nested qubits · the QFT phase-ruler

each shell rotates at HALF the angle of the one outside — 360→180→90→45→22.5→0 · this is the Quantum Fourier Transform

the nest · {layer, angle, count}

l0 · 360° · 1I — whole turn (home)
l1 · 180° · 2Z — coarsest bit
l2 · 90° · 3S
l3 · 45° · 4T
l4 · 22.5° · 5 ×2√T — finest, paired
l5 · 0° · 6I — fully read (home)

what the nesting DOES

a phase written in binary, read one bit per shell. each shell halves the angle to catch the next finer bit — a ruler for phase. six shells = six bits. this is the engine of Shor\u2019s factoring: turn "find the period" into "read this phase," then nest it out.

🧒 toddler corner — a ruler made of spins

picture rulers inside rulers. the biggest has one mark: a whole turn. inside it, a ruler with the turn cut in half. inside that, cut in half again — quarter, eighth, sixteenth. each nested ruler measures the leftover the bigger one couldn\u2019t, twice as finely. line them all up and you can read a spin\u2019s exact angle bit by bit, from coarse to fine. that\u2019s what these nested qubits do — they\u2019re a ruler built of spins, and reading the ruler is how a quantum computer factors giant numbers. the finest ruler (the 22.5 one) is the shakiest, so you keep two of it. you start at a whole turn and finish at zero-left-to-measure: you\u2019ve read the whole number.

Nested qubits as the Quantum Fourier Transform: each shell rotates at half the phase angle of the shell outside it, reading a phase bit by bit.