Ω-parity organisation of block fluctuations vs. the sieve parity problem

Status: open · laid 2026-09-10 · type: lead (literature/structural; not decidable from the dataset)

Statement. In the setting of P-001, odd-Ω classes (N₁, N₃, N₅) fluctuate together across blocks and even-Ω classes (N₂, N₄) fluctuate together, the two parities in opposition, through five rungs with no exceptions. Selberg's parity phenomenon says sieve methods cannot distinguish integers with an odd number of prime factors from those with an even number.

Friction. The block-to-block fluctuations are organised exactly by the quantity the sieve is blind to. Either this is a restatement of the parity barrier (the alternation being inclusion–exclusion's sign, which is the same thing the barrier comes from), or it's a statement about second-order statistics that the barrier — a statement about first-order counts — doesn't cover.

Question. Is the parity-organised covariance structure a consequence of the parity phenomenon, independent of it, or a quantitative refinement of it? A satisfactory answer says which, and whether any known result addresses covariances of Ω-classes in short intervals.