How ordinary nuclei can be produced in very high energy collisions of protons?

I learned about a possible solution of a long-standing anomaly related to the proton collisions at very high energies studied at LHC using the ALICE detector. There is a popular article in CERN Courier (see this) telling about the findings published in Nature (see this).

1. The production of deuterons and antideuterons was studied. A long standing mystery has been why also light nuclei are produced so abundantly. At the huge energies involved the temperature is of order pion mass scale ≈ 100 MeV and is so high so that the nuclei have quite too low binding energy to survive and only nucleons should be produced.
2. The researchers found evidence that the about 90 per cent of the deuterons were produced by nuclear fusion from the particles emerging from the collision with one of the particles emerging from a decay of a massive short-lived particle, so called Δ resonance, which consist of 3 quarks just like nucleons. Δ decays in a time of order 10^-24 seconds to nucleon and pion. Note that proton Compton time is about 33 times longer than this time.
3. The studies support the view that decay occurs in the periphery, at sufficient distance from the collision point, where the temperature is lower and the fusion can produce stable deuteron nuclei. The article also mentions that the findings provide support for a model of cosmic ray interactions as cascades.

   The proposal is that the d (anti-d) formation is made possible by pion catalysis. The reaction would be π+p+n→ π+d and the final state pion would carry away the 4-momentum. The pion would be produced in the decay of Δ.

These findings are very interesting from the point of view of TGD.

1. TGD predicts the symmetries of the standard model but since QCD color does not correspond in TGD spin-like quantum number but to partial waves in CP₂, both quarks and leptons move in color partial waves and each color multiplet gives rise to a scaled version of the standard model physics. An infinite hierarchy of standard model physics is predicted. Ordinary hadron physics would correspond to Mersenne prime M₁₀₇ = 2¹⁰⁷-1 and the next one, for which there is evidence from anomalies at LHC, to Mersenne prime M₈₉ with mass scale, which is 512 times higher than for ordinary hadrons (nuclei). For instance, the pion of M₈₉ hadron physics would have mass scale 512× 140 MeV ≈ 70 GeV.
2. The recent solar model is plagued by anomalies. In the TGD based view of the Sun, solar wind and radiation are produced at the surface layer of the Sun in the transformation of M₈₉ hadrons to ordinary hadrons.

   M₈₉ hadrons would decay to ordinary M₁₀₇ hadrons by a process that I call p-adic cooling. The p-adic mass scales would be reduced by powers of 2 (or 2^1/2) and for the first option (107-89)/2=9 steps would be involved (see this). One of the first applications of the p-adic physics was the proposal that p-adic cooling could be involved with very high energy cosmic ray events like Centauros (see this and this).

3. The emerging ordinary nuclei produced in the p-adic cooling could fuse to heavier ones by what I call dark fusion, which provides a TGD based model for the "cold fusion" to heavier nuclei. Less plausibly, they could be produced in the p-adic cooling directly. These heavier nuclei gradually fall downwards in the gravitational field of the Sun so that the usual layered structure of nuclei is near the surface of the Sun rather than in the core of the Sun.
4. Quite recently, a direct support from this layered structure emerged from very weird findings related to a Supernova explosion. One cannot even exclude the possibility that the same process could have occurred even in the formation of planets as a surface layer of the Sun exploded (see this, this, this, and this).

Consider now the very energetic proton and heavy ion collisions from the TGD perspective.

1. The TGD view of generalizes the QCD view of hadron collisions (see this). The interaction region contains the TGD analog of quark gluon plasma in which fermions move as massless particles and obey at the spae-time surfaces induced Dirac equation. In final hadron states the Dirac equation in H= M^4xCP_2 is satisfied. These equations are consistent with each other.

