The Lipschitz factor relating the type radius of the transmitted (V₁, X)-block to the width of the three entropy bands it has to pin. The sum runs over the whole alphabet, so letters carrying no ambient mass are budgeted for as well.

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The type radius at which the transmitted (V₁, X)-block has to be pinned for the output bands to hold at radius ε. It is a computed term of ε rather than a further parameter, so the signatures downstream carry one radius only. It is strictly smaller than ε, and its amplification by martonBandConst stays strictly inside ε/2, for every ensemble.

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Whatever auxiliary word v₁ and input word x the encoder transmits, as long as their empirical type is pinned to the ambient (V₁, X)-law at radius martonStrongRadius, the channel output leaves the pair (v₁, y₁) outside the weakly jointly typical set with probability at most tol, for every block length past a threshold depending on ε and tol alone.

The threshold is uniform in v₁, x and hence in the code, which is what lets the receiver-1 error decomposition consume it after the encoder's selection has already distorted the law of the transmitted words. The hypotheses on pV, K, W are the Markov-kernel regularity of the ensemble; no full-support assumption is needed. Both properties a free reference law would have to assume are structural here: it is a probability law because the ensemble is built from a probability measure and two Markov kernels, and its (V₁, X)- and (V₁, Y₁)-marginals are consistent because both are marginals of the same compProd chain. A letter of zero ambient mass therefore needs no separate treatment — it enters the type radius and the band constant like any other.

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The hypothesis of marton_condAEP_jointlyTypical is met by the ambient ensemble itself with probability tending to one: an i.i.d. (V₁, X)-block is type-pinned at the strong radius. This certifies that the conditional AEP is not vacuous — it is the ambient counterpart of the pinning the encoder's selection has to preserve.

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The complement reading of marton_condAEP_jointlyTypical.

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The type radius at which the selected auxiliary pair has to be pinned for the transmitted (V₁, X) block to be pinned at martonStrongRadius. The two are separated by the alphabet size because the conditional mean of a letter statistic of the input is an average of the auxiliary type against the input kernel, so a type deviation of the pair is amplified by the number of auxiliary letters before it reaches the input.

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The transmitted (V₁, X) block inherits the type pin of the selected auxiliary pair: drawing the input word letterwise from K applied to a pair whose joint type is pinned at martonCoveringRadius leaves the pair (v₁, x) outside the strongly typical set of radius martonStrongRadius with probability at most tol, uniformly in the selected pair.

The pin the hypothesis supplies is the full empirical type of the pair, which is finer than the (V₁, X)-type the conclusion pins, and the ambient ensemble meets it with probability tending to one at every positive radius, so the statement is not vacuous.

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The receiver-1 error term of a selected auxiliary pair, averaged over the input tier: whatever type-pinned pair the encoder selects, drawing the input word from K and passing it through the channel leaves (v₁, y₁) outside the weakly jointly typical set with probability at most tol.

This is the form the error decomposition of Marton.ErrorAnalysis consumes, since the threshold is uniform in the selected pair and hence in the code.

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