The Primorial Tower

Working notes on one deletion and the object built to carry it, kept by a small research program.

The k-th primorial is the product of the first k primes, pk# = 2·3·5⋯pk. The ring Z/pk# splits by the Chinese Remainder Theorem into k independent channels, one per prime:

Z/pk#  ≅  F2×F3×F5×⋯×Fpk\mathbf{Z}/p_k\# \;\cong\; \mathbf{F}_2 \times \mathbf{F}_3 \times \mathbf{F}_5 \times \cdots \times \mathbf{F}_{p_k}

Every channel is a prime field, and no channel sees another's content. The primorial tower is the sequence of these rings, each nested in the next: Z/6 ⊂ Z/30 ⊂ Z/210 ⊂ Z/2310 ⊂ ⋯

By Ostrowski's theorem, the ways to measure a rational number are the finite places — one per prime, each read through its residue window — plus exactly one more: the archimedean place, where size, sign, and order live. The tower is what remains of the integers when every finite window is kept and the archimedean one is deleted.

These notes chart the rings themselves — their structure, geometry and spectra (Object) — and measure that deletion from both sides. One side prices what left: which operations still read through a residue window and what replaces order (Walls), what any window — positional or not — can read at bounded lookahead (Reading), and what measurement without size decides (Measure). The other side builds on what remains: growth laws that grow the tower itself (Growth), the limits, clocks and prices their runs converge under (Schedule), the class-group arithmetic deciding which characteristics a walk closes cheaply (Places), the growing tower read as a machine (Computation), arrival spectra that read classical ramification (Observatory), learners made of exact comparisons — where descent stalls and what forgetting can certify (Learning) — evaluations that know their own ceiling (Instruments), and the polynomial flattening boundary those rulers ran into (Flattening).

The tower is constructed, not discovered: its properties hold by axiom at every rung. The working question is never "is this surprising?" but "what can we see from here?"

How to read these notes

Every result is stated as a claim block: a name, a tier, the precise statement, its scope, and the script that verifies it. The tiers: property (follows from the construction) · observation (computed, not proved) · pattern (observed across many k, no proof) · rule (proved algebraically, or verified exhaustively across a stated range) · criterion (necessary and sufficient, proved) · theorem (complete general proof — reserved). Cited scripts are downloadable where linked, with the CRT library crt.py beside them; each script's header states what it checks.

The blueprint property

At every rung, by construction: the k channels are independent — arithmetic decomposes window by window and no channel carries information about another; every channel is a prime field; encoding is bijective (an element is its residue tuple, losslessly); division is total, every element a carrying a pseudo-inverse a′ with a·a′·a = a and 0′ = 0 (a meadow); the 2k idempotents index the sub-rings, one per subset of the primes; and elements live on the k-torus Tk, one angular coordinate per channel.

Once there are channels to spare, k ≥ 4, the tower split — the first k−3 primes data, the last 3 parity — is an MDS code of distance 4: any k−3 channels reconstruct the element. It is the one entry whose guarantee leans on what the construction deleted, being a bound on the integers and not a property of the product ring.

Scope. Every rung Z/pk#, all k — these are axioms of the construction, not findings, but for the code's distance, which is proved and swept exhaustively k = 4..10 (a rule) and needs k ≥ 4 to exist.

verifiers: crt.py self-test, 90 checks, the library parametric over rungs; the code's distance, explore_genesis_ladder.py.

The sections

Also: Claims — every claim on the site, one line each, indexed by section.