Branches

Every integer n ≥ 2 is owned by its largest prime factor. Remove one occurrence of that prime and you get the canonical cofactor u = n / LPF(n); repeat, and the integer walks down a branch through its factor layers L = Ω(n) until it reaches a prime in layer 1. Primes are the roots; composites hang from them.

The exact part is that at every layer a branch has definite lower and upper bounds — the branch-layer envelope. The empirical part is how the population inside those bounds is distributed: where the clumps sit, how the terminal paths compare after normalization, and how the picture changes as N grows. The four labs below draw these views; the Envelope Law page carries the proof.

• Branch Envelope Law

  The terminal-branch envelope law, stated and drawn. For a branch owned by terminal prime p, the first member in layer k is p · 2k−1 and the last is pk; every other member sits between them. Pick the terminal prime, choose how many layers to complete, and switch between ordinary and logarithmic position. Nothing on this page is an observation; the proof is in section 3 of the research journal.

  exact

  A short page. Read it first if you want the exact bounds the other three labs live inside.

  Open original prime-branch-envelope-geometric-law.html

• 3D Branch Space V3

  Three views of exact integers: complete branch layers, two long segments compared on one shared scale, and a prime-power wavefield in which every power period is a separate lane whose contributions add up. Rotate the charts to separate coordinates that overlap in two dimensions. Interval builds run in a background worker and can be cancelled.

  exactempirical

  About 3 MB and computation-heavy at large N; best on a desktop browser. Keep the default N for a first look.

  Open original prime-branch-space-3d-labV3.html

• Branch Rays

  The complete terminal branch of a chosen seed prime and the rays that lead into it: parent edges follow n → n / SPF(n), direct rays go straight from n to its terminal prime. Choose the horizontal coordinate — the integer itself, log n, the log cofactor, or branch arrival rank — to see which normalization makes branches comparable, and read the within-branch gap fingerprint.

  exactempirical

  Open original prime-branch-ray-lab.html

• Branch Clumps

  A large, measurable view of the largest-prime-factor columns. Every integer appears exactly once: Ω gives its layer, log LPF fixes its column, and log n fixes its height inside that layer. Watch how the clumps in each layer grow and shift as the build extends.

  exactempirical

  Open original prime-branch-clump-wave-lab.html

Notation used on these pages

LPF(n)

Largest prime factor. It owns the branch. (Some software uses a similar abbreviation for the least prime factor; that is not the convention here.)

SPF(n)

Smallest prime factor.

Ω(n)

Number of prime factors counted with multiplicity; the factor layer. For 12 = 2² · 3, Ω = 3.

ω(n)

Number of distinct prime factors. For 12, ω = 2.

u = n / LPF(n)

Canonical cofactor: remove one occurrence of the largest prime.

Full terminology and claim boundary