This submission gives direct geometric proofs that every finite tree is outerplanar, that every finite star is outerplanar and planar, and that every two-terminal series-parallel graph is planar. It also proves that a finite graph is outerplanar if and only if it has neither a $K_4$ nor a $K_{2,3}$ minor. The tree and star drawings place vertices on the unit circle using a rational parametrization.

For trees, induction removes a leaf and then inserts it beside its neighbour in a gap between the existing circle parameters. An affine functional for the new chord separates it from all edges with disjoint endpoints. A tangent functional rules out vertices inside edges. The star proof is also given separately: all its edges share the centre.

For series-parallel graphs, an induction keeps the terminals at the ends of a horizontal segment and the other vertices in the triangle above it, with a positive height bound. Explicit affine maps join drawings in series. Vertical compression separates parallel components. The proof checks all vertex and edge intersections, including a terminal edge shared by both components, and requires no excluded-minor or straightening theorem.

For the outerplanar characterization, induction splits disconnected graphs and graphs with a cut vertex into smaller pieces. Acyclic connected pieces use the tree construction. In the remaining case, a longest cycle contains every vertex: an outside component would either extend the cycle or produce a $K_{2,3}$ minor. A pair of alternating chords would produce a $K_4$ minor, so the cycle order gives the required circle drawing. Conversely, connected minor branch sets preserve circular noncrossing order, and neither forbidden graph admits such an order. This follows the direct cycle argument in Leander's On the bunkbed conjecture, Theorem 14, with explicit proofs of the decompositions, minor witnesses, and geometric steps.

These five proofs discharge the original statements in Planar Graph Classes (Lax68), without changing its definitions or assuming any open characterization theorem.

The submission also formalizes Diestel's equivalence between containing a $K_5$ or $K_{3,3}$ minor and containing a subdivision of one of those graphs. Three-terminal branch sets are replaced by tripod paths. Four-terminal branch sets either give a four-arm fan or split into two connected pieces that expose a $K_{3,3}$ minor. This combinatorial bridge is unconditional. It yields the exact original Wagner characterization using Kuratowski's straight-line characterization as its sole statement assumption. Each proof is added to the archive after kernel validation.

Together with the existing Lax68 proofs, finite-tree outerplanarity also settles finite-tree planarity and path outerplanarity and planarity. Thus all Lax68 statements labeled as theorems are proved in the combined proof network. Series-parallel planarity and the outerplanar excluded-minor characterization close two statements labeled $opn$ unconditionally. Wagner's theorem now has a checked proof conditional on Kuratowski's theorem. Kuratowski's straight-line characterization, which includes the straightening step, remains unproved; discharging it will also remove Wagner's remaining dependency.

As groundwork for Kuratowski, this checkpoint also proves Diestel's three-connected edge-contraction lemma (Lemma 3.2.4), constructs the contracted graph and its minor model, and proves preservation of Kuratowski-freeness. It gives an explicit straight-line drawing for every graph with at most four vertices and proves that sufficiently small perturbations preserve a finite straight-line drawing. These are auxiliary results; the geometric induction step and the full Kuratowski characterization remain unfinished.