Indexing the archive…
Your Universe of Digital Possibilities
Every trade can trigger more trades. Below one trigger per trade the cascades die out; near one, a single order can set off a crash. Run the market twice from one seed, publish the tape to one of them, and measure its reflexivity from the tape alone.
The rate of events right now: an exogenous base μ, plus a decaying kick from every past event. Each kick integrates to n, the branching ratio — the mean number of events one event sets off — and fades on the time scale 1/β.
Each immigrant brings 1 + n + n² + … events in all, so the long-run rate is the exogenous rate divided by 1 − n, and the endogenous share of activity is n. At n = 1 the series diverges: there is no finite rate, and no amount of news is needed to keep the market busy.
The size of one immigrant’s family when every member has Poisson(n) children. Below n = 1 its tail is cut off exponentially; at n = 1 the cut-off vanishes and P(s) falls only as s−3/2.
The Sandpile’s law, met from the other side. Below n = 1 Borel’s law has an exponential cut-off and a typical family size; at n = 1 the cut-off goes and the tail is a pure power, P(s) ~ s−3/2 — the critical branching process, where the rare giant cascade is the tail of the everyday.
The seventh prophecy is the market reading its own tape. A published trade is a signal, the signal triggers trades, and the trades are published. The Bubble’s chartists read a chart; these algorithms read every print. Below one reaction per event the feedback is finite and measurable — the branching ratio, the market’s reflexivity — and its families are The Sandpile’s avalanches by another name: at n = 1 Borel’s law turns into a power law. Next, The Proxy asks what happens to a number once people start aiming at it.