Critical-Scale Clustering and the Breakdown of Universality for Near-Maximum Phase Measures of Exponential Sums

This paper studies parameter families of finite exponential sums in which two tie endpoints of a near-maximum level set approach at the critical rate δ_r ≍ qr-1/2. The endpoint separation, half-bowl width, and competitor-kernel decay length then coincide. The cluster trio is rigid: its three exponents are determined by the level and endpoint positions, while coefficient ratios and other competitors require separate control. In a realized family, six rescaled kernels carry the interaction through one microscopic phase. Under explicit kernel-convergence hypotheses, the second-order term has a joint limit given by the excess sublevel length of a limiting profile, decomposed into a separated-endpoint sum and an interaction term. For an explicit four-term family, the hypotheses are verified with uniform explicit constants for every admissible phase lock. Two families have identical spectra and corresponding coefficient moduli at every radius but different critical limits. Along an unlocked calibration, phase dependence yields non-existence of the critical limit for specified parameters; certified intervals quantify the gap, and failure persists on an open parameter set. For a positive zero-lock family, upper and lower bounds describe the excluded shared arc. Full exclusion never occurs, and nonempty exclusion requires an additional parameter condition. The formal endpoint-anchored cap cannot be realized by a family satisfying the cluster hypotheses. High-precision numerical measurements on actual exponential sums compare with the profiles and exhibit phase dependence.