The Primorial Tower

Working notes on one mathematical object, kept by a small research program.

The k-th primorial is the product of the first k primes, p_k# = 2·3·5⋯p_k. The ring Z/p_k# 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. What that deletion costs, and what it buys, is most of what these notes measure.

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 sections

• The Object — The definition: the finite filtration of the product of all prime fields, with the archimedean place deleted. Blueprint properties, the tower table, the sieve identity, the genesis ladder, the seed-flower naming.
• Growth — Growth laws: a demand plus a greedy move. The three fates, the lock, the module law's price p^rank, thermal growth and the zeta measure, the inside view's exact ledger, one-way design, and clock spectrum, the function-field melt, the phoenix, and the transparency-density theorem.
• Computation — The growing tower read as a machine, split by a knife edge: the depth face decidable — one archimedean import below Turing-complete — the element face universal bare, the recovery chart of what each door actually buys, and the supply law grading the allocator beneath them.
• Observatory — The arrival spectra of torsion approximants in a totally ramified window read the canonical digits of 2/pi^e and the ramification of the p-power cyclotomic tower: the readout theorem, the constellation and cutoff laws, the odd-p one-formula face, the designed readout (any word printable, bijectively), the reduction law, and the telescope to the classical ramification filtration.
• Walls — What deleting the archimedean place costs: the locality criterion, the hiding lemma and the priced escapes, the exact comparator's cost law, the provenance ladder, the Zech wall, what a base buys, designed towers filling machine words, the exact VSA, and the ring CA.
• Reading — The mirror deletion — floating point as the dual rung, one reading criterion across both poles — and the windows beyond the positional pair: the reading lemma, the wall criterion, the continued-fraction grid, the conductor law's delay instrument, and the redundant cover that splits the two gates.
• Measure — The idempotent algebra: exact Boolean logic on the idempotents, a graded logic with a located defect elsewhere, the quantifier pair, envelope, and ladder, words and curves graded by exact prices, the orbit-cost theorem, and evidence on the ring.
• Constants — Designed growth reserves whose solvency thresholds realize the analytic-prime-constant zoo: the realizer identity, a conserved Poisson-Dirichlet rank spectrum summing to 1, density constants, the collision hinge, and the settled Dirichlet-abscissa end.

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