Complete terminal branch and its rays
For branch 2 this is exactly 2, 4, 8, 16, … . A parent edge follows n → n/SPF(n); a pink ray goes directly from n to its terminal prime LPF(n).
Branch landing sequence
successive members of the same branchHorizontal placement uses the selected x-coordinate; height is Ω(n). This never connects unrelated consecutive integers.
Within-branch gap fingerprint
gⱼ = nⱼ − nⱼ₋₁Both the raw gap and the scale-adjusted gap gⱼ/log(nⱼ) are retained in the export.
Twin-prime three-body ray comparison
Each row is normalized by its own terminal seed, so differently sized branches can be compared without confusing numerical scale with shape.
Exact ancestry of p, p+1, p+2
candidate trio onlyThe two primes terminate at themselves. The middle composite follows the canonical smallest-factor removal path to its largest prime factor.
Neighboring branch atlas
equal spacing by prime index π(p)Seven adjacent prime branches are shown with equal vertical spacing. This removes the expanding numerical gaps between primes from the geometry.
Selected branch ray records
| arrival | n | terminal p | cofactor n/p | Ω(n) | SPF(n) | parent | previous gap |
|---|
1 · What the picture is
Every point is one real integer. Points sharing the same largest prime factor belong to one terminal branch. Lines join either a number to its actual parent or directly to its terminal prime.
2 · Exact equations
parent(n)=n/SPF(n)
b(n)=LPF(n)
Rₚ(N)={n≤N:LPF(n)=p}
gₚ,ⱼ=nₚ,ⱼ−nₚ,ⱼ₋₁
3 · What is constructed versus tested
Branch membership and every ray are construction-driven by factorization. The experimental question is whether normalized landing and gap shapes transfer between branches or twin-prime neighborhoods in a frozen forward test.