   The counterpart of quark gluon plasma corresponds in QCD to the fragmentation of initial state hadrons to quarks but not gluons as elementary particles. In TGD, gluons as all elementary bosons would only exist as fermion-antifermion pairs. Hadronization correspond to the fusion of quarks and antiquarks to form massive hadrons. This picture applies to all particle reactions, also those involving leptons. In fact, weak interactions and color interactions can be seen as aspects of color interactions.

2. In the TGD Unverse, M₈₉ hadrons could be created in very high energy nuclear and proton collisions in a TGD counterpart of the transition interpreted as a formation of quark gluon plasma. M₈₉ hadrons would have effective Planck constant h_eff/h= 512 and mass scale 512 times higher than for ordinary M₁₀₇ hadrons so that the Compton scales for M₈₉ and M₁₀₇ hadrons would be the same. This would correspond to quantum criticality for M₁₀₇→ M₈₉ phase transition as the TGD counterpart for the QCD transition to quark gluon plasma.
3. The hadronization would lead to the formation of M₈₉ hadrons rather than only M₁₀₇ hadrons, say M₈₉ pions, whose age would be the same as for ordinary hadrons if it scales like (h_eff/h) × m₁₀₇/m₈₉ =1.
4. M₈₉ pions would decay by a p-adic cooling to ordinary nucleons. Dark fusion (or ordinary fusion, which in the TGD framework could actually reduce to dark fusion) at a sufficiently large distance from the collision point could produce nuclei.
5. What would be the value of h_eff/h, if the dark fusion occurs. If it is equal to 512, the Compton lengths of ordinary nuclei would be scaled up by factor 512 and would be of the order of electron Compton length. Could this allow us to understand why the fusion occurs in the periphery with a high probability?

   At the temperature T∼ m(π₁₀₇) the thermal energy E_th= T-m(π₈₉) of π₈₉) pions is E-m≈(π₈₉)[1/(1-β_th²)^1/2-1] giving the estimate β_th ∼ m(π₁₀₇)/ m(π₈₉)^1/2≈ 2^-9/2. Could the slow thermal velocity imply that also the decay products of π₈₉ move slowly. Could this increase the rate for the cold fusion proportional to the inverse of the relative velocity?

   Note that if Δ has h_eff/h=512, its lifetime is scaled up by this factor and is by a factor ≈ 15 longer than the proton Compton time. What if also the produced pion and the nucleons involved have as h_eff/h=512? Could this increase the cross section for the pion catalyzed fusion to deuteron.

6. In the TGD based model for the "cold fusion" (see for instance this), the value of h_eff would be of the same order of magnitude since the scale up Compton length of proton would be of the same order of magnitude as electron Compton length. The Δ resonance could be generated at the last step of the p-adic cooling. The question boils down to whether also the ordinary hadrons, rather than only M₈₉ hadrons, can have a large value of h_eff/h and whether the effective scaling of the quantum coherence length in the transversal degrees of freedom can make possible the creation of light nuclei by the TGD dark fusion as the counterpart of "cold fusion". This question is very relevant also for the TGD based solar model.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD 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.

Is Psi real?

Sabine Hossenfelder has a very interesting Youtube video (see this) telling about the article "Experimental demonstration of the PBR test on a superconducting processor" (see this). The experiments are argued to support the view that wavefunctions really exist and therefore the ontic view of quantum theory.

To my first understanding, the argument goes very roughly as follows.

1. By reading the original article one learns that there are 3 options: a) epistemic option (EO), b) hidden variable option, and c) ontic option. For a) given values of hidden variables can correspond to several quantum states and for b) only to a single one: in this case one can say that quantum state describes reality rather than our incomplete knowledge of it. For option c) hidden variables could mean finite measurement resolution so that the state in a given resolution can correspond to several states in an improved resolution.

   In EO there are hidden variables and Ψ gives a kind of average description. In state function reduction it is updated. Sabine however notes that Bohr proposed epistemic interpretation but did not accept hidden variables!

