Equilibrium

Price is free to swing. The float is not.

Market cap
Price
Burned
Float
The equilibriumsampling

Abstract

A memecoin is a body in motion and nothing else. No cashflow, no book, no promise, only a crowd pushing a price one way and then the other. Every protocol built on one has treated that motion as the problem to be managed. Equilibrium treats it as the drive. A pendulum is pushed by whatever happens to be pushing it and loses a fixed share of every swing to the escapement that turns the hands; the loss is small, it is taken each period, and it never comes back. Here the swing is volume, the share is β = 25% of the fees a block claims, and the loss is a purchase that is burned. What the design holds fixed is that ratio rather than any level:κ = φβ = 7.5 basis points of everything that trades leaves the float, at any size, at any price. As a map on the float, T(S) = (1 − φβv)S: fee rate, share, turnover. Price cancels out of it algebraically rather than approximately, and λ = 1 − φβv is below one whenever a single trade occurs, so the map is a contraction with one fixed point. So the word does two jobs here: a ratio the mechanism is held to, visible above as a line the state rides and is pulled back onto, and a fixed point of the map, at zero, approached rather than reached. That is the only claim about direction on this page. The rest is about where the rule runs: it belongs in the transfer path, as a Token-2022 hook fired by every movement of the token; it is not there, because a pump.fun mint carries no hook and none can be added afterwards, so it runs beside the path on a period that shortens as the fees quicken. What is checkable is marked checkable. What is a model of the thing is called a model.

Drive and damping

Two quantities decide what an oscillator does. One is what pushes it, which sets how far it travels. The other is what takes energy out of it, which sets what happens to that distance over time. A system with a drive and no damping keeps whatever amplitude it is given and tells you nothing; the interesting behaviour is in the second term.

A memecoin is the most heavily driven instrument on this chain and ordinarily has no second term at all. Enormous energy goes in as volume and all of it leaves again: to the venue, to routers, to searchers, to the next holder. The supply is indifferent to every bit of it. This protocol adds the damping term and puts the receipt on chain.

The analogy is load-bearing in exactly one place, and it is the place that matters. A fixed fractional loss per period integrates to an exponential envelope, in a pendulum and here, for the same reason: the loss is proportional to the thing it is taken from. Everything else about pendulums is decoration and is labelled as such below.

The map

Let S be the float. Each block claims the fees that have accrued, spends a fixed fraction of that claim buying the token, and destroys what it bought:

T(S) = S − φβV/P(1)

with φ = 0.0030 the creator fee rate, β = 25% the share, V the volume in the interval and P the price it filled at. Substitute the definitions of turnover and market value, V = vM and M = PS, and the price disappears:

T(S) = (1 − φβv) S,   λ = 1 − φβv(2)

A linear contraction, and the entire dynamical content of the coin. The product κ = φβ = 7.5 bps appears in it as a unit: it is the fraction of every unit of volume that leaves the float, and the only quantity in (2) that is a property of the design rather than of the market. Everything below argues about v, or admits where the map is applied from.

The equilibrium

The word does two jobs, and the useful one is not the one people expect. Start with the ratio. Two quantities accumulate as this runs: what the protocol has claimed in fees, and what it has spent buying its own float to destroy. The design fixes the second as β of the first, so the pair traces a line of slope β through the origin, and the chart at the top of this page is that line with the state on it.

A ratio is a stronger thing to hold than a level. It is scale-free, so it means the same at any size and at any price, and it is checkable by dividing one lifetime total in the ledger by another rather than by trusting a description. Levels are what a market decides; proportions are what a mechanism can promise.

Nothing keeps the state exactly on the line, which is the strip under the chart. A claim can be too small to survive a swap, so a period sometimes spends less than it owes and the point falls below the line. Measured from the first claim the deviation is one-sided, since the protocol can owe the fire and cannot overdraw it, and it is restoring, since what was not spent is carried into the next period that can afford it. The state therefore runs as a sawtooth against the line, never more than one minimum swap away from it and pulled back each time. That is equilibrium as a mechanism has it: not a state of rest, but a deviation that is bounded and always corrected. The strip is drawn from the start of its own window rather than from the first claim ever made, so it can open square and then show an older debt being paid down as an excursion the other way; the lifetime ratio under it is the figure that carries the claim.

The second sense is the fixed point of (2). Solve S = λS and for any λ below one there is exactly one solution, at zero. An equilibrium in this sense is not where a swinging thing spends its time: a pendulum is almost never at the bottom of its arc, it is moving fastest there and nowhere near it on average. What it gives you is the direction of everything that is not conserved. After n periods the float is S0λn, and λ = 1 exactly when v = 0, which is a market with no trades in it and a pendulum in a vacuum.

The two senses meet in κ. Because the ratio is held, the coupling is a constant: 7.5 basis points of every unit of volume is bought and burned, so ten thousand SOL through the venue removes 7.50 SOL of float whatever the price was while it happened. Hold a proportion and the level follows from it. Promise a level and you have to defend it against the market, which is the failure mode of every mechanism that has tried.

