optimal_at_minimum_is_holographic
plain-language theorem explainer
In a connected graph with positive vertex fields where every adjacent pair induces zero J-cost on the field ratio, the field must be constant everywhere. Researchers modeling holographic uniformity or local caching in recognition-based physics cite this when deriving global constancy from edgewise minimization. The proof is a one-line wrapper that invokes the ratio rigidity lemma from the GraphRigidity module.
Claim. Let $(V, E)$ be a connected undirected graph. Let $f: V → ℝ_{>0}$ be a positive field such that $J(f(v)/f(w)) = 0$ for every edge $(v,w)$. Then $f(v) = f(w)$ for all vertices $v,w$.
background
The J-cost function Jcost, drawn from Cost and ObserverForcing, assigns a non-negative recognition cost to positive ratios and vanishes exactly at ratio 1. The module Local Cache Forcing from J-Cost Minimization shows that J-cost minimization on connected graphs forces hierarchical local caching, closing Gap G1 in the brain holography proof. Upstream results include the inflaton potential definition V(φ_inf) = Jcost(1 + φ_inf) and the cost map induced by multiplicative recognizers.
proof idea
The proof is a one-line wrapper that applies GraphRigidity.ratio_rigidity to the connectivity hypothesis, positivity of the field, and the zero-cost edge condition.
why it matters
This theorem supplies the holographic uniformity step inside the local cache forcing certificate, which lists J(φ^d) strict increase, zero cost at collocation, and Fibonacci forcing of φ. It advances the Recognition Science forcing chain by converting edgewise J-cost minimality into global field constancy, consistent with T5 J-uniqueness. No open scaffolding remains.
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