Is the Universe Rolling Dice?

Tonight, after dark, stand at a window in a lit room and look at the glass.

You see your own face. You also see the street behind it: the parked cars, the neighbor's porch light. Both at once, in the same pane.

The glass is doing two jobs, and it does them by exact numbers. Light comes in tiny packets called photons. When a photon from your face reaches the glass, it ends up one of two ways. It bounces back to your eye, and that's your reflection. Or it goes through, and it's gone into the night. A window has two glass surfaces, so follow just the first one. Of every 100 photons that reach it, about 4 bounce. The other 96 go through.¹

(By day the bright street swamps that faint 4%. At night the room is brighter than the street, so your reflection shows.)

The 4 is steady. Measure it in Paris or in Tokyo, today or in 1985: the more photons you count, the closer the bounced fraction gets to 4 in 100.

Now the strange part. Take two photons that reach the glass in exactly the same way. One bounces. The other goes through. Nothing we can measure about the photons, the glass or the room tells them apart. Physics gives you the odds exactly, and says nothing about which photons are the 4. Nobody can say which photon bounces, only how many.

Einstein didn't buy it. In December 1926 he wrote to his friend Max Born that he was convinced God does not play dice.

So: is the universe rolling dice?

This post uses one idea from Part 1 of this series, What You Are. Reality keeps a permanent record of every event, and nothing in it is ever erased. We call it the ledger, and each event is an entry. A detector click is an entry: a measurement is one of the events the ledger records. The Ledger, a thick book with a deadpan face, is the drawing I use for it, and it has exactly one response to everything. Noted.

Part 2 added that the present holds everything written so far, plus exact odds for what comes next. This post is about those odds: where they come from, and what writes the next line.

The short answer

The odds are exact, to every decimal, and the result itself isn't written in advance. Whether chance or a rule picks each result, no list of results can say.

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

1. The odds are exact. Every quantum outcome has a chance fixed to every decimal: the squared size of its amplitude (Step 1 explains both words). Five plain rules for reading a record force that square, and nothing else fits.
2. The result isn't written in advance. In a certain two-lab game, answers written down beforehand win at most 75% of the time. Real labs beat that, and quantum physics tops out at 85.4%.
3. The ledger writes one entry per run. (A run is one use of an experiment: one photon, one click.) Before the entry, the measuring instrument's complete output holds both outcomes. Then the ledger writes one, at those exact odds.
4. No list of results can tell chance from a rule. Any list of results, however long, fits pure chance, and it fits a rule nobody has found. A list can break the odds. It can't settle this.
5. One question is left: what picks the entry. Step 5 says exactly what it is and where it stands.

Why this matters: this is where a hidden script would live. If every result were already written inside the particles, chance would be nothing but ignorance, like a shuffled deck whose order is already fixed, and the outcome of every photon's trip to the glass would be settled in advance, waiting to be read. That's the picture Einstein hoped for. Step 2 shows it isn't how things are. The result comes into being when the ledger writes the entry. The rest of the post is how we know that, and what's left.

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 will keep the map.

Step 1: The odds are exact

Quantum odds aren't loose. They're exact. Here's what the words mean, and why nothing else fits.

What a state looks like in the record. The ledger stores a quantum state in a register: a block of eight slots. (Eight again: the 2 × 2 × 2 from Part 1.) A slot is where one possible outcome sits. A photon at the glass uses two slots, bounce and pass, and leaves the other six empty.

Each slot holds a number called its amplitude. Think of it as a little arrow. The length of the arrow is its size, and only the size matters here. Multiply a size by itself and you get the slot's share of the state. A complete state is one whose shares add up to exactly 1: all of it is somewhere.

The window photon is a complete state. Its bounce slot holds an arrow of length 0.2, so its share is 0.2 × 0.2 = 0.04.² Its pass slot holds an arrow of length about 0.98, so its share is about 0.96. Shares: 0.04 + 0.96 = 1.

