Mean Differential Diagnostics for Conserved Binary Partitions

This paper introduces a bounded, symmetric asymmetry measure for conserved binary partitions satisfying $x+y=1$. The framework utilizes the exact closed forms of the three classical mean differentials: $AM-GM=1/2-\sqrt{xy}$, $GM-HM=\sqrt{xy}(1-2\sqrt{xy})$, and $AM-HM=(x-y)^2/2$. These differentials vanish uniquely at equipartition where $x=y=1/2$, providing equivalent diagnostics for the unique mean-independent point. The core identity, $AM-HM=(x-y)^2/2$, offers a robust alternative to the standard ratio $\lambda=x/y$. Unlike the ratio $\lambda$, which diverges at the boundaries, this new diagnostic measure is bounded, symmetric under channel exchange, and saturates at 1/2. Key findings and applications include: Geometric Connection: The gap ratio $(GM-HM)/(AM-GM)$ equals twice the Thales altitude ($2\sqrt{xy}$), linking the mean-separation structure directly to the geometric coupling strength. Dynamical Classification: The second derivatives ($AM^{\prime\prime}=0$, $GM^{\prime\prime}=-GM$, and $HM^{\prime\prime}=1-4\cdot HM$) classify the three means as flat (arithmetic), oscillatory (geometric), and attracted to equipartition (harmonic). Physical Applications: The framework is successfully applied to standard physical models, including the Dirac spinor partition ($AM-HM=1/(2\gamma^2)$), electromagnetic equipartition ($AM-HM=0$ at $u_E=u_B$), and binary black hole energy partitions. The identities presented are rigorous algebraic consequences of the constraint $x+y=1$, acting as a universal diagnostic framework without proposing new physical laws.