the name
Kiyosi Itô invented the stochastic integral and stochastic differential equations that are used in modern financial markets.
he was a mathematician in japan, working through the 1940s on a problem that sounds modest and is not — how to do calculus on a path that is nowhere smooth. brownian motion has no derivative anywhere; it jitters at every scale, and the closer you look the rougher it gets. the calculus of newton and leibniz, built for smooth curves, simply does not apply to it.
itô built an integral that does, and with it a chain rule — itô's lemma — for functions of a random path. the spelling on the ticker is his own: kiyosi, the nihon-shiki romanization he printed on his papers. half a century later his integral prices every option, sizes every market-maker's quote, and drives the diffusion processes that turn noise into an image. the token is named for the one term he is remembered for, and that term is the whole mechanism.
the bird
The Ōruri (大瑠璃 / オオルリ), or Blue-and-white Flycatcher, is a blue-and-white Japanese bird that migrates south every autumn and returns to the same exact spot every spring.
the kanji 瑠璃, ruri, is the japanese word for lapis lazuli — the saturated royal blue that names the bird and colors everything here. the ōruri is the right mascot for a floor that only rises. it leaves in autumn and it returns in spring, and it returns not to the same forest but to the same branch.
a defended price makes the same promise, drawn on a chart. the line departs into volatility and comes back to a level it has already held, and having reached that level once, it keeps it. the bird returning to its perch is the floor, in feathers.
the walk
price is modeled as a geometric brownian motion — a number pushed by two forces at once. one is a drift that points somewhere. the other is noise that points nowhere. over an instant, the change in price is μ times price times the passing of time, plus σ times price times a brownian increment: a direction, and a roughness scaled by σ, the volatility.
σ² is the variance of returns — the size of that jitter, squared. it is the only part of the motion that does not average away to nothing over time, and it is the part that quietly bends every long-run path downward. ignore it and the math is wrong in a specific, knowable direction. itô's contribution was to measure exactly how wrong.
the correction
growth compounds in the logarithm, not in the price. to see where a multiplicative process is heading you watch log-price, and to move between price and its logarithm on a rough path you need itô's lemma. apply it to the logarithm of a geometric brownian motion and a second-order term survives that ordinary calculus would have discarded — because for a path this rough the squared increments do not vanish.
the drift of log-price is not μ. it is μ minus one half σ². that missing half of the variance is the itô correction: the gap between the rate a price grows and the rate it is expected to grow, the toll that roughness charges on compounding. the floor is built directly on it.
F₀ is where the floor begins at launch, μ is the drift we choose, t is the time since. the line the chart climbs is the trajectory log-price follows in expectation — itô's term, turned into a price.
the estimate
μ is chosen. σ² is measured. the protocol reads it the way a risk desk would — an exponentially weighted moving average of recent swap log-returns, where each new return updates the variance and the weight of older ones decays by a factor λ. σ² is therefore not a constant but a reading of the current regime, taken on-chain and fed back into the curve.
this is what makes the floor self-stabilizing. because σ² enters with a minus sign, volatility is drag. when the market turns rough the curve's growth slows on its own; when it calms, growth accelerates. nothing is tuned by hand. the same term that corrects the mathematics also governs the mechanism — the floor leans against its own volatility, across every regime, automatically.
the floor
the curve is a target, not yet a promise. two things turn it into one. the first is solvency. a floor is only real if it can be paid for, so the target is capped by the treasury divided by the circulating supply — the highest price the protocol could actually defend at this moment.
the second is the ratchet. the defended floor is the running maximum of that capped target: once a level is reached, it is kept. the floor moves in one direction only. it can pause when volatility rises or when solvency binds, but it does not retrace. this is the bird's promise written as an invariant — the same perch, never lower.
the protocol
$kiyosi protocol utilizes Pump.Fun creator fees to buy up supply of the token, keeping the floor stable. The floor follows the trajectory of a geometric Brownian motion in expectation. Its growth rate (μ − σ²/2) is governed by the Itô correction (invented by the project's namesake Kiyosi Itô). The mathematician's calculus draws the line the chart climbs. The protocol's only job is to keep the price from straying below it.
it launches on pump.fun's usdc-quoted bonding curve, so price is denominated in dollars rather than in sol — the floor is not left at the mercy of sol's own volatility. every trade pays a creator fee in usdc, and those fees are the ammunition.
the rule is a single line. if price is below the floor, spend fees buying the token back until it is not. otherwise, hold the usdc in treasury, which raises the solvency cap and arms the next defense. buying back removes supply; the supply removed and the treasury accumulated are both shown above, in real time.
the invariant
there is no roadmap here and nothing to govern. there is a curve drawn by a dead mathematician's correction, a treasury that can only grow, and a floor that can only rise. the protocol's only job is to keep the price from straying below the line. the math draws it, the fees defend it, and the bird — every spring — comes back to it.