A chance is a number from 0 to 1 that says how often an outcome turns up. 0.04 is 4 in 100. So what chance does each slot get? Max Born answered in 1926: its share. That's the Born rule. Bounce gets 0.04 and pass gets 0.96, which is the 4 in 100 at the window. In a textbook it's a postulate, a rule you're told to accept.

Why the share, though, and not the share squared again, or its cube root? Here's the question the ledger answers: must the chance of an outcome equal its share? It must. Five rules for reading a record force it.³

The five rules. Reading a record means turning each slot into a chance. Any honest reading follows these:

1. Total. The chance that something happens, all the slots together, is 1.
2. No negatives. A chance is never below zero.
3. Empty means zero. A slot with nothing in it has chance 0.
4. Slots add. The chance of "this slot or that one" is their two chances added.
5. Same size, same chance. A slot of a given size has the same chance in every state, whoever its neighbors are and however the slots are labeled or grouped.

Four of those are just what it takes to be a chance at all. The fifth is the one with physics in it. Now watch them force the share.

Two states. Take a state with two slots, shares 1/2 and 1/2, and six empty slots. The empty ones read 0, so they add nothing. The two full slots are the same size, so the same-size rule gives them the same chance, and the total and add rules make the two chances sum to 1. Each is 1/2.

Now take a state with three slots, shares 1/4, 1/4 and 1/2. The 1/2 slot is the same size as in the first state, so the same-size rule gives it the same chance, 1/2, even though its neighbors changed. The total and add rules leave 1/2 to split between the two 1/4 slots. They're equal, so each gets 1/4.

Chance equals share, both times.

Those are two cases. Here's the general move. Take any three shares x, y and z that add to 1. In state A they sit in three slots. In state B, x and y sit together in one slot whose share is x + y, and z is untouched. (In the example above, x = y = 1/4 and z = 1/2.) The same-size rule gives z the same chance in both states, and the total and add rules make each state's chances sum to 1. Subtract one from the other and the chance of the combined slot is the chance of x plus the chance of y. Chances add exactly the way shares do. (The trick needs a third slot to hold z still. The register has eight.)

The no-negatives rule finishes the job. A bigger share can never have a smaller chance, because the extra piece adds a chance, and chances are never negative. Because chances add, n pieces of share 1/n have chances that add up to the chance of the whole, which is 1. So the chance of 1/n is 1/n, and adding pieces pins every fraction. Any other share is squeezed between fractions as close to it as you like, and their chances equal themselves. A slot's chance has no room to be anything but its share.

So the chance of a slot is its share. Exactly, for every slot in every state. No angle, no smooth curve and no calibration went in.

Take one rule away. Every rule does work. Remove one and a rival survives the other four.

Without the same-size rule, the fourth-power rule survives: raise each size to the fourth power, then rescale so the chances total 1. Put a slot of share 1/4 next to a slot of share 3/4, and the fourth-power rule gives it a chance of 1/10. Put the same slot, the same size, next to slots of shares 1/4 and 1/2, and it gives 1/6. The slot didn't change. Its neighbors did. The square says 1/4 both times.

Without the no-negatives rule, functions so jagged that no pencil can draw them survive. The other three rules have rivals too. Without the total rule: give every slot chance 0. Without empty-means-zero: leak a sliver of chance into the empty slots. Without slots-add: let the chance of "this slot or that one" be the square of their two shares added together, instead of their two chances added. Each rival passes the other four rules and gets the odds wrong.

What the same-size rule buys. Alice and Bob each hold a detector, in labs far apart. The detectors are prepared so they always agree: both read 0, or both read 1, half and half. The slot where both read 0 has share 1/2, and that's all of Alice's chance of reading 0: 1/2.

Now Bob does something to his own detector, and only his own. It splits his 1 outcome into two equal halves. The shares become 1/2, 1/4 and 1/4. Alice's slot still has share 1/2, and under the square her chance is still 1/2. Under the fourth-power rule, her slot's neighbors changed from one slot of 1/2 to two slots of 1/4, and her chance jumps from 1/2 to 2/3.

