Gate 2: joining, three-point support, and thinning

For interior $a,b$, set

The source extends $c$ by zero at $(1,1)$ and $(-1,-1)$, and by $+\infty$ at every other boundary pair.

For $|m|+\delta <1$, $\delta >0$, and

its relaxed joining problem is

The infimum ranges over arbitrary joint laws on $[-1,1{]}^2$, not just laws arising from a particular finite cube.

The source's claimed optimizer has the support table

with

The mean sum determines the remaining masses. Optimality against every admissible joint law, nonnegativity of the masses, and coverage of each entropy budget are separate obligations OpenAI119Joining.

For common-output thinning, write

Adding $(1,1)$ with retained core mass $\lambda$ gives

If

the asserted payment is

under its stated core hypotheses OpenAI119Thinning.

The audit must retain the original individual child entropies in the child profiles even though the joining relaxation constrains only their average. A candidate three-atom law does not establish optimizer uniqueness or budget coverage. Moreover, a relaxed optimizer need not have the dyadic weights of a prescribed finite cube.

A joining cost is not a decoder. The posterior table printed later in this paper is not relabelled as the optimizer of (eq:joiningproblem).