Prime Branch Clump Wave Lab

A large, measurable view of the largest-prime-factor columns. Every integer appears once; Ω gives its layer, log(LPF) fixes its column, and log(n) fixes its height inside that layer.

PrimeMind · branch field

All Ω layers · largest-prime branches

L1 at bottom · wheel to zoom · drag to pan · click a clump

observed clumpsselected prime branchexact first-possible-height guide

Prime-column inspector

Click a column or enter a prime. Counts show how that branch grows and then shrinks across Ω layers.

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largest prime factor p

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log-column x

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integers in branch

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highest occupied layer

Layer	Count	First n	Last n
Build the field to inspect a branch.

Plot coordinates

x(n)=log(P⁺(n))/log(N)
layer(n)=Ω(n)
z(n)=log(n)/log(N)

P⁺(n) is the largest prime factor. Ω(n) counts all prime factors with repetition. z is the height inside the layer.

Exact clump beginning

nₘᵢₙ(p,k)=p·2ᵏ⁻¹

The lowest integer in prime column p and layer k uses p once and fills every remaining factor position with 2.

Measured clump size

Cₚ,ₖ(T)=#{n≤T : P⁺(n)=p, Ω(n)=k}

This is the number of blocks in one prime-column/layer clump. Its rise and fall is measured directly rather than inferred from apparent thickness.

Construction-driven structure

The clump starts must rise because p·2ᵏ⁻¹ rises. The highest possible layer also falls as p increases: k≤1+⌊log₂(N/p)⌋. The log coordinate turns these exact boundaries into straight guides.

Wave-field test

The unresolved signal is the measured clump count and span after removing those exact boundaries. Repeated peaks, troughs, phase relationships, or stable residual frequencies would be evidence beyond the construction itself.