Order observable effects even when tensor values are independent

Consider an operation that returns the current integer counter and then increments it modulo eight. Two calls starting from zero must return zero and one, leaving the state at two. If a compiler incorrectly treats this operation as a pure constant-producing node, common-subexpression elimination could replace the second call with the first result. The numerical tensor graph may then look simpler while the observable sequence has changed. A state input and output make the dependency explicit rather than asking the optimizer to infer hidden global behavior.

I/O introduces similar obligations, with additional external uncertainty. Deleting a log operation because its tensor result is unused can remove a promised audit event. Duplicating a request can repeat a side effect. Moving an operation above a conditional can execute it when the source program did not. MLIR's side-effect and speculation rationale distinguishes effect information from permission to speculate: even an operation without ordinary memory writes may trap, have undefined behavior for some inputs, or fail to terminate under its semantics.

With an implicit mutable pseudorandom generator, two draws can depend on the order in which generator state is consumed. Reordering or recomputing them may change both outputs and later state. With a pure keyed design, the key is an explicit input: using the same key and same distribution operation can intentionally reproduce the same result. Distinct logical draws require the documented key-derivation or splitting discipline. Reusing a key is not the same as advancing an implicit global generator once more.

JAX documentation describes explicit keys and effect-ordering mechanisms, but this is not a claim that every framework operation is pure or that arbitrary Python effects are supported inside every transform. Separate host tracing effects, supported runtime effects, and external state. For checkpointing or recomputation, ask which operations may run again and what state must be restored. Matching the final loss once is a weak oracle when the transformation has silently consumed a different amount of state or repeated an external action.

An ordering token can carry an otherwise absent dependency between effectful operations. It does not magically make an external service transactional or impose a universal order across independent processes. The scope of runtime synchronization and compiler ordering must be stated. An analysis can permit disjoint effects to proceed independently when their contracts allow it, while preserving required order for writes to one resource. This is more precise than either treating all tensor nodes as pure or serializing every operation without examining its effects.

The modular lab makes one explicit state sequence inspectable. Reset, set both modulus m=8 and additive step a=3, predict residues zero, three, six, one, four, then take four Steps. Reset and set both m=8,a=1 to compare a different transition contract. Each Step advances one addition and the initial residue is zero. The lab does not generate random numbers, emit I/O, model concurrency, or execute effect tokens; it only shows why a specified sequence of state reads should not be silently replaced by repeated reads of the initial state.