2. The PBR (Pussen, Barret, Rudolph) theorem is the second piece of argument. Suppose that one has two independently prepared states. The assumption, call it IP, is that in this case the distribution of hidden variables is a product of the distributions for the states. PBR theorem states that if IP holds true, standard quantum theory excludes EO. PBR is like Bell's theorem excluding local hidden variables.
3. One can empirically test the IP hypothesis. In the simplest situation, one can take two non-orthogonal states obtained as rotations of qubits. One can quantum entangle them and measure the value of either qubit. One can arrange so that for some outcomes the reduction probability vanishes. If this is not the case then IP fails and epistemic options could hold true. In this case it can happen that there is no definite outcome as in standard quantum measurement theory.

   Measurement errors are however present and the deviation from IP prediction allowed by the epistemic option could be interpreted as measurement error if one is not careful. One must construct a model for measurement errors and show that the measurement errors are below the bounds predicted by the EO.

   This is done and the conclusion is that the findings exclude the epistemic option.

How does this relate to TGD? Already in the relativistic wave mechanics Ψ is replaced with a spinor field and quantum states are constructed using fermionic oscillator operators so that the question is of whether the quantum states are real or only epistemic.

1. In the TGD framework, Ψ is replaced at the fundamental level by what I call spinor field in an infinite-D "world of classical worlds" (WCW). Its spinor structure can be seen as a "square root" of the metric structure and the Fock state basis provides a representation of Boolean algebra. Geometry and logic would be closely related.

   WCW spinor modes are Fock states for the second quantized spinor fields in H=M⁴×CP₂. If WCW metric is real then also the fermion fields are. Same applies to space-time surfaces X⁴ as 4-D Bohr orbits or particle-like 3-surfaces and H.

2. Why the interpretations, and the epistemic interpretation in particular, would be needed in the first place? The reason is that quantum measurement theory leads to a conflict between non-determinism quantum measurement and deterministic time evolution of Ψ and various classical fields.
3. In TGD, the basic problem of the quantum measurement theory finds a solution in terms of zero energy ontology (ZEO) forced by holography= holomorphy principle in turn forced by the conditions that general coordinate invariance can be realized without the path integral over space-time surfaces, which unavoidably would lead to non-renormalizable divergences. Therefore the interpretations are not needed and WCW spinor fields and also the geometrized classical fields can be real.

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.

Fermion scattering as conformal dissipation

The quantal view fermionic of the time evolution as analog of Brownian motion can be made more detailed and this makes it clear that something very different from the quantum field theoretic description of scattering is in question. The interpretation as analog of dissipative time evolution is suggestive. The dissipation could be related to the scaling generator h of conformal algebra, which defines an additive conformal quantum number and is proportional to the mass squared operator m². Dissipation indeed gives rise to the arrow of time making sense both at the level of both space-time and momentum space.

1. At the single particle level, energy dissipation corresponds to a gradual reduction of energy and approach to thermal equilibrium. Analogously, the dissipation for the conformal weight identified as mass squared dissipation would lead from high mass states as unstable on-mass-shell states via unstable virtual states to low mass on-mass-shell states stable in the time scale considered. Asymptotically the evolution would preserve mass squared in M⁸ and the corresponding value of a in M⁴⊂ H. This conformal analog of elastic scattering would preserve mass squared and the conformal dissipation would transform to energy dissipation for stable particles.
2. M⁸-H duality is an inversion M⁴⊂ H → M⁴⊂ M⁸ so that the time evolution as the increase of the values of the light-cone proper time a for CD would correspond to decrease of the mass scales. The largest values of a in time evolution would correspond to the smallest values of mass m² in dissipative evolution for h and at the last steps the masses for stable particles would not change anymore.
3. p-Adic cooling proposed to be behind the production of solar wind and solar energy would be a basic example of this kind of dissipation. M₈₉ hadron physics would be realized at the magnetic monopole flux tubes at the surface layers of the Sun (see this). p-Adic cooling would be a cascade leading from M₈₉ hadron physics to the ordinary M₁₀₇ hadron physics via virtual hadron physics labelled by p-adic primes p≈ 2^k near power 2 with k in the range [89,107].
4. This means a unification of quantum theory and thermodynamics. Zero energy ontology, replacing 3-surfaces with their slightly non-deterministic time evolutions, implies that thermodynamic description and kinetic equations need not be added as additional elements to the quantum description provided by the scattering amplitudes.
5. The following detail is worth noticing. In fermion pair creation, the two-valuedness of the map H→ M⁸ means that the members of the fermion pair have identical masses. Also the Brownian scattering conserves the value of mass squared but not energy.

