Let $X$ be a random vector in $\mathbb{F}_2^m$, and let $X'$ be an independent copy. We prove that, for every $\varepsilon>0$, there is a subspace $G$ with $\dim G\le 2^{320(1+\varepsilon)/\varepsilon}H(X)$ such that $H(q_GX)\le(1+\varepsilon)(H(q_GX+q_GX')-H(q_GX))$. In particular, the quotient entropy is at most $(1+\varepsilon)(H(X+X')-H(X))$. The coefficient is independent of the ambient dimension, and its exponential order in $1/\varepsilon$ is necessary. The structural result is a logarithmic rigidity estimate for finite weighted families of laws with averaged subspace entropy sparsity. A common variational quotient controls both a family and its convolution family without decreasing the original family's normalized mean entropy defect. Posterior families preserve averaged sparsity, and a fresh-index argument combines a local-to-global estimate with the entropic Marton theorem to increase the defect ratio at a fixed multiplicative sparsity cost. We also obtain two-input almost-additivity and concentration of moderate-doubling sets on subspace cosets at the same exponential scale. The constants are not optimized.

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