We study the prior-free zero-error private-coin external information complexity $B(f)$ of finite two-party functions, with correctness on every input. An elementary rectangle/entropy argument gives $B(\mathsf{AND})=\log_2 3$. A complete classification of Boolean $3\times3$ matrices leaves, up to input permutations and output complementation, only the two-overlapping-rectangles function $F$ without an exact value. An entropy potential and a finite rational protocol give $2.1212661667060486842\ldots\le B(F)<2.130998318$. We also determine its deterministic external information and zero-error communication costs. A separate product-prior certificate and internal-information additivity yield $R_0^{\mathrm{pub}}(F^{\times t})\ge2.021991187178562\ldots\,t$, strictly exceeding the fractional-cover rate $2t$.
We study bounded-error randomization and majority operators in structural communication complexity. We give a communication-complexity formulation of Tarui's theorem and identify the bounded-error closure of weakly unbounded-error communication with a distributional perturbation of logarithmic margin complexity. Retaining the Boolean guarantee in Tarui's probabilistic polynomials yields the stronger inclusion $\mathsf{PH}^{\mathrm{cc}} \subseteq \mathsf{ZP}\mathbin{\cdot} \mathsf{PP}^{\mathrm{cc}} \subseteq \mathsf{BP}\mathbin{\cdot} \mathsf{PP}^{\mathrm{cc}}$. We also define a majority operator satisfying $\mathsf{PP}\mathbin{\cdot} \mathsf{P}^{\mathrm{cc}} = \mathsf{PP}^{\mathrm{cc}}$ and use Sasaki's majority formula to prove that $\mathord{\oplus}\mathsf{P}^{\mathrm{cc}} \subseteq \mathsf{PP}\mathbin{\cdot} \mathsf{PP}^{\mathrm{cc}}$. For the inner-product function modulo two, both the outer index and each inner protocol have logarithmic cost. Together with Forster's lower bound, this gives an explicit witness to the non-idempotence of the majority operator and to $\mathsf{PP}\mathbin{\cdot} \mathsf{PP}^{\mathrm{cc}} \not\subseteq \mathsf{UPP}^{\mathrm{cc}}$. Both the bounded-error closure and the two-level majority class are contained in $\mathsf{PSPACE}^{\mathrm{cc}}$.