synapse_cost
plain-language theorem explainer
Synapse cost quantifies the recognition penalty between two firing rates as the J-cost of their ratio. Workers on Hebbian covariance or the local-cache theorem in Recognition Science cite this when deriving minimal-cost neural wiring. The definition is a direct one-line application of the J-cost operator to the ratio.
Claim. For firing rates $f_u, f_v$, the synapse cost equals $J(f_u / f_v)$, where $J(r) = (r + r^{-1})/2 - 1$.
background
J-cost is the cost function induced by a multiplicative recognizer on positive ratios, equivalently the cost of any recognition event. In the Local Cache Theorem module this supplies the energetic penalty for uncorrelated firing. The module establishes that caching reduces total access cost under axioms A1-A3 and that the optimal partition recurrence forces the golden ratio.
proof idea
One-line definition that applies the J-cost function to the ratio of the two firing-rate arguments.
why it matters
The definition supplies the explicit cost term used in the Hebbian covariance result of the same module and in the broader claim that local minds are inevitable. It instantiates the J-uniqueness step of the forcing chain (T5) inside an information-theoretic setting and directly encodes the statement that any deviation from ratio 1 raises cost.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.