Sets of $p^2$ points in $\mathrm{AG}(4,p)$ determining no complete line of directions, and extendable additive codes without additive extensions

Abstract: Alderson conjectured that for every prime $p$ (i) every set of $p^2$ points of $\operatorname{AG}(4,p)$ determines all points of some line of $\operatorname{PG}(3,p)$, and (ii) every extendable additive $(\nu,2,d)_{p^2/p}$ code admits an additive extension. Over non-prime fields $\mathbb F_q$ the analogue of (i) fails, and that of (ii) for square $q$; over prime fields (i) was known only for $p\le3$. We show that both parts fail for infinitely many primes, and at the first open prime $p=5$ we settle the case of graphs of maps $\mathbb F_p^2\to\mathbb F_p^2$. Our counterexamples are product graphs built from base-$\lceil p^{1/3}\rceil$ digits. For every prime $p>4804^3$ their direction sets contain no complete line, by a lattice-point estimate uniform in the slope and a quotient-set argument. Evaluating an exact criterion by computer shows that they already work for every prime $29\le p<20000$ except twenty primes below $223$; the smallest of them, $841$ points in $\operatorname{AG}(4,29)$, can be checked by hand. Weighting each line by its number of undetermined directions turns every such graph into an extendable additive code without an additive extension; applied to Alderson's examples, the weighting answers a question of his: the analogue of (ii) fails over every non-prime field. At $p=5$ every $25$-point graph determines a complete line: symmetry and elementary counting reduce this to $50$ propositional formulas, whose unsatisfiability is certified by the formally verified checker cake_lpr. A counterexample at $p=5$ would leave between $55$ and $103$ directions undetermined. Hence the least primes violating (i) and (ii) lie between $5$ and $29$.