consecutive squares
2026-10-16 · PDF - Paper 9. The Exact Form of the Obstruction
An identity for the twin count with no error term, and why an ordinary sieve cannot cross it
2026-10-16 · PDF - Paper 10. Four Tests of the Cell System
Jacobsthal, almost-primes between squares, shared cofactors, and collisions
2026-10-16 · PDF - Paper 11. What a Continuation Would Have to Supply
The external ingredient, the limitations, and the routes already tried
2026-10-16 · PDF - Paper 12. Paired Almost Primes in Square Windows and in Short Intervals
A dimension-two sieve on the same window, with certified coefficients, the distance to four factors, and a weight that reaches them
2026-10-16 · PDF
Text CC BY 4.0 · code MIT