Sorry Doesn't Rewind the Tape

You promised to feed your neighbour's cat on Monday. You forgot.

On Tuesday you turn up with double food. By Friday the cat is glossy, enormous, and behaving as if Monday never happened, which is what cats do best. Your neighbour comes home to a perfect cat.

So: is Monday fixed? Is it, in the end, as if it never happened?

Most of us feel that it is, or nearly. After a repair people say "no harm done," and the more thorough the repair, the more it feels as if the whole thing never happened. A perfect repair feels like rewinding a tape and recording over the bad bit.

It isn't that. Reality isn't a tape. It's a ledger, a record that only takes new lines, like the page in the drawing above. Nothing that happens is ever erased. The reason comes from physics, below. The small world with numbers that follows shows exactly what a permanent record means for making things right.

The short answer

No. Monday is not fixed, and nothing you do on Tuesday can fix it. The past still exists. Monday's empty bowl is written down, and nothing that happens afterwards reaches back to change it.

What Tuesday does change is what happens next: whether the cat gets through the week. One more thing can change after the fact: what you owe. If your neighbour says "forget it" and you take it, the debt is gone. The miss is still there.

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

1. The ending forgets. Two histories can finish in exactly the same state, with the same payments made, and still differ in what happened on the way. The difference lives in the history, not in the ending.
2. Zero is not safe. Paying what you owe and finishing with nothing left is not the same as being fine. Going on takes a margin above zero.
3. Repair the future, not the past. After a miss, no way of finishing the story makes the whole story right. A repair that puts everyone back on their feet does exist, and the miss stays written beside it.
4. Forgiveness is an event. Two exchanges can have identical notebooks, one for each person, and still differ in what's owed: 1 in one, nothing in the other. Forgiveness is a new entry that changes what's owed, and in this exchange only the person who is owed can give it.
5. Making it right has one shape. When a correction is owed between two people, there is exactly one way to carry out just that correction, and nobody else is touched.

Why this matters: everyone has broken something and tried to put it right. "Making it right" sounds like a feeling with no edges. It has edges, and you can count them. It can change what each person has (a repair), what is owed (a release), and how lopsided the relationship between two people is (a correction). It never changes what happened.

It also shows what forgiveness is. Forgiving has something to work on only because the past is still there. If sorry were a rewind, there would be nothing left to forgive.

Why the past still exists

Reality keeps a permanent record of every event, the ledger. Nothing in it is ever erased, and no new event reaches back and edits an old one. Here is why.

In known physics. The basic laws of physics never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly. Everything that happened is still carried, in full, by the world as it is now: run the laws backward in a calculation, and every earlier state comes back out. That calculation undoes nothing in the world. It shows that no detail of the past is ever lost. That is why physicists treat the black hole information problem as a real problem: losing information would break the laws.¹

In this work. Inside the record the same holds by its own rule: writing a new entry never alters an earlier one.² So the record only grows. Each new entry is added, and every earlier entry is kept exactly as it was.

What an entry is comes from a separate fact. Each entry is a comparison: one thing told apart from another, as every measurement and every perception is, including your reading of these letters. Telling two things apart needs information about both of them, so every entry is written against what came before.

The route

I'll stop after each step to zoom out, so you always know where we are and why the next step comes next.

Step 1: The ending forgets

Two histories can end in exactly the same place and still be different histories. A rule that judges a history by its ending alone can't tell them apart.

The cat story is the everyday version. The exact version needs a world small enough to check with a pencil, so: two people, Alice and Bob. Time in their world runs in intervals, and you can picture each one as a day. In every interval, Alice either feeds Bob or waits, which means she doesn't cook.

Alice has promised to feed Bob, so that he keeps going. Bob counted on her, so he's running low: he spent his last interval working instead of feeding himself. Here are the rules of their world.

• Each of them has a reserve: what they have left to keep going on. (The drawings show it as a fuel gauge.) Alice starts with 2 units. Bob starts with 1/4 of a unit.
• Every interval costs each of them a quarter unit of upkeep, just to stay in working order.
• A meal costs Alice a quarter unit of work and gives Bob half a unit.
• At the end of every interval comes a checkpoint, where the promise is checked. To pass, both of them need more than zero left: something to go on with into the next interval.
• The example runs for four intervals, and Alice can cook at most two meals in all.

The numbers are this example's own, chosen so the arithmetic fits on a napkin. The results below are exact for this world. Real life has more people, more promises and numbers that come out of real bodies, and carrying the exact results over to all of that is work still ahead.³

Run two histories and watch Bob's reserve. In the first, Alice feeds in interval 1 and waits in interval 2. In the second, she waits and then feeds.

Interval 1, Alice feeds. Alice: 2, minus 1/4 for upkeep, minus 1/4 for the cooking, leaves 3/2. Bob: 1/4, plus 1/2 from the meal, minus 1/4 for upkeep, leaves 1/2.

