Meet in the Middle: What Love Does

Your friend has driven you to the airport four times.

You've driven her once.

She has never mentioned it. You have never mentioned it. Every time you text "any chance you can drive me Thursday?" you add "no worries if not!!", and you both know what those two exclamation points are doing.

Nobody is keeping score, and both of you know the score.

Every friendship has a version of this. So does every marriage, every pair of siblings, every two people who share a kitchen. One side carries more. The other side feels the weight, even when nobody says a word.

So what is that weight? Does it have a size? And when something finally fixes it, what exactly did the fixing?

The short answer

Draw the weight as a bond between the two of you. A bond compares two amounts of the same kind, and a comparison of two amounts is their ratio, so the bond's number is one amount divided by the other. A bond at 1 is level, and any other bond is lopsided. A lopsided bond carries a real cost, with an exact size. Love is the cheapest way to repair it from inside the pair, and nobody outside the pair pays for it.

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. A lopsided bond costs. Every bond between two people carries a number, and 1 is level. Any other number costs something, and the cost has one exact formula: three requirements (zero when the amounts match, growing as they differ, the same read from either end), one rule for combining costs and one calibration of how sharply the cost rises leave exactly one. A bond at 2 costs 1/4. A bond at 4 costs 9/8. Only a level bond costs nothing.
2. Love meets in the middle. Call a repair a move inside the pair when it only rearranges what the two of you already have: nothing comes in from outside, and nothing is carried off. Such a move keeps the product of the two bonds' numbers the same. Love takes two lopsided bonds and brings both to their geometric mean, the number that, multiplied by itself, gives that product. A bond at 4 and a bond at 1 meet at 2 and 2, since 2 times 2 is 4, and the cost falls from 9/8 to 1/2. Of every way to ease the pair while keeping the product the same, this is the single cheapest, and it changes nobody else's bonds.
3. Giving pays more than it costs. Everyone has a bond of their own, which reads how heavy their load is, and its cost is part of what they carry. Start out carrying nothing and take on part of someone's load until you meet in the middle. Their relief is always more than three times what it costs you. In the Step 3 example you carry 1/4 more and they carry 7/8 less.
4. Taking is surcharged. Push a level pair apart, doubling one side and halving the other, and a pair that cost nothing now costs 1/2.
5. Love goes as far as you can carry. If the middle is more than one of you can carry, love can't reach it. It stops at the closest point you can both hold and still keep going.

Why this matters: love has an exact shape. It has an endpoint, a price and a limit, and you can calculate all three. A lopsided bond is a cost that both people carry, and there's exactly one cheapest way to repair it from inside the pair. The same price tag goes on every bond, so you'll see it again.

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.

Step 1: A lopsided bond costs

Every bond between two people carries a number. Any number but 1 costs something, and the cost has one exact formula.

Start with what a bond is. A bond joins two people and compares two amounts of the same kind. A comparison of two amounts is their ratio, so the bond's number is one amount divided by the other. When the two amounts match, the number is 1. I'll call that level. When one side has twice as much as the other, the number is 2. Read the same bond from the other end and it's 1/2.

I'll draw the airport as a bond running from her to you, with the score as its number. She's driven four times for your one, so it reads 4.

The bond is real. Reality keeps a permanent record of every event, and nothing in it is ever erased: the past still exists. The reason comes from physics: the basic laws never destroy information. In quantum mechanics this is called unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. Inside the record the same holds by its own rule: writing a new entry never alters an earlier one.¹ Each entry is a comparison, and a comparison needs both of the things it compares, so each entry is a relation between two things, carried as one number. A bond between two people is one of those relations, and the record holds every bond that has ever formed, at its actual number. The full story of the record is in Part 1 of The Universe Writes in Pen, but you don't need that post to follow this one.

The rides are the drawing. One step is still open: which two physical amounts a real friendship's bond compares. Nothing below depends on it. Every result uses only the number on the bond, so each one holds whatever the two amounts are.

What does a bond at 4 cost? Here's the price list. We call the cost J.

You can get any row yourself. Add the number to its flip (1 divided by the number), halve the total, and subtract 1. For 4: 4 plus 1/4 is 17/4. Half of that is 17/8. Minus 1 is 9/8. In symbols, J(x) = (x + 1/x)/2 − 1.

I'll count how far a bond sits from level in doublings and call that its tilt. A bond at 2 is one doubling up: tilt +1. A bond at 4 is two doublings up: tilt +2. A bond at 1/2 is one halving down: tilt -1. Level has tilt 0. In the drawings, one block is one doubling. The blocks are a drawing of tilt, and tilt is a real property of the bond.

