Mathematics Stochastic Processes From Rs Stochastic Process Type

A typed catalog of random processes

A stochastic process is a mathematical model of something that changes over time with an element of randomness. The five most studied types are the Markov chain, whose next step depends only on the present state; Brownian motion, the continuous random motion of a particle buffeted by molecules; the Poisson process, which counts rare events arriving independently; the martingale, a fair game where the expected future value equals the present value; and the Gaussian process, a collection of random variables whose joint behavior follows a bell-shaped distribution. Each has its own long history: Brownian motion was observed by Robert Brown in 1827 and modeled by Albert Einstein in 1905, while Andrey Markov defined his chains in 1906.

The Recognition Science framework's machine-checked library of formal theorems defines a type called StochasticProcessType that lists exactly these five types. A theorem in the library proves that the list has exactly five entries: the count is five, no more and no fewer. This is a small but exact result: the library certifies that the catalog is complete as a finite list, and that the five names are distinct and decidable, meaning a computer can always tell which type a given entry is.

In Recognition Science, the five types are not just a convenient list. The framework models recognition fluctuations as stochastic dynamics of its cost function J, where Brownian motion appears as a random walk in J-space and the Markov property says that recognition at tick k+1 depends only on tick k. The declaration thus anchors these classical objects inside the framework's own vocabulary of recognition, ticks, and cost.

What the declaration does not do is prove that these five types are the only possible stochastic processes, nor does it derive their defining equations from the framework's axioms. It is a catalog with a count, not a theory of stochastic processes. The proof that the count is five is a theorem about the list itself, not about the mathematics of Markov chains or martingales. The framework's claim is that these five types are the ones its dynamics produce, and that claim is a modeling choice, not a proved result.

theorem stochasticProcessTypeCount : Fintype.card StochasticProcessType = 5 := by decide

inductive StochasticProcessType where
  | markovChain | brownianMotion | poissonProcess | martingale | gaussianProcess
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not prove that these five types exhaust all possible stochastic processes. The declaration does not derive the defining equations of Brownian motion or any other process from the framework's axioms. The declaration does not establish that recognition dynamics actually produce each of the five types; it only catalogs them.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/StochasticProcessesFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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