Interval 1, Alice waits. Alice: 2, minus 1/4 for upkeep, leaves 7/4. Bob: 1/4, no meal, minus 1/4 for upkeep, leaves 0.

Interval 2 works the same way, with the choices swapped. Here are both histories side by side. Watch the last column, where they meet, and Bob's interval 1 numbers, where they split:

After interval 2 the two histories are identical. Alice has 5/4, Bob has 1/4, one meal has been eaten, and each of them has paid a quarter in upkeep twice.

After interval 1 they differ. Bob has 1/2 in the first history and nothing in the second. Zero doesn't pass the checkpoint, so the first history meets checkpoint 1 and the second misses it.

How we know. Check every number with a pencil. The rest is one plain fact about rules: a rule that reads only the ending gives the same answer to the same ending. These two endings are identical, but the verdicts differ: checkpoint 1 is met in one history and missed in the other. So no rule that reads only the ending can get both right. The same goes for a rule that reads only the list of upkeep payments, because that list is identical too: a quarter each, twice. What separates the two histories is when Bob had what he needed.

The ledger keeps every event in order, so it keeps the part the ending forgets.

Zoom out. Step 1 is done. A history holds facts its ending forgets, and the ledger keeps the history. Next, and why: the number that decided checkpoint 1 was Bob's reserve, and it read zero. Zero looks like a number you can shrug at. It isn't.

Step 2: Zero is not safe

Going on takes more than zero left. Paying the upkeep, like paying a bill, isn't enough.

Look at Bob's first interval in the second history. He starts with 1/4. His upkeep is 1/4: each interval pulls each person out of balance, and paying the upkeep brings them back. Strain is how far out of balance something is. Bob pays, so he's back to even, and there's no strain anywhere. A meter that reads only strain says everything is fine, because strain measures imbalance, not what's left over. And he has exactly nothing left. (Zero here means only this: nothing left for the next interval.)

Technically paid. Now look how thin that is. Run one more interval with no meal. The Bob who was fed in interval 1 has 1/2, pays his quarter and still has 1/4. The Bob who wasn't has nothing to pay with, so his upkeep goes unpaid and the strain stays on him, costing a quarter (Step 4 explains the quarter).

It doesn't even take a whole missing meal. Go back to Bob's first interval, where he had exactly the 1/4 he owed. Say a hundredth of a unit had gone missing before the upkeep was due. He'd have been a crumb short, and the bill would have bounced. Give Bob a margin, some amount left over after the bill, and he can absorb any crumb up to the size of the margin. With a margin of exactly zero, no crumb is small enough.

How we know. It isn't a quirk of these numbers. In this world, going on through every checkpoint is exactly the same thing as having a margin at every checkpoint: some positive amount you can lose, at every interval, and still pay. With a margin at every checkpoint, you get through every checkpoint, because a checkpoint asks only for more than zero left after the bill. And if you get through every checkpoint, you have a margin at each one, because getting through means more than zero was left after the bill. A reserve of exactly zero has no margin, so any loss, however small, breaks it. That's why the checkpoint asks for more than zero, not just enough to cover the bill.

The everyday version: anyone who has ended a month at exactly zero knows this. Every bill paid, and one surprise fee away from a bounce.

Zoom out. Step 2 is done. Being fine isn't "the bill is paid." It's "something is left." So Bob at zero is a miss, even though every payment went through. Next, and why: a miss is written down. Can anything Alice does next fix it? That's the question we opened with: is Monday fixed?

Step 3: Repair the future, not the past

After a miss, no way of finishing the story makes the whole story right. A repair that puts everyone back on their feet exists anyway.

Back to the second history, where Alice waited in interval 1. Nothing in physics stopped her: waiting was possible. It was also a miss: Bob reached checkpoint 1 with zero, so the promise failed its first check, and that check is written down.

The natural objection: "But didn't Alice fix it in interval 2?" There are two questions hiding in that.

Can Alice finish the story so that all four checkpoints come out met? No. Checkpoint 1 is already written, and it reads zero. Whatever Alice does from here gets added after it, and nothing added after a line changes that line. So the list of ways to finish the story with all four checkpoints met is empty. Not short. Empty.

Can Alice finish the story so that Bob is back on his feet and every checkpoint still ahead is met? Yes: feed, feed, wait. Alice feeds in interval 2 and again in interval 3, then waits in interval 4. That's the exact version of "double food on Tuesday": two meals in a row. Run it, and read the Verdict column:

Checkpoints 2, 3 and 4 are all met. And look at the ending: Alice 1/2, Bob 1/4, two meals eaten. That is exactly the ending of the history where nobody slipped (feed, wait, feed, wait).

