The Bare Binary Biconditional: Its Quantum-Calculus Correspondence and Implications for the Equal-Weight Envariance Gap

This paper presents a cross-disciplinary synthesis connecting bare binary symmetry, half-probability, Bernoulli product measure, Lebesgue measure, integration, and existing symmetric binary quantum measurement. Within the Bernoulli family, it establishes an exact correspondence between exchange invariance of a two-outcome law, the value p = 1/2, Lebesgue measure under dyadic pushforward, and almost-sure convergence to half-frequency. It then addresses the equal-weight envariance gap by stating an explicit physical-bareness premise, supported by but not derived from the cited foundational results, and shows that the value this premise selects is the same one the correspondence independently distinguishes. The contribution is the connection of these independently developed mathematical and physical results into a single bounded structure. It also presents the binary-Lebesgue construction as a clearer, exact replacement for the familiar but misleading Zeno picture of calculus and limits.