Is it possible to measure cognitive entanglement negentropy somehow?

The discussion with Ville-Einari Saari and Claude inspired a blog post related to the measurement of entanglement negentropy as a measure for the level of cognitive consciousness. In the following I try to articulate the basic ideas more precisely.

Entanglement negentropy as a measure of conscious information is not the negative of the ordinary entanglement entropy but sum over p-adic contributions obeying however the same kind of formula as the Shannon entropy. For a given p-adic prime p, the logarithms of probabilities are replaced by integer value p-based logarithms of their p-adic norms. This requires that the entanglement probabilities are rationals or belong to the extension of rationals.

Assume that the entanglement probabilities are measured somehow. The problem is that they cannot be known with an infinite precision and the approximation as a rational number can lead to very different outcomes for the negentropy. For instance, multiplying the probabilities with a rational r=m/n very near to unity such that m and n are very large integers, can change the sum of the p-based logarithms dramatically. The reason is that real and p-adic topologies are very different. The power pⁿ for large n approaches zero in p-adic sense but to infinity in real sense.

Measurement of the amount of conscious information is in question and it is not surprising if problems emerge if one starts from real numbers which are essentially measures for magnitude: consciousness cannot be weighed.

The first question is of course whether cognitive entanglement negentropy is useful in any way? This seems to be the case. If one takes the number theoretical physics predicted by TGD as a correlate for cognitive consciousness seriously, one can see the effects due to the reduction of negentropy at a qualitative level. In absence of metabolic energy feed needed to increase the values of h to h_eff, h_eff spontaneously decreases and the negentropic resources are reduced. The level of consciousness is reduced and the system gets tired or even loses consciousness. This can be seen as a direct qualitative support for the notion if subjective existence is accepted as something real.

What is clear is that if the cognitive measurement problem can be solved it must be carried out in the number theoretic framework. At least to me this means that notions like field body, zero energy ontology, and number theoretic physics are taken seriously. For the sake of simplicity, consider in the sequel rational probabilities. One can also consider the possibility that the probabilities are always rational: this would conform with the way how they are estimated experimentally, at least in real number based physics by repeated measurements.

1. As far as the approximation as rationals is considered, only the p-based logarithms appearing in the expression of negentropy are problematic. The integer of the lowest power of p is sensitive to the approximation as a rational. Could some additional physically motivated assumptions allow to eliminate this sensitivity? And could one restrict the number of primes involved?
2. Suppose approximate values for the probabilities have been somehow deduced as rational numbers by performing measurements for a cognitive ensemble. The estimates for the probabilities P_k= m_k/n_k are rational. The integers in m_k and n_k can be developed to powers series in powers for a given prime p_i and the integer exponent of the lowest power of p_i determines the norm of m_k and n_k.

   If the actual probabilities P_k are rational numbers P_k=m_k/N, only a finite number of p-adic primes matter since the p-adic norms of numerator and denominator of r= m/n are equal to 1 for large primes and p-based logarithm vanishes. One should be able to identify for a given probability reliably the prime, which appears as the lowest power in the expansion.

3. Canonical identification, crucial for the interpretation of p-adic mass calculations (see this and this), provides an attractive way to fix the p-adic norm assigned to the real probability. Canonical identification I: ∑ x_kp^k→∑ x_kp^-k maps p-adic numbers in a continuous way to real numbers. The inverse of I is for a finite number of the pinary digits two-valued. The reason is that the p-adic numbers -1_p=(p-1)/(1-p) and 1/p are mapped to the same real number p. Assuming that the number of the pinary digits is finite, the image of a real number is unique. Note that it is absolutely essential that rationals (and even reals) are mapped to p-adics: if the integers m and n in r=m/n are mapped separately by canonical identification one encounters the non-uniqueness problem caused by finite accuracy.

   This raises the possibility that one could, at least formally, assign cognitive negentropy also with ordinary probabilities, even with association probabilities associated with language models. If one can assign a useful information measure to these probabilities, one is forced to ask whether the system involved could have rudimentary consciousness?

Consider an actual cognitive measurement (whatever it could mean!).

1. The assumption that the experimenter can control the total number N of measurements looks unrealistic since cognitive entanglement is in question so that standard kind of measurement is simply impossible. It is not possible to put the mind on a scale.
2. The assumption that a measurement in the standard sense is possible indeed leads to problems. For the actual measurement n_k would correspond to the total number N of measurements so that one has P_k= m_k/N. The problem is that the prime decomposition of N is highly sensitive to its value and changes dramatically in N→ N+1. A technical way to avoid these problems is to assign p-adic norms to the probabilities by canonical identification. This option looks rather convincing.
3. The alternative way to get rid of this sensitivity is to assume that N is not under the control of experiment and the probabilities are deduced in some other way than by performing a measurement for a cognitive ensemble.
4. Could time series of measurement, whose duration cannot be controlled by the observer be considered. Could the number of loci of non-determinism for the Bohr orbit somehow determine the number N of cognitive measurements? If so, the geometric duration of the Bohr orbit would determine the value of N and the probabilities P_k.

   p-Adic length scale hypothesis for which the holography = holomorphy vision leading to a generalization of p-adic number fields to their functional counters suggests that favored values for N are primes or powers of prime.

Assuming that one is not satisfied with the technical solution of the problem, could the assumptions about the measured cognitive system help?

1. The number of p-adic primes associated with m_k and n_k in P_k=m_k/n_k are finite and they have a decomposition to a finite number of primes p_i. A reasonable assumption is that the integers m_k and n_k can be taken to be as small as possible. This conforms with the frequency interpretation of P_k. This would help to make the approximation as rationals more unique and for instance multiplication by a rational, which is a ratio of very large integers and near to unity is not allowed.
2. I have proposed the notion of multi-p p-adicity (see this and this) motivated by the need to define interaction vertices for particles characterized by different p-adic primes. Multi-p p-adicity would be related to the world of the "classical worlds" (WCW) expected to have a spin glass type structure having a decomposition to regions with ultrametric topology characterized by a p-adic primes.

   In the interfaces of the regions of WCW with different values of p-adic prime p, multi-p p-adicity would prevail and mean that the integers involved have expansion in powers of integer n: the primes p_i dividing n would define p-adic primes p_i associated with the multi-p p-adicity. This assumption would give very strong constraints on the p-adic expansion of probabilities and the lowest power for each p_i could be highly unique for the integers m_k and n_p in P_k= m_k/n_k. The assumption that the integers m_k and n_k have expansion in powers of the same integer n would make the rational approximation highly unique.

3. Negentropy Maximization Principle (see this), which states that the number theoretic evolution tends to maximize algebraic complexity and therefore the maximal value of the negentropy, suggests a possible (admittedly ad hoc guess): determine the rational approximation from the condition that the negentropy is maximized!

See the article The recent view of TGD inspired theory of consciousness and quantum biology or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.