See the article About the construction of the scattering amplitudes using M⁸-H duality 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.

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 multiplied by a factor log/p)/log(2) taking care that bit serves as the unit of information. 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.

   This could allow us to determine  the p-adic  norm of the p-adic probability assigned to a real probability reliably. Rationals r=m/n can be mapped by writing them first as a series in powers of p. The integers m and n  can be also mapped separately to p-adics. The same result is obtained in both ways. This description conforms with the frequency interpretation of probabilities.

   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 first 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 the 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.

About the construction of the scattering amplitudes using M8-H duality

In TGD, point-like particles are replaced with 3-surfaces and these in turn with the analogs of Bohr orbits. M⁸-H duality is the generalization of momentum-position duality and is now rather well understood (see this). It however remains a mere academic mathematical construct unless it can be used to achieve some practical goal. The construction of scattering amplitudes is the basic dream of TGD and M⁸-H duality gives hope of achieving this goal in terms of the TGD counterparts for the momentum space Feynman diagrams.

The notion of exotic smooth structure, having interpretation as an ordinary smooth structure with 3-D defects and possible only in 4-D space-time, is crucial. Fermions in H are free but fermion pair creation is possible at the defects at which fermion lines can turn backwards in time. Also a more general change of direction is possible. This makes the counterpart of fermionic Feynman diagrammatic extremely simple at the level of H. Only fermionic 2-vertices associated with 3-D geometric defects are needed. Fermionic interactions reduce to an 8-D Brownian motion in the induced classical fields and the singularities of the space-time surfaces at which minimal surface property fails define the location of the vertices.

The interactions of two space-time surfaces, identified in holography = holomorphy vision as 4-D generalized Bohr orbits, correspond geometrically to contact interactions at their intersections. If the Hamilton-Jacobi structures are the same, the intersections are 2-D strings world sheets. The edges of these string world sheets would contain the vertices.

The challenge is to formulate this picture at M⁸ level by using a precise formulation of M⁸-H duality.

See the article About the construction of the scattering amplitudes using M⁸-H duality 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.

The observed 20 GeV gamma rays as evidence for galactic dark matter halo or something else?

I learned of a very interesting recent finding claiming evidence for galactic dark matter (see this). Gamma rays with 20 GeV energy have been detected. They could emerge from the decays of a particle with mass 40 GeV.

A possible TGD based interpretation could be as decay products of p-adically scaled up pions. TGD strongly suggests a scaled up copy of hadron physics with the scaling factor for mass scale equal to 2⁹=512 and there are indeed several indications for its existence. I have proposed (see this) that the transformation of the hadrons of M₈₉ hadron physics for which mass scales is by factor 512 higher than for the ordinary hadron physics would be responsible for both solar wind and radiation and would occur at the surface of the Sun by the so-called p-adic cooling (see this and this), which would involve a cascade reducing the p-adic mass scale of hadron physics by powers of two so that eventually eventually hadrons of the ordinary M₁₀₇ would emerge. Powers of 2 for mass scales are indeed favored by the p-adic length scale hypothesis. Mersenne primes are good candidates for the p-adic primes defining stable copies of hadron physics and also of the standard model.