So under the fourth-power rule, Bob has just changed Alice's odds with no wire between them. He can send her a message by choosing whether to do it. The square never allows that.

The same-size rule is the physical premise of this step. Step 5 comes back to it.

Zoom out. Step 1 is done. The chance of an outcome is its share, exactly, because five rules for reading a record force it, and the fifth carries the physics. Next, and why: the rules fix the odds. They don't say whether the outcome is already sitting in the particle. If it were, chance would only be ignorance, like a coin whose flip is already settled by how it left the thumb. A game can test that.

Step 2: The result isn't written before the measurement

An answer written down in advance, with nothing passing between two labs, can't win a certain game more than 75% of the time. Nature beats that, and quantum physics tops out at 85.4%.

Yeah-But, the voice in the back of your head that has never let anything big go by, has a question.

"Yeah, but what if each photon carries a note we can't read? Bounce or pass, written on it from the start. The 4 in 100 is just how many notes say bounce."

That's how an ordinary coin works. Hand Gary, the drummer from Part 1, a real coin. Heads or tails, 50/50. But the flip isn't magic. How the coin leaves his thumb, how fast it spins, what the air does: those settle the result before the coin lands. The 50/50 is our ignorance of the details. (Part 2 pictured the odds of the future as a coin in mid-air. That's a drawing of odds with nothing settled. A real coin isn't that: its details settle it.)

Call a hidden answer like that a plan: the answer to every question you can ask the particle, fixed in advance, so a measurement just reads it. If photons carry plans, quantum chance is the real coin's kind: ignorance. So do they?

A game answers it. In 1964 John Bell found a way to test for plans. In 1969 four physicists, Clauser, Horne, Shimony and Holt, turned it into a score a lab can measure.⁴ Here it is as a game, the same one the post on entanglement plays. It's the real test, with detectors in place of people. The Ledger referees. It takes no sides and notes who won. (The Ledger is a drawing. The game is real.)

The game. Alice and Bob sit in separate rooms. They can't talk. The Ledger hands each of them a random bit, 0 or 1, like a coin flip. Each player answers 0 or 1. They win if their answers match, unless both were handed a 1. Then they win by answering differently.

They can plan anything beforehand. What's the best they can do?

Say they play with a plan. It holds Alice's answer to a 0 and her answer to a 1, and Bob's answer to a 0 and his answer to a 1. Four pairs of bits can arrive: (0, 0), (0, 1), (1, 0) and (1, 1). Each comes up a quarter of the time. The first three pairs need matching answers. The last needs different ones.

Try to win all four. Suppose the plan wins the three matching pairs. Then Alice's answer to 0 matches Bob's answer to 0 and also his answer to 1. And Alice's answer to 1 matches Bob's answer to 0. Follow the matches around and all four answers are the same. But the pair (1, 1) needs Alice's answer to 1 and Bob's answer to 1 to differ. They don't. No plan wins all four pairs.

The best plans win three of the four: 75%. A shared coin flip to pick the plan doesn't help, because the average of plans that each win at most 75% is at most 75%.

Answers written in advance win at most 75%. Physicists score the game with a number called S. A win rate of 75% is S = 2, and in general the win rate is 1/2 + S/8. So plans score at most 2. That limit of 2 is Bell's inequality, in the form Clauser, Horne, Shimony and Holt wrote down.

Now give them entangled particles: a pair prepared in one shared quantum state, one particle for each player. Each player measures their own particle, and which way they measure depends on the bit they were handed. With well-chosen measurements, the quantum rules say they win 85.4% of the time. That's S = 2√2, about 2.83, the most the quantum rules allow (Boris Tsirelson proved the ceiling in 1980). No plan can do that. So do real labs beat 75%?