None of this is a statement about price. Price is a coordinate and the map does not touch it; the float is a denominator and the map only subtracts from it. A reader who wants the second fact to imply the first is welcome to, and should notice that this page does not.

The escapement

β is the share taken per period, and the one place this design has been wrong in a way worth recording. It is taken of the claim, never of the vault’s balance. A share of the balance re-spends the retained remainder on every later block, which compounds into spending all of it: a clock drawing on its own weight rather than on the swing, which runs fast and then stops. Taken of the claim, the share is bounded by construction, and across any number of blocks the protocol spends exactly 25% of everything it has ever collected.

Exactly, including the blocks too small to act on. A quarter of a few thousandths of a SOL will not survive a swap, and dropping those claims would let the realised share drift quietly below β forever, so instead they are owed: every claim adds its share to a credit, every purchase draws the credit down, and a run of blocks too small to spend accumulates into one that is not. The remaining 1 − β is genuinely unspent. Not staked, lent, paired or converted, just SOL sitting where it landed, which is what pays for the next period’s transactions and absorbs a run of failures.

equilibriumβ = 25% of the claimbought, then burnedamplitude × √(1−β) = 0.871 − β retainedpays for the next periodthe claim · 0.30% of volume
Figure 1One period. The escapement takes its share of the swing and never of the weight that drives it: β of what the block claimed, not β of what the vault holds. Energy goes as amplitude squared, so a quarter of the energy is √(1−β) = 0.87 of the arc, which is why the inner sweep is closer to the outer one than a quarter would suggest. The shaded sliver is the only part of this diagram that does not come back.

Where the rule should live

Damping in a real oscillator acts at every instant; an escapement acts once a period and approximates it. This protocol is the second kind, and that is the part of the design which is unfinished rather than clever. Token-2022 offers a transfer hook: an extension naming a program the token program invokes on every transfer, both parties in scope. A token carrying one does not have behaviour bolted beside it, it has behaviour in the transfer path, sampled at every trade rather than integrated over intervals of a clock. No period to tune, because there is no interval. No keeper able to be late or selective, because no keeper is in the path.

It is not there, and the reason is one RPC call away. pump.fun mints under Token-2022, TokenzQdBNbLqP5VEhdkAS6EPFLC1PHnBqCXEpPxuEb, with an empty extension set: no transfer hook, no transfer fee, no permanent delegate, no freeze authority. Extensions cannot be added to a mint after creation. So the map is applied from outside the path.

That empty set is a trade rather than an oversight. A hook buys continuity and costs composability: pools, routers and wallets must each support it or refuse the token, and many refuse hook mints outright. A token no venue will list is perfectly damped and has nothing driving it.

The period

Applied at intervals, the interval is the last decision, and a constant is the wrong answer. A quiet market and a violent one would wait the same number of seconds, so the quiet one spends transactions on dust and the violent one leaves revenue in a vault while the thing it is meant to act on is happening. The period is therefore chosen per block from the measured rate at which fees arrive, in lamports per second, read from the protocol’s own claims rather than from any reported volume: the version nobody can inflate by trading somewhere this earns nothing. Nine rungs, in seconds:

1 · 2 · 3 · 5 · 8 · 13 · 21 · 34 · 55(3)

Each rung is roughly 1.6 times the one below it, which is what puts a factor of fifty inside nine steps and keeps the jump between neighbours small enough that a market moving between them does not visibly change behaviour. Heavy flow takes the fast end and a dead market settles onto the slow one. The floor is two seconds rather than one because a period is three transactions, claim, buy and burn, and the next is armed only once the last confirms, so periods never overlap and never race each other for the same balance. Below 0.003 SOL nothing is claimed at all. The burn takes the vault’s entire holding rather than the purchase, so a period that buys and then fails cannot strand inventory: the vault ends every settled period holding none of the token.

The envelope

Holding the ratio across many periods gives the fractional contraction per unit time as the coupling times the turnover, and nothing else:

dS/S = −φβv dt,   S(t) = S0 e−φβvt(4)
Turnoverλ per dayFloat removed dailyHalf gone in
1×0.999250.075%924 days
5×0.996250.375%185 days
20×0.985001.500%46 days
1× turnover87% of the float left at 180 days5× turnover51% of the float left at 180 days20× turnover7% of the float left at 180 days
Figure 2The same rule at three turnovers over half a year. The band is the float, e−φβvt, computed from the fee rate, the share and the turnover and from nothing else. The wave inside is drawn at the same frequency in all three panels because it is there to show what the band bounds: it is not a price, and nothing on this page forecasts one. Only the width of the band is a claim.

Proposition. λ ≤ 1 for all v ≥ 0, with equality only at v = 0; and λ depends on no price.

Proof. φ, β and v are non-negative by construction, and subtraction is the only operation (1) performs on S: create_v2 discards the mint authority in the transaction that uses it, so no account can issue a token, this protocol included. Price enters (1) once in the denominator and once through M = PS, and cancels.