Uniform fiber bound for the (V₁, Y₁) jointly typical set: whatever the received word, the V₁-block mass of the codewords jointly typical with it is at most exp(−n (I(V₁; Y₁) − 3ε)).

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Averaged alias bound at receiver 1. For an arbitrary output law ν, drawing the alias codeword independently of ν from the V₁-block law makes the probability that it is jointly typical with the received word at most exp(−n (I(V₁; Y₁) − 3ε)).

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Uniform fiber bound for the (V₂, Y₂) jointly typical set: whatever the received word, the V₂-block mass of the codewords jointly typical with it is at most exp(−n (I(V₂; Y₂) − 3ε)).

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Averaged alias bound at receiver 2. For an arbitrary output law ν, drawing the alias codeword independently of ν from the V₂-block law makes the probability that it is jointly typical with the received word at most exp(−n (I(V₂; Y₂) − 3ε)).

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A Marton subcodebook: for each message and each covering index one length-n auxiliary codeword. The covering index is the over-provisioning that lets the encoder pick a jointly typical pair of auxiliary codewords; unlike the message it is not decoded.

The subcodebook law: p-i.i.d. over all M · M' · n auxiliary letters, read as a codebook whose alphabet is the set of length-n words.

The encoder's covering choice for one message pair: an index pair whose two auxiliary codewords are jointly strongly typical, falling back to (0, 0) when the two subcodebook rows contain no such pair. It reads only the two rows addressed by the message pair, which is what keeps the codewords of every other row independent of the transmission.

Strong typicality — rather than the weak typicality the decoders use — is what pins the empirical type of the selected pair, and hence, through the input kernel, the empirical type of the transmitted (V₁, X) block that the receiver-1 conditional AEP consumes. The selection radius ε_cov is therefore a parameter of its own, independent of the decoding radius.

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The first auxiliary codeword actually transmitted for the message pair m.

The second auxiliary codeword actually transmitted for the message pair m.

The input codebook law conditional on the two subcodebooks: the input word of the message pair m is drawn letterwise from K applied to the selected auxiliary pair, independently across message pairs. This is where the input randomization of a general kernel K is placed, the deterministic BroadcastCode.encoder being fixed only after a pigeonhole step.

Receiver-1 decoder: return the unique message owning an auxiliary codeword jointly typical with the received word, falling back to ⟨0, hM₁⟩ when there is none or more than one. Only the message is required to be unique — the covering index is quantified existentially, so a second typical index inside the transmitted row is not an error.

Receiver-2 decoder, the mirror image of martonMessageDecoder₁.

Bundle two subcodebooks and an input codebook into a BroadcastCode: the input codebook is the encoder, and each receiver runs its own message decoder over its own subcodebook.

Receiver-1 error decomposition at a fixed message pair: the error probability is at most the probability that the transmitted auxiliary word fails to be jointly typical with the received word, plus the sum, over the auxiliary codewords of every other message row, of the probability that one of them is jointly typical with it. Quantifying the covering index existentially is what keeps a second typical index inside the transmitted row off the error event, so the alias sum ranges over the rows of the other messages only.

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The message-averaged form of marton_errorProbAt₁_le_bonferroni: averaging the pointwise decomposition over all message pairs bounds the receiver-1 average error probability by the mean of the transmitted-pair term and of the alias sum. The bound holds at every selection radius ε_cov, which is what lets the covering radius be chosen independently of the decoding radius.

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Averaged alias bound over the full three-tier Marton ensemble. An alias codeword taken from a different message row than the transmitted one is jointly typical with the received word with probability at most exp(−n (I(V₁; Y₁) − 3ε)). The covering choice reads only the transmitted rows, so the alias row stays independent of the transmission even though the choice depends on the codebook; and because the fiber bound behind marton_alias₁_slice_avg_le is uniform over received words, the law of the received word never has to be identified. The estimate reads the selection only through the fact that it yields a probability law on input words, so it is insensitive to the radius and to the typicality notion the selection tests.

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Receiver-2 error decomposition at a fixed message pair, the mirror image of marton_errorProbAt₁_le_bonferroni: the error probability is at most the probability that the transmitted auxiliary word fails to be jointly typical with the received word, plus the sum, over the auxiliary codewords of every other message row, of the probability that one of them is jointly typical with it.

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The message-averaged form of marton_errorProbAt₂_le_bonferroni: averaging the pointwise decomposition over all message pairs bounds the receiver-2 average error probability by the mean of the transmitted-pair term and of the alias sum.

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Averaged alias bound over the full three-tier Marton ensemble at receiver 2, the mirror image of marton_random_codebook_alias₁_le. An alias codeword taken from a different message row than the transmitted one is jointly typical with the received word with probability at most exp(−n (I(V₂; Y₂) − 3ε)). The ensemble is summed in the same order as at receiver 1, the alias row now sitting in the inner subcodebook tier.

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