Bell's Theory with no Locality assumption: putting Free Will at work

We prove a version of the Bell's Theorem that does not assume Locality but only the Effect After Cause Principle (EACP) according to which for any Lorentz observer the value of an observable cannot change because of an event that happens after the observable is measured. Since the EACP is compatible both with Locality and with Non-Locality, Locality cannot be considered as the common cause of the contradictions obtained in all versions of Bell's Theory. By definition, all versions of Bell's Theorem assume Weak Realism according to which the value of an observable needed in the discussion of Bell's Theorem is well defined whenever the measurement could be made and some measurement is made. As a consequence of our results, Weak Realism becomes the only hypothesis common to the contradictions obtained in all versions of Bell's Theory. This work indicates that it is Weak Realism, not Locality, that needs to be negated to avoid the contradictions in microscopic Physics associated to Bell's Theory, at least if one refuses as false the de Broglie-Bohm Hidden Variable theory because of its essential violation of Lorentz invariance. This paper completes with much more details the genuine Bell Theorem part of a previous paper. That paper also offered a treatment of the GHZ entanglement, a treatment which did not suffer from the lack of clarity of the definition of the EACP.