The Genealogy Theorem for Monic Prime‑Generating Polynomials

We formulate and prove a Genealogical Theorem for prime-generating monic integer polynomials evaluated on the non-negative integers. The key result states that any such polynomial with run length L > 1 admits a unique parent with run L − 1 under a canonical shift of the argument. This induces rooted genealogical trees whose leaves correspond to structural maxima for the prime run length. We provide full formal proofs, structural corollaries, and computational examples, including a monic quartic with run L = 38 and a monic cubic with run L = 31 in the canonical setting. Another example is the quartic polynomial of length L=49.