Arithmetic Primitive-Orbit Rigidity in Stationary CCR Polynomials: Shift Companions, Prime Divisors, and Bernoulli Ladders

The paper develops the reverse prime-to-CCR direction of our arithmetic transport research, asking when the Nörlund–Stirling shadow of a later stationary CCR column contains a genuinely new irreducible factor or root orbit not inherited from earlier columns. It establishes exact trailing-source interpolation limits and a shift-companion identity that reduces adjacent-column common factors to a self-translation gcd, yielding structural degree obstructions. It then proves source-independent nonsharing for several families of sufficiently long Frobenius orbits. Using a classical large-Zsigmondy-prime theorem together with new exact polynomial arguments, it establishes rational predecessor-factor noncoverage at every odd order, and derives cofinite good-prime and positive-density primitive-root consequences at fixed order. An explicit coefficient-content criterion decides exceptional good primes order by order; exact finite certificates establish polynomial noncoverage at every good prime for odd orders through 71. Bernoulli-coordinate ladders and primitive p-adic root examples complete the picture. Its main novelty is a unified arithmetic theory of source-uniform primitive-orbit rigidity and effective predecessor noncoverage. The stronger all-order, every-good-prime assertion remains open, and no physical automaton-realizability theorem is claimed.