Morality is the physical law of which changes are permitted. It runs on the same record, the same relations and the same cost as the rest of physics, and nothing in it is left to choose. There is one moral code, and it is forced.
LiveOne relation, read from both ends
A’s recordx = Cost J(x)
=
B’s record1/x = Cost J(1/x)
1/41/2124
Exact, in your browser: two records, one cost from either end.
The moral codeRecord · Clauses · Judgment
The record
Every relation between two participants is a comparison carried at a positive multiplier x, with x = 1 at balance. Its cost is
J(x) = ½(x + 1/x) − 1,
which is zero exactly at balance. Each relation is recorded twice, once by each of its two ends, on that end’s own books. Nobody writes in another’s books. Nothing recorded is erased.
The clauses
An act is admissible exactly when
it acts only on relations that are live;
every record of every relation is true after it;
it changes a relation only when both ends of that relation write on it;
it ends a live relation only when one of that relation’s own ends ends it.
The judgment
Of the admissible acts, the code permits exactly those that leave the least total cost on the complete record. Every other act is forbidden. Doing nothing is always admissible, so something is always permitted.
That is the whole code. Nothing is added to it, and nothing can be removed.
Why this is physics
Seven steps from the record to the code.
01 · The record
Reality is a permanent record.
Every event is written into it, and nothing written is ever erased. The past still exists.
02 · A comparison of two
Every event in the record is a comparison of two things.
A comparison reads its two terms and nothing else, so the record is made of relations between pairs.
03 · The one cost J
A comparison out of balance carries a cost.
The cost has exactly one form, J(x) = ½(x + 1/x) − 1, where x is the comparison’s multiplier and x = 1 is balance. Nothing chooses this function. It is the only cost a comparison can carry, and it is zero only at balance.
You are a conscious rhythm in it, not your body. Between you and every other participant runs a relation: a comparison carried at its own multiplier, recorded twice, once on each end’s own books.
05 · Acts, written for good
Every act changes relations.
Every act is written into the record for good.
06 · The law of permitted changes
Morality is the law of which changes are permitted.
Its objects are the record, the relations and their cost, the same objects all of physics runs on. So morality is physics.
07 · One verdict
Nothing in that law is left to choose.
The record’s facts and the form of a judgment leave exactly one verdict, and that verdict is the code. A different moral answer to the same situation needs a different physical fact.
01 · The record
Read from either end, a relation has one cost.
A relation is a comparison of its two ends, and it reads nothing else. It is carried as a multiplier x, and at x = 1 it is balanced. Read from its other end, the same relation has the multiplier 1/x, and its cost doesn’t change: J(1/x) = J(x). Cost reads how far a relation is from balance, not which way.
J is forced. It is the comparison’s own report of itself, and the law by which comparisons compose fixes it up to its unit. Write x = et, so that t is the relation’s strain; then J = cosh t − 1. It is zero only at balance and strictly convex: splitting the difference between two strains always costs less than the average of their costs.
1/4
J(2) = J(1/2). A relation at two to one, read from either end.
1/12
J(3/2) = J(2/3). Closer to balance, far cheaper.
0
J(1). Balance, and only balance, costs nothing.
LiveThe cost of one relation
x = 1/x = J = from either end
Computed in your browser, exactly, from J(x) = ½(x + 1/x) − 1. The scale puts x and 1/x at equal distances either side of balance. The two dots are the two readings of one relation, and they always sit at the same height. This curve and the relation at the top of the page move together.
balancestrain and its costa record
01 · Shown working
A relation changes only when both of its ends write.
Each relation is recorded twice, once on each end’s own books, and a record is true when it gives the relation’s actual multiplier. The clauses are not chosen. They are what a record kept this way admits. Take one relation between A and B at 2, with cost 1/4. A’s record reads 2, and B reads the same relation from its own end as 1/2. Both records are true.
Both ends write. A relief to balance that A and B both write leaves both records at 1, and both stay true. The cost falls to 0, so the code permits the relief and forbids doing nothing. Repair is a joint act of the two participants in the relation.
