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Source. Sections 1, 3, 6–8 and the references, pp. 125–130 (published original).

Graph inputs from Paths, Trees and Flowers

The weighted paper explicitly uses the earlier paper's Sections 4 and 7. Their complete original arguments are now available at their canonical source. They remain external to this six-page article.

• Section 4.5 identifies the odd near-perfect blossom and stem in a planted tree with an added edge between two outer vertices.
• Section 4.7 grows a planted tree by adding a matched outside pair, or produces an augmenting tree, a flowered tree, or a Hungarian tree.
• Section 4.13 contracts a flowered tree to a planted tree whose new node is outer.
• Section 4.14 lifts a matching through remembered odd-circuit blocks using the compatible near-perfect matching at each prescribed attachment.
• Section 7.2 replaces an inner pseudovertex by the even circuit arc between its matching and nonmatching attachment children; equal children use the zero-edge arc.

The remembered-contraction conventions and the separate unweighted refinement are in contraction_structure and refinement_7_3. The present source supplies new weighted caps, reduced-edge identities and event rules. The weighted expansion proof checks the extra tightness condition; a bare citation of the unweighted lemma would not supply it.

Finite-dimensional foundations

The nearest-point argument for Theorem P uses compactness of the convex hull of a finite set, which is the continuous image of its compact finite-dimensional coefficient simplex. It expands the needed separation argument locally. The Section 6 limit uses that a bounded sequence in a finite-dimensional real space has a convergent subsequence. Closed equalities and inequalities are preserved by that limit. These are exact finite-dimensional compactness inputs, not an infinite matching or choice theorem.

Section 3 states general real linear-programming strong duality: for a finite real matrix $A$ and real vectors $b,c$,

$$\max\{c^\mathsf Tx:x\ge0,\ Ax\le b\} =\min\{b^\mathsf Ty:y\ge0,\ A^\mathsf Ty\ge c\}$$

when the finite extrema exist. Its general proof is not given. The source uses it in its surrounding converse discussion. The complete constructive proof here instead exhibits an optimal matching and a feasible dual of the same value, with weak duality proved by direct summation. General strong duality is therefore not an unproved essential step in the matching-polytope proof.

Historical and deferred material

The references to Birkhoff and von Neumann concern doubly stochastic matrices and the assignment problem. Kuhn's Hungarian method, König's bipartite theorem and Tutte's factor theorem are historical comparisons. Their alternative proofs are not reconstructed or used as hidden weighted-matching inputs.

Witzgall–Zahn's different cardinality algorithm is a separate paper. Section 8's two degree-capacity polyhedra and algorithm are announced for another source and remain unproved here. The source's conceptual work count has not been promoted to a checked implementation or bit-complexity theorem.

The earlier cardinality LP is a separate, weaker deduction. Exactness for that one objective does not imply the real-weight theorem proved in this source. No source-specific formalization or local formal build is claimed.