Fixed Points and Recursion

Fixed points and recursion describe self-referential definitions, repeated behavior, and systems whose outputs feed future inputs.

A fixed point is a value that is unchanged by a function:

f(x) = x

In modeling, fixed points appear when a whole is defined by a recursive equation or when behavior stabilizes under repeated application.

Examples:

A behavior can be described by recursive equations over time.
A process can define its next state in terms of prior state and incoming events.
Retry and recovery loops continue until success, rejection, exhaustion, or compensation.
Projections may be rebuilt by replaying all source events until the projected state reaches the same result as incremental updates.
Recursive system structure appears when processes spawn sub-processes or observers route to observers.

Fixed-point thinking helps distinguish productive recursion from accidental loops. A retry policy without limits, backoff, idempotency, or recovery semantics is not a meaningful fixed point; it is an uncontrolled loop.