This paper develops a single bipartite entanglement diagnostic and follows it from elementary quantum information into holographic gravity. For a pure state of any Schmidt rank, a natural "balance" quantity built from the elementary symmetric polynomials of the Schmidt spectrum turns out to be exactly half the I-concurrence, its mean hierarchy is Maclaurin's and Newton's inequalities, and—via Newton's identities—it is interchangeable with the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield-double models, a closed-form distinction between the maximal-entanglement and first-law (linearized-Einstein) points on the probability simplex, and a Gaussian continuous-variable lift are given. The diagnostic is then lifted to the eternal BTZ black hole, where the concurrence layer is shown to be intrinsically finite-dimensional (it saturates in the continuum) while the moment/relative-entropy layer becomes genuinely geometric. The relative entropy of a single interval is obtained in closed form, S=c3[u26−ln⁡(sinh⁡u/u)]S=\frac{c}{3}[\frac{u^2}{6}-\ln(\sinh u/u)] S=3c[6u2−ln(sinhu/u)]: its first law, an elementary positivity proof, and a quantum-Fisher/bulk-canonical-energy identification follow, and a full reconstruction from the gravitational symplectic flux yields a rigid series in even zeta values resumming to the Ryu–Takayanagi area. The two-interval case is worked out both holographically (mutual information with a phase transition) and for the free fermion via the resolvent (smooth, theory-dependent), making explicit where the universal single-interval law gives way to operator-content dependence. All novel results are stated as propositions/theorems with proofs and verified numerically.