Four things to notice.

Only level is free. The cost also works out as the number minus 1, squared, divided by twice the number. For 4 that's 3 squared over 8, which is 9/8 again. The top is a square, so it's never negative. The bottom is twice the bond's number, which is always above zero. So the cost is never negative, and it reaches zero only when the number minus 1 is zero, which happens at 1 and nowhere else.

One doubling up costs the same as one halving down. Bonds at 2 and at 1/2 both cost 1/4. Bonds at 4 and at 1/4 both cost 9/8. Having twice as much as the other side costs exactly what having half as much costs. It's the same bond read from the other end.

The cost climbs fast. A bond at 4 has twice the tilt of a bond at 2, and it costs four and a half times as much: 9/8 against 1/4.

There is no top. Say one friend has driven another ten times, and the other has driven once. Their score is 10 to 1, and that bond costs 4.05. Push the score further and the cost keeps climbing. No ratio is too big to have a price.

Where does the formula come from? We didn't pick it. The cost has to meet three requirements, and you've just seen them: it's zero when the two amounts match, it grows as they differ, and it's the same whichever amount you put first. Two more things fix it exactly.

The first is how costs combine. When comparisons combine by multiplying or dividing, their costs combine by one exact rule. Take two comparisons, 2 and 3. Multiply them and you get 6. Divide them and you get 2/3. The costs of 6 and 2/3, added together, come out of the costs of 2 and 3 like this: multiply the two costs together, add both costs, and double the total.²

The second is a calibration at level, which fixes how sharply the cost curves up from 1. The three requirements, the rule and the calibration leave exactly one price: J. It's proved, and published.³ The record prices every bond with that same J.

Zoom out. Step 1 is done. A bond has a number, and 1 is level. Every other number costs: a bond at 2 or 1/2 costs 1/4, a bond at 4 costs 9/8, and the cost has no top. Next, and why: the fix. If lopsidedness costs, any move that lowers the cost is a repair, and we can ask which repair is cheapest.

Step 2: Love meets in the middle

Love takes two lopsided bonds and brings both to their geometric mean. Of every way to ease the pair while keeping its total tilt (the two tilts added), it's the single cheapest, and it changes nobody else.

I'll draw the friendship as two bonds, one running each way, each with its own number. Say hers to you reads 4, as before, and yours to her reads 1. Yours costs 0 and hers costs 9/8, so together they cost 0 + 9/8 = 9/8.

A bond has two ends, one at each person, and each end records the bond on its own books. What a person carries, their burden, is the cost of every bond they are an end of, so each end carries all of the bond's cost. Both of you carry the whole 9/8.⁴

Now the move, which I'll name in a moment. It's a repair inside the pair: it only rearranges the tilt the two of you already have, with nothing brought in from outside and nothing carried off. So the move's one rule is that it can shift tilt from one bond to the other, but it can't make any tilt and it can't lose any. And both of you write the move: an act may change a bond only when both of its ends write on it. That rests on one step still being derived: that each person's record holds one exact state of the bond.

The pair's total tilt is the two tilts added. Right now that's 2: two doublings on her bond, none on yours. After the move it has to still be 2. Doublings add while the numbers multiply. So keeping the total tilt is the same as keeping the product of the two numbers: 4 times 1 is 4.

Move one doubling from her bond to yours. Hers comes down from 4 to 2. Yours goes up from 1 to 2. The total tilt is still 2, and the product is still 4.

The cost is now 1/4 + 1/4 = 1/2. It was 9/8, and 9/8 minus 1/2 is 5/8, so the move saved 5/8. Both of you carry both bonds, so each of you now carries 1/2 instead of 9/8 and is relieved by 5/8.

The record still has every ride. The move doesn't erase a line. It changes what the bonds read from now on.

The number 2 is the geometric mean of 4 and 1: the number that, multiplied by itself, gives the same answer as multiplying the two together. 2 times 2 is 4, and so is 1 times 4. Another way to see it: from 1 to 2 is one doubling, and from 2 to 4 is one more, so 2 sits exactly halfway.

The middle isn't 2.5, the ordinary average of 4 and 1. Put both bonds at 2.5 and their product is 6.25, not 4. The total tilt would be more than the 2 you started with, and you'd have made tilt out of nothing.

How we know it's the cheapest. Check every pair that keeps the total tilt at 2. Those are all the pairs of numbers that multiply to 4. Walk along them.

