Axiomatic shared-medium coordination for stigmergic systems

Stigmergic coordination has many medium-specific formalizations. This paper formulates a medium-agnostic comparison layer whose comparison object is the abstract enabled-response signature. For metadata refinements, we prove that a coarse quotient is response-adequate exactly when each quotient fiber is response-aligned. When that fails, there is a canonical coarsest response-adequate repair, and every response-adequate repair must retain at least the corresponding fiberwise number of response classes. On the dynamic side, quotient-compatible matched writes preserve one-step behavior, freshness-conditional actions yield a one-step non-liftability obstruction, and the matched one-step result lifts to finite serial traces. These results are instantiated for a tuple-space medium and a timestamped virtual-stigmergic dataspace. The result is a medium-agnostic comparison framework with one explicit positive/negative theorem cycle; richer pathwise and concurrent semantics remain outside scope.