So both are true at once, in one history: the repair worked, and the miss happened. Neither cancels the other. They're two lines in the ledger. If Alice says sorry, that's a third line, added after both.

Here's the everyday version. A drummer misses beat one of a four-beat bar and nails beats two, three and four. Everyone in the room still heard the gap where beat one should have been.

How we know. The arithmetic is in the table. Here's the rule behind it. Bob's reserve at a checkpoint is 1/4, plus 1/2 for each meal so far, minus 1/4 for each interval so far. It has to be above zero. So he needs at least 1 meal by checkpoint 1, 1 by checkpoint 2, and 2 by checkpoints 3 and 4. Waiting first gives him 0 meals by checkpoint 1, and that's the miss. The repair gives him 1, 2 and 2 meals by checkpoints 2, 3 and 4, which is enough.

The empty list is plain logic: a later move can't change the reading at an earlier checkpoint. And the repair is a real, possible way to finish from where Alice and Bob stand. Repair looks forward from the state things are in. It never had a way to look back.

Zoom out. Step 3 is done. A miss is permanent and repair is still possible, and both sit in the same history. Repair changes what each person has from here on. Next, and why: what is owed is a different fact from what each person has, and it has its own rules. That's where forgiveness lives.

Step 4: Forgiveness is an event

Forgiveness is a new event that changes what is owed. It changes nothing about what happened.

The natural question: "So forgiving is just forgetting, politely?"

No. Forgetting would be an edit. Forgiving is an entry: a new line in the ledger. Here's the exact version, in the same world, on a different afternoon.

Alice offers Bob a meal on credit: the price is 1, to be paid later. Bob accepts, and Alice delivers the meal. Now Bob owes 1.

Each of them keeps their own notes of the deal, which I'll draw as a notebook. Alice wrote "Bob: 1" in hers. Bob accepted the deal, but he never wrote anything in his. These notebooks are private notes, not either person's own record of the relationship in the ledger, so when a release both of them write leaves Alice's notebook unchanged, her notebook isn't a false record.

Now two worlds. In World A nothing else happens. In World B Alice says "forget it" and releases the whole 1, which means she lets Bob off all of it, and Bob takes the release.⁴

Open the notebooks. They're identical in the two worlds, page for page: Alice's says "Bob: 1" and Bob's is blank. But what's owed is 1 in World A and 0 in World B.

How we know. A rule that reads only the notebooks has to give the same answer for two identical sets of notebooks. The right answers here are different, 1 and 0. So no rule that reads only the notebooks, however clever, gets what's owed right in both worlds.

What does tell the worlds apart is what happened, written down event by event. In this exchange, five kinds of event decide what's owed: an offer, an acceptance, a delivery, a payment and a release. Once the meal is delivered, what's owed is the price, minus what's been paid, minus what's been released. In World A that's 1. In World B it's 1 minus the 1 that was released, which is 0. A release can be partial: if Alice releases half, what's owed is 1/2.

The release is a new line at the bottom of the ledger. It doesn't edit the offer, the acceptance or the delivery. Alice still offered, Bob still accepted, and the meal was still delivered. Forgiveness changed the total at the bottom and left every line above it alone. It is written in the same ink as every other line.

What release does to their relationship. The ledger writes every relationship as a bond: a ratio, one side divided by the other, that reads 1 when the two sides are even. Like what's owed, the bond is read from what happened, not from either notebook. In this example each side starts at 1, a baseline this example chooses. The meal Alice delivered on credit adds its price of 1 to her side, so her side reads 1 plus 1, which is 2. Bob's side still reads 1. The bond is the ratio of the two sides, 2 divided by 1, so it reads 2. Both of them wrote that deal, Alice the offer and Bob the acceptance, and an act may move a bond only when both of its ends write on it.

Every bond has a cost, and the cost measures imbalance. It meets three requirements: it is zero at balance; it grows as the two sides differ; and it is the same whichever side comes first, so a bond at 2 and a bond at 1/2 cost the same. Two more things pin it down: the rule for how the costs of combined comparisons add, and one calibration of how sharply the cost rises near balance.⁵ With the rule and the calibration, those three requirements leave exactly one formula. For a bond at x, the cost is J(x) = (x + 1/x)/2 − 1. Check it: at x = 1 it is (1 + 1)/2 − 1 = 0; at x = 2 it is (2 + 1/2)/2 − 1 = 1/4; at x = 1/2 it is the same 1/4. So this bond costs 1/4. That's the same quarter as the upkeep. In this example each interval stretches each person's own bond to twice its balanced size (the doubling is one of the chosen numbers), and putting it back costs exactly this 1/4. The ledger has one cost curve, one rule for what any imbalance costs.

