eˣ

Exponential tokens: a price floor that compounds with time and multiplies with volume, defended by the coin’s own fees.

tokens

0 seeding a floor

above floor

0/0

none defending

highest floor

×1.00

compounded since seeding

volume absorbed

drives every floor's exponent

burns

0 in the last 24h

into burns

0.00

0 floor defends

fees claimed

0.00

creator fees, all coins

in reserve

0.00

claimed, waiting to buy

abstract

A token’s price is a consensus that can evaporate in a block. This paper describes a second number that cannot: a floor, published for every coin, defined by a closed-form equation, and monotone in both of its inputs. The floor compounds continuously in time at rate r and multiplicatively in cumulative volume with scale V*, so that its logarithm is the sum of the coin’s age and its usage. An engine attached to each coin claims the creator fees its trading throws off and spends them in one of two regimes. Above the floor it drips, buying small random amounts and burning them. Below the floor it defends, committing a fixed fraction of everything it holds to buying the coin back and burning the result. The floor is not a guarantee; it is a schedule for where the coin’s own money is spent hardest. We give the formula, its two clocks, the engine’s state machine, the dynamics of a defense and the accounting invariants the engine never breaks.

1The formula

Every coin launched here carries one equation:

floor(t) = F₀ · e^r·t · e^{V(t) / V*}(1.1)

symbol	meaning	value
F₀	where the floor starts	80% of the first price the engine observes
t	time since the floor was seeded	days, continuous
r	the time rate	0.05 per day
V(t)	cumulative trading volume since seeding	US dollars, monotone
V*	the volume scale	$250K

Taking the logarithm separates the two forces cleanly. The exponent is a sum, and each term only ever grows:

ln( floor / F₀ ) = r·t + V(t) / V*(1.2)

Because t and V are both non-decreasing, so is the floor. There is no input that lowers it: not a sell-off, not a quiet week, not the passage of a year. The floor is a running record of how long a coin has existed and how much it has been used, compressed into a single number with a ratchet.

fig. 1 · The exponent is a sum. Time contributes a straight ramp of r = 0.05/day; volume contributes a staircase that climbs one unit for every $250K traded. Exponentiate the total and you get the floor.

2Two clocks

Each term has a natural unit. Time alone doubles the floor every ln 2 / r ≈ 13.9 days. Volume alone doubles it every V*·ln 2 ≈ $173K traded. A coin therefore runs on two clocks at once, and its floor ticks forward on whichever is faster.

d ln(floor) / dt = r + v(t) / V*(2.1)

Here v(t) is the instantaneous volume rate. The floor’s growth rate is the time rate plus the coin’s turnover measured in units of V*. A coin trading $250K a day adds a full unit to its exponent daily, twenty times what time contributes. Two coins with identical launches can end a month three orders of magnitude apart.

fig. 2 · Two clocks. A coin that barely trades still compounds on time alone: ×4.5 in 30 days. A coin trading $40K a day adds 0.16 to its exponent daily and ends the month at ×545. Log scale: straight lines are exponentials.

3Why exponential

A linear floor is easy to reason about and useless in practice: attention arrives in bursts, so any floor that grows at a constant rate is either irrelevant during a run or punitive during a lull. The volume term is built to be scale-free instead. Every additional $250K has the same proportional effect, so the floor responds to a coin’s hundredth thousand dollars of volume exactly as it did to its first, relative to where it already stands.

volume traded	exponent	floor ×
$0	e^0	1.00
$250,000	e^1	2.72
$500,000	e^2	7.39
$1,000,000	e^4	54.60
$1,500,000	e^6	403
$2,000,000	e^8	2,981

fig. 3 · The volume term, e^V/V*, on linear axes. Each $250K step multiplies by e, so the curve spends most of its length looking flat and then leaves the page. The first four steps together move the floor less than the fifth one does on its own.

The consequence is that the floor does not trail a successful coin at a fixed distance. It accelerates toward it. At some volume the floor catches the price, and from then on every dip that reaches it meets the engine buying.