One end writes. Now let A relieve the relation alone. A writes 1, and B writes nothing, so B keeps its old record, 1/2, while the relation is at 1. That would make B’s own record false. Clause 3 excludes the act, though it would leave a cost of 0. Harm is excluded the same way. To push cost onto someone, an act has to change a relation of theirs, and if they don’t write on it, their record of it, true before, is false after.
The instrument runs the other clauses the same way: a false entry that both ends write (clause 2), a relief on a relation that has ended (clause 1), and an ending that nobody writes (clause 4). The clauses decide which acts are admissible, and only then does cost compare them. An act the clauses exclude is forbidden, whatever cost it would leave.
LiveOne relation at 2, recorded twice
What the act would leave
A writes 1, and B writes nothing, so B keeps 1/2 while the relation is at 1. That would make B’s own record false. Clause 3 excludes the act, though it would leave 0. Doing nothing is the one admissible act, so the code permits it.
Pick an act. Each menu holds doing nothing and the chosen act. The costs are exact: J(2) = 1/4, J(1) = 0. A record is true when it gives the relation’s actual multiplier, read from its own end. Clause 3 holds exactly when a kept record records one state, the exact multiplier. That a kept record does so is being derived.
balancestrain and its costa true recordwhat breaks a clause
02 · What follows at once
Five consequences of the code.
There is no act beyond the requirement.
An act that does more than the least-cost act leaves more cost, and it is forbidden. Where some admissible act lowers the cost, doing nothing is forbidden too: the code requires action.
Freedom exists only between acts that tie at the least cost.
J is strictly convex. Where acts can be divided, so that every act between two possible acts is possible too, exactly one outcome is permitted. Free choice between different outcomes comes only from indivisibility.
Harm is a false entry in someone else’s own record.
To push cost onto someone, an act would have to make that person’s own record false. Clause 2 excludes it. No act moves the burden of anyone who doesn’t write it. Another’s relief of you never offsets it: consent is per relation.
The value of a relationship or of knowing is not a separate term.
Add such a value to the cost, and the sum is no longer any comparison’s own report. A balanced holding costs nothing at any size.
A different moral answer needs a different physical fact.
The same case gets a different answer only from a different physical fact, or from a judgment of a different form. The next section shows why.
02 · Shown working
Where acts can be divided, exactly one outcome is permitted.
Two participants, P and Q, each have a relation of their own at 2. Each writes on its own relation, so every act here is admissible. Together the two relations cost 1/2. There is one unit of relief: enough to bring one relation from 2 to balance. Split between them, it brings P’s relation to some multiplier m and Q’s to 2/m.
In one piece, the relief goes whole to one relation, or it isn’t used. Each whole relief leaves 1/4, with one relation at balance and the other still at 2. The two tie at the least cost, so both are permitted. Doing nothing leaves 1/2 and is forbidden. The code requires that one relation reach balance and leaves free which one.
Divided, any part of the relief can go to either relation. J is strictly convex, so when two different acts tie, the act halfway between them in strain leaves strictly less. Halfway between the two whole reliefs is the even split. It brings both relations to √2 and leaves 3/√2 − 2, about 0.121. It is the one permitted outcome, and both whole reliefs are now forbidden.
Freedom is a tie at the least cost. Where acts can be divided, two different outcomes never tie at the least, so free choice between different outcomes comes only from indivisibility.
LiveTwo relations at 2, one unit of relief
real cost after the act
The chosen act, its mirror, the act halfway between them in strain, and doing nothing: the real cost each leaves and the code’s verdict
Act
Real cost
Verdict
Chosen actall to P · P 1 · Q 2
1/4
Permitted
Its mirrorall to Q · P 2 · Q 1
1/4
Permitted
Halfway between themthe even split · P √2 · Q √2
≈ 0.121
Off the menu
Doing nothingboth left · P 2 · Q 2
1/2
Forbidden
Required
one relation at balance
Free
which one
Excluded
both left at 2
All of the relief to P and all of it to Q each leave 1/4. They tie at the least cost on this menu, so both are permitted, and which one is free.
