How Did the Universe Begin?

Before anyone has moved a single piece, the rules of chess have already settled something. White has exactly 20 legal first moves. Each of the eight pawns can step one square or two (16 moves), and each of the two knights can jump to one of two squares (4 more).

Now open the record of any chess game ever played. Line one says which of those 20 was chosen. The rulebook never says. It can't. A rulebook lists what's allowed. What was played is a separate fact, and only the scoresheet has it.

Chess is a drawing of the real mechanism here. The laws are the rulebook. The record is the scoresheet. On the first line of reality's scoresheet the same thing happens: the laws allow a short list of first moves, and one of them gets written down.

The list is short. Reality's rulebook allows exactly six first moves.

So how did the universe begin? With one mark in the Ledger, the smallest change there is, written in one of those six places. This post is about that line, because everything that has happened since is written under it. What was it? What did it cost? And what picked it?

Last time, in Did Time Have a Beginning?, we found the page that line one goes on. It's blank: nothing compared, nothing owed, cost zero. It's the first page because the record counts events that happened, and no count of events goes below zero. The Ledger sat in front of it with its pen up and nothing to note.

If you haven't met the Ledger, it's the permanent record of everything that happens, drawn as a thick book with a deadpan face. It has no eraser, and it has exactly one response to any entry: Noted. (Part 1 introduced it.)

The Ledger keeps two lists. The first is the entries written. What they add up to is a small table called the books: three rows, two counts in each row, six counts in all. Each count says how many entries of its kind have been written, and every entry adds one mark to one count. The second list is the comparisons made. Step 3 gets to that one.

This post covers the first entry. The hot, expanding universe astronomers measure is a bigger story, and I'll mark the exact spot where what we know stops.

The short answer

Here's the whole answer in six steps. The rest of the post takes them one at a time and shows how we know each one.

1. The first entry is one mark. The smallest change the books can make is one mark on one count. The first thing that ever happened is that change.
2. It had six places to go. The books have six counts: three rows, two sides each. So there are exactly six possible first marks, and each one leaves different books.
3. It went in with its reverse, and it cost something. A mark records a comparison, and every comparison goes in with its reverse, so what's owed nets to zero. The pair costs 2J(r): twice J(r), the cost of one comparison at ratio r (the ratio says how lopsided it is). That's more than zero for every ratio except 1. At ratio 2 the pair costs 1/2. After that the total never comes down.
4. No law picks which of the six. All six are allowed, all six cost the same, and the laws say nothing more. The record holds which one it was.
5. The first complete moment takes at least eight ticks. Each tick writes one mark. A complete moment touches all eight corners of a cube and comes home, and the fastest trip is 8 ticks: 2 × 2 × 2. It can only finish on an even tick.
6. Where this stops. How these first entries become the hot, expanding universe astronomers measure isn't derived yet.

Why this matters: it says what a beginning is made of. The first thing that ever happened is about as small as a thing can be: one mark in a table of zeros, entered with its reverse, at a cost. It's an entry like every entry since, with the same bookkeeping. And one thing about it isn't settled by any law: which of the six it was. That's last time's split again, now at line one. The laws say what's allowed. The record holds what happened.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got, and why the next step comes next. The Ledger keeps the map.

Step 1: The first entry is one mark

The first thing that ever happened is the smallest change the books can make: one mark on one count. Here's how we know it's the smallest.

Start with the blank books: three rows, two counts in each, every count at zero. (I'll call the two counts in a row the left count and the right count.) The Ledger has no eraser, so marks only get added. They never come off.

The first entry has to change those zeros. So what's the smallest change that counts?

First, what a mark is. A mark records one distinction: one thing told apart from another. Telling two things apart is a comparison, and a comparison has a ratio: how many times bigger one side is than the other. From last time, a comparison at ratio x costs J(x) = (x + 1/x) / 2 - 1. Say a mark records a comparison at ratio 2. Then it costs J(2) = 1/4.

No marks isn't a change. One mark is. Two marks are more, and the cost shows it. Add a second mark:

• On a different count: the costs add: 1/4 + 1/4 = 1/2. That's twice the cost of one mark.
• On the same count: the ratios multiply, because each mark multiplies the ratio by 2: 2 × 2 = 4. And J(4) = 9/8. That's four and a half times the cost of one mark, for twice the marks.

However you place them, two marks cost more than one. So one mark on one count is the cheapest change that changes anything, and every other change that only adds marks costs strictly more. That holds at every ratio except 1, not only at 2.¹

That's what a tick is: the smallest change the books can make, one mark. (A tick is a step in order. It isn't a number of seconds.) The first entry is the first tick's mark: one tally stroke in a table of zeros.

