Per-letter mutual-information superadditivity under input independence: I((X₁, X₂); Y) ≤ I(X₁; Y | X₂) + I(X₂; Y | X₁). This is the hsub well-formedness hypothesis of mac_avgPentagon_mem_convexHull; it is a universal geometric fact about the product input, threaded here exactly like the existing hac/hbc corners mac_macInfo₁/₂_le_macInfoBoth. Proved by the two chain-rule decompositions I((X₁, X₂); Y) = I(X₂; Y) + I(X₁; Y | X₂) and the identity I(X₂; Y | X₁) = I(X₂; Y) + I(X₁; X₂ | Y) (the I(X₁; X₂) = 0 term drops under the independent product input), so I(X₂; Y) ≤ I(X₂; Y | X₁) and the claim follows.

Membership of the shrunk rate point of a single code: for a length-n two-user code with 2 ≤ M₁, 2 ≤ M₂ and ⌈exp (n Rⱼ)⌉ ≤ Mⱼ, if the uniformly-shrunk rate point (R₁(1−Pe) − log2/n, R₂(1−Pe) − log2/n) (with Pe the average error probability) is in the first quadrant, then it lies in the closed convex hull of all per-input pentagons. Combines the finite-n Fano bounds with the geometric gateway mac_avgPentagon_mem_convexHull and the per-letter identifications mac_condMI_eq_macInfo₁_at and friends.

Interior case of the converse: for strictly positive rates, an achievable pair lies in the closed convex hull of the per-input pentagons.

Finite-n Fano corner bound for user 1: the single user-1 corner inequality log |M₁| ≤ ∑ᵢ I(X₁ᵢ; Yᵢ | X₂ᵢ) + h(Pe₁) + Pe₁ log(|M₁| − 1), obtained directly from mac_converse_bound₁ and mac_singleletterize_bound₁ on the canonical ambient measure, without routing through the two-user mac_converse_from_code. Requires only 2 ≤ M₁; user 2 enters only through NeZero M₂, so this survives the M₂ = 1 axis degeneracy that blocks the joint converse.

Finite-n Fano corner bound for user 2, symmetric to mac_converse_from_code_bound₁: requires only 2 ≤ M₂, surviving the M₁ = 1 axis degeneracy.

Membership of the shrunk rate point when user 2 is silent (R₂ = 0): the analogue of mac_converse_rate_mul_one_sub_errorProb_mem_of_ceil_exp_le for the point (R₁(1−Pe) − log2/n, 0), which uses only the user-1 Fano bound (mac_converse_from_code_bound₁, needing just 2 ≤ M₁) plus per-letter nonnegativity, so it survives the M₂ = 1 degeneracy.

Membership of the shrunk rate point when user 1 is silent (R₁ = 0), symmetric to mac_converse_rate₁_mul_one_sub_errorProb_mem_of_ceil_exp_le.

Converse on the single-user axis with user 2 silent: for a strictly positive rate R₁ achievable with R₂ = 0, the pair (R₁, 0) lies in the closed convex hull of the per-input pentagons. Uses the user-1-only finite-n Fano bound mac_converse_from_code_bound₁ (which needs only 2 ≤ M₁, and thus survives the M₂ = 1 degeneracy of the axis), then takes the Fano → 0 limit as in the interior case.

Converse on the single-user axis with user 1 silent, symmetric to mac_timesharing_converse_rate₁: uses the user-2-only finite-n Fano bound mac_converse_from_code_bound₂ (needing only 2 ≤ M₂, surviving the M₁ = 1 degeneracy), then takes the Fano → 0 limit.

Time-sharing converse for the MAC: every achievable first-quadrant rate pair lies in the closed convex hull of the union of all per-input pentagons macPentagon p₁ p₂ W over probability inputs p₁, p₂. Assembled by casework on whether each rate is zero or positive: the interior case uses the Fano → 0 limit mac_timesharing_converse_interior, the origin (0,0) lies in any pentagon, and the two axis cases reduce to the single-user Fano corner via mac_timesharing_converse_rate₁ and mac_timesharing_converse_rate₂.

Clamping a rate pair into the first quadrant does not change achievability. MACAchievable depends on the rates R₁, R₂ only through the message-count thresholds ⌈exp (n Rⱼ)⌉ ≤ Mⱼ, and ⌈exp (n R)⌉ = ⌈exp (n (max R 0))⌉ for every block length n (for R < 0 both ceilings equal 1, since exp (n R) ∈ (0, 1]). This lets the antisymmetry argument fold a negative achievable rate back onto the axis.

Full first-quadrant characterization of the MAC capacity region (Cover–Thomas): for a channel whose transition probabilities are all strictly positive, the operational capacity region, intersected with the first quadrant, equals the closed convex hull of the union of all per-input pentagons macPentagon p₁ p₂ W over probability inputs p₁, p₂. The ⊆ half is the converse (mac_timesharing_converse, with negative rates clamped back to the axis via mac_achievable_clamp_iff); the ⊇ half is achievability (mac_achievability_region_allprob, whose pentagons already lie in the first quadrant).