### What does Negentropy Maximization Principle really say?

There is something in NMP that I still do not understand: every time I begin to explain what NMP is I have this unpleasant gut feeling. I have the habit of making a fresh start everytime rather than pretending that everything is crystal clear. I have indeed considered very many variants of NMP. In the following I will consider two variants of NMP. Second variant reduces to a pure number theory in adelic framework inspired by number theoretic vision. It is certainly the simplest one since it says nothing explicit about negentropy. Second variant says essentially the as "strong form of NMP", when the reduction occurs to an eigen-space of density matrix.

I will not consider zero energy ontology (ZEO) related aspects and the aspects related to the hierarchy of subsystems and selves since I dare regard these as "engineering" aspects.

** What NMP should say?**

What NMP should state?

- NMP takes in some sense the role of God and the basic question is whether we live in the best possible world or not. Theologists asks why God allows sin. I ask whether NMP demand increase of negentropy always or does it allow also reduction of negentropy? Why? Could NMP lead to increase of negentropy only in statistical sense - evolution? Could it only give potential for gaining a larger negentropy?

These questions have turned to be highly non-trivial. My personal experience is that we do not live in the best possible world and this experience plus simplicity motivates the proposal to be discussed.

- Is NMP a separate principle or could NMP be reduced to mere number theory? For the latter option state function would occur to an eigenstate/eigenspace of density matrix only if the corresponding eigenvalue and eigenstate/eigenspace are expressible using numberes in the extension of rationals defining the adele considered. A phase transition to an extension of an extension containing these coefficients would be required to make possible reduction. A step in number theoretic evolution would occur. Also an entanglement of measured state pairs with those of measuring system in containing the extension of extension would make possible the reduction. Negentropy would be reduced but higher-D extension would provide potential for more negentropic entanglement. I will consider this option in the following.

- If one has higher-D eigenspace of density matrix, p-adic negentropy is largest for the entire subspace and the sum of real and p-adic negentropies vanishes for all of them. For negentropy identified as total p-adic negentropy strong from of NMP would select the entire sub-space and NMP would indeed say something explicit about negentropy.

** The notion of entanglement negentropy**

- Number theoretic universality demands that density matrix and entanglement coefficients are numbers in an algebraic extension of rationals extended by adding root of e. The induced p-adic extensions are finite-D and one obtains adele assigned to the extension of rationals. Real physics is replaced by adelic physics.

- The
**same**entanglement in coefficients in extension of rationals can be seen as numbers is both real and various p-adic sectors. In real sector one can define real entropy and in various p-adic sectors p-adic negentropies (real valued).

- Question: should one define total entanglement negentropy as

- sum of p-adic negentropies or

- as difference for the sum of p-adic negentropies and real etropy. For
**rational entanglement**probabilities real entropy equals to the sum of p-adic negentropies and total negentropy would vanish. For extensions this negentropy would be positive under natural additional conditions as shown earlier.

- sum of p-adic negentropies or

** State function reduction as universal measurement interaction between any two systems**

- The basic vision is that state function reductions occur all the for all kinds of matter and involves a measurement of density matrix ρ characterizing entanglement of the system with environment leading to a sub-space for which states have same eigenvalue of density matrix. What this measurement really is is not at all clear.

- The measurement of the density matrix means diagonalization of the density matrix and selection of an eigenstate or eigenspace. Diagonalization is possible without going outside the extension only if the entanglement probabilities and the coefficients of states belong to the original extension defining the adele. This need not be the case!

More precisely, the eigenvalues of the density matrix as roots of N:th order polynomial with coefficients in extension in general belong to N-D extension of extension. Same about the coefficients of eigenstates in the original basis. Consider as example the eigen values and eigenstates of rational valued N× N entanglement matrix, which are roots of a polynomial of degree N and in general algebraic number.