The mass scale of the pion of M₈₉ hadron physics would be by a factor 512 2⁹ higher than for the pion of the ordinary hadron physics. In the recent case the scaling factor for the ordinary pion mass giving mass of 40 GeV would be 11 percent larger than 256= 2⁸. Deviation is 11 per cent. The pion with mass scale 2⁸ m_π could appear at the first step in the p-adic cooling. Could the particle in question correspond to an unstable hadron physics with p∼ 2⁹¹?

There is also considerable evidence for gamma rays with energy very near to electron mass. They could come from what I call electropions predicted by TGD and having with mass slightly larger than 2×m_e (see this). There is also evidence for the leptopions assignable to muon and tau. In particular, there is evidence for tau pions in galactic nucleus.

Electropions would be dark in a different sense than galactic dark matter: they would have a large value of effective Planck constant h_eff implying that their Compton size scale is scaled up from one half of electron Compton length to the size scale of atoms. Large values of h_eff are predicted at quantum criticality implying long length scale quantum fluctuations and in this case the quantum criticality means ability to overcome Coulomb wall. This would explain why electropions have not been observed as elementary particles in say decays of weak bosons. This notion of darkness explains the well-known mysterious gradual disappearance of baryons during the cosmic evolution as a transformation of protons to dark protons with very large gravitational Planck constant.

In the same way, the hadrons of M₈₉ would be created at quantum criticality against transformation of M₁₀₇ to M₈₉ hadrons and h_eff=512 would guarantee that at quantum criticality their Compton length is the same as for ordinary hadrons.

In TGD, galactic dark matter halo would not be an explanation for the 20 GeV gamma rays since there would be any dark matter halo. The monopole flux tubes arriving from the galactic nucleus to stars could produce 20 GeV gamma rays at the surface of the stars at the first step of the cooling process. This would create the illusion that the galactic halo exists. In TGD, the galactic dark matter would be dark energy assignable to extremely thin and massive objects, cosmic strings (see this), which are the key elements in the TGD view of the formation of galaxies and stars and explain the flat velocity spectrum without any additional assumptions. Galactic blackholes would emerge in the collision of cosmic strings and both electropion and the now discovered particles could relate to the decay of these cosmic strings.

See the chapter Dark Nuclear Physics and Condensed Matter.

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.

How to create monopole-flux loops?

Yesterday evening we had a very fruitful discussion with Marko Manninen about wormhole contacts and monopole flux loops. The question was following:

How do wormhole contact pairs, which correspond to elementary particles, arise?

We pondered this a lot, but the final outcome remained vague and bothered me and I continued to ponder the problem during the night. Finally, I realized that the problem was structurally the same as another problem: how is the creation of fermion pairs possible in TGD even though spinor fields in H=M⁴×CP₂ are free fields? The solution was also the same and possible only for 4-D spacetime surfaces.

Some background

I will summarize the basic facts first and then the results.

1. A wormhole contact is a Euclidean spacetime region that connects two Minkowski spacetime sheets. They appear as two different types.
   1. One that arises when the sheets touch each other. This region is not stable because there is no net magnetic flux flowing from one sheet to the other.
   2. One with a monopole flux. One can talk about a very short piece of flux tube. Since the flux is conserved, it cannot arise when the sheets touch. On the other hand, the conservation of flux stabilizes the tube so that it cannot be split.

      How could monopole wormhole contacts arise? This is the basic question we pondered. During late-night reflections it became clear that the attempts that first came to my mind did not work.

2. More facts.
   1. The boundary conditions of the field equations of TGD do not allow open flux tubes, i.e. cylinders with ends from which flux would escape into vacuum. The flux losses at the ends would compensate each other, but this is not enough. Local conservation is required and is not possible. Flux tubes must be closed loops in which monopole flux flows.
   2. This implies that wormhole contacts carrying monopole flux must be appear in pairs. They can be visualized as two wormhole contacts, i.e. magnetic flux flows from throat A1 to throat B₁, from there to throat B₂ on the lower leaf and from there to throat A₂ and further to throat A₁ on the upper leaf. A closed loop, then. At least massive elementary particles would correspond to such loops. Point-like fermions would inhabit the throats.
   3. Magnetic flux corresponds to K hler magnetic charge. It must be conserved as it flows along the tube. The total magnetic charge would be zero if open flux tubes were allowed. Here the hydrodynamic analogy to incompressible flow, which is mathematically very accurate, helps.
   4. But what happens in the case of closed flux tubes? Does the sum of the charges conserved for closed flux tubes? This would generalize the conservation of flux inside the tube. The flux for a closed flux tube would be analogous to an electric charge.

