Genealogical Dynamics of Monic Prime-Generating Polynomials

This work introduces a genealogical structure for monic polynomials with integer coefficients generating consecutive prime numbers on the non-negative integers.

The paper studies the discrete dynamical behavior induced by the canonical polynomial translation

f(n)↦f(n−1)

showing that the genealogical child of a canonical monic polynomial can only:

• increase the prime-run length by one unit, or
• collapse immediately to L=0

This produces finite linear genealogical structures bounded by collapse states and naturally induces a discrete forest organization of prime-generating polynomials.

The paper introduces:

• the concept of prime run length L(f),
• genealogical children and parents,
• genealogical trees,
• the Genealogical Bifurcation Theorem,
• the Inverse Genealogical Theorem,
• the Discrete Forest Principle.

A complete computational genealogy of the cubic monic polynomial with $L=31$ is presented as an explicit example of the theory.

The work combines elements of experimental number theory, polynomial translations, and computational mathematics to propose a new structural interpretation of prime-generating polynomials.