Question: Is state function reduction number theoretically forbidden in the generic case? Could entanglement be stable purely number theoretically? Could NMP reduce to just this number theoretic principle saying nothing explicit about negentropy? Could phase transition increasing the dimension of extension but keeping the entanglement coefficients unaffected make reduction possible. Could entanglement with an external system in higher-D extension -intelligent observer - make reduction possible?

- There is a further delicacy involved. The eigen-space of density matrix can be N-dimensional if the density matrix has N-fold degenerate eigenvalue with all N entanglement probabilities identical. For unitary entanglement matrix the density matrix is indeed N×N unit matrix. This kind of NE is stable also algebraically if the coefficients of eigenstates do not belong to the extension. If they do not belong to it then the question is whether NMP allows a reduction to subspace of and eigen space or whether only entire subspace is allowed.

For total negentropy identified as the sum of real and p-adic negentropies for any eigenspace would vanish and would not distinguish between sub-spaces. Identification of negentropy as as p-adic negentropy would distinguish between sub-spaces and´NMP in strong form would not allow reduction to sub-spaces. Number theoretic NMP would thus also say something about negentropy.

I have also consider the possibility of weak NMP. Any subspace could be selected and negentropy would be reduced. The worst thing to do in this case would be a selection of 1-D subspace: entanglement would be totally lost and system would be totally isolated from the rest of the world. I have proposed that this possibility corresponds to the fact that we do not seem to live in the best possible world.

** NMP as a purely number theoretic constraint? **

Let us consider the possibility that NMP reduces to the number theoretic condition tending to stabilize generic entanglement.

- Density matrix characterizing entanglement with the environment is a universal observable. Reduction can occur to an eigenspace of the density matrix. For
**rational**entanglement probabilities the total negentropy would vanish so that NMP formulated in terms of negentropy cannot say anything about the situation. This suggests that NMP quite generally does not directly refer to negentropy.

- The condition that eigenstates and eigenvalues are in the extension of rationals defining the adelic physics poses a restriction. The reduction could occur only if these numbers are in the original extension. Also rational entanglement would be stable in the generic case and a phase transition to higher algebraic extension is required for state function reduction to occur. Standard quantum measurement theory would be obtained when the coefficients of eigenstates and entanglement probabilities are in the original extension.

- If this is not the case, a phase transition to an extension of extension containing the N-D extension of it could save the situation. This would be a step in number theoretic evolution. Reduction would lead to a reduction of negentropy but would give potential for gaining a larger entanglement negentropy. Evolution would proceed through catastrophes giving potential for more negentropic entanglement! This seems to be the case!

Alternatively, the state pairs of the system + complement could be entangled with observer in an extension of rationals containg the needed N-D extension of extension and state function possible for observer would induce reduction in the original system. This would mean fusion with a self at higher level of evolutionary hierarchy - kind of enlightment. This would give an active role to the intelligent observer (intelligence characterized by the dimension of extension of rationals). Intelligent observer would reduce the negentropy and thus NMP would not hold true universally.

Since higher-D extension allows higher negentropy and in the generic case NE is stable, one might hope that NMP holds true statistically (for rationals total negentropy as sum or real and total p-adic negentropies vanishes).

The Universe would evolve rather than being a paradize: the number theoretic NMP would allow temporary reduction of negentropy but provide a potential for larger negentropy and the increase of negentropy in statistical sense is highly suggestive. To me this option looks like simplest and most realistic one.

- If negentropy is identified as total p-adic negentropy rather than sum of real and p-adic negentropies, strong form of NMP says something explicit about negentropy: the reduction would take place to the entire subspace having the largest p-adic negentropy.

## 7 Comments:

Matti, you are deeply embedded in and familiar with your own writings/ideas. (for example, after all these years, I still have no idea what 'P-adic' and 'adele' mean or refer to mathematically).

Can you give 3 explicit examples (physical systems) where you see NMP occurring? Can you describe where NMP displays in those 3 examples - and - real mechanisms of NMP in them? The specific active processes of behaviors?