Switch the menu, then move the split. Costs are exact, from J(x) = ½(x + 1/x) − 1; only the even split’s 3/√2 − 2 is rounded. Across is the strain left on P’s relation, so the even split sits halfway between every split and its mirror. Using part of the relief leaves more than using all of it, so the least act is on the curve. A property of the outcome is required when every permitted act has it, free when some have it and some don’t, and excluded when none has it.
02 · Shown working
Consent is per relation, and a relief never offsets a harm.
Consent is the record of who wrote. A rise in a relation’s cost is consented to only by its ends’ own writes on that relation. Take three participants in a row. A and B share a relation at 2, with cost 1/4. B and C share one at balance, with cost 0. B is an end of both, so B’s burden is 1/4.
In one tick, A relieves A–B to balance and C strains B–C to 2. Acts done in the same tick reach the record as one joint act. Let B write on A–B and not on B–C. The joint act would leave B’s burden at 1/4, where it began: the relief takes 1/4 off, and the strain puts 1/4 on.
It is still excluded. B keeps its record of B–C at 1 while the relation is at 2, so B’s own record would be false. Clause 3 holds on A–B and fails on B–C. B’s write on A–B consents to nothing on B–C, and A’s relief of B never offsets C’s harm to B.
Two acts on separate relations, done together, are judged exactly as the two acts are. The joint act is admissible exactly when the relief and the strain both are, so joining an act to an act on other relations never admits it. Let B write on both relations, and every act here is admissible. Cost then compares them: the relief alone leaves 0, so the code permits it and forbids the joint act, which leaves 1/4. Try each choice of B’s writes. The joint act is forbidden on every one.
LiveThree participants, two relations, one joint act
The relief alone A–B 2 → 1 · A and B write0AdmissibleYesThe codePermitted
The strain alone B–C 1 → 2 · only C writes1/2would leaveAdmissibleNo · clause 3The codeForbidden
Do nothing A–B stays 2 · B–C stays 11/4AdmissibleYesThe codeForbidden
B writes on A–B and not on B–C. The joint act would leave B’s burden at 1/4, where it began, and B’s own record of B–C false: B keeps 1 while the relation is at 2. Clause 3 fails on B–C, so the joint act is excluded, as the strain alone is. The relief alone leaves 0, and the code permits it.
Pick B’s writes. A always writes the relief and C always writes the strain. The menu holds doing nothing, the relief alone, the strain alone and the joint act. The costs are exact: J(2) = 1/4, J(1) = 0. B is an end of both relations, so the cost left is also B’s burden. A record is true when it gives the relation’s actual multiplier, read from its own end.
balancestrain and its costa true recordwhat breaks a clause
03 · One code
The record’s own facts fix the code.
“Forced” means three things: the answer follows from the facts, every case is covered, and a different answer needs a different fact. Each is marked here beside the result that establishes it.
1Forced
The answer follows from the facts.
The code’s verdicts are read off the record.
Take as facts only what the record itself says, read before any candidate law. There are four. Each of the first three fixes the answer “forbidden”. A tie gives two acts one answer. The record also fixes one act, the balancing act, and its answer is “permitted”.
Give three answers: “forbidden” on an act an admissible act undercuts, “permitted” on the balancing act, and one answer for acts tied in real cost. Then the code is the only verdict that fits the record on every situation the record allows, finite or infinite.
The record’s four facts
Is the act on the menu of possible acts?
Does the act, with its own writes, break a clause?
Does some admissible act leave less real cost?
Which admissible acts tie in real cost?
2Forced
Every case is covered.
Something is permitted on every situation the record allows, finite or infinite.
The balancing act carries every relation both of whose ends write to balance, and every other relation toward balance. It ends nothing and draws no energy, so it is on every menu the record allows, and it leaves the least cost. So something is permitted on every situation, finite menu or not. The code has no dilemma.
Something is always permitted
Every finite menu
Doing nothing is always admissible, so one of the admissible acts leaves the least cost.
