Each input bit is independently deleted without an erasure marker; the exact capacity is unknown for every nontrivial deletion probability.
Open
\(0.1221(1-d)<C_{\mathrm{del}}(d)\le0.3578(1-d)\)
Channel capacities and open gaps.
A community-maintained registry of channel capacities, open gaps, and Lean statements.
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A causal relay assists a source, but decode-forward and the cut-set bound do not coincide in general.
Open
\(R_{\mathrm{DF}}\le C\le R_{\mathrm{cut}}\)
The capacity region for two arbitrary broadcast receivers remains unknown outside important ordered subclasses.
Open
\(\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C_{\mathrm{BC}}\subseteq\mathcal R_{\mathrm{UV}}\)
Two transmitter-receiver pairs interfere, and the exact capacity region is unknown in general.
Open
\(\mathcal R_{\mathrm{HK}}\subseteq\mathcal C_{\mathrm{IC}}\subseteq\mathcal R_{\mathrm{outer}}\)
A multiple-unicast instance where the exact nonlinear symmetric capacity is separated from the known linear answer.
Open
\(\frac5{13}\le C_{\mathrm{sym}}\le\frac{11}{28}\)
For a finite confusability graph, the regularized independence number defines capacity but is difficult to compute or characterize.
Open
\(\Theta(G)=\sup_{n\ge1}\alpha(G^{\boxtimes n})^{1/n}\)
A registry, not a proof monorepo
Capacity Atlas keeps canonical channel specifications, controlled tags, and versioned Lean statements in one reviewable repository. Substantial proofs live in dedicated repositories and link back by immutable commit.
The statement-first organization, external-proof links, and simple browse interface are inspired by Google DeepMind's Formal Conjectures. See the acknowledgements for the exact design influences and citation.