Each rung of the primorial tower works the same way: the CRT decomposes every element into independent prime-field residues. The demos below use k = 7 (Z/510,510) — 7 channels, small enough to explore exhaustively. For example, 42 decomposes as (0, 0, 2, 0, 9, 3, 8) — its remainders mod 2, 3, 5, 7, 11, 13, 17.
Constants with structural significance. Click to see their decomposition.
Pick two numbers. Their sum or product decomposes channel-by-channel: the CRT of (a + b) equals the channel-wise sum of CRT(a) and CRT(b), and the same for products. 100 + 200 = 300 in the ring is (0,1,0,2,1,9,15) + (0,2,0,4,2,5,13) = (0,0,0,6,3,1,11), each position reduced mod its own prime.
The decomposition is a bijection. Pick any 7 residues — exactly one number in Z/510,510 has that tuple. The formula uses idempotents: elements that are 1 in one channel and 0 in all others. Multiply each residue by its channel's idempotent, add, reduce. For example, the residues (0, 1, 1, 1, 1, 1, 1) reconstruct to 255,256.
Sums of these seven elements give one idempotent per subset of channels — 27 = 128 in all, each a projector that keeps its channels and zeroes the rest. They are exactly the fixed points of squaring: e² = e.
The decomposition has exact limits, and they are as structural as the decomposition itself. Every question of size — sign, comparison, overflow — is channel-computable at no rung with two or more channels, provably, and no per-channel recoding brings it into view. Multiplication has a second exact coordinate system — exponents, the slide-rule move — with its own tax, and a third does not exist: the ladder ends at two rungs. The Walls page charts all of it — the walls, their unification, the size a channel can see, and what replaces the ordering of the integers.