LocalCache

formal source

 133/-- A local cache: a nonempty connected subgraph with a cost function.
 134    The cache is "at J-minimum" when all internal edge costs vanish.
 135    This models a brain as a hierarchical J-cost cache (Local Cache Theorem). -/
 136structure LocalCache (V : Type*) where
 137  adj : V → V → Prop
 138  graph_connected : ∀ u v : V, Relation.ReflTransGen adj u v
 139  field : V → ℝ
 140  field_positive : ∀ v, 0 < field v
 141  cache_nodes : Set V
 142  cache_nonempty : cache_nodes.Nonempty
 143  at_J_minimum : ∀ v w, adj v w → Jcost (field v / field w) = 0
 144
 145/-- **HOLOGRAPHIC CACHE FROM GCIC**: Any local cache on a connected graph
 146    at J-minimum is holographic: every node in the cache determines the
 147    entire global field, and in particular the boundary encodes the bulk.
 148
 149    This derives Bentov's claim that the brain is a hologram. -/
 150theorem holographic_cache_from_gcic {V : Type*} (cache : LocalCache V)
 151    (v_cache : V) (_ : v_cache ∈ cache.cache_nodes)
 152    (w : V) : cache.field w = cache.field v_cache :=
 153  ratio_rigidity cache.graph_connected cache.field_positive
 154    cache.at_J_minimum w v_cache
 155
 156/-- Corollary: all cache nodes have the same field value. -/
 157theorem cache_nodes_uniform {V : Type*} (cache : LocalCache V)
 158    (v w : V) (_ : v ∈ cache.cache_nodes) (_ : w ∈ cache.cache_nodes) :
 159    cache.field v = cache.field w :=
 160  ratio_rigidity cache.graph_connected cache.field_positive
 161    cache.at_J_minimum v w
 162
 163/-! ## Part 4: Surface Area Scaling in D=3
 164
 165In D=3, the boundary of a connected region in ℤ³ scales as R² (surface area),
 166while the volume scales as R³. Since the holographic property says the boundary