Mathematics Stochastic Processes From Rs Stochastic Process Type Count

Five canonical types

Stochastic processes are families of random variables indexed by time, used to model phenomena that evolve with uncertainty: stock prices, queue lengths, radioactive decay. Five types appear across mathematics and physics so often that they form a standard toolkit. A Markov chain forgets its past beyond the present state. Brownian motion is the continuous random walk of a particle jostled by molecules. A Poisson process counts events that occur independently at a constant average rate. A martingale is a fair game: its expected future value equals its current value. A Gaussian process is a collection of random variables whose finite subsets are jointly normal.

The framework's machine-checked library of formal theorems defines an inductive type with exactly these five constructors, and proves by direct computation that its cardinality is 5. The theorem stochasticProcessTypeCount states Fintype.card StochasticProcessType = 5. The proof is a single decide tactic: the type is finite, the constructors are distinct, and the count is immediate. No axioms beyond the standard three are used, and no sorry appears. A certificate structure packages this fact for downstream use.

In Recognition Science, the five types are not merely a list. The framework models recognition fluctuations as stochastic dynamics of a cost function, the price a system pays for recognizing an event. Brownian motion becomes a random walk in cost space. The Markov property becomes: recognition at tick k+1 depends only on tick k. The count of five is then a structural claim: the framework's ledger of recognition events supports exactly these five canonical stochastic process types, no more and no fewer.

What the theorem does not claim is just as important. It does not assert that these five types are the only stochastic processes in mathematics, nor that every real-world random phenomenon must be one of them. It does not derive the types from the cost function; it defines them and counts them. The identification of Brownian motion with a random walk in cost space is a modeling choice, not a proved equivalence. The theorem is a precise, narrow fact about a finite enumeration, and its value lies in the clean foundation it provides for the framework's stochastic vocabulary.

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

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

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

structure StochasticProcessesCert where
  five_types : Fintype.card StochasticProcessType = 5

What this page does not claim

The theorem does not prove that these five types are the only stochastic processes in all of mathematics. The identification of Brownian motion with a random walk in cost space is a modeling choice, not a proved equivalence. The theorem does not derive the five types from the cost function; it defines and counts 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)

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