4The engine

Every coin launched here is created through pump.fun with its creator set to a fresh wallet that only the engine controls, so its creator fees accrue where the engine can claim them. A process called the keeper visits each coin on a short cycle and runs one pass:

fig. 4 · One pass. Everything before the diamond happens to every coin every time; the diamond is the only branch. Whichever side runs, the pass ends in a burn and the loop starts again.

step	what happens
claim	Collects the creator fees pump.fun has set aside, as SOL on the bonding curve or wrapped SOL after graduation. Claims under 0.005 SOL wait.
shave	Tops the coin's own wallet back up to 0.01 SOL so it can pay its own gas. Everything else joins the pool.
observe	Reads price and hourly volume. Volume is integrated into V(t); the first price ever seen seeds F₀.
drip	Price at or above the floor: buy a random $1 to $2 and burn it.
defend	Price below the floor: buy 25% of the pool, at most $50, and burn it.
burn	Destroy exactly the tokens the buy delivered, measured as the balance after minus the balance before.

5Defending the floor

The decision rule is a single comparison, made every pass with fresh data. Let P be the claimed but unspent pool and p the market price:

spend = p ≥ floor ? U($1, $2) : min(0.25 · P, $50)(5.1)

Dripping is deliberately small and random, so routine buying never lands on a predictable beat. Defending is deliberately heavy, so that the money a coin has accumulated during good hours is concentrated on the hours when the price has fallen through its own schedule.

fig. 5 · A defended dip, simulated. The floor (blue) rises regardless of the market. While the price (dark line) sits above it the engine drips; each orange mark is a pass where the price was under the floor and the engine spent 25% of its pool buying. The floor does not stop the dip. It decides where buying gets heavy.

Repeated defense drains the pool in two phases. While 25% of the pool exceeds the cap, each pass spends exactly $50 and the pool falls linearly. Once the pool is below $200, each pass spends a quarter of what remains:

P_k+1 = P_k − min(0.25·P_k, $50)(5.2)

fig. 6 · Spending down a $1,500 pool with consecutive defends. While 25% of the pool is more than $50, the cap binds and the pool falls in a straight line. Below $200 it switches to geometric decay and never quite reaches zero, so a coin always keeps some ammunition for the next dip.

The geometric tail means a pool is never spent to zero by defense alone. Some ammunition always remains, and every new claim refills it.

6The loop

Put together, the pieces form a cycle with no step that runs in reverse. Volume is simultaneously the input that raises the floor and the source of the fees that defend it. Burns remove supply permanently. A higher floor and a smaller supply are, in turn, the reasons to keep trading.

fig. 7 · The loop. Volume raises the floor and pays the fees; the fees buy and burn; a rising floor and a shrinking supply are the reasons to trade again. No step requires anyone’s permission, and none of them runs in reverse.

7Invariants

The engine is narrow on purpose. Four properties hold on every pass, for every coin:

invariant	statement
fees only	Spend is bounded by claims, up to swap slippage: Σ spent ≲ Σ claimed − Σ shaved. A claim is the wallet's balance after minus before, so no other SOL is ever touched.
exact burns	Burned = tokens after the buy − tokens before it. Anything the wallet already held is never burned.
no sells	There is no code path that sells. Tokens move in one direction: into a burn.
monotone floor	The floor is a function of t and V(t) only, both non-decreasing. Nothing the market does can lower it.

S_n+1 = S_n − b_n, b_n ≥ 0(7.1)

Supply S after the nth burn: a sequence that can only fall.

8What the floor is not

It is not a guarantee. The floor is a target, and the only thing defending it is the coin’s own creator fees. A large enough sell-off can exhaust the pool and leave the price underneath. Because the floor keeps rising regardless, a coin below its floor is simply a coin whose engine buys every time new fees arrive, at the heaviest rate the rules allow.

The keeper holds the wallet keys, so the rules are enforced by a process rather than by a contract. Everything it does is public: every claim, every buy and every burn is recorded with its transaction, and each token’s page shows its floor, its price, the components of its exponent and every burn as it lands.