Rank-Based Bounds on the Band-Edge Response of Gaussian Ball States: Conditional Convergence Statements, Scheme Independence, and Numerical Tests on a Lattice Scalar Field

This paper studies the modular data of Gaussian ground states of a free harmonic field on a periodic lattice with positive mass regulator, restricted to ball-shaped regions. The entropy response of a region to weak band-limited probes is analyzed through the symplectic spectrum of the reduced covariance data. The paper establishes an exact factorization of the modular data through a boundary Schur complement and derives rigorous bounds on the number of symplectic gaps above a threshold. It proves a rank-based bound on the weighted first-order entropy response, extends the analysis to finite perturbation strength, and demonstrates scheme independence of the main results. The exact statements apply to arbitrary finite positive covariance matrices and do not depend on lattice structure, translation invariance, dimensionality, or kernel profile beyond the restricted covariance data. Conditional power-law bounds are derived for classes of Gaussian ball states satisfying explicitly stated hypotheses. A dichotomy theorem shows that rank-based envelope methods alone cannot establish convergence to the continuum conformal prediction. Instead, any successful proof of such convergence must resolve individual mode contributions. Extended-precision numerical calculations are performed for a two-dimensional lattice model with positive mass regulator. The smallest resolved symplectic gaps reach values as small as 5.3e-29. The numerical results confirm the exact theorems and illustrate the limitations of current convergence strategies. Several open problems are identified, including the absence of lower bounds on the smallest gap, the lack of a massless counting theorem, and the unresolved convergence of the response ratio to the continuum prediction. The work contributes rigorous finite-dimensional results on Gaussian modular data, clarifies which assumptions are required for convergence programs based on entanglement equilibrium, and delineates both the possibilities and limitations of rank-based approaches.