Michel Planat

False Chebyshev Primes: Prime-Phase Non-Equidistribution and a Weighted Local-Time Law

Let ψ be the Chebyshev prime-counting function and π(x) the number of primes up to x. The Chebyshev-prime condition K(p)=li(ψ(p))−li(ψ(p−1))<1 is not equivalent to the sign rule ψ(p)−p>0. We prove that its leading threshold is the midpoint correction ψ(p)−p>12logp, with an explic...

Painlevé Dynamics and the Origin of Life: A Universal Chemical Pathway

This paper proposes that the origin of life may be understood as a cascade of dynamical integration events governed by Painlevé transcendental equations—a class of nonlinear differential equations arising widely in physics, from quantum mechanics to general relativity. Four prebi...