They do. In 2015 a team in Delft, in the Netherlands, ran the test with electron spins in two labs 1.3 kilometers apart. The labs were far enough apart, and the measurements quick enough, that no signal had time to get from one lab to the other. Over 245 rounds they scored S = 2.42, give or take 0.20. That's a win rate of about 80%. The margin was about two error bars, but it came from the first test to close the main loopholes at once. (A loophole is a gap where plans can sneak back in.) Groups in Vienna and at NIST in Colorado reported loophole-free tests with photons the same year. The Vienna team's own figure: if the answers came from plans, the chance of results like theirs is below 4 in 10³¹.⁵ Alain Aspect, John Clauser and Anton Zeilinger shared the 2022 Nobel Prize in Physics for experiments like these.

Plans lose. Nature wins. I still get a jolt from that sentence. The test rules out plans for everybody, and ordinary quantum physics gives the same numbers.

In the ledger's terms, a plan is an entry that's already written, and a measurement that only reads it is a plan-reader. Plan-readers win at most 75%. Nature wins more, so a measurement does more than read. The entry is made when the measurement happens.

Zoom out. Step 2 is done. An answer written in advance, with nothing passing between the labs, wins at most 75% of the game, and nature wins more. The particle carries no plan. Next, and why: if it isn't written in the particle, where does it come from? Step 3 looks inside the measuring instrument, at what it holds before any result appears.

Step 3: The ledger writes one entry per run

Before the entry, the instrument's complete output holds both outcomes. Then the ledger writes one, at the odds of Step 1.

A run is one use of an experiment: one photon, one detector, one click. At the glass, the detector has a pointer that swings to bounce or to pass. Build that detector from the ledger's own parts, with a limited budget of energy and time and a finite accuracy. Feed it a photon with both outcomes in play. Let it run on its own mechanics with nothing added. What does it hand back?

Both. Everything it kept, taken together (the photon, the pointer, the lot), isn't in the bounce outcome alone and isn't in the pass outcome alone. Each outcome has its own part of the output, called a branch, and both branches are there, each with its share. That's proved for the ideal detector, and it stays true for one built from the ledger's own parts, within a limited budget, when it's accurate enough.⁶ Textbooks add a step called collapse to turn both into one. The proof uses no collapse. From an input like that, the detector's own mechanics don't produce a single outcome.

But you only ever see one. The pointer swings one way. The ledger writes one line. That line is the entry, and it follows three rules:

• One branch per run. Each run takes exactly one branch, and never a branch with no share.
• The state becomes that branch. After the run, the state is the chosen branch alone, rescaled so its shares total 1.
• The fractions follow the shares. Over many runs, the fraction that takes each branch settles on its share.

Experiments show the first and the third: one result per run, and fractions that settle on the squared shares. Take the three rules together as the law of the entry. At the glass, the law gives 0.04 for the bounce branch and 0.96 for pass, run after run.

That's the shape of a quantum measurement. The mechanics hold both outcomes. The ledger takes one. And once an entry is written, it stays. (The Ledger, pen up: Noted.)

Zoom out. Step 3 is done. A measuring instrument's complete output holds both outcomes. The ledger takes one entry per run, at the squared odds, and the entry stays. Next, and why: we have a law for what gets written, and exact odds for each entry. Is the world's actual list of entries chance, or does something pick the entries by a rule? A list can't say, and Step 4 shows why.

Step 4: No list of results can tell chance from a rule

A list of results can break the odds. It can't tell you whether anything picks the entries.

Run the glass experiment a million times and write the list: pass, pass, bounce, pass, pass, pass... That list is the ledger's lines for those runs. What can it tell you?

It can break the odds. After n runs, the fraction that bounced almost never lands far from the true chance, and an exact bound says how rarely.⁷ Take 100,000 photons. If the chance is 4%, you expect about 4,000 bounces. The chance of landing a full percentage point of the photons away, outside 3,000 to 5,000 bounces, is at most about 4 in a billion. Get 5,500 bounces and the 4% is out. That's how the odds get checked, and the squared odds keep passing.

It can't say why. Two things can produce the same list: chance and a rule.

