Planar k-uniform states have maximally mixed reductions on every k-site interval of a ring. We introduce planar k-purity—the mean purity of those intervals—as a faithful cost function for this simultaneous constraint. For a Haar-random pure
Planar k-uniform states have maximally mixed reductions on every k-site interval of a ring. We introduce planar k-purity—the mean purity of those intervals—as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of n parties with local dimension p, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every 1\le k\le\lfloor n/2\rfloor. In the balanced qubit case, k=\lfloor n/2\rfloor and N=2^n, the variance is asymptotic to 20/(3nN^2) for even n and 6/(nN^2) for odd n. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through n=10 validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately 2.83 times larger at n=10. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.