Residual Compression in the P -Chain Family Exact image comparison for x² + x + 1 and x⁴ + x² + 1 over prime fields

Let P (m) = m² + m + 1. Its square-input subfamily P (k²) = k⁴ + k² + 1 appears naturally in the identity P (k²) = P (k-1) P (k) = (k² - k + 1) (k² + k + 1). In this note we compare the residual images of the two associated polynomials A (x) = x² + x + 1 and B (x) = x⁴ + x² + 1 = A (x²) over prime fields. We prove that for every odd prime p, |ImFp (A) | = (p+1) /2, and for every prime p, ImFp (B) is a subset of ImFp (A). Our main result shows that this inclusion is strict for every prime p >= 5: ImFp (B) is a strict subset of ImFp (A). Thus the family P (k²) is residually compressed inside the ambient family P (m) over every odd prime field other than F₃. We also include exact computational data for all primes p <= 199, fully consistent with the theorem.