cache_nodes_uniform
plain-language theorem explainer
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.
Claim. Let $C$ 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
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.
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