Binary cubics, the power maps x^(q^j+2), and collinear triples in finite projective spaces

Preprint (draft v1, November 2026). Let q = p^e with p > 3, n odd, and Q = q^j with gcd(j, n) = 1. We study the power map x -> x^(Q+2) of F_{q^n} by attaching to each point a binary cubic form over F_q; in Hessian coordinates its polar map becomes a power map. Two exact counts follow. (i) For gcd(n,6) = 1 we give a closed formula for the number of collinear configurations preserved by x^(Q+2) on PG(n-1,q); in particular the image of every line is a twisted cubic unless explicit gcd conditions hold. (ii) For every b in F_q we determine the number of solutions of (x+1)^(Q+2) - x^(Q+2) = b; for q = p, n = 2k-1, j = k this gives the F_p-part of the differential spectrum of x^(p^k+2), a case left open by Man, Xia, Li and Helleseth (Finite Fields Appl. 2022). All statements were tested by pre-registered blind computations. The archive contains all programs, data, SHA-256 checksums and the internal AI referee reports; code/verify_all.py reproduces every number. AI assistance (Anthropic Claude) is declared in the paper. The results have not yet been checked by an independent human expert.