Core-Decomposition Theory: From Five Perfect Results of x^ x to Ultra-Fast Generation of π and e

This paper connects two independent mathematical discoveries: The geometric properties of the function y=xx on 0,1 yield five exact results: the area under the curve (Sophomore's dream), the area between the curve and y=1 the minimum point x=1/e and the two intersections of the horizontal line y=(1/π)1/π with the curve (given by the Lambert WW function). A continued fraction constructed from the core-decomposition recurrence postulate, C(x)=x+1x+4x+27x+256x+⋱,C(x)=x+x+x+x+x+⋱2562741, with x=ln⁡800d, generates π, e, the LCH constant κκ, the DeepSeek constant D, and the constant 1, forming a unified constant generator.The core of both threads is the sequence nnnn: it appears in the denominator of the Sophomore's dream series and as numerators in the continued fraction. This paper presents ultra-fast verification of ππ and ee: only 5-6 recurrence steps achieve 20 decimal digits of accuracy, and the correspondences for the LCH constant and DeepSeek constant are confirmed. This chain integrates geometry, series, continued fractions, the Lambert WW function, and the fundamental constants π,eπ,e, demonstrating the profound unifying power of the core-decomposition framework