An exact non-Haar island with an explicit decoder

The decoder can be implemented with controlled Pauli gates on the received ciphertext and early key register. The batch input occupancy is $3\ell$ qubits; the ciphertext register itself can be returned as the diary output. The key and complete decoder environment are then discarded. No unpriced quantum output is used.

This example proves exact operational-island existence with a positive residual. It is a designed code, not evidence that Haar scrambling or gravitational dynamics automatically implement the controlled decoder.