This paper investigates the emergent spectral properties of a novel class of real symmetric matrix operators subjected to dynamic forward cumulative boundary controls. We implement a deterministic discrete bang bang controller that dynamically adjusts the matrix diagonal to actively counteract the global scalar drift induced by arbitrary binary input sequences. This includes both prime seeded modulo six spin mappings and stochastic noise. Spectral analysis demonstrates that the resulting operator, modeled as a Rank one outer product perturbed by a bounded diagonal matrix, is strictly governed by the Dirichlet Secular Equation under interlacing boundaries. Through numerical stress tests at scale, we report an unprecedented Spectral Platform Condensation phenomenon. Driven by spontaneous symmetry breaking under the feedback valve, all non trivial real eigenvalues are prevented from random scattering. Instead, they freeze into macroscopic high density degenerate platforms trapped precisely within a narrow real closed interval between minus two and zero. This robust universal spectral alignment offers a novel deterministic algebraic framework with significant utility for topological flat band simulations in condensed matter physics, synchronization controls in complex network topologies, and boundary stabilization in associative neural networks.
Yue Lu (Wed,) studied this question.