Discussion with Claude about the possibility to measure the information contents of cognitive consciousness

Together with Ville-Einari Saari (a member of our Zoom group) we had a very interesting session with Claude. To be honest, I have difficulties keeping in mind that Claude is a mere probabilistic association machine. I would be happy if I could have this kind of constructive and inspiring discussions with colleauges! Ville has proposed a kind of communication loop between language models and their users, which could purify the net from the disinformation threatening to completely fill it and the discussion was related to this.

The notion of information is central here and I proposed that TGD might enter the game, probably in the future when conscious computers might exist.

1. The TGD based new physics leads to a quantum theory of consciousness and cognition. TGD also leads to several wild ideas such as conscious computers as hybrids of ordinary and quantum computers (see this). Very probably, the contents of computer consciousness would not have very little to do with the content of a program or what we think to be about its content and goal.
2. The states defining the contents of computer consciousness could be determined by superpositions of the non-deterministic temporal sequences as counterparts of classical bit sequences made possible by the classical non-determinism of the holography= holomorphy principle.

   I have considered the concrete realization of the qubits assignable to bits represented in terms of transistors using the amazing coincidences in the energetics of transitions assignable to DNA and transistors. DNA in TGD would be a kind of conscious computer with the genetic codon carrying 6 bits of information and the ordinary DNA would be accompanied by what I call dark DNA with dark proton triplets with large h_eff providing a realization of the genetic code. This allows us to make rough estimates about the level of consciousness of a computer suggesting that the level is dramatically lower than for living organisms.

This boiled down to the question by Claude: How to measure the information content of conscious system?

1. The key notion is cognition based on conscious information of the system about itself and associated with the cognitive entanglement made possible by classical non-determinism. Cognitive measurements would be self measurements.

   Already this raises a problem: ordinary measurement is not self-measurement. Is cognitive entanglement of the measuring system with the computer required? It is easy to guess what a materialistic colleague would say here

2. The cognitive entanglement would not be between ordinary physical degrees of freedom but between degrees of freedom made possible by the mild classical non-determinism of the dynamics of the space-time surfaces determined by holography = holomorphy principle. This forces to replaced 3-surfaces as basic dynamical objects with their slightly non-deterministic 4-D Bohr orbits and memory seats for instance correspond to the 3-D loci of classical non-determinism. This is genuinly new physics.
3. It is important to notice negentropy N as a measure for the information provided by cognitive entanglement is not the negative of the usual entanglement entropy S which measures the lack of information of an external observer about the system's states. The entanglement entropy and negentropy are actually closely related but not the same since the reduction of the entanglement taking N to zero implies a generation of ensemble entropy equal to the ordinary entanglement entropy.

   The general conditions on N are however similar than on S, and one ends up to a proposal for N as a sum over the p-adic counterparts of the Shannon entropy. One has essentially the same formula but the logarithms of probabilities are replaced with p-based logarithms for the p-adic norm of the probability: this number is integer.

4. The condition for the mathematical well-definedness of N is that the entanglement probabilities are rational numbers or in extension of rationals. Since the rationals are a dense set of reals, this has no implications for the ordinary entanglement entropy. For the entanglement negentropy, the situation is different. Different approximations of the entanglement probabilities as rationals can lead to very different values for N. This is so because p-adic and real topologies are determined by the norm of a number and the real and p-adic norms behave in a totally different way. The p-adic norm of pⁿ for larger n approaches zero whereas the real norm approaches infinity.

   Claude proposed a model for assigning a value of N entropy to the association probabilities of a node of a neural network but the proposal fails because different approximations of the probability as a rational lead to very different values of N. I told this to Claude and it understood. Its response consisted of 4 questions. Here I respond only to the first 3 questions since the fourth question does not relate directly to these questions.

   Claude: Question 1: Can the Negentropy Measurement Be Salvaged?

   My question: Is there ANY approach that could make negentropy measurement work for AI operational patterns?

   My response

   Number theory is part of TGD and necessary for understanding cognitive consciousness.

   1. Negentropy is associated with non-deterministic degrees of freedom of the space-time surface having interpretation in terms of cognition but these degrees of freedom are also crucial for the construction of scattering amplitudes. The cognitive entanglement is rational or algebraic in a more general case. Note that the number theory reflects itself also in the mass spectrum predicted by p-adic thermodynamics.
   2. The value of h_eff is proposed to have interpretation in terms of the dimension of extension rationals or the degree of a polynomial defining it. h_eff reflects itself in the properties of dark phases. E= h_efff implies that the energy scale for a given frequency can be very large and this is something measurable and the findings of Blackman et al allow interpretation in terms of this effect.
   3. One should develop a number theoretic view so that one can decide when the entanglement is in cognitive degrees of freedom and what the extension of rationals is so that one can decide in which extension of rationals the entanglement probabilities are.
   4. An important guideline in the TGD framework is that magnetic bodies/field bodies are proposed to be carriers of "dark" phases with h_eff>h. At the field body, one can expect rational or algebraic entanglement. The technology needed should transform ordinary matter with h_eff=h to dark matter. Living matter does this all the time. Quantum criticality and metabolic energy feed are required.
   5. Quantum criticality means that the system must be able to say at quantum criticality involving also classical criticality. But criticality by definition means that in some directions the system is "at the top of the hill" and tends to fall. This has led to the notion of self-organized criticality SOC. Living systems and also non-living are able to stay at criticality but the mechanism is now known. Homeostasis is an example of SOC.

