Fractal Quantum Holographic Paradigm - Rényi Entanglement Entropy Scaling in Graph States

Within the FKGP (Fractal Quantum Holographic Paradigm) framework we propose and experimentally verify a scaling law for the second-order Rényi entanglement entropy S2(A) in graph states of varying topological dimension D. The key result is S2(A) = rank_GF2(Gamma) · ln 2, where the GF(2)-rank of the A-B boundary adjacency matrix increases by exactly one at each step of the dimensional ladder. On IBM Heron r2 (ibm_fez) with 65,536 shots per circuit, four steps are confirmed: D=2.00 (ΔSNR=76.9σ), D1=2.05 (34.3σ), D2=e (7.8σ), and D=3 (5.1σ). All detections exceed the 5σ threshold. Cosmological hypotheses are advanced: D1 = 2 + Ω_b (baryon fraction) and D2 = e (informational efficiency). This paper includes a complete reproducibility appendix (Batches 1–10): all QASM circuits, measured P₀ values, noise corrections, and ΔSNR calculations.