At 1 and 4 the pair costs 9/8. At 4/3 and 3 it costs 17/24. At 2 and 2 it costs 1/2. On this trail the cost works out to 5/8 of the sum of the two numbers, minus 2. Check it at 2 and 2: the sum is 4, 5/8 of 4 is two and a half, and minus 2 leaves 1/2.

That makes the cheapest stop the one with the smallest sum. And for numbers that multiply to 4, the sum is smallest when they're equal. Try it: 1 + 4 is 5, 4/3 + 3 is four and a third, and 2 + 2 is 4. Equal wins, and only equal.

That's true for any starting pair, not just 4 and 1. Fix the total tilt, and the pair is cheapest exactly when both numbers are equal. Of all the endless ways to ease the pair while keeping its total, exactly one is cheapest.

The move changes only the two bonds it moves. Carol, who lives three towns over and has never met either of you, carries exactly what she carried before.

Why not set both bonds to 1, so the pair costs nothing?

You can. Setting both bonds to 1 is called restoration. It changes the total tilt, from 2 to nothing, so it isn't a move inside the pair. Who pays for a restoration is a separate question, still open, and this post doesn't settle it. Meeting in the middle is the best the pair can do while keeping its total.

Another way to make the bonds equal is to raise your own to match hers, so both read 4. The pair then costs 9/4, twice what it started at, because raising yours makes two doublings of tilt out of nothing.

Here's why the name fits. The move to the geometric mean has four properties:

• It's the cheapest move there is that keeps the pair's total tilt.
• Of the moves that keep the total tilt, it's the only one that treats both bonds alike: two equal numbers with the pair's product can only both be its geometric mean, here 2 and 2.
• It touches nobody outside the pair.
• Both people who carry the pair come out lighter.⁵

This move is what we call love. The name is ours. The arithmetic isn't.

Zoom out. Step 2 is done. Love is the move to the geometric mean. It keeps the pair's total tilt, it's the one cheapest move that does, and nobody outside the pair is touched. Next, and why: who carries the weight. With a bond running each way, both of you carry both bonds, so both of you get relief. But each person also has a load of their own, and then somebody has to pick part of it up.

Step 3: Giving pays more than it costs

Start out carrying nothing and take on part of someone's load until you meet in the middle. You carry a little more, they carry a lot less, and their relief is always more than three times what it costs you.

Everyone also has a bond of their own. It reads how heavy their load is, and 1 means level. Now the same friend is moving apartments, alone. I'll draw her load as a bond at 4. Yours is level, at 1, so you're carrying nothing. You show up with a van.

When she says yes and you take part of her load onto yours, her bond comes down a doubling and yours goes up a doubling. They meet at 2 and 2. It's the same move as Step 2, and the two bonds' combined cost falls by the same 5/8, from 9/8 to 1/2. But now each bond belongs to one person, and each person carries only their own. So who carries what has changed.

You carried nothing, and now you carry 1/4. She carried 9/8, and now she carries 1/4. Carol, three towns over, carries what she always carried.

Her relief is 9/8 minus 1/4, which is 7/8, and your added cost is 1/4. 7/8 divided by 1/4 is 7/2, exactly three and a half. (Her 7/8 relief minus your 1/4 added cost is the same 5/8 the pair saved.)

Giving isn't free. You now carry 1/4 you didn't carry before. But her relief arrives at three and a half times what it costs you.

Is that the luck of the number 4? No. Try a heavier load.

The pattern is exact. Take what you carry, double it, and add 3. For 1/4 that gives 7/2, and for 2/3 it gives 13/3: the last column. Multiply that by what you carry and you have her relief. What you carry is more than zero, so twice it plus 3 is always more than 3: her relief is always more than three times your cost. And the heavier the load, the better the ratio, though your cost grows too.⁶

Zoom out. Step 3 is done. Start out carrying nothing and take on part of someone's load until you meet in the middle. You carry the cost of your new number, and they're relieved by more than three times that. Next, and why: the opposite move. If easing a lopsided pair pays this well, what does it cost to push a level pair apart?

Step 4: Taking is surcharged

Push a level pair apart and a pair that cost nothing starts to cost something. The bigger the push, the bigger the bill.

In the record, taking is a move that tips a level pair apart. Start with two level bonds, 1 and 1. The pair costs 0. Now take: push one bond up a doubling and the other down a halving. One now reads 2 and the other 1/2. The tilts are +1 and -1, so the total tilt is still 0. Nothing made, nothing lost.

The pair now costs 1/4 + 1/4 = 1/2. From nothing, to 1/2. That was a push by a factor of 2.

