On the Sum of p-adic Negentropies & Real Entropy

Abstract

It is shown that, for rationals entanglement probabilities, the real entropy equals to the sum of p-adic negentropies. However, this is no more true for entanglement probabilities in an extension of rationals inducing a finite-dimensional extension of p-adic numbers. A possible interpretation is that at the lowest level of the evolutionary hierarchy defined by the extensions of rationals the p-adic negentropy serving as a measure for conscious information (trivial extension) equals to real entanglement entropy. For algebraic extensions, this is no more true and Negentropy Maximization Principle suggests that total p-adic negentropy is, in general, larger than real entropy. Alternative (not so attractive), interpretation is that negentropy also includes the negative real contribution so that, for rational entanglement probabilities, the total conscious information would vanish. Large p-adic negentropy, however, tends to be accompanied by large real entropy which conforms with the vision of Jeremy England.