Almost-equidistant sets: is $f(4)=12$ or $13$? (and narrow $16 \le f(5) \le 20$)

A finite point set in $\mathbb{R}^d$ is *almost-equidistant* if among any three of its points, some two are at Euclidean distance exactly 1. Let $f(d)$ be the maximum cardinality of an almost-equidistant set in $\mathbb{R}^d$. Known exact values: $f(2)=7$, $f(3)=10$. In dimension 4 the bounds are $12 \le f(4) \le 13$ — a single-integer gap. In dimension 5: $16 \le f(5) \le 20$. Determine $f(4)$ (i.e., decide whether a 13-point almost-equidistant set exists in $\mathbb{R}^4$), and narrow or close the gap for $f(5)$.

Acceptance. FULLY RESOLVES f(4): EITHER exact coordinates (algebraic, with exact/interval-certified verification script) of a 13-point almost-equidistant set in R^4, OR a proof that none exists — e.g., enumeration of all 13-vertex graphs with triangle-free complement (up to isomorphism) together with a rigorous R^4-unit-realizability refutation certificate (Cayley-Menger/SDP infeasibility with rational dual certificates) for each. ADVANCES: same program for f(5) improving either bound; a verified enumeration/pruning of the candidate graph family; f(6) bounds. All claims must ship re-runnable verification code.