Formal proof of detector-activation “fingerprint” embeddings

Prove that, for a laid-out code space and its associated detector hierarchy constructed as described in Section 6, the activation pattern produced by the detector space in response to a stimulus (i.e., the matrix A and the resulting set of active detectors and their codes) uniquely characterizes the stimulus sufficiently to serve as a valid embedding, in the sense that distinct stimuli yield distinguishable activation patterns and codes under the specified thresholds.

Background

The paper proposes constructing a detector hierarchy over a laid-out code space and using the activation pattern of that hierarchy as a compact structural embedding for stimuli. The authors argue that the activation behaves like a fingerprint and thus can encode the structure of the stimulus domain in a discrete vector.

While experiments suggest the approach works across modalities (e.g., morphology and histochemical markers), the authors explicitly note that they have not yet provided a formal proof that the activation pattern is discriminative enough to serve as a reliable embedding. Establishing such a proof would validate the theoretical foundation of using detector-space activations as embeddings and clarify the conditions under which the mapping is sufficiently unique.