Congruences, p-adic Numbers, and Elementary Number Theory from Three Primitives

Every theorem of elementary number theory is a statement about cylinder sets andperiodic orbits in the Tree of Continua C.The central observation is this: a congruence a ≡ b (mod n) is precisely the statementthat the integers a and b — periodic orbits in Per(C) — lie in the same cylinder set ofthe partition of Z induced by n. Congruence is the same primitive applied to the residueclass structure.From this single identification, the entire edifice of elementary number theory unfolds:the Chinese Remainder Theorem is the product structure of independent cylinder setpartitions; Fermat’s little theorem is the finiteness of the multiplicative group of cylindersets; Euler’s theorem extends this to composite moduli; primitive roots are single periodicorbits that generate the full cylinder set group; Hensel’s lemma is depth lifting — thesame compatible family construction that builds Hilbert space; and the p-adic integers arethe IPG limit of the compatible family of residue classes, placing p-adic analysis entirelyinside C.No axioms of number theory are imported. Every result is a theorem about thecylinder set structure of Per(C), derived from Fthe three primitives — same, different,opposite — and the partition identity Vd (x) = s Vd+1 (xs ).