Multiple access channel — time-sharing converse (convex-geometry gateway) #

The pure convex-geometry core of the two-user MAC time-sharing converse. An achievable rate pair (R₁, R₂) bounded, coordinate-wise, by the time averages of a family of per-letter pentagons lies in the convex hull of the union of those pentagons.

This file currently provides only the geometric gateway lemma mac_avgPentagon_mem_convexHull; the measure-theoretic gaps (code→ambient bridge, weak-converse limit extraction, per-letter identification) are handled elsewhere.

Note on hypotheses #

The gateway lemma requires both a i ≤ c i and b i ≤ c i. These are the two single-user mutual-information bounds I(X₁;Y|X₂) ≤ I(X₁,X₂;Y) and I(X₂;Y|X₁) ≤ I(X₁,X₂;Y) in the MAC application. Without b i ≤ c i the statement is false: with n = 2, a = (0,4), b = (4,0), c = (0,4), the point (0,2) satisfies every remaining hypothesis yet the union of pentagons collapses onto the x-axis, so (0,2) is not in the hull.

Module structure #

Umbrella of the Shannon/MultipleAccess/TimeSharingConverse/ family, re-exporting:

• TimeSharingConverse.Bridge — the convex-geometry gateway mac_avgPentagon_mem_convexHull, pentagon well-formedness, the code → ambient bridge, rate extraction, and per-letter information transport.
• TimeSharingConverse.Assembly — the converse headline mac_timesharing_converse, assembled from the Fano → 0 limit, the rate-point construction, and the axis casework.