That one mark is the cheapest change is proved. That the first entry is one mark, recording one comparison, is a premise. It's adopted, not derived.

The Ledger writes it down. Noted. It's the first Noted. there has ever been.

Zoom out. Step 1 is done. The books have six counts, and the smallest change they can make is one mark on one count. A tick is exactly that. Next, and why: which count? The number six matters, and Step 2 shows where it comes from.

Step 2: It had six places to go

At the start of reality there are exactly six possible first marks, and each one leaves different books.

Count the places. Three rows with two counts each make six counts, so there are six places for the first mark. Here they are, each one starting from the blank books.

Six isn't an accident of how I drew the table. It's the cube again.

Each row has a yes/no: is its gap odd? (A row's gap is the difference between its left count and its right count.) A mark on either count of a row changes that gap by one, so the row flips between even and odd. That's one flip of one yes/no: one step along one road of the cube from Why Space Has Three Dimensions.

The blank books have every gap at zero, which is even, so they sit at corner 000. Three roads leave that corner, one for each row. And two marks run down each road, one for each side of the row, because a mark on either count changes the gap by one. Three roads, two marks each: six.

Different marks make different books. Put the 1 on row 1's left count and you get one page. Put it on row 1's right count and you get another. The six first pages are all different.²

Does the cost pick one? No. The cost of a comparison depends only on how lopsided it is, never on which count it lands on. All six cost the same.

Yeah-But, the voice in the back of your head from Part 1, clears his throat. "Yeah, but why six? Chess gets 20."

Chess gets 20 from its pieces. Reality gets 6 from its books: three rows, two sides each. And three rows because space has three directions, which the space post works out.

Zoom out. Step 2 is done. The first mark has six places to go: three roads out of corner 000, two marks on each, all at one cost, all leaving different books. Next, and why: a mark is more than a tally stroke. It records a comparison, and every comparison goes in with its reverse. Step 3 shows the pair and what it costs.

Step 3: It went in with its reverse, and it cost something

A mark records a comparison, and every comparison goes in with its reverse. The pair costs 2J(r), which is more than zero for every ratio except 1. After that the total never comes down.

Here's where the Ledger's second list comes in. Besides the entries written, it keeps the comparisons made. The first mark records one comparison, and that comparison goes on the comparisons list.

Write it as one amount, A, set against another, B, at some ratio. Say 2: one is twice the other. That's A to B at ratio 2.

Last time you met the rule that comes with every comparison. It goes in with its reverse, B to A at ratio 1/2. The two cancel on balance, so nothing is owed. The net flow, the amount left owing, stays zero. (The reverse goes on the comparisons list. It isn't a second mark: the books still show one.)

And each one costs something. J(2) = 1/4. J(1/2) = 1/4 too, because J gives a comparison and its reverse the same cost. So the first entry, with its reverse, costs 1/4 + 1/4 = 1/2.

In general the pair costs 2J(r), where r is the ratio.³ It's more than zero for every ratio except 1, because J is zero only at 1. At ratio 1 the two sides are equal and nothing has been told apart. Every comparison that tells two things apart costs something.

So the first thing that ever happened was a bookkeeping entry, filed with its mirror image. Reality's opening act was paperwork.

It also ended something. The blank page's cost of zero lasts only until the first comparison that tells two things apart. It had a short career.

From here every comparison, with its reverse, adds the same way: 2J(r) for its ratio r. J is never negative, so a comparison adds something or adds nothing and never takes anything off. Make three pairs at ratio 2 and the total goes 0, 1/2, 1, 3/2. Make a pair at ratio 1 and it stays put.

The total starts rising at the first entry and never goes down.

What was the ratio of the very first comparison? The cost law doesn't set it. J gives a cost for every ratio and picks none. I use 2 because the arithmetic is friendly. Whatever it was, the first entry with its reverse cost 2J(r).

Zoom out. Step 3 is done. The first entry records a comparison that goes in with its reverse, so the net flow stays zero. The pair costs 2J(r), which is 1/2 at ratio 2, and the total has never come down since. Next, and why: six places to go, one cost for all of them. So what decides which one? Step 4 asks the laws.

Step 4: No law picks which of the six

The laws allow all six first marks at the same cost and say nothing more. Which one was written is a fact of the record, like the first move of a game.

Go back to the chess scoresheet. The rulebook allows 20 first moves and says nothing about which one gets played. Hand reality's laws the blank books and ask the same question.

Three exact facts answer it.

All six marks are allowed. Each is one tick of the books, and nothing in the laws forbids any of them.

All six cost the same. The cost depends on the ratio and not on the count, so the cost can't prefer one.

