holographic_cache_from_gcic
plain-language theorem explainer
Any local cache at J-minimum on a connected graph has constant field value: the field at one cache node equals the field at every other vertex. Neuroscientists citing Bentov and holography researchers would reference this as the forced consequence of GCIC on brain-like caches. The proof is a direct term application of the ratio_rigidity lemma to the cache's connectedness, positivity, and zero internal J-cost conditions.
Claim. Let $C$ be a local cache on vertex set $V$: a nonempty connected graph equipped with a positive real field $f:V→ℝ$ such that $J(f(v)/f(w))=0$ for every adjacent pair. For any cache node $v$ and any vertex $w$, $f(w)=f(v)$.
background
A LocalCache is a nonempty connected subgraph whose field is at J-minimum, meaning every internal edge satisfies Jcost(field v / field w)=0. This structure models a brain region as a hierarchical J-cost cache. The module derives brain holography from RS first principles: T5 J-uniqueness forces GCIC graph rigidity, which yields the local-global information theorem that every connected subgraph vertex determines the global field. The derivation chain continues to boundary-encodes-bulk and surface-area scaling in D=3.
proof idea
The term proof invokes ratio_rigidity from GraphRigidity, supplying the cache's graph_connected predicate, field_positive predicate, and at_J_minimum condition together with the two vertices w and v_cache. No further tactics or reductions are required.
why it matters
This declaration completes the step from GCIC Graph Rigidity to the Holographic Cache Property in the module's chain, directly supporting the master certificate brain_holography_inevitable. It supplies the mathematical content for Bentov's claim that the brain is holographic and feeds downstream results on info_scales_with_boundary and partial_removal_preserves_info. The result sits at the intersection of T5 J-uniqueness and the Local Cache Theorem.
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