Theorem: Exact conditional weighted representation

Under the block and endpoint contracts, the marked tree conditioned on $N=2n+1$ nodes produces exactly the weighted $n$-edge map law proportional to $w(M)$. Every $n\ge 1$ is attainable.

Consequently all tree and channel theorems above apply to this independently specified enriched map model.

Proof. A block with $m$ edges has $2m$ child slots. Thus

The probability factor for a prescribed nontrivial block is

A leaf contributes $1/\Phi (v)$. Their product is

The first two factors are constant at fixed $N$, proving the law. Because $p_0{p}_2>0$, a full binary tree with $n$ internal nodes has positive probability for every $n$.