3 Layer L1a: Hepp trees
The multiscale bookkeeping structure of the paper’s §5.2, on the plane carrier with order-forgetting automorphisms (DESIGN.md §5.5).
A Hepp tree is a plane rooted tree in which every internal node has at least two children (carrier PlaneTree; leaves are identified by their position paths), together with, as separate data: a marking \(N\colon \) branching nodes \(\to 2^{\mathbb {N}}\), strictly increasing toward the root, and multiplicities \(m\colon \) leaves \(\to \mathbb {N}_{\ge 2}\). Derived notation: \(\mathcal{B}\), \(\mathcal{L}\), \(\gamma _{\mathfrak n}\), subtrees, least common ancestors. Automorphisms are order-forgetting: recursive child permutations preserving subtree shape (and marks, for \(\mathrm{Aut}(\mathcal{T},N)\)); concretely accessed through recursive canonicalization. The finite group \(\mathrm{Aut}(\mathcal{T})\), its action on leaves and markings, and the stabilizer description of \(\mathrm{Aut}(\mathcal{T},N)\) are part of the checked definition layer.
For a marked Hepp tree: injections \((z_{\mathfrak l})\colon \mathcal{L} \to \mathbb {Z}^4_M\) with \(\lvert z_{\mathfrak l} - z_{{\mathfrak l}'}\rvert \ge \tfrac 12 N_{{\mathfrak l}\vee {\mathfrak l}'}\) and the linked-children condition (b); realizability \(Z \leftrightarrow (\mathcal{T}, N)\); the multiset/multiplicity variants.
Every finite \(Z \subseteq \mathbb {Z}^4_M\) is realized by some marked Hepp tree with \(1 {\lt} N_{\mathfrak n} \lesssim M\) (greedy scale construction); every paired vector \((y_j)\) is realized in the multiplicity sense with even multiplicities.
The master reduction of the counting sum (5.5): fix the tree (\(C^n\) choices by 3), the multiplicities and the ordering data; the symmetry-factor denominator; and the word-sum bound (5.10) in the normalization of 13, with \(\mathrm{pairedWordSum}\) on the left and \(\big(\prod _{\mathfrak l}(m_{\mathfrak l}/2)!/m_{\mathfrak l}!\big) \mathrm{paperSum}\) on the right exactly as printed.
The sufficient parent-function injection estimate \(\sum _{\mathcal H}\prod _i (1 + w_i)^{d(i)-1} \le \big(q + \sum _i w_i\big)^{q-1}\) used in Step 5 of the proof of 12. This replaces the paper’s exact weighted Cayley identity by the weaker conclusion actually needed in (5.24)–(5.25), obtained from an injection into parent functions.
The orbit-stabilizer lower bound (5.12) for the number of markings realizing a given embedding, and the embedding-counting upper bounds (5.13)–(5.14) with automorphism factors \(\lvert \mathrm{Aut}(\mathcal{T})\rvert / \lvert \mathrm{Aut}(\mathcal{T},N)\rvert \) and \(\prod _{\mathfrak n} N_{\mathfrak n}^{4(\gamma _{\mathfrak n}-1)}\) (and the \(\langle z_0 - z_1\rangle ^{-4}\)-refined variant). Six-step proof as in the paper’s §5.3.