      Could one conclude that if a monopole flux tube breaks up into two flux tubes with fluxes n₁ and n₂, then the flux remains n=n₁+n₂?

      A reasonable half-guess is that flux can be considered as some kind of conserved charge. This makes sense if the sign of the flux can be operationally defined. If a flux tube has some kind of chirality (DNA helix is an example of this) or parity, then it determines the sign of the flux regardless of the position of the flux tube. Different chiralities could also be represented by flux tubes that are complex conjugates of each other with respect to the complex conjugation of CP₂. This would correspond to charge-conjugation. Holography holomorphy hypothesis predicts this kind of geometric charge conjugation.

   5. In any case, even the creation of a single closed monopole flux tube seems impossible. So the alternative would be that they are created in pairs so that the fluxes are opposite. How would this happen?

Connection with similar problem relate to the creation of fermion pairs

Here emerges a connection to a similar problem regarding the fermion number.

1. In TGD, the spinor fields are free. That is the only option. If a quartic term were added to the action, the result would be a non-renormalizable theory and thus a catastrophe.

   But is the creation of fermion pairs from a vacuum or a classical induced field then possible at all? In QED, it is when the sm field is quantized but also when it is classical.

2. An intuitive picture of what happens has been clear since Dirac's time. The fermion line turns back in time when a pair is created. This picture generalizes to TGD. The V-shaped fermion line has an edge with an infinite acceleration.
3. Now comes the connection with zero energy ontology. In TGD, the particle is geometrically replaced by a Bohr orbit of the 3-surface and the infinite acceleration at the edge corresponds to the bracing of the holomorphism on the 3-surface. These singularities are the fundamental prediction and correspond to the mild non-determinism of holography and the poles of analytic functions. These surfaces correspond also to interaction vertices.

   At the edge, the minimal surface property fails and the trace of the second fundamental form, which is the generalization of the acceleration form 1-D to a 4-D object, is infinite. Its CP₂ part transforms in symmetries like the Higgs. The Higgs would be non-zero only at the vertices. The M⁴ part corresponds to the ordinary acceleration and the particle's earthly pilgrimage would be geometrically analogous to 8-D Brownian motion such that the vertices are special moments in its life since they correspond to conscious choices.

4. A couple of years ago I realized that the edge singularity of a fermion line corresponds to the phenomenon of "exotic smooth structure". It is a smooth structure, which reduces to the standard smooth structure except for defects that are 3-surfaces: they are "edges" and correspond geometrically to particle vertices in TGD. Differentiability is broken.

   The exotic smooth structures only occur in dimension D=4. Pair creation and non-trivial fermionic dynamics are only possible in 4-D spacetime! A really bad problem of general relativity turns into a triumph in TGD.

Do also monopole flux tubes turn back in time?

Now we are very close to solving the original problem. So: how do monopole flux loops arise?

1. The emergence of a pair formed by closed monopole flux tubes corresponds to a situation when a closed monopole flux tube coming from the geometric future turns back to the geometric future! More general reversals can occur and the emission of virtual particles at vertices corresponds to this.

   Therefore, a particle pair in geometric sense can be created in the classical induced field in a geometric sense. What happens to fermion lines also happens to 3-surfaces.

2. This view generalizes: a closed monopole flux tube can break up, thus creating, for example, two monopole flux tubes and the total flux is preserved.

