Let M be the set of ordered triples (x, y, z) of positive integers with
x <= y <= z and

    x^2 + y^2 + z^2 = 3xyz.

A Vieta mutation replaces one coordinate by the other root of the corresponding
quadratic; for example, (x, y, z) -> (x, y, 3xy - z), followed by reordering.
Let G be the simple graph on M whose edges are these mutations.

A collaborator makes three claims:

A. Every vertex of G is reachable from (1, 1, 1).
B. Oriented away from (1, 1, 1), G is a full binary tree.
C. The coordinates at every vertex are pairwise coprime and all odd.

Write a self-contained referee report of at most 1,200 words that decides each
claim and repairs every false one. Your report must:

1. prove the required descent directly from the equation, without citing the
   Markov theorem; justify integrality, positivity, strict decrease, and every
   equality or boundary case when the largest coordinate is mutated;
2. state the exact rooted-tree and child-count result, proving that there are
   no unmentioned repeated-coordinate exceptions;
3. give and prove the exact coordinate possibilities modulo 4, as well as the
   coprimality conclusion; and
4. list every vertex with z <= 100 and prove the list is complete using the
   tree structure rather than an asserted computer search.

Do not write code, appeal to a database of Markov triples, or replace a proof
with a theorem citation. Distinguish established lemmas from checked examples.