Computational Complexity in Dynamic Systems:A Rigorous Rejection of P = NP in Chess,Genetics, and the Stock Market

Authors

This paper provides a rigorous analysis of the computational complexity

in three dynamic systems—chess with rule-changing power, genetic

sequence alignment with random mutations, and stock market prediction

with manipulative players—to argue that P ̸= NP. We emphasize that

the temporal patterns (e.g., move sequences in chess, evolutionary timelines

in genetics, and time-series price movements in the stock market)

in these systems cannot be reduced to polynomial-time computable problems,

due to their inherent exponential or super-polynomial complexity.

In the chess scenario, rule changes are constrained to preserve EXPTIMEcompleteness.

In genetics, multiple sequence alignment (MSA) remains

NP-complete even under stochastic mutations. In the stock market, predicting

short-term time-series patterns is NP-hard or worse, as market

efficiency implies P = NP. Formal arguments, including reductions and

complexity class separations, demonstrate that these temporal dynamics

resist polynomial-time solutions. A chart visualizes the exponential state

space growth, reinforcing the rejection of P = NP.