Suppose the supplied full break characters have labelled plane quotients and satisfy the one-vertex scalar-pushout assembly contract, initially with \(C=1\). Their source representation has dimension \(p^d=N\).
For odd \(p\), the actual leading subspace has dimension \((p-1)p^{d-1}\ge2d\). For dyadic \(d\ge3\), use the host quotient dimension \(N-2\ge2d\). Choose the declared symplectic embedding into the actual host.
An orthogonal splitting \(W=V_{\rm src}\perp V_{\rm src}^{\perp}\) gives
A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-\(N\) image-algebra projector. The retained induction/recovery theorem recovers the same \(\Lambda\) and then the same \(\pi\).
Scope
For \(p>3\), genuinely supplied plane data are required: a native minimal rank-\(p\) quotient has dimension \(p-1\) and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.