single_vertex_suffices
plain-language theorem explainer
In a local cache at J-minimum on a connected graph with positive field values, the field is constant, so any single vertex determines the value at every other vertex. Researchers modeling brain information storage via holographic principles in recognition-based physics would cite this to bound removable vertices by connectivity alone. The proof is a direct one-line application of the ratio rigidity lemma to the cache's connectivity, positivity, and zero-cost edge conditions.
Claim. Let $C$ be a local cache on vertex set $V$, consisting of a connected graph with positive field $f:V→ℝ^+$ such that $J(f(v)/f(w))=0$ for adjacent $v,w$. Then $f$ is constant: $f(w)=f(v_0)$ for all $v_0,w∈V$.
background
A LocalCache is a nonempty connected subgraph equipped with a positive real field whose internal edges satisfy zero J-cost. The structure models a brain region as a J-cost-optimal cache whose internal ratios sit at the unique minimum of the J function. The module derives brain holography from T5 (J-uniqueness) through GCIC graph rigidity, which forces constant fields on connected zero-cost graphs, to the local-global information theorem and surface-area scaling in D=3.
proof idea
One-line wrapper that applies the ratio_rigidity lemma to the graph_connected, field_positive, and at_J_minimum properties of the cache, instantiated at the supplied vertices w and v₀.
why it matters
This result supplies the single-vertex case of the local-to-global step in the derivation chain from T5 through GCIC rigidity to the holographic cache property. It is invoked directly by the master certificate brain_holography_inevitable, which concludes that brain holography is forced once D=3 and surface-area scaling are in place. The theorem closes the partial-removal resilience claim in the module's chain.
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