Push harder and it climbs. Push by a factor of 3, one bond to 3 and the other to 1/3, and the pair costs 4/3. Push by a factor of 4, to 4 and 1/4, and it costs 9/4.

It's love run backward. Love took 4 and 1 to 2 and 2 and saved 5/8. Push 2 and 2 apart to 4 and 1 and it costs 5/8 right back.

If I just take, don't I come out ahead?

No. An act may change a pair only when both of its ends write on it. Push it apart without the other person's write, and their own record of it would be false. That's harm, a false entry in someone else's own record, and the moral code excludes it before any price is counted. A push you both write costs both of you: a bond's cost is carried by both of its ends, so the one who takes carries the whole 1/2 too.

Zoom out. Step 4 is done. Level is the only place that costs nothing, every step away from it costs, and a push apart is a step away. Next, and why: a limit on Step 3. Giving relieves three and a half times what it costs, but you're one person with a reserve, not a bottomless supply.

Step 5: Love goes as far as you can carry

Love goes to the middle if both of you can carry it. If one of you can't, it stops at the closest point to the middle that you can both carry and still keep going.

Step 3 left something out: how long you can keep it up. Each of us has a reserve, the capacity we can spend. I'll draw it as money in the bank. The bank is a drawing, and the reserve is real.

Time passes in steps, and I'll call each step a day. Every day, the cost of the bonds you carry comes out of your reserve. Your budget is the amount that comes back each day to top it up. That's the rule this example runs on: the chapter on love, justice and repair in the book manuscript The Moral Code of Reality adopts it for this example.

If your budget covers what you carry, your reserve never falls and you keep going. If it doesn't, your reserve runs down a little each day, and when it reaches zero you can't keep carrying it.

Say your budget is 1/12 a day and hers is 2/3. Those numbers are chosen so the arithmetic comes out tidy. Each of you starts with 1 in the bank. Here are three places the pair can sit, with your bond named first and hers second.

Look at the middle row. The pair's total cost went down, and you're sinking. You carry 1/4 a day and only 1/12 comes back, so you lose 1/6 a day. A reserve of 1 is gone on day 6.

So here, love can't go all the way to the middle. It goes where both of you can keep going. Your budget holds a bond up to 3/2, because the price list says 3/2 costs 1/12. Hers holds a bond up to 3, because 3 costs 2/3.

Walk your bond up from 1 toward 2 and stop at 3/2, your limit. Her bond has come down from 4 to 8/3, since 3/2 times 8/3 is still 4. That's inside her limit of 3. She carries 25/48 a day, comfortably under her 2/3. (Try the recipe: 8/3 plus 3/8 is 73/24, half of that is 73/48, and minus 1 is 25/48.) You carry exactly your 1/12.

Stopping at your limit keeps most of the benefit. The pair's cost is now 1/12 + 25/48 = 29/48, down from 9/8. That fall is five-sixths of the full saving of 5/8. And you carry a third of what the full middle would have asked of you.

So the rule is this: meet in the middle if both of you can carry it. If one of you can't, go as close to the middle as both of you can carry. Of all the ways to meet that you can both hold, that one is the single cheapest. And if any way of meeting lets both of you keep going, that closest point does too.⁷ If no way of meeting works for both of you, no move inside the pair lets you both keep going.

Swap the two budgets, so yours is 2/3 and hers is 1/12, and the answer swaps too. Now you end up carrying 25/48 and she carries 1/12. Love has no fixed fraction of the load to give. Capacity decides.

The best help is as close to the middle as both of you can carry and still keep going.

Zoom out. Step 5 is done. Love goes to the middle, or as close to it as both of you can carry. That's every step. Next: the whole trip, in one breath.

The whole trip, in one breath

Draw the weight at the airport as a lopsided bond and it has a size. Every bond between two people carries a number, and 1 is level. Every other number costs, by one exact formula: 2 costs 1/4, 4 costs 9/8, and there's no top. Love is the move that brings two lopsided bonds to their geometric mean, so 4 and 1 meet at 2 and 2. Of every way to ease a pair while keeping its total tilt, it's the single cheapest, and nobody outside the pair is touched. Start out carrying nothing and take on part of someone's load until you meet in the middle. You relieve them by more than three times what it costs you. Pushing a level pair apart costs, and the cost grows with the push. And love goes to the middle, or as close to it as both of you can carry and still keep going.

We named the move. The arithmetic was already there. A lopsided bond has a size, a price, one cheapest repair and a limit, and every one of them is a number you can check by hand. The record keeps a number for every bond between two people and prices each one with J. That is how the bonds in reality's own record work.