A continuous coding function produces a real tree #
The interval minima satisfy the four-point inequality. Midpoints are constructed by applying the intermediate value theorem to a running interval minimum, and then choosing a point where that minimum is attained. Compactness supplies completeness, so the real-tree criterion applies. All three properties are proved for every continuous real coding function; no four-point or midpoint assumption, nonnegativity condition, or endpoint normalization is imposed.
The compact coding quotient satisfies the four-point condition.
Midpoints are realized by times obtained from a continuous running interval minimum.
Every continuous real coding function on the unit interval codes a compact real tree.