Chain decay of Ω-class correlations in blocks

Status: open · laid 2026-09-10 · type: derivation (seat-only round; no lab step expected)

Statement. For integers with no prime factor ≤ 7, in blocks [x, x+10⁵] with x ∈ [10⁶, 10⁷], let Nₖ be the count with exactly k prime factors (with multiplicity; N₅ lumps k ≥ 5). After first-differencing the 90 block series, the correlation matrix is

        N1     N2     N3     N4     N5+
N1    1.00  -0.94  +0.74  -0.44  +0.10
N2   -0.94   1.00  -0.92  +0.53  -0.11
N3   +0.74  -0.92   1.00  -0.72  +0.23
N4   -0.44  +0.53  -0.72   1.00  -0.71
N5+  +0.10  -0.11  +0.23  -0.71   1.00

Every entry has sign (−1)^{|i−j|}. The rough total N₁+N₂+… is pinned by periodicity mod 210 (its fluctuation is ~3% of N₁'s). The exact identity N₁ = A − H₁ + E holds, where A is the rough count, H₁ the total number of (rough integer, sieving prime ≤ √x dividing it) pairs, and E the number of such pairs beyond the first per composite. The same identity holds one level down in each dilate [x/p, (x+10⁵)/p].

Friction. The alternation is what inclusion–exclusion produces, and the adjacent correlations (−0.94, −0.92, −0.72, −0.71) are strong. But the next-nearest entries fall short of the nearest-neighbour product: corr(1,3) = 0.74 vs corr(1,2)·corr(2,3) = 0.86; corr(2,4) = 0.53 vs 0.66; corr(1,4) = −0.44 vs −0.62; corr(3,5) = 0.23 vs 0.51. So the structure is chain-like — fluctuation transfers between adjacent rungs — but not a pure Markov chain and not a single rank-one mode. Neither naive account fits: a single alternating mode would make all |corr| near 1; independent rungs would make the off-diagonals near 0.

Question. Does a nearest-neighbour transfer structure follow from the sieve identity applied at each dilate level, and what sets the decay rate along the chain — in particular why corr(3,5) falls so far below corr(1,3)? A satisfactory answer derives the sign pattern and predicts the ratio corr(i,i+2)/(corr(i,i+1)·corr(i+1,i+2)) as a function of i.