Jamie

One other point to mention. I wish you would refrain from mentioning "god" or "best possible world" remarks. Such notions mix metaphysics in with mathematics you seem to be searching for, and the two ways of thinking do not cross-interpret well. Metaphysics can only ask questions -- it is notional, un-testable, and provides no 'answers'.

If you are searching for "meaning" in the mathematics ... or topics like 'reincarnation' ... every science person reading your work will walk away. - JUst my opinion - but an important one for you to consider.

Response to first comment. p-Adics and adeles are standard mathematics of last century. For instance, Groethendieck tried to unify real and p-adic homology theories. Physicists started to work with them slightly before me (I started around 1993 or so).

There are lot of books about them and it is easy to find them in web. Any book about advanced number theory talks also about p-adic numbers and adeles which combine real and extensions of all p-adic number fields associated with a given extension of rationals into bigger structure.

More about first comment. Concerning NMP the applications are at this moment at general level. I am building the theory rather than applying it in the sense of Big Science. I am architect rather than craftsman. I am extending the existing theory. My position is comparable to that of Dira or von Neumann.

NMP and ZEO provide just general solution to the poorly understood problems of quantum theory itself and possibility to formulate theory of conscious entities free of mathematical contradictions.

a) NMP in ZEO would provide the basic variational principle behind quantum measurement theory and consciousness. It produces as special case ordinary quantum measurement theory and this we encounter if every quantum measurement. The basic motivation is to get rid of the basic logical paradox of quantum theory: non-deterministic state function reduction is in conlict with deterministic unitary process.

This problem disappears and outcome is theory of consciousness and understanding of the relationship between subjective and geometric time and prediction that the arrow of time is not the same always. Also understanding about what would happen in biological death and totally surprising prediction that biological death follows by reincarnation as time-reversed conscious entity. This is prediction - not an input from Easten philosophies. This is general understanding, not numerical predictions.

b) The possibility of two arrows of time was realized in living matter already by Fantappie long time ago. Phase conjugate laser rays obeying second law in reversed time direction is one example. Spontaneous self-organization of building bricks of molecules to bigger molecules could be second example.

c) NMP would be behind evolution by forcing the increase of amount of negentropy carried by negentropic entanglement: this involves p-adic numbers in absolutely essential manner. Negentropic entanglement would be correlate for cognition, understanding, information, and maybe also positively colored emotions like love. DNA would be source of negentropic entanglement. In the case that this entanglement is maximal it can be cloned unlike generic entanglement and DNA replication would involve replication of information.

d) NMP and ZEO are especially interesting concerning behaviours. In TGD zero energy state corresponds at space-time level time evolution for 3-surface. Geometric correlate for a behavior rather than state/structure: function as 4-D structure. NMP forces self-organization of behaviours to optimal ones. Self-organization theory for biosystem becomes self-organization theory for its behaviours. This provides again a totally new vision about genes. These are extremely general predictions and I am sorry that I cannot provide detailed models for specific examples.

All these predictions relate to something that standard physics does not allow to say anything: what happens in quantum measurement, how to described observer in quantum theory (as conscious entity), how to describe cognition (p-adics and adeles), what happens in life and what after biological death, what is a measure for conscious information.

I use "god" just in order to make clear that thinkers have encoutered same problems but in different disguise. The only difference between ancient thinkers and us is that we have refined mathematics and natural sciences to use. The problem is that we have also enormous arrogance which tends to make these tools useless.

I have certainly no intention to mix metaphysics with mathematics. Mathematics for me is much more that deterministic algorithts or Newtonian clockwork model for Universe. The basic element of consciousness theory is

to bring in ontology totally new level allowing to speak about free will and consciousness. In TGD one can say that one level of reality consists of mathematical objects: zero energy states and consciousness is associated with quantum jumps between them re-creating the Universe. This is something totally new but consistent with mathematics - and making conscious mathematical understanding possible! The only goal is to find quantum physical correlates for self,

qualia, thoughts, emotions, or even meaning. Mathematical formulas are zombies and one cannot assign "meaning" to them.