Each leaves different books. Put the mark on row 1's left count and the books read one way. Put it on row 3's right count and they read another. So the choice isn't a formality. The six possible first lines are six different lines, and the laws have nothing more to say about which one gets written.⁴

It's the same at every tick. From any books, the cheapest next marks are the same six, at one cost. A history is the list of marks in the order they were written, and the choices multiply. After tick 1 there are 6 possible histories. After tick 2 there are 6 × 6 = 36, and after tick 3 there are 216. After tick 8 there are 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 = 1,679,616.

Keep going, and by tick 103 the number of possible histories passes 1 followed by 80 zeros, which is about the number of atoms in the observable universe. The rulebook is vast. The scoresheet is one path through it.

Those are counts of what the laws allow. They don't give odds for any history. Count them, don't weigh them.

"Yeah, but who picks?"

Not a law. The laws allow all six, charge the same for each and say nothing more, so they can't prefer one. Which mark was written is a fact of the record, in pen like every other fact in it. I can't tell you which of the six it was, and no law can either.

That's last time's split again, now at line one. The laws say what's allowed. The record holds what happened. And it repeats at every tick: the laws allow six again, and the record holds one.

Zoom out. Step 4 is done. Six first marks, one cost, and nothing in the laws to choose between them. The record holds which one. Next, and why: one piece of the beginning is left to count. One mark is one tick, so how many ticks does the first complete moment need?

Step 5: The first complete moment takes at least eight ticks

Each tick writes one mark. A complete moment touches all eight corners of the cube and comes home, and that takes at least eight ticks. It can only finish on an even tick.

Here's the cube from the space post. It has eight corners, one for each combination of the three yes/no's, and twelve roads. Each tick is one step along one road. A complete moment is one trip that touches every corner and ends back where it started.

The trip has to land on the seven other corners and then land back home: eight landings. A tick makes one landing. So no complete moment fits in fewer than eight ticks. Eight is enough. A trip that visits every corner exactly once and comes home is called a tour, and the drawing shows one.

How many ways can the books make that trip? Exact counts:

• 12 tours. From 000 there are exactly 12 tours: six different loops, each walked in either direction.
• 3,072 histories. Each step of a tour can be written with either mark on its road, so every tour has 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 written forms. 12 × 256 = 3,072.

Of the 1,679,616 histories the laws allow for the first eight ticks, 3,072 make the trip. That's about one in every 547.⁵ Again, a count and not a bet.

Even ticks only. Every tick flips one yes/no. Count the yeses. That count goes up or down by one at every tick, so it's even after an even number of ticks and odd after an odd number. Home is 000: no yeses, an even count. So the trip can only end on an even tick. A first complete moment can land on tick 8, tick 10, tick 12 and so on. It can't land on tick 9.

So the first complete moment comes no sooner than tick 8. Ticks 1 through 7 write entries, and none of them completes a moment. The laws allow a complete moment at tick 8. They don't say it happened then, so "at least eight" is the exact claim. How long a tick lasts in seconds takes one more input, and nothing in this post uses one.

That's the eight from Part 1 and the space post: 2 × 2 × 2, one doubling for each direction of the cube.

Zoom out. Step 5 is done. Each tick writes one mark. A complete moment visits all eight corners and comes home. The fastest trip takes eight ticks, and only even ticks work. Next, and why: all five steps are done. Time to mark where what we know stops.

Where this stops

That's the first entry: one mark, six places, a cost, no law to choose it, and a first complete moment no sooner than tick 8. Here's the edge of what we know.

On last time's grid, the separate test world of clocks and light, the blank page doesn't stay blank. It has zero energy and sits above the lowest state, and the gap pays for the first ticks and the first light.⁶

Astronomers trace the expansion of space back about 13.8 billion years to a hot, dense state, and they call it the Big Bang. How the first entries connect to that state isn't derived yet: not the heating, not the expansion, not the first atomic nuclei, not the cosmic microwave background.

The whole trip, in one breath

The record starts as a blank page: nothing compared, nothing owed, cost zero. The first thing written is the smallest change there is, one mark on one of six counts, with the comparison it records entered with its reverse. The pair costs 2J(r), more than zero for every ratio except 1, and the total never comes back down. The laws allow all six first marks at one cost and say nothing more, so which one was written is a fact of the record. The first complete moment takes at least eight ticks, 2 × 2 × 2, and lands only on an even one.

That's how reality starts to be written: one mark, one cost, no law to choose it. Every entry since goes in the same way, with its reverse, adding to a total that never goes down. The dinosaur's yawn, the asteroid and your seventh birthday party are all written below line one, in the same ink.

(The Ledger would like you to know that line one is the only entry it ever wrote with nothing above it. It says that's why it wrote it neatly.)