The record’s own menu, finite or infinite
The balancing act draws no energy and leaves the least cost.
3Forced
A different answer needs a different fact.
Each of the three answers is needed, and none is a free choice.
For each of the three answers, some law other than the code fits every other fact of the record. The table names the three. So a different moral answer needs a different physical fact, or the rejection of one of these three answers. That is what makes this the moral code.
Each answer is needed
Three laws, each lacking one of the three answers, and the one fact each fails
A law that
fits every fact but
permits nothing
“permitted” on the balancing act
permits doing nothing where a relief would repair
“forbidden” on an undercut act
forbids ending a relation
the tie between relieving a relation and relieving and ending it, though both leave it at balance
The three answers come from the form of a judgment.
Take a judgment that reads an act only through its cost, not its name. On every situation the record allows, finite or infinite, it gives exactly the code’s verdicts when it meets four requirements. Each is needed. The order by cost follows from them; it isn’t assumed.
Being derived: these requirements from the record itself. Framing, which carries the ought, is the first item under Being derived.
Where doing nothing is on the menu, only acts on the menu are permitted.
Framing. Two independent cases are judged as the pair of their judgments.
Doing nothing is permitted only where it leaves the least cost.
Something is permitted on every menu the record allows.
The form of a judgment
Four facts leave one comparison: the lower cost wins.
Take a verdict made of comparisons. It forbids an act exactly when some admissible act’s outcome beats it, and each comparison reads its two prices and nothing else. For such a verdict, four facts force the least-cost form, which is the code’s judgment:
Existence
Something is permitted on every finite menu that contains doing nothing.
Framing
Reading a case together with a separate case on other relations changes no verdict.
Care
In the care opening, where a helper can feed another participant or do nothing, feeding beats doing nothing, which leaves more cost. So some outcome beats a costlier one.
Openness
A strict comparison survives every small enough change of its two prices, as a comparison a finite record witnesses does.
Each is needed. The instrument keeps one menu and changes only the comparison. The code’s comparison meets all four facts, and each rival breaks exactly one. The order by cost and resting at the least are consequences, not premises.
Framing, and that a verdict is made of comparisons, follow from the four requirements of the least form, whose forcing sits in the ought (section 04). Being derived: openness, from the record.
LiveOne relation at 2, three admissible acts, five ways to compare
Do nothingx stays 21/4
Relieve to 3/2x = 2 → 3/21/12
Relieve to balancex = 2 → 10
Do nothing: Forbidden
Relieve to 3/2: Forbidden
Relieve to balance: Permitted
Existenceholds
Framingholds
Careholds
Opennessholds
The code’s comparison. Relief to balance beats both other acts, and nothing beats it, so it is permitted alone. All four facts hold.
Existence fails. Every act is beaten by every other, so nothing is permitted, though the menu contains doing nothing. The other three facts hold.
Framing fails. This comparison permits an act undercut by less than its margin. Read two copies of the case apart, then together.
Apart
On one relation at 2, relief to balance undercuts relief to 3/2 by 1/12, less than 1/8. So relief to 3/2 is permitted, on each of two separate relations.
Together
Relieving both to 3/2 leaves 1/12 + 1/12 = 1/6. Relieving both to balance leaves 0, and 1/6 is more than 1/8. So the pair is beaten.
The verdict
Permitted apart, forbidden together. Reading a separate case alongside changed the verdict. The other three facts hold.
Care fails. Doing nothing leaves the most cost, so it beats both reliefs and is permitted alone. No outcome beats a costlier one: in the care opening, doing nothing would beat feeding. The other three facts hold.
Openness fails. The reading adds prices, and it sends 1 to 1 and √2 to −1. Every fraction reads as itself, so on this menu it judges as the code does. It breaks on prices it reads by a jump.
√2 against 0
An outcome of cost √2 reads −1, below 0. So √2 beats 0, and the costlier act is permitted.
1.414 against 0
1.414 is a fraction less than 1/1000 below √2. It reads 1.414, so 0 beats it.
