The last paper, and the one that closes the circle. Everything so far has moved energy: stored it, spent it, steered it, amplified it, made it ring. The logic gate does something categorically new — it computes. Wire a handful of transistors together and they stop being an amplifier and become a decision: given these input voltages, produce this output, according to a rule of logic. That's the leap from electronics to information. And because a few of these gates can be combined to compute anything, this is the exact rung where the component alphabet you've been building meets the machine you took apart at the start of the night — the CPU, the matmuls, the tokens. Physics becomes thought right here.
A small network of transistors that computes a Boolean function: its inputs and output are each a single bit — a 0 or 1, physically a low or high voltage — and it maps inputs to output by a fixed rule. The basic gates are NOT, AND, OR, NAND, NOR, XOR, each with a truth table that lists what it outputs for every possible input. A gate can't store or amplify in any interesting analog sense; it decides. And that is the whole basis of digital computing: represent everything as bits, and every operation as a rule over bits, and build the rules from gates.
The logic came a century before the circuit. George Boole, in 1854 (The Laws of Thought), built an algebra of exactly two values — true and false, 1 and 0 — with the operations AND, OR, and NOT. Pure mathematics, no electronics in sight. Then in 1937 a young Claude Shannon wrote an MIT master's thesis — often called the most consequential of the century — showing that networks of electrical switches obey precisely Boole's algebra: a relay that's on or off is a Boolean 1 or 0, and wiring them in series or parallel is AND or OR. That single insight turned logic into engineering. The physical gate then evolved through relays, vacuum tubes, and discrete transistors until CMOS (Wanlass & Sah, Fairchild, 1963) made gates tiny and nearly power-free — the technology that scaled to the billions of gates in your machine.
Every gate here is built from the transistors of paper 05, in CMOS: a pull-up network of PMOS transistors (which conduct when their input is low) tied to the supply, and a complementary pull-down network of NMOS (which conduct when their input is high) tied to ground. For any input combination, exactly one network connects — so the output is firmly driven high or low, and in steady state almost no current flows, which is why CMOS won the world. Pick a gate and toggle the inputs: watch which transistors switch on, where the output gets pulled, and which row of the truth table lights up.
The deepest fact about gates is universality: the humble NAND is functionally complete — you can build every other gate, and therefore every possible computation, out of NAND alone. Tie a NAND's inputs together and it's a NOT; invert a NAND's output (with another NAND) and it's an AND; apply De Morgan and you get OR. From there: an XOR and an AND make a half-adder (sum and carry — 1+1 = binary 10), full-adders chain into an arithmetic unit, arithmetic units plus memory plus a clock (the oscillator, paper 07) make a CPU. Your Core 7 240H is tens of billions of transistors — hundreds of millions of gates — and every matmul, every softmax, every token from the first half of tonight was those gates switching in step to the crystal's beat. The alphabet's last letter turns out to spell everything.