LocalCache
plain-language theorem explainer
The LocalCache structure assembles a connected graph on vertices V together with a positive real field whose adjacent ratios lie at J-cost zero. Recognition Science researchers cite it when deriving that any local brain region encodes the full global ledger state. The definition directly packages connectivity, positivity, nonempty cache nodes, and the zero-cost edge condition to serve as the hypothesis carrier for rigidity results.
Claim. A structure on a type $V$ consisting of an adjacency relation $adj : V → V → Prop$, the connectedness condition that the reflexive-transitive closure of $adj$ relates every pair of vertices, a field map $f : V → ℝ$ satisfying $f(v) > 0$ for all $v$, a nonempty subset $C ⊆ V$ of cache nodes, and the J-minimum condition that $J(f(v)/f(w)) = 0$ whenever $adj(v,w)$ holds.
background
In Recognition Science, J-cost is the function $J(x) = ½(x + x^{-1}) - 1$ with unique zero at $x = 1$ (T5). The module models neural tissue as a local cache: a nonempty connected subgraph whose internal edges incur zero J-cost, forcing the field to be constant throughout the component. This encodes the Local Cache Theorem that every vertex inside the cache determines the global field value.
proof idea
The declaration is a structure definition that packages the six fields: adjacency, reflexive-transitive connectivity, positive field, nonempty cache nodes, and the at_J_minimum predicate on adjacent pairs. No proof body is present; the structure acts as the hypothesis carrier for downstream applications of ratio_rigidity.
why it matters
LocalCache supplies the central carrier for the Local Cache Theorem and feeds brain_holography_inevitable, holographic_cache_from_gcic, cache_nodes_uniform, info_scales_with_boundary, and partial_removal_preserves_info. It closes the step from GCIC graph rigidity to surface-area scaling in D=3, confirming that accessible information equals boundary size rather than volume and thereby derives Bentov's holographic-brain claim from T5 plus connectedness.
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