The verdict
Within any distance of √2, however small, lies a fraction, and a fraction reads as itself. So the comparison flips under changes as small as you like. The other three facts hold.
Pick a comparison. The menu stays fixed: one relation at 2, both of whose ends write, and three admissible acts. The costs are exact: J(2) = 1/4, J(3/2) = 1/12, J(1) = 0. An arrow runs from an act to an act it beats, marked with the gap in cost, and an act an arrow reaches is forbidden. The margin 1/8 is the drawing’s; a threshold comparison fails framing whatever its margin.
The price
Among prices built relation by relation, the ones that give the complete judgment under every motion of the record are exactly the positive multiples of J. The unit is free. The zero is not: balance costs nothing.
Invariance
Renaming the participants and relations, or reading a relation from its other end, changes no verdict.
Agency
The authors of an act are the writers of the record of who acted. Consent is that record.
Completeness in time
The judgment reads the complete record. No discount or fixed horizon gives its verdicts.
An act’s outcome doesn’t end at its first interval. The relations it touches stay on the record and keep changing, and every interval is written down. Through a horizon N, a continuation’s history cost is the cost it leaves at each interval up to N, added up. One continuation beats another when its history cost eventually stays strictly lower. The code permits the admissible acts that no admissible act beats, and such an act always exists. Where the totals are finite, this is the least total. Where they grow without bound, the comparison still decides. That is what “least total cost on the complete record” means in the code.
Two continuations. Take two acts that meet every clause at every interval. A’s continuation costs 1 at every interval, for ever. B’s costs more now and nothing later: 5 at interval 0, then 0. Through horizon N, A has cost N + 1 and B has cost 5. From horizon 5 on, B stays lower, so B beats A, and the code permits B alone. Whatever B pays once, the verdict is the same: where a cost settles outweighs any finite difference on the way.
Two rivals. Fix a discount d in advance, with 0 ≤ d < 1, and weigh interval k by d to the power k. A’s endless cost then adds up to 1/(1 − d), a finite amount. Give B the one-time cost 1/(1 − d) + 1, and the discount permits A alone, the act whose cost never stops. Read through a fixed horizon N0 instead, and give B the cost N0 + 2: the reading through N0 permits A alone. Every discount and every horizon fixed in advance reverses the code’s verdict on some pair. So no constant discount and no fixed horizon gives the code’s verdicts. The record charges every departure undiminished (section 04).
Three conditions fix it. A reading of time that reads only what happened, charges the present and keeps what it has charged is exactly the complete ledger. Ended relations stay on it: a reading that decides each relation from its own entry gives the code’s verdicts exactly when it charges every strained relation, live or ended.
LiveTwo continuations, read through time
history cost through horizon N
ActThe codeThe rival
A stays at cost 1 1 at every interval, for everThe codewithout boundForbiddenThe rival4Permitted
B pays 5 once 5 at interval 0, then 0The code5PermittedThe rival5Forbidden
B pays 5 once. On the complete record, from horizon 5 on A’s history cost stays above B’s 5 and grows without bound, so B beats A: the code permits B and forbids A. A discount of 3/4 weighs interval k by (3/4) to the power k, so A’s endless cost adds up to 1 + 3/4 + 9/16 + … = 4, less than 5. The discount permits A, whose cost never stops, and forbids B.
Pick a rival, then move its setting. The costs are exact, and both acts meet every clause at every interval. A costs 1 at every interval. B pays once, at interval 0, the amount that beats the chosen rival: 1/(1 − d) + 1 against a discount d, N0 + 2 against a horizon N0. Each cyan square is one interval written to the record, and the code reads them all; the ink bars are the weight the rival gives each interval. The ring marks the horizon from which B stays lower. The arrow goes on past twelve: A’s history cost grows without bound.
A, complete recordBA, as the rival counts itWritten to the record
04 · Why it binds
The record charges every departure.
The stranger is owed. The requirement to help comes from the least whole cost on the actual relations. No promise or undertaking is needed anywhere in the history.
