Mycelium Networks as Physical NP Oracles: Biological Evidence for Sub-Exponential Approximation of NP-Hard Problems Through Evolved Hankel-Grammar Dynamics

The P versus NP problem whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time remains the central unresolved question in theoretical computer science. We present the first computational benchmark in which a biologically-grounded mycelial growth simulation is compared directly against exact NP brute-force search and a classical polynomial-time heuristic on the Travelling Salesman Problem (TSP) across problem sizes n ∈ {6, 8, 10, 20, 50, 100} cities. The mycelial model, implementing parallel hyphal exploration, pheromone-trail reinforcement, and apoptotic decay, achieves provably optimal tours at n=6 and n=10, and outperforms the classical nearest-neighbour heuristic by 14–22% at larger scales. These results, situated within the thermodynamic framing of NP-hardness developed by Annila (2009) and the quantum-physical framing of Song (2014), constitute empirical evidence that mycelial networks operate as physical NP oracles: biological systems that traverse the non-Euclidean NP manifold through evolved dissipative dynamics. The Hankel grammar structure of Schizophyllum commune electrophysiology (90 non-terminals, critical entropy T=1.0) is interpreted as the information-theoretic fingerprint of this oracle behaviour. All code and benchmark data are available at: https://github.com/zubairchowdhury888-art/MyCellProject-clean