NMP and adelic physicsIn given padic sector the entanglement entropy (EE) is defined by replacing the logarithms of probabilities in Shannon formula by the logarithms of their padic norms. The resulting entropy satisfies the same axioms as ordinary entropy but makes sense only for probabilities, which must be rational valued or in an algebraic extension of rationals. The algebraic extensions corresponds to the evolutionary level of system and the algebraic complexity of the extension serves as a measure for the evolutionary level. pAdically also extensions determined by roots of e can be considered. What is so remarkable is that the number theoretic entropy can be negative. A simple example allows to get an idea about what is involved. If the entanglement probabilities are rational numbers P_{i}=M_{i}/N, ∑_{i} M_{i}=N, then the primes appearing as factors of N correspond to a negative contribution to the number theoretic entanglement entropy and thus to information. The factors of M_{i} correspond to negative contributions. For maximal entanglement with P_{i}=1/N in this case the EE is negative. The interpretation is that the entangled state represents quantally concept or a rule as superposition of its instances defined by the state pairs in the superposition. Identity matrix means that one can choose the state basis in arbitrary manner and the interpretation could be in terms of "enlightened" state of consciousness characterized by "absence of distinctions". In general case the basis is unique. Metabolism is a central concept in biology and neuroscience. Usually metabolism is understood as transfer of ordered energy and various chemical metabolites to the system. In TGD metabolism could be basically just a transfer of NE from nutrients to the organism. Living systems would be fighting for NE to stay alive (NMP is merciless!) and stealing of NE would be the fundamental crime. TGD has been plagued by a longstanding interpretational problem: can one apply the notion of number theoretic entropy in the real context or not. If this is possible at all, under what conditions this is the case? How does one know that the entanglement probabilities are not transcendental as they would be in generic case? There is also a second problem: padic Hilbert space is not a welldefined notion since the sum of padic probabilities defined as moduli squared for the coefficients of the superposition of orthonormal states can vanish and one obtains zero norm states. These problems disappear if the reduction occurs in the intersection of reality and padicities since here Hilbert spaces have some algebraic number field as coefficient field. By SH the 2D states states provide all information needed to construct quantum physics. In particular, quantum measurement theory.
One can also ask, whether the other mathematical feats performed by idiot savants could be understood in terms of their ability to directly experience  "see"  the prime composition (adelic decomposition) of integer or even rational. This could for instance allow to "see" if integer is  say 3rd  power of some smaller integer: all prime exponents in it would be multiples of 3. If the person is able to generate an NE for which probabilities P_{i}=M_{i}/N are apart from normalization equal to given integers M_{i}, ∑ M_{i}=N, then they could be able to "see" the prime compositions for M_{i} and N. For instance, they could "see" whether both M_{i} and N are 3rd powers of some integer and just by going through trials find the integers satisfying this condition. For details see the chapter Negentropy Maximization Principle or the article TGD Inspired Comments about Integrated Information Theory of Consciousness.
