cache_nodes_uniform
All vertices in the cache nodes of a LocalCache at J-minimum share identical field values. Researchers deriving holographic brain models or local-global information theorems in Recognition Science would cite this corollary. The proof is a direct term application of the ratio rigidity lemma to the cache's connectedness, positivity, and minimum-cost edge conditions.
claimLet $C$ be a local cache on vertex set $V$. For any vertices $v, w$ in the cache nodes of $C$, the field value at $v$ equals the field value at $w$.
background
In the Brain Holography from GCIC module, a LocalCache is a nonempty connected subgraph equipped with a positive real-valued field and vanishing J-cost on internal edges. This structure models a brain region as a J-cost-optimal ledger cache, with the at_J_minimum condition enforcing zero internal defect. The module derives holographic properties from T5 J-uniqueness through GCIC graph rigidity to boundary encoding of bulk information.
proof idea
The proof is a one-line term wrapper that applies ratio_rigidity to cache.graph_connected, cache.field_positive, cache.at_J_minimum, v, and w.
why it matters in Recognition Science
This corollary supplies the uniform field step required for holographic_cache_from_gcic and info_scales_with_boundary in the module derivation chain. It feeds the surface-area scaling argument in D=3 and the partial-removal resilience result. The uniformity follows from the Recognition Composition Law and the phi-ladder calibration of J-cost.
scope and limits
- Does not establish uniformity for caches outside J-minimum.
- Does not apply when the subgraph fails connectedness.
- Does not quantify information loss under node removal.
- Does not address time-dependent cache evolution.
formal statement (Lean)
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 :=
proof body
Term-mode proof.
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
167encodes the bulk, information accessibility scales with boundary (surface area),
168not volume. -/
169
170/-- Surface area of a region: number of boundary vertices.
171 Defined abstractly as the boundary set cardinality. -/