hyperbola

Hyperbolic tokens: a price floor that barely moves, then bends, then goes vertical. Time and trading carry every coin toward its asymptote while its own fees buy it back and burn it, every 10 seconds.

y = 1/x · every coin has an asymptote

coins

5 above floor

nearest asymptote

0.67%

floor at ×1.01

volume absorbed

$3.6K

every $5.0K adds 0.01 to p

burned

0.51

67 buybacks · 67 today

abstract

A token’s price is a consensus that can evaporate in a block. This paper describes a second number published for every coin: a floor, defined by one closed-form equation, that can only rise. The floor is a hyperbola, F₀ / (1 − p), where p is the coin’s progress toward an asymptote. Progress accumulates from two sources that add: the passage of time, and the dollars traded. Far from the asymptote the floor is nearly flat; close to it, the floor climbs faster than any fixed growth rate can. Separately, an engine attached to each coin claims the creator fees its trading throws off and, every 10 seconds, spends a small slice of them buying the coin and burning it on arrival. We give the formula, the shape of the curve, how progress accumulates, why a hyperbola is the right shape, the engine’s two loops, and the invariants it never breaks.

1The formula

Every coin launched here carries one equation:

floor = F₀1 − p , p = tT + VV*(1.1)

symbol	meaning	value
F₀	where the floor starts	80% of the first price the engine observes
p	progress toward the asymptote	0 at launch, capped at 0.99
t	time since the floor was seeded	days, continuous
T	the time horizon	60 days
V	cumulative trading volume since seeding	US dollars, never decreases
V*	the volume scale	$500K

Read it from the inside out. The coin’s age is measured in units of T, its trading in units of V*, and the two fractions are added to give p. The floor is then F₀ divided by what is left: 1 − p, the coin’s distance from its asymptote. Neither t nor V can fall, so p cannot fall, so the floor cannot fall. No sell-off, quiet week or bad day lowers it.

2y = 1/x

Write x for the distance left, 1 − p. The floor in units of F₀ is then the oldest curve with an asymptote there is:

floor / F₀ = 1 / x, x = 1 − p(2.1)

The rule that follows is easy to hold in your head: every time the distance halves, the floor doubles. Going from p = 0 to 0.5 halves the distance and doubles the floor. Going from 0.5 to 0.75 halves it again and doubles it again. So does 0.75 to 0.875. Each doubling takes half as much progress as the one before, which is why the curve looks asleep for most of its length and then goes vertical.

fig. 1 · The floor multiple 1 / (1 − p) on linear axes. Through the first three quarters of the way it climbs only to ×4; the next fifth takes it to ×20, and the line leaves the frame. The cap at p = 0.99 stops it at ×100, just short of the asymptote at p = 1, where the curve would be vertical.

p	distance left	floor ×	of the ×100 cap
0	1	×1	1%
0.25	0.75	×1.33	1.33%
0.5	0.5	×2	2%
0.75	0.25	×4	4%
0.9	0.1	×10	10%
0.95	0.05	×20	20%
0.98	0.02	×50	50%
0.99	0.01	×100	100%

The last column is the point. Three quarters of the way to the asymptote, a floor has reached 4% of its final height. Half of the multiple lives in the last two percent of progress, between 0.98 and 0.99. The cap at p = 0.99 is what keeps the arithmetic finite: at that point the floor stops at ×100 and stays there.

3How progress accumulates

Progress has two sources, and they add rather than multiply. Time contributes a constant 1/60 ≈ 0.0167 a day, whether or not anyone trades. Volume contributes 0.01 for every $5K traded. In rates:

dp / dt = 1 / T + v(t) / V*(3.1)

Here v(t) is the dollar volume per day. Time alone would carry a coin to its cap in 59 days; volume alone would do it after $495K traded. A real coin gets both, in whatever mix its market produces: a slow ramp with a staircase on top, one step for each burst of attention.

fig. 2 · Progress is a sum. Time lays down a straight ramp of 1/60 a day; volume stacks a staircase on top of it, one step per burst, $500K of trading worth a full unit. This simulated coin reaches the cap on day 38, where time alone would have taken until day 60. Right axis: what each level of p means for the floor.

Because the two terms add, a coin that trades steadily at v dollars a day has a date with its asymptote that can be written down in advance:

t* = 0.99 / (1 / T + v / V*)(3.2)

volume a day	p per day	reaches the cap in
none	0.0167	59 days
$1K	0.0187	53 days
$5K	0.0267	37 days
$10K	0.0367	27 days
$25K	0.0667	15 days
$50K	0.1167	8.5 days
$100K	0.2167	4.6 days

Every coin has an asymptote. Volume decides when it arrives, and time guarantees that it does. A coin nobody trades still gets there in about 59 days; a coin trading $50K a day gets there in about a week.

fig. 3 · Every coin has an asymptote; volume decides the date. A coin trading $1K a day reaches the cap around day 53. One trading $15K a day gets there on day 21. Both spend most of the journey under ×3, and both make most of their gain in the last week. Log scale on the floor.

4Why a hyperbola

A floor is a promise about the future, and its shape decides when the promise is tested. We wanted three properties, and the hyperbola is the simplest curve that has all of them.

A slow, honest start. A young coin’s price is noise. A floor that sprints away from it in the first days would be a fiction, so the hyperbola spends its first half doing almost nothing: ×2 at p = 0.5, a pace any coin with a pulse can keep up with.