"Best possible world" has clear meaning when I use it (a little bit jokingly): in the best possible world NMP would allow always only the maximal negentropy gain: this is a precisely defined statement mathematically. One could consider also the possibility that the gain is not maximal and this might be used to explain why conscious entities do not seem to behave completely rationally always ("God" would allow freedom to do sin).

One cannot avoid the analogies for the basic questions of spiritual world views when one talks seriously about consciousness. Only materialistic could do this but he has already given up the attempt to say anything interesting about consciousness. I have a humbler attitude: people at different times are talking about same things but using different words. I do not identify myself as physicists or mathematicians since this would narrow down my scope fatally and I would become only a specialistist getting pleasure from hunting of crackpots.

As I told I am not searching for explanation for terms like re-incarnation. TGD based quantum meaurement theory predicts it in a well-defined sense. After experience of almost forty years, I am course extremely well aware that an average colleague refuses to even read TGD. This I must just accept.

Average colleague applies algorithms to make the building. I see myself as the architect who plans the building: there is enormous intellectual gap which is extremely difficult to bridge. TGD is at least century before its time but eventually no other option remains than to accept it. But this is not my problem. I am just a messanger and usually the messangers are killed - especially so if the message is especially good;-).

Immediate short reply - thank you for the math-world re-directions about 'p-adics'. I have found internet articles~explanations - bookmarked, copied, printed several, to fill the hole in my knowledge bank. My first brief impression is that p-adics help define a smoother mathematical continuum. I am interested in examining relationship to a possible math architecture of "diagonally nested exponents numberlines" - which one to use as relativistic first reference-base, and, if it is reasonable to navigate to any of them optionally ; and then: 'cybernetic information re-coding' . . . if such navigation choices are made ; can information be invariant under such cybernetic navigation shifts. Key is: L'Hopital's Rule.

Will read your other responses now, Matti. respond to them after reading. Thank you, Jamie

p-Adics are number theory a tool of simplifying Diophantine equations. One can find p-adic solution very often as p-adic power series. If the power series has finite number of terms then also real solution exists. One might say that p-adic topology is much rougher than real topology and this makes possible this simplification. Indeed p-adic numbers are not well-ordered expect by p-adic norm determing by the lowest power in its pinary expansion. Interpreted as real numbers as such most of them are infinite. p-Adic differential calculus exists. Integral is problematic and brightests mathematicians have worked with this.

In TGD framework the algebraic continuation of rational number based physics to various number fields allows to define the needed integrals so that they are number theoretically universal. This leads to the notion of monadic manifold having in preferred coordinates a discretization in an extension of rationals inducing those of p-adic number fields. In rel case one has manifold but the open sets are labelled by these preferred algebaic points. In p-adic context one can also assign to each point a smooth p-adic continuum so that one obtains differential calculus also now. Definite integral is defined in terms of discretization. Platonic solids provide a good example: they define algebraic discretizations of sphere. This generalizes to coset spaces naturally since they have natural preferred coordinates by their symmetries.

In TGD adeles can be seen as Carteisian product of reals and of extensions of various p-adic number fields iduced by extension of rationals. One can say that at the level of imbedding space reality and p-adicities intersect through the algebraic points defining the discretization of imbedding space. This can be seen as cognitive representation: the fact is indee that we can do numerics only using rationals and in more abstract sense in terms of algebraic numbers in some extension of rationals. At the end of calculations we perform rational approximation for the algebraics. The intersection of this discretization with space-time surface does same at the level of space-time surfaces.

I have strong gut feeling that p-adic calculus has a lot to give to computation. One problem is the correspondence between reals and p-adics and canonical identification and identification of common algebraic points is one possible solution to this problem.

Matti

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