Partition Topology Kernel
Abstract
Inseparability in a readout partition topology is exactly equality in the readout kernel.
Theorem 1.1 (Partition-topology inseparability is equality of readouts).
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/ObservationTopology/PartitionTopologyKernel.partition_inseparable_iff_kernel (✓ std3). ∎
Source. Repository-derived.
Commentary.
The partition topology is induced by the readout into a discrete coordinate space. Every open set is therefore a union of readout fibers.
Equal readouts place two states in the same fiber, so no open set can distinguish them.
If the readouts differ, the preimage of the singleton containing the first readout is open and contains exactly one of the two states. Thus topological inseparability is equivalent to kernel equality.
Literature scope: Pauly (2016), recorded in D5/L/ConceptDynamics/pauly2016represented, develops Sierpinski-valued observations and represented spaces. For the discrete codomain used here, equal kernels determine the partition topology. General initial topologies can have the same inseparability relation but different open tests, while effective observation additionally depends on the chosen representation.
Quantitative observation interface: Eftekhari et al. (2018), recorded in D5/L/ConceptDynamics/eftekhari2018embedology, distinguishes topological recovery from stable delay embedding. Consequently, equality of observable partitions alone does not control geometric conditioning or sensitivity to measurement noise.