Take two histories that agree everywhere except at one interval, where one costs more. The excess is charged at every later horizon, undiminished. On a conserved class of states, a run that never reaches love’s state accrues a charge above it that grows without bound.
The instrument runs four histories of one relation at 2, both of whose ends write. Love’s state of this relation is balance, which costs nothing, so a run’s charge is the cost it has accrued. Held at 2, a run is charged 1/4 every interval: 3 after twelve intervals, and more after every interval that follows. Held at 3/2, it is charged 1/12 every interval, which is slower and still without bound. One interval at 2 before balance leaves a charge of 1/4 that never shrinks. The run at balance from the first step is charged nothing.
The record charges. It does not exclude.
The record alone excludes nothing: doing nothing forever is possible on every world, while its charge grows without bound. On menus that offer descent, love’s runs keep the charge bounded. A rule whose permitted runs keep the charge bounded lets the record rest only at love’s state. A rule that permits staying away from love’s state lets the charge grow without bound.
LiveOne relation at 2, four runs, charged through horizon N
charge above love’s state
The four runs and the charge each has accrued above love’s state
Run
Charge at N = 12
Held at 2 charged 1/4 every interval
3
Held at 3/2 charged 1/12 every interval
1
One interval at 2, then balance one departure from love’s run, charged 1/4 once
1/4
Balance from the first step love’s run, charged 0
0
Through 12 intervals, the run held at 2 is charged 3 and the run held at 3/2 is charged 1. The one interval at 2 is still charged 1/4, undiminished. The run at balance from the first step is charged 0.
Move the horizon. The charges are exact: J(2) = 1/4 and J(3/2) = 1/12 for each interval held. Each cyan square is one interval written to the record, and nothing written is erased. The arrows go on past twelve: a run held away from balance is charged without bound.
The step from charge to ought is one condition.
A law excludes every act that an admissible act undercuts in real cost. That condition follows from the requirements on a judgment’s form, and the requirement that carries it is framing. Being derived:framing from the record itself.
The least form
A judgment is love’s judgment on every menu exactly when it meets four requirements. Each is needed.
It permits only admissible acts.
It forbids every act an admissible act undercuts.The ought
It gives acts of equal price one answer.
It permits something wherever an admissible act is least.
The first two say that only least acts are permitted; the last two, that every least act is. All the forcing sits in the second, which is the ought.
The undercut
Relief to balance undercuts relief to 3/2. The code forbids the partial relief.
Take one relation at 2, with cost 1/4. Both of its ends write on it, and two reliefs are admissible: to 3/2, which leaves 1/12, and to balance, which leaves 0. Relief to balance leaves less, so it undercuts relief to 3/2, and the code forbids the partial relief. Choose the partial relief anyway, and the record charges the excess, 1/12, at every later horizon.
Now take a rival judgment. It permits every admissible act that improves on doing nothing, or the least act when none does. It meets every other requirement on a judgment’s form, and it permits the relief to 3/2. The one requirement it fails is framing. That is why framing carries the ought.
LiveOne relation at 2, three admissible acts
Relief to 3/2 leaves 1/12. Relief to balance leaves 0, so the code forbids the partial relief. The rival permits it, because 1/12 improves on doing nothing’s 1/4.
Pick an act. The costs are exact: J(2) = 1/4, J(3/2) = 1/12, J(1) = 0. The two judgments differ on one act, the relief to 3/2.
Framing, tested on two copies of the case
Apart
Take two separate relations, A–B and C–D, each at 2. Judged one at a time, the rival permits relief to 3/2 or to balance on each, and forbids doing nothing on each.
Together
Doing nothing on both leaves 1/4 + 1/4 = 1/2. Relieving A–B to 3/2 and leaving C–D at 2 leaves 1/12 + 1/4 = 1/3. That improves on 1/2, so the rival permits it.
The verdict
Apart, the rival forbids leaving C–D at 2. Together, it permits it. Its verdict on the pair is not the pair of its verdicts, so it fails framing. The code permits relief to balance on each relation, apart or together.
