Outer N construction — max-error closed form #

Part file split from ShannonTheoremMaxError. Constructs the outer N₀ that simultaneously controls the TV smoothing error and the smooth-channel achievability.

Outer N construction #

For any R < capacity W and ε > 0, we build an N₀ such that for every n ≥ N₀ we can simultaneously:

• pick δ_n ∈ (0, δ_B] with 2 n δ_n < ε/2 (so the TV bound contributes ≤ ε/2);
• build a code at the smooth channel W_smooth δ_n with max-error < ε/2.

The construction:

1. The uniform smooth-capacity bound gives p_full := pSmooth p₀ δ_p (full support) with R < I_lb < I(p_full; W_smooth δ) for all δ ∈ (0, δ_B].
2. Choose an interior rate R' := (R + I_lb)/2 < I_lb for the closed-form average-error code; we then upgrade to max-error at rate R via the subcode trick (giving max-error ≤ 2 · avg < ε/2).
3. Choose δ_n := min(δ_B, ε/(16(n+1))) and check 2 n δ_n < ε/4 < ε/2.
4. Bound V_Y(δ_n), V_Z(δ_n) ≤ const + 2 · (log(n+1))² via the closed-form pmfLog bounds, using 1/δ_n ≤ (1/δ_B + 16/ε)·(n+1) and (a+b)² ≤ 2a²+2b².
5. The closed-form channelCodingSmoothMinN is then O((log(n+1))²); exists_N_log_sq_plus_const_le_n produces the outer N₀.