A violent finish. The same curve that is patient early is ruthless late. From 0.9 to 0.99 the floor goes from ×10 to ×100, a tenfold move in the final tenth. A coin that has earned its progress gets a floor that finally means something.

A finite horizon. A curve that grows by a fixed percentage has no end: it compounds gently forever, and there is never a day when it is done. A hyperbola has a place, p = 1, where it stops being a slope and becomes a wall. That makes the floor a story with a last chapter: you can see how far away it is, and what will bring it closer.

fig. 4 · Same start, same initial slope. A straight line never bends. A curve that grows by a fixed percentage bends gently and forever, and has no point where it is done. Early on the three are hard to tell apart (at p = 0.3 they read ×1.30, ×1.35 and ×1.43); then the hyperbola leaves both behind at a known place, p = 1. That place is the horizon.

5The 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 runs two loops for every coin, side by side.

The claim loop runs every 15 seconds. It collects whatever pump.fun has set aside for the creator, SOL on the bonding curve and wrapped SOL after graduation, and unwraps the wrapped SOL. It keeps 0.01 SOL in the wallet for gas and rent; everything else joins the pool.

The buy loop runs every 10 seconds. It takes a small, varied slice of the pool and spends it on the coin:

one buy every 10 s, spend ≤ pool(5.1)

That is the whole rule. The buy does not look at the price, and it does not look at the floor. Above or below, it buys the same way. Varied sizing keeps the buying off a predictable beat; a fixed cadence means a coin with fees is never more than 10 seconds from its next buyback. The floor is a tracked, published target, not a trigger.

fig. 5 · Two loops that never wait on each other. The claim loop fills the pool every 15 s; the buy loop drains it every 10 s in small, varied slices, with no reference to price or floor. Each buy transaction also burns what the previous buy delivered. A transaction that fails or is dropped is refunded to the pool, so nothing is counted twice.

step	what happens
claim	Every 15 s. Collects the creator vault, SOL or wrapped SOL, and unwraps it. Vaults under 0.005 SOL wait until they are worth a transaction.
shave	Tops the coin's wallet back up to 0.01 SOL so it always pays its own gas. Only the remainder is added to the pool.
observe	Reads price and hourly volume. Volume is integrated into V; the first price ever seen seeds F₀. The floor is recomputed and recorded.
buy	Every 10 s. A small, varied slice of the pool, never more than it holds. An empty pool waits for the next claim.
burn	In the same transaction as the next buy: destroy exactly the tokens earlier buys delivered, and nothing the wallet held before.
confirm	Tracked in the background. A transaction that fails or is dropped is refunded to the pool, so nothing is counted twice.

6Burns accumulate

At one buyback every 10 seconds, a coin whose fees keep up gets 360 buybacks an hour, each one burned on arrival. The engine never spends what it has not claimed, so a quiet coin buys back in bursts as fees arrive and a busy coin buys back on every tick. Either way the supply moves in one direction:

S_n+1 = S_n − b_n, b_n ≥ 0(6.1)

fig. 6 · Thirty minutes of one coin, simulated: 180 passes, one every 10 s. Each bar is a buyback, small and never quite the same twice; the gaps (27 of them) are passes where the pool was dry and waited for the next claim. When trading picks up in the middle, fees arrive faster and the gaps close. The line is the running total sent to the burn, and it only goes up.

The buyback is small on purpose: too small and too frequent to front-run, and no single one moves the chart. The pool drips into the market; it never dumps into it. What the drip does is turn a coin’s trading into a steady, visible subtraction from its supply, around the clock, with every burn on chain.

7The loop

Put together, the pieces form a figure eight through a single node. Volume pushes p toward the asymptote, and the floor rises with it. Volume also pays the creator fee, which the engine spends on the coin and burns. A higher floor and a smaller supply are, in turn, reasons to keep trading. Time keeps pushing p forward even when the market goes quiet, so the floor never stalls.

fig. 7 · Two loops through one node. On the left, volume pushes p toward the asymptote and the floor rises with it; time pushes p too, even when nobody trades. On the right, volume pays the creator fee, the fee buys the coin, the coin is burned, and a smaller supply is one more reason to trade. Nothing in either loop runs backwards.

8Invariants

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

invariant	statement
fees only	Spend is bounded by claims: Σ spent ≤ Σ claimed − Σ shaved. A claim is what the creator vault reported as claimable, so no other SOL is ever touched.
exact burns	Burned = tokens the buys delivered. 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 only, both non-decreasing. Nothing the market does can lower it.
blind buys	Buy timing and size never depend on price or floor. Nobody can trade against a trigger that does not exist.

9What a token page shows

Every coin has a page that shows the engine at work, live. Nothing is summarised away:

you see	what it means
floor	F₀ / (1 − p) today, plotted against the price, with the curve still ahead of it.
progress	p, split into its time and volume parts, and how far is left to the asymptote.
claimed	Every creator fee collected, in SOL, all time, and how much was kept back for gas.
buybacks	Every buy, its size and its transaction, as it lands.
burn curve	Tokens destroyed over time and the share of the supply gone for good.
market	Price, hourly and daily volume, and when the coin was born.

10What the floor is not

It is not a guarantee. The floor is a target the engine tracks and publishes; it does not size the buys, and the only money behind the coin is its own creator fees, spent in small, steady buybacks. A price can sit below its floor, and as the floor nears its asymptote it is likely to. What the floor does is state, in one public equation, how much time and trading a coin has accumulated and how close it is to the end of its curve.

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 progress, its price and every burn as it lands.