Pairwise independence of any BC coordinate selector under the ambient measure.

Channel fold on the (U, X, Y₁) axes: the (U, X, Y₁)-block law of a finite set T equals the cloud/satellite/channel average of the β₁-projected channel mass. Receiver-1 analogue of bc_chan_fold_UY₂_set, obtained from the master fold by projecting the pair output to β₁.

Receiver-1 correct-triple averaged swap for the E0 event: the two-tier random-codebook average of the correct-triple atypical event equals the joint (U, X, Y₁)-block law of the atypical set. Receiver-1 analogue of bc_random_codebook_E0₂_swap.

The receiver-2 E0 error term vanishes: the correct-cloud atypical (U, Y₂)-block mass tends to 0 by the two-variable joint AEP (jointlyTypicalSet_prob_tendsto_one).

The receiver-1 E0 error term vanishes: the correct-triple atypical (U, X, Y₁)-block mass tends to 0 by the three-variable joint AEP (macJointlyTypicalSet_prob_tendsto_one).

Per-codebook averaging bound for receiver 2: the .toReal of the receiver-2 average error probability of the deterministic code bcCodebookToCode cU cX is at most the uniform average of the two-event Bonferroni bound (bc_errorProbAt₂_le_bonferroni).

Per-codebook averaging bound for receiver 1: the .toReal of the receiver-1 average error probability of bcCodebookToCode cU cX is at most the uniform average of the three-event Bonferroni bound (bc_errorProbAt₁_le_bonferroni3).

Monotonicity of the two-tier nonnegative-weighted codebook average.

Additivity of the two-tier weighted codebook average.

Pulling a constant scalar out of the two-tier weighted codebook average.

Interchanging a finite index sum with the two-tier weighted codebook average.

Receiver-2 aggregation: folding the per-message two-event Bonferroni bound together with the two evaluated swaps (E0 mass A, wrong-cloud exponent e2) bounds the two-tier weighted codebook average by the closed form A + (M₂ − 1) · e2.

Receiver-1 aggregation: folding the per-message three-event Bonferroni bound together with the three evaluated swaps (E0 mass A, wrong-satellite exponent eb, wrong-cloud exponent ec) bounds the two-tier weighted codebook average by the corresponding closed form.

Two-codebook average bound for receiver 2: the random-codebook expectation of the receiver-2 average error is at most the (vanishing) E0 mass plus the wrong-cloud exponent.

Two-codebook average bound for receiver 1: the random-codebook expectation of the receiver-1 average error is at most the (vanishing) E0 mass plus the wrong-satellite (E_b) and wrong-cloud (E_c) exponents.

Abstract two-tier pigeonhole. Nonnegative outer weights wU summing to 1, and for every outer index a nonnegative inner-weight family wX cU summing to 1, whose weighted double average of val is ≤ B, force some index pair with val cU cX ≤ B. (If every pair had val > B the weighted average would strictly exceed B.)

Two-tier pigeonhole for the superposition random code: if the random-codebook expectation of the summed per-receiver errors is ≤ B, some deterministic cloud/satellite codebook pair achieves the summed error ≤ B. Bounding the sum lets a single codebook meet both receivers' targets simultaneously.

Base data-processing Markov chain (U, X) → Y₁ → Y₂ for the degraded broadcast joint law: under physical degradedness the degraded output Y₂ is a stochastic function of Y₁ alone (via the degrading kernel Q), hence conditionally independent of the cloud/input pair (U, X) given Y₁. @audit:ok

Superadditivity under degradedness: for a physically degraded channel X → Y₁ → Y₂, the joint information I((U, X); Y₁) dominates the sum of the two per-receiver informations I(X; Y₁ ∣ U) + I(U; Y₂). Chain rule I((U, X); Y₁) = I(U; Y₁) + I(X; Y₁ ∣ U) plus data processing I(U; Y₁) ≥ I(U; Y₂). This makes the receiver-1 joint-decoding rate sum R₁ + R₂ < I((U, X); Y₁) follow automatically from the two corner constraints. @audit:ok

The receiver-1 wrong-cloud rate slack (E_c) vanishes: with the joint AEP gap I((U, X); Y₁) − (R₁ + R₂) − 3ε > 0 and non-negative rate 0 ≤ R₁, the wrong-cloud prefactor (⌈exp(nR₂)⌉−1)·⌈exp(nR₁)⌉ times exp(n(−I((U, X); Y₁) + 3ε)) falls below any tolerance for large n. The 0 ≤ R₁ hypothesis is essential: for R₁ < 0 the ceil ⌈exp(nR₁)⌉ floors at 1 instead of shrinking like exp(nR₁), so the negative slack the gap allocates to the R₁ factor is not delivered and the prefactor diverges. The bound is stated over abstract rates so that a superposition scheme may instantiate it at either the message rates or the subcodebook rates; every such instantiation runs at a strictly positive rate, so the precondition is met.

Achievability half of the degraded broadcast channel coding theorem, in the superposition inner-bound form of Cover–Thomas Elements of Information Theory. Over a physically degraded broadcast channel W with cloud law pU and conditional input kernel K, any rate pair strictly inside the auxiliary-variable region

• R₁ < I(X; Y₁ ∣ U) (= bcInfo₁, the strong receiver), and
• R₂ < I(U; Y₂) (= bcInfo₂, the degraded receiver)

is achievable: for all large enough block lengths n there is a BroadcastCode whose two per-receiver average error probabilities are both below any prescribed ε' > 0. The proof is the two-tier superposition random-coding argument; degradedness X → Y₁ → Y₂ is a structural precondition ensuring the receiver-1 joint-decoding rate sum is met automatically. @audit:ok