the integers, read through every residue window at once, with the size window deleted
There are many exact ways to read a whole number. Its remainder mod 2 is one — a single bit, even or odd. Its remainder mod 3 is another, mod 5 another: one residue window per prime, each blind to all the others. And there is exactly one more reading, the familiar one: size — how big a number is, its sign, which of two is larger. The primorial tower keeps the residue windows — all of them at once — and deletes the size reading. The ring Z/(2·3·5·…·pk), built on the product of the first k primes, reads the first k windows simultaneously: the Chinese Remainder Theorem splits it into one independent prime-field channel per window. One ring per k, each nested in the next — that sequence is the tower.
Read the windows in the order of the primes and you are running the sieve of Eratosthenes: “remove the multiples of p” is exactly “read the window at p and discard what reads zero.” The sieve, in algebraic form, is the tower's prime lens — the same object is a structure worth studying and a machine for studying primes.
A worked split: 17 read through the first three windows is (1, 2, 2) — remainder 1 mod 2, 2 mod 3, 2 mod 5 — and no other number from 0 to 29 reads that way: in Z/30 the three windows together are the number. Arithmetic splits the same way: 17 + 5 = 22 is (1, 2, 2) + (1, 2, 0) = (0, 1, 2), each window reduced on its own, none seeing another.
This site is an exercise in minimal emergence — how much structure you can get from the fewest moving parts. The tower is constructed, not discovered, and the interesting question is never “is this surprising?” but “what can you see from here, and what can you build with it?”
Independent channels (zero mutual information between windows — a theorem, not an approximation), bijective arithmetic, built-in error correction (from k = 7 onward the last three channels act as parity and the code is maximum distance separable — the ring is the code), total division among the sieve's survivors, torus geometry. All of it holds by construction, at every rung. One problem with 510,510 possibilities becomes seven problems with at most 17 possibilities each.
The deletion is structural, and it is measurable. Every question of size — sign, comparison, overflow — is computable in no channel, provably, and no per-channel recoding brings it into view; even six channels of seven leave the sign of an element essentially a coin flip. Multiplication has a second exact coordinate system with its own price; a third does not exist. The Walls page charts these limits exactly — they are as structural as the decomposition itself, and they say precisely what arithmetic without size looks like.
Taken in prime order, the tower is the sieve, and it sees ahead of itself: transparency sorts new primes into ones that add complexity and ones that only add capacity, and sub-rings name primes they have never met. The Tower page is the lens itself — every sieve step a channel, the first 14 rungs, the two growth modes; the Prediction page shows how far it sees.
Every demo computes live in your browser — type a number and watch it split into channels, corrupt a channel and watch the ring heal it, pick primes and see what they predict. The same structure runs in silicon and over radio: FPGA pipelines proving millions of exhaustive checks, an ESP32 mesh catching every injected error over the air. Every page also reads complete without JavaScript: the worked examples are in the text, the demos let you make your own.
Claims here state their strength: by construction (follows from the definition), computed (verified over a stated range), or measured (read off real hardware).