A Rank-One Obstruction for Odd-Primary Wood Equivalences

Let p be odd, \(h=(p-1)p^{r-1}m\) with \(r\ge1\) and \(m\gt1\), and \(H=C_{p^r}\) a subgroup of the height-h Morava stabilizer. For Honda Morava E-theory over \(\mathbb F_{p^h}\), put \(A=E^{hH}\). We prove that no finite p-local spectrum F satisfies A smash F equivalent to E as A-modules. More generally, if A smash F is equivalent to a nonzero finite wedge of q copies of E, then \(q\ge2\). No parity condition on the cells is required. The proof combines Ravenel's classical strict-stabilizer \(H^1\) calculation, cyclotomic valuations, Galois base change, and an elementary modular-representation flag bound. An abstract rank-two matrix example shows the flag bound is optimal as a group-theoretic statement, without asserting a topological realization. The result concerns external finite-spectrum Wood equivalences only. Internal A-module filtrations, optimal realizable multiplicities, and the \(m=1,r\gt1\) cases are not resolved; the broad source question AIM-ALGEBRAIC_GEOMETRY-0035 remains partially solved. This is an unrefereed preprint with AI assistance disclosed in ordinary prose. Classical inputs are credited; independent review, formal verification and absolute priority are not claimed.