Theorem: Haar subcode decoupling

For the fixed code dimension $d$,

Consequently,

For $0<\alpha <1$, with probability at least $1-\alpha$,

These are whole-code, reference-preserving bounds.

For fixed $d$ and $D\to \infty$, ${\Lambda }_B\to 0$ whenever $b/r\to 0$. More generally, the sufficient regime is $d^{3}b/r\to 0$. This last dimension dependence is a conservative certificate, not an optimal random-code threshold.

Proof. Appendix app:fourth proves the fourth-moment identity (eq:HS). The operator acts on a space of dimension $db$, so

Cauchy–Schwarz in expectation, followed by Lemma lem:benchmarks, gives

Markov's inequality gives the leakage statement, and Proposition prop:ksw gives recovery.

Since

the stated sufficient regimes follow. At $b=1$, the expression vanishes exactly.