Now the release. It changes what's owed, so it moves the bond: the bond reads 2 minus the amount released. How what's owed sets the bond is still being derived; here we read it as counts from a baseline of 1, the example's choice. That's 3/2 after releasing half, and 1 after releasing the whole thing. Half a release leaves a cost of (3/2 + 2/3)/2 − 1 = 13/12 − 1 = 1/12. After a full release the strain on the bond falls from 1/4 to nothing, for both of them. Because it moves the bond, the release, like the deal, is written by both of them: Alice gives it, and Bob takes it. That a release needs Bob's write as well as Alice's rests on one step still being derived: that each person's record holds one exact state of the bond. The release leaves the notebooks alone, so Alice's still says "Bob: 1", which is no longer what's owed, until she updates her own notebook to match.

Who can give the release. In this exchange, only Alice can, because she's the one who is owed. If Bob writes "I release myself," it doesn't count as a release, and what's owed stays at 1. If Bob writes "I owe nothing" in his own notebook, his notebook changes and what's owed doesn't. That's a rule this exchange is built with. Deriving it from deeper physics is still being worked out.

"Release" here means one thing: the person owed gave it, and the one who owes took it. Whether they wrote it freely and knowing everything is a different question, and this result doesn't settle it.

The same holds for Alice's promise to feed Bob. After Alice skips the first meal, Bob can release her from the rest of the promise, and she takes the release. That clears what's required of Alice from then on. It doesn't cook the missed meal, and it doesn't erase the missed checkpoint: Bob's reserve stays where the omission left it.

Zoom out. Step 4 is done. Forgiveness is a new event that changes what is owed. The notebooks don't change when it's written, and in this exchange only the person owed can give it. It leaves what happened exactly as it was. Next, and why: the last piece. When a correction is owed between two people, what does carrying it out look like, and who else has to change?

Step 5: Making it right has one shape

When a correction is owed between two people, there is exactly one way to carry out just that correction, and nobody else's bonds change.

Take two people and the two bonds between them, one in each direction: Alice's bond to Bob and Bob's bond to Alice.⁶ Say Alice's reads 1/2 and Bob's reads 2. They're lopsided in opposite directions, and the gap between them is a factor of 4, because 2 divided by 1/2 is 4. The case we'll work is one where what happened shows this gap is owed a correction, so it has to close. How much is owed is read from what happened, as in Step 4, where identical notebooks gave different answers. How that amount turns into this correction is a link still being worked out. The shape of the correction itself is exact.

What does closing it take? Multiply Alice's bond by some number a and Bob's by some number b. Two requirements decide a and b.

• Keep the pair's total. The total is the product of the two bonds, not the sum. Bonds are ratios, and ratios combine by multiplying, so lopsided by 2 one way and by 2 the other way nets out to even: 1/2 times 2 is 1. A correction only moves things from one side to the other. If it also added to the pair or took from it, that would be a second act, with its own questions about who owes it. Multiplying the bonds by a and b multiplies that total by a times b, so keeping the total means a times b is 1.
• Close the gap. The two bonds have to end up reading the same: a times 1/2 has to equal b times 2. That makes a four times b, so a divided by b is 4.

Only one pair of numbers does both:

1. a times b is 1.
2. a divided by b is 4.
3. Multiply the two: the bs cancel, leaving a times a equals 4.
4. So a is 2, because 2 times 2 is 4, and then b is 1/2.

Double Alice's bond, halve Bob's, and both read 1.

Meet Carol. She has a bond of her own, reading 2, that has nothing to do with this pair. It reads 2 before the correction and 2 after. Carol notices nothing, and that's the point: nobody outside the pair is touched.

How we know. The two requirements leave exactly one answer, and the arithmetic above is the whole argument. It works the same way for a correction of any size: one bond takes half of it and the other takes the opposite half. (To halve a multiplication, take its square root: half of a factor of 4 is a factor of 2, because 2 times 2 is 4.)

Corrections behave like entries in the ledger. Two corrections on the same pair add up. An opposite correction undoes one, as a new line after it, and both lines stay written. A correction of nothing changes nothing.

Justice, in the exact sense, is the correction that is owed, carried out exactly, with no one else carrying any of it. That is what this shape is.

Carrying out just the correction that is owed has exactly one shape, and nobody else's bonds change.

So, is Monday fixed?

No. Monday is exactly where you left it, in pen, and so is Tuesday. That's the bad news and the good news in one sentence: the miss is permanent, and so is the repair. Nothing you do right is erased either.

That's how reality is built. Nothing in it is ever erased, because the basic laws of physics never destroy information and the record only grows. So the past can't be edited, and the only thing the ledger ever does is take a next line. The numbers show exactly what that means for making things right: every apology, every repair, every "forget it" and every correction is one more line, none of them reaches back to change a line above it, and every one of them is still there.

Next time: what does love do to a lopsided bond? The cheapest way to even two people out, while keeping what they share, has exactly one answer. It changes no one outside the pair.