Cyclotomic Prime Extractors

We develop explicit prime-recovery formulas from the values ${\mathrm{\Phi }}_n(2)$ of cyclotomic polynomials. Binary divisibility patterns detect repeated prime factors and identify the least prime divisor of a squarefree index, while small corrections to ${\log }_2{\mathrm{\Phi }}_n(2)$ allow successive recovery of the distinct prime factors. Specializing the index gives identities for prime products and the least prime above a given integer. The same mechanism extends from finite factorizations to an infinite prime sequence: a normalized limit of cyclotomic values along the odd primorials defines a real constant $\mathrm{\Omega }=0.25061403238015047218\dots$, from which every odd prime can be recovered by a recursive rounding rule.