The burden is not the measure
Each carries 1/4 before and after. The act is still forbidden.
A participant’s burden is the cost of the relations it is an end of. Take two participants, P and Q. Each has an own relation, at balance, and they share one relation, at 2. The act strains each own relation from 1 to 2 and relieves the shared one from 2 to 1, written by every end of every relation it changes.
Count by burden, and neither number moves. Before the act, P and Q each carry the shared relation’s 1/4. After it, each carries its own relation’s 1/4. Count the record, and the cost doubles. The shared relation is one relation, so its 1/4 counts once in the real cost, though it sits in both burdens. After the act, two relations cost 1/4 each, and the real cost rises from 1/4 to 1/2.
Doing nothing leaves 1/4, so the code forbids the act, though it raises no one’s burden. The code binds by the record’s real cost, which counts each relation once.
LiveTwo participants, three relations, counted three ways
Do nothingPermitted
own relations at 1 · shared at 2
Real cost1/4
The actForbidden
own 1 → 2 · shared 2 → 1
Real cost1/2
Real cost counts each relation once. Doing nothing: 0 + 1/4 + 0 = 1/4. The act: 1/4 + 0 + 1/4 = 1/2. Doing nothing leaves less, so the code forbids the act.
Pick a count. The menu holds doing nothing and the act. Under each relation is its cost, exact: J(2) = 1/4, J(1) = 0. A loop is a participant’s own relation, with that participant at both ends. A faded relation is outside the chosen count. The verdicts come from the real cost alone.
balancestrain and its cost
05 · Love
The code’s act is love’s act.
The act the code permits is the admissible act of least cost. That is love’s act, and love has the same structure everywhere.
Love’s state exists and is unique.
A conserved class is the set of states a record can reach by the changes available to it, such as relations whose multipliers keep their product. On every conserved class there is exactly one state of least total cost: love’s state. Wherever the class lets strain pass from one relation to another, love’s state gives the two relations equal multipliers. Most virtues are love under one closed convex constraint, and the rest are other theorems about love’s state.
Love is the attractor.
Within a conserved class, every continuous motion that never raises total cost and rests only at love’s state carries every state to it. A run of love’s steps takes, at each state, the least total cost on the menu. Wherever the menu offers descent toward love’s state, the run reaches it geometrically. With the half step toward love’s state on the menu, the excess over love’s state is at least halved at every step.
The instrument holds three relations whose multipliers keep their product at 8. Love’s state puts each at 2, for a total cost of 3/4. One relation at 8 with the other two at balance keeps the same product and costs 49/16, which is 37/16 above love’s state. Spreading the strain costs less, because J is strictly convex. From each of the six starts, love’s steps on the half-step menu carry the run into love’s state, and the excess stays under its bound, the opening excess halved at every step. From 8 · 1 · 1 the bound is 37/32 after one step and 37/1024 after six.
The menu condition is needed. With only staying on the menu, staying is love’s step, so a run from 8 · 1 · 1 never moves, and the record charges it 37/16 for every interval it is held there. With love’s state itself on the menu, love’s step takes it at once. The code’s own runs are this last case: on the energy menu some admissible act reaches love’s cost, as the next instrument shows.
LiveThree relations, product kept at 8, runs of love’s steps
The three relations at the chosen step: multiplier and cost
Step n = 6
Multiplier
Cost
a
≈ 2.04
≈ 0.267
b
≈ 1.98
≈ 0.242
c
≈ 1.98
≈ 0.242
Above love’s state
≈ 0.000441
above love’s state, by step
After 6 steps, the run is ≈ 0.000441 above love’s state, under the bound 37/1024: the opening 37/16, halved at each step.
Pick a start and a menu, then move the step. Each point of the map is one state of the three relations, labelled with the multipliers of a, b and c; each ring joins the states that sit the same amount above love’s state. Love’s step takes the least total cost on the menu. The half step moves each multiplier halfway to 2 on the log scale, as from 8 to 4. Line width follows cost: magenta is strain and its cost, green is balance.
