Ratio-Energy Rigidity on Connected Graphs Forces Holographic Information Distribution: A Machine-Verified Proof with Application to Neural Architecture

We prove, in machine-verified Lean 4 (41 theorems, zero sorry), that the ratio energy CG[x] = ∑(v,w)∈E J(xv/xw) on any connected graph G forces field constancy at its unique minimum: every vertex carries the same value, so any local region determines the global state. We extend this to a quantitative perturbation bound: J(r) ≤ δ implies (r−1)² ≤ 8δ, and show the exact result is recovered continuously at δ=0. Separately, we prove that J-cost monotonicity forces optimal pattern allocation to be local (caching theorem), and that self-similar access structures fix the hierarchy ratio at the golden ratio φ. Applying these graph-theoretic results to the brain—modeled as a connected subnetwork whose dynamics approximately minimize J—we derive that (i) the boundary of any cortical region determines the information accessible from it (boundary-encodes-bulk), (ii) information accessibility scales with boundary size (surface area in D=3), and (iii) partial removal preserves information access provided connectivity is maintained. The latter prediction is consistent with hemispherectomy data. We distinguish this ratio-rigidity constancy (all vertices equal at the minimum) from Pribram–Gabor holography (interference-pattern encoding), and identify the perturbation structure as the physically relevant regime where distributed, non-trivial information coexists with approximate global coherence. Four falsifiable predictions are given, including a surface-area–vs.–volume test for fMRI decoding accuracy.