Residual Compression in the P -Chain Family Exact image comparison for x^2 + x + 1 and x^4 + x^2 + 1 over prime fields

Let P(m) = m^2 + m + 1. Its square-input subfamily P(k^2) = k^4 + k^2 + 1 appears naturally in the identity P(k^2) = P(k-1)P(k) = (k^2 - k + 1)(k^2 + k + 1). In this note we compare the residual images of the two associated polynomials A(x) = x^2 + x + 1 and B(x) = x^4 + x^2 + 1 = A(x^2) over prime fields. We prove that for every odd prime p, |Im_Fp(A)| = (p+1)/2, and for every prime p, Im_Fp(B) is a subset of Im_Fp(A). Our main result shows that this inclusion is strict for every prime p >= 5: Im_Fp(B) is a strict subset of Im_Fp(A). Thus the family P(k^2) is residually compressed inside the ambient family P(m) over every odd prime field other than F_3. We also include exact computational data for all primes p <= 199, fully consistent with the theorem.