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:
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, with the limits, clocks and prices they converge under
(Growth), the growing tower read as a machine (Computation), arrival
spectra that read classical ramification (Observatory), and learners
made of exact comparisons — where descent stalls, what forgetting can
certify, and evaluations that know their own ceiling (Learning).
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.