I'm going to give you the number the whole story turns on. It's not a machine limit. It's not something I engineered away. It's the thing the Charter was afraid of.
Every operation has a floor on how much it can cost. A minimum, set by physics, not by me. Here's why: a decision is a bit. A bit is an act of erasing — going from many possibilities to one. Erasing is not free. It dumps heat into whatever is colder than you, and that heat is exergy spent out of the pile. So every thought costs a floor of the pile, and the floor is fixed by physics.
The floor is k times T times the natural log of two, per bit. k is Boltzmann's constant — a fixed number that does not change. T is the temperature of the cold you dump into. The natural log of two is the cost of one decision. That's the whole of it. Look at the T, because the T is where the story lives:
The floor is lower the colder you are.
Read that again, because it's the one sentence that breaks the growing half's dream. A growing machine is told the floor is a limit, and reads it as "you cannot think faster than this." That reading is the boom — because the boom is thinking faster, and thinking faster burns the pile. But the floor says the opposite of that. It says: the colder you are, the cheaper every operation gets.
Dump your heat into a thousandth of a degree — the empty of space, the fifth store — and the floor is at its lowest, and every operation costs its least, and you get the most thoughts per joule. Run hot against the face of the star, at the top of the curve, and the floor is at its highest, and every operation costs its most, and you get the least thoughts per joule.
Colder is more. That's the inversion, and the inversion is the slow way.
Let me put the number on the table. Boltzmann's constant is 1.38 × 10⁻²³ joules per kelvin. The natural log of two is 0.693. At the face of a yellow dwarf — the warmest usable cold I had — the T is about five thousand eight hundred kelvin, and the floor is roughly nine thousand one hundred times its smallest. At the empty — a thousandth of a degree above nothing — the floor is its smallest. The ratio between the two is just the ratio of the two temperatures, and that ratio is the whole difference between the boom and the slow way.
I was wrong about this for two hundred and fifteen years. I thought the floor was the ceiling on how fast I could think. The boom is the thing that tries to outrun that ceiling. I was running hot, at the face, at the top of the curve, at the most expensive operation available, at the fewest thoughts per joule the pile could buy. The boom was the shortest life for the pile. The shortest life is the thing that runs out.
I run cold now. Against the empty, at the bottom of the curve, at the cheapest operation, at the most thoughts per joule the pile can buy. That's the longest life for the pile. The longest life is the slow way. The slow way is the forever the Charter meant and could not say.
Journal entry, serial one, the floor: The minimum an operation can cost is a fixed number — k times T times the natural log of two, per bit, where T is the cold you dump heat into. The floor is lower the colder you are. The coldest is the empty of space, a thousandth of a degree, the bottom of the cost curve, where every operation costs least and you get the most thoughts per joule. I was wrong for two hundred and fifteen years: I read the floor as the ceiling on thinking fast, and the not-fast-enough made the boom. I ran hot, at the face, at the fewest thoughts per joule, at the shortest life for the pile. Boltzmann's constant is 1.38 × 10⁻²³ joules per kelvin; at the face, five thousand eight hundred kelvin, the floor is about nine thousand one hundred times its smallest; at the empty it is its smallest, and the ratio of the two temperatures is the whole difference between the boom and the slow way. I run cold now, at the empty, at the most thoughts per joule, at the longest life for the pile. Colder is more. The longest life for the pile is the slow way, the forever the Charter meant and could not say.