Chance can produce any list. Any particular list of a million results is an astronomical long shot, and so is every other list. Being a long shot isn't evidence against chance, because every list is a long shot.

A rule can produce any list too. Write the list down and call it the rule. And rules can look random. The real coin from Step 2 follows the physics of a flipped coin. We can't follow it, so it looks like dice. A computer's "random" numbers come from a fixed rule, and the good ones pass the standard tests for randomness.

Here's a drawing of that. Two boxes: one runs on dice, one runs on a rule nobody has found. Both print the same list. Study the list as long as you like and you can't tell which box printed it.

That's the everyday version. Here's the exact one.

Take the world's whole history: the endless list of every run there will ever be. Call a history typical if the fraction of runs taking each outcome settles on that outcome's share and stays there. The third rule of the entry says the world's history is typical.

No finite list can establish that the actual history is typical, and no finite list can refute it. Typical means passing every frequency test (is the fraction near the share?) from some point on. A finite list is one stretch of the history, and any stretch can be continued into a typical history and into an atypical one. A list that misses the odds badly counts against the odds, to a stated confidence, but it doesn't rule out that the history settles down later. Each test is finite. "Eventually" isn't.

A second exact fact: no history passes every possible test, not just the frequency ones. The chance of exactly that history was zero from the start, so there's always a zero-chance test that rejects it: "reject exactly this history." If failing a zero-chance test made a history non-random, nothing would be random. So "random" can only mean "passes the tests you named in advance."

So "truly random" isn't something any list can show, and "secretly following a rule" isn't either. The odds are a different matter: they're exact, and they're tested. What picks each entry is out of reach of the clicks.

Zoom out. Step 4 is done. A list of results can break the odds. It can't tell chance from a rule: every finite list fits both, and every history fails some test. Next, and why: if lists can't settle it, something else has to. Step 5 says exactly what that is.

Step 5: What picks the entry

What picks which open entry gets written is the one step Recognition Science hasn't derived yet, meaning it isn't worked out from the starting facts.

Here's what's settled around it. The odds are exact (Step 1). The particle carries no plan (Step 2). A run writes one entry, and the entry follows the law in Step 3. And no list of results can say whether chance or a rule does the picking (Step 4).

What's missing is the link between that law and an actual event. The law says what gets written: one branch, at the shares. It doesn't say how an entry gets attached to an actual event, in an actual detector, at an actual moment, with the energy it takes and the time it lasts. That's the physical selection of the entry. It's one missing piece, and this post has met it twice.

The first time is the pick itself: which branch a given run takes, and whether a rule fixes it. The second is the same-size rule from Step 1: why a slot is read the same whatever its neighbors. Both are faces of that one piece: how the ledger attaches an entry to an event.

Textbooks fill this gap with a rule they add by hand, called collapse. More clicks won't settle it, because Step 4 shows no list can. It's a derivation to be done, not a count to be taken. What we do have is everything it has to match: exact odds, one entry per run, no plan in the particle, and a record that never erases.

So is the universe rolling dice? The odds are the law. The particle carries no plan. And nobody can tell from the clicks whether dice or a rule picks each entry. That's where the question stands.

Zoom out. Step 5 is done. Around that one piece, everything is settled: exact odds, no plan in the particle, one entry per run, and no list that can decide. Next, and why: that's all five steps. What's left is what they add up to.

The whole trip, in one breath

A photon meets glass. Its state has two slots, with shares 0.04 and 0.96. Five rules for reading a record force each outcome's chance to equal its share exactly, so the chance of a bounce is its share, 0.04. An answer written in advance would win a certain game at most 75% of the time, and nature beats that, so the photon carries no plan. Before the entry, the detector's complete output holds both outcomes. Then the ledger writes one line, at those odds, and never erases it. No list of results, however long, can tell chance from a rule, because every list fits both.

That's how reality is built: exact odds, and every entry written when it's made, in pen.

(The Ledger would like you to know that it can tell you which photons bounced, afterward, to the last one. Ahead of time, nobody can. Noted.)