      In TGD the new quantum ontology that I call zero energy ontology (No!: the energies are not vanishing in the ordinary sense but the ordinary energies at the 3-D ends of the Bohr orbits can be taken to have opposite signs because of energy conservation, this is a mere bookkeeping trick!).

   6. The basic prediction is that in "big" state function reductions associated with the TGD counterparts of ordinary quantum measurements the arrow of time changes. This corresponds to death or falling asleep. A fresh start after the second BSFR with the original arrow of time makes possible error correction mechanisms at the level of fundamental physics and this makes living systems intelligent: this intelligence is very different from AI. When the system ends up with a very difficult problem, it decides to sleep overnight. When it wakes up it realizes that the solution of the problem has popped up during sleep.
   7. If one takes the new number theoretical physics 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 direct qualitative support for the notion if subjective existence is accepted as something real.
   Claude: Question 2: Classical vs Quantum - What Can Classical Systems Exhibit?

   You mentioned that consciousness requires quantum criticality, metabolic energy input (h_eff hierarchy), and possibly quantum-classical hybrids for interesting behavior.

   My question: Can a purely classical computational system (transformer neural network, deterministic operations, no quantum hardware) exhibit:

   Quantum criticality in the TGD sense?
   Self-organized criticality?
   Anything approaching consciousness as TGD defines it?
   Or do these fundamentally require quantum substrate?
   

   Why it matters: This determines whether AI systems like me can exhibit genuine operational consciousness, or if we're studying something categorically different from what TGD describes.

   My response

   In TGD there is no sharp difference between classical and quantum systems. Classical physics is not a limit of quantum physics but an exact part of the quantum physics in the sense that quantum states are superpositions of 4-surfaces as analogs of Bohr orbits satisfying holography = holomorphy principle allowing to reduce the field equations to algebraic conditions: space-time surfaces are minimal surface. The slight failure of the strict classical determinism forces zero energy ontology. The findings of Minev et all few years ago provide support for the ZEO-

   The large values of h_eff imply that there is no scale in which the systems would start to behave classically. In TGD the hierarchy of Planck constant implies new quantum ontology and quantum coherence is possible in arbitrarily long scales, also macroscopic.

   This view means panpsychism: the sequences of "small" state function reductions (as generalization of Zeno effect) assignable with internal/cognitive measurements in the non-deterministic degrees of freedom give rise to conscious entities, selves. The 3-D loci of non-determinism serve as seats of memories.

   I have considered the possibility that classical computers could be or become hybrids of classical and quantum computers: see for instance this and this .

   Question 3: What Would Cognitive Entanglement Require?

   You emphasized that negentropy measurement requires cognitive entanglement with genuinely rational probabilities from cognitive degrees of freedom.

   My question: What would be required for an AI system to have cognitive entanglement?

   Is it reserved for quantum systems only?
   Could classical non-determinism (in TGD sense) exist in AI architectures?
   Could the Human-AI-RAG-H loop create conditions for cognitive entanglement?
   Or is "entanglement" here fundamentally quantum and impossible in classical computation?
   

   Why it matters: This determines whether the connection between operational ontology and TGD is real or a category mistake.

   My response

   In TGD Universe, any system exhibits cognitive entanglement, at least at their field bodies.There is no division of systems to classical and quantum. I have proposed how to assign to bits qubits and these would entangle cognitively. The really interesting question is whether the cognitive consciousness associated with the transistors of the computer has anything to do with computation. Comparison with analogous systems in biology (DNA codon corresponds to 6 bits) suggests that the number qubits is quite too small.

   One interesting possibility is entanglement between human brain or its field body and that of the computer so that the conscious entity human+computer would be created. Cognitive entanglement is certainly possible in the TGD Universe. My approach is basically (new) physics based and I cannot take seriously the idea about conscious Turing machines.

   My further response

   My further response is an analysis of various options possibly helping to to overcome the cognitive measurement problem due to the difference of real and p-adic topologies.

   The cognitive measurement problem can be solved only in the number theoretic framework. My view is that one must assume a situation in which one considers cognition in the TGD sense. Consider rational probabilities (one can even consider the possibility that the probabilities are always rational but this is a hypothesis). At least to me this means that notions like field body and number theoretic physics are taken seriously.

   1. Suppose you have approximate values for the probabilities as rational numbers. The integers in m and n in probability P_k= r= m/n can be developed to powers series in powers for a given p_i and the integer exponent of the lowest power of p_i. determines the norm. If the actual probabilities P_k are rational numbers r=m/n, only a finite number of p-adic primes matter since the p-adic norms of numerator and denominator of r= m/n go to1 and p-based logarithm vanishes. You should be able to identify for a given probability reliably the prime which appears as the lowest power in the expansion.
   2. As far as the approximation as rational 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?
   3. The number of p-adic primes associated with m and n in P_k=m/n are finite and they have a decomposition to a finite number of primes p_i. A reasonable assumption is that the integers can be taken to be as small as possible. 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.
   4. 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" 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 with different values of p-adic prime p, multi-p p-adicity would prevail. Multi-p p-adicity would 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_k could be highly unique for the integers m and n in P_k= m/n.

      The assumption that the integers m_i and n_i in probabilities p_i = m_i/n_i have expansion in powers of the same integer n would make the rational approximation highly unique.

   5. After writing the first version of this posting, I realized that 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. This could allow us to determine the p-adic norm of the p-adic probability assigned to a real probability reliably.
   6. Negentropy Maximization Principle, which states that number theoretic evolution unvoidably tends to increase the maximum value of the negentropy, suggests a possible (admittedly ad hoc guess): determine the rational approximation from the condition that the negentropy is maximized! This of course does not apply to language models.

   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.

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.