Lattice reduction smooths the Gram-Schmidt profile, and we use majorization to describe the local swap mechanism behind that smoothing. In this language, each non-degenerate Lovász swap acts as a T-transform on the log-norm profile. As a consequence, every strictly Schur-convex measure of profile spread decreases at such a swap. Two structural consequences follow. First, the worst-case GSA envelope admits a variational interpretation. It is the unique minimum-variance profile compatible with the Lovász gap geometry, so its slope is determined by the LLL parameter alone. Second, the realized swap trajectory satisfies an exact telescoping identity for variance dissipation. The same viewpoint also helps organize deep-insertion heuristics. It suggests a thermal family of Schur-convex scoring rules, motivates adaptive selection within that family, and leads to two concrete selectors: Thermal-Adaptive, which reduces operation counts relative to SS-GG on flat profiles in our benchmarks while recovering SS-GG on $q$-ary inputs, and Geodesic Deep-LLL, which reduces equivalent-swap counts on structured lattices in our benchmarks at higher wall-clock cost.

翻译：格约化平滑了Gram-Schmidt剖面，我们利用优势化来描述这一平滑过程背后的局部交换机制。在此框架下，每个非退化的Lovász交换相当于对对数范数剖面施行T-变换。因此，任何严格Schur凸的剖面扩散度量都会在此类交换下递减。这一结论隐含两个结构性推论：首先，最坏情况下的GSA包络具有变分诠释——它是与Lovász间隙几何相容的唯一最小方差剖面，因此其斜率仅由LLL参数决定；其次，已实现的交换轨迹满足一个精确的方差耗散伸缩恒等式。同一视角也有助于梳理深度插入启发式策略，它建议采用一族热力学式的Schur凸评分规则，激发在该族规则内的自适应选择，并衍生出两种具体选择器：Thermal-Adaptive（在基准测试中，相对于SS-GG，该选择器在处理平坦剖面时减少了操作次数，同时在处理q元输入时恢复SS-GG的性能）和Geodesic Deep-LLL（在基准测试中，该选择器在结构化格上减少了等价交换次数，但代价是更高的墙钟时间）。