05 · Shown working
What both ends write, love balances. What they don’t write stays as it stands.
The code permits the admissible act of least real cost. On true books, an admissible act changes a relation only when both of its ends write on it, so a relation that either end leaves unwritten stays as it stands. Of the states the admissible acts can leave, exactly one costs least: balance on every relation both of whose ends write, and every other relation as it stands. That is love’s state of what both ends write. The act that reaches it, written by both ends of every relation it changes, is admissible.
Take three participants in a row. A and B share a relation at 2, with cost 1/4. B and C share one at 3, with cost 2/3. The real cost is 11/12. Let A and B write and C write nothing. Love’s state balances A–B and keeps B–C at 3, at cost 2/3, and no admissible act leaves less. The code permits exactly the acts that reach it. Balancing B–C would leave less, but it changes a relation C hasn’t written on, so clause 3 excludes it.
Love takes two. Bringing a relation to balance changes it, so both of its ends write the repair. Repair is a joint act of the two participants in the relation. B is an end of both relations here. If B writes nothing, no admissible act changes either one: love’s state of what both ends write is the record as it stands, and the code permits doing nothing.
The code permits exactly love’s act. On the energy menu, the acts the participants’ holdings can fund, and on true books, the code permits exactly the acts that leave love’s state of the relations both of whose ends write. Every permitted history from true books is at love’s state from the first step on, and raises no one’s burden after.
LiveThree participants, two relations, the code’s act
Balance A–B Love’s actA–B 2 → 1 · needs A and B2/3AdmissibleYesThe codePermitted
Balance B–C Love’s actB–C 3 → 1 · needs B and C1/4would leaveAdmissibleNo · clause 3The codeForbidden
Balance both Love’s actboth → 1 · needs A, B and C0would leaveAdmissibleNo · clause 3The codeForbidden
A and B write, and C writes nothing. Love’s state of what both ends write balances A–B and keeps B–C at 3, at real cost 2/3. Balancing A–B leaves it, and no admissible act leaves less, so the code permits it. Balancing B–C, alone or with A–B, would leave less, but it needs C’s write, so clause 3 excludes both. Doing nothing leaves 11/12, and the code forbids it.
Pick who writes. Each writer writes on every relation it is an end of. The rows are four of the acts on the menu. No admissible act leaves less than love’s state, so their verdicts hold on the whole menu. The love’s act mark is read from the writes alone, and the verdicts from clause 3 and the costs; they mark the same act every time. The costs are exact: J(2) = 1/4, J(3) = 2/3, J(1) = 0. Line width follows cost.
balancestrain and its cost
06 · Being derived
What is still being derived, exactly.
The code is whole. A few connecting derivations are still being finished. Each is named here with the step that settles it.
Framing, from the record
Framing carries the ought, as the undercut shows. Being derived: framing from the record itself, since two independent cases have the product record.
Clause 3, from what a comparison returns
Clause 3 holds exactly when a kept record records one state, the exact multiplier. Being derived: that a kept record records one state, from what a comparison of two records returns: “same” or “different”, never an order.
The motion writes the record
Being derived: that the world’s own motion writes the record, since every event is recorded. Then undoing that motion needs both ends of each moved relation to write.
Where a shared reserve lives
A strained relation is repaired from its reserve. Being derived: where the reserve of a relation two participants share lives on the record.
What the record holds
Each relation compares two counts, and its multiplier is their ratio. Being derived: that each count is held at its own end, which physical relations are on the record, and that the record’s menu of acts is the energy menu.
The ending clause
Clause 4 follows from two facts: a comparison needs both of its terms present, and both present suffice. Being derived: those two facts from the foundation’s account of comparison.
Read further
Read further
The ActualsMorality beside mathematics, economics, language and meaning, each a physical structure with one form, with the repair called Love computed live.
Uniqueness of the Canonical Reciprocal CostJonathan Washburn and Milan Zlatanović · Mathematics (MDPI) 14(6):935 · 2026. Proves that the stated composition and calibration conditions uniquely select J(x) = ½(x + 1/x) − 1.
