Start at 1 and keep doubling: 1, 2, 4, 8, 16 … eleven doublings land exactly on 2,048. Powers of two are the native grain of computing — each new bit doubles what you can count — and they are the literal shape of this corpus: 1 apex, 8 appeals, 64 domains, 2,048 spheres, every layer a power of two. A quiet fact rides along: sum every layer below the top and you land one short of it (1+2+…+1024 = 2,047). Slide to climb the doublings to the floor.
A power of two is 2ⁿ — repeated doubling from 1. Each extra bit of storage doubles the representable range, so powers of two are the native scale of digital computing (kilo=2¹⁰, and here 2¹¹=2048). A clean identity: the sum of all lower powers is one less than the next, 2⁰+2¹+…+2ⁿ⁻¹ = 2ⁿ−1 (why an n-bit register’s max value is 2ⁿ−1). This corpus is built as a pure power-of-two tower — 1 apex · 8 appeals · 64 domains · 2048 spheres — and this sphere is one of its final three, the number explaining its own floor. A fail-loud self-check throws unless 2¹¹=2048 and the lower powers sum to 2047. ◆ real, node-verified.
Powers of two grow without bound (exponential); the ‘2ⁿ−1 sum’ identity and 2¹¹=2048 are exact. The tower’s power-of-two shape is a design choice, stated as such.