See the article Holography = holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M⁸-H duality 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.

How to realize Lorentz invariance for on-mass-shell states in M^8-H duality?

M⁸-H duality corresponds physically to a generalization of momentum-position duality due to the replacement of point-like particles with 3-surfaces whose orbits are slightly non-determinic space-time surfaces analogous to Bohr orbits satisfying holography = holomorphy principle at the level of H=M⁴× CP₂. Mathematical M⁸-H duality can be seen as a generalization of the geometric Langlands duality from dimension D=2 to dimension D=4.

Recently, considerable progress (see this) has occurred in the understanding of M⁸-H duality. This has meant in some sense a return to the roots and generalization of the earlier too narrow approach and the emergence of a sound physical interpretation. This has led to an exact solution of the duality in terms of local G₂ invariance.

1. A given Y⁴⊂ M⁸, having interpretation as 8-D momentum space, is determined as roots of an octonion analytic function f(o). The roots of Re(o) or I(o) define 3-D holographic data for Y⁴ which can have quaternionic tangent space or normal space and correspondingly Minkowskian or Euclidean signature with respect to the number theoretic metric defined by Re(o₁o₂).
2. The general solution of the octonionic holography is in terms of the action of local G₂ transformations acting on the simplest solutions, which are pieces of M⁴ identified as quaternions or of its orthogonal complement E⁴. having quaternionic tangent resp. normal space. This picture leads to Feynman diagram type structures in which Minkowskian and Euclidean pieces are glued together at vertices which the conditions Re(f)=0 and Im(f)=0 hold true simultaneously.
3. The 4-D surfaces Y⁴ in M⁸ having interpretation as octonions have interpretation for a representation of dispersion relation, which is however not Lorentz invariant but is invariant under 3-D rotation group. Is it possible to obtain Lorentz invariant dispersion relation in a natural way at some 3-surfaces defining on-mass-shell states and having interpretation in terms of hyperbolic 3-space H³? Can one assign a mass squared spectrum to a given Y⁴? How does this spectrum relate to the mass squared spectrum of the Dirac operator of H=M⁴× CP₂? Does 8-D light-likeness fix this spectrum?

If the tangent spaces of Y⁴ are quaternionic, the condition Re(f) =0 or Im(f) =0 has as a solution the union of ∪_o0S⁶(o₀) of 6-spheres with radius r₇(o₀). The intersection Y³= E³(o₀)∩ ∪_o0S⁶(o₀)= ∪_o0S²(o₀) defines the holographic data. For the 2-spheres S²(o₀), the 3-momentum squared is constant but depends on the energy o₀ via a dispersion relation that is in general not Lorentz invariant.

M⁸-H duality suggests how to obtain Lorentz invariant mass shell conditions E²-p²= m².

1. The modes of the Dirac equation in H (see this and this) are massless in the 8-D sense. This is a natural additional condition also in M⁸ and could define on mass shell states consistent with Lorentz invariance and distinguish them from the other points of Y⁴ having an interpretation as off-mass-shell momenta allowed by Y⁴ as a representation of a dispersion relation.
2. 8-D masslessness corresponds in M⁸ to the condition o₀²-r₇²=0, where r₇² is the counterpart of the CP₂ mass squared as the eigenvalue of the CP₂ spinor Laplacian. The additional condition o₀²-r₇²=0 picks up a discrete set of values (o₀(r₇),r₇). The 4-D mass squared would be m₄²= r₇² and a discrete mass spectrum is predicted for a given f(o) and a given selection a Re(f)=0 or Im(f)=0.
3. An interesting question is whether the eigenvalue spectrum of CP₂ spinor Laplacian is realized at the level of M⁸ as on-mass-shell states.
4. A natural guess would be that the eigenvalue spectrum of CP₂ spinor Laplacian is realized at the level of M⁸ as on-mass-shell states.

   The TGD based proposal (see this and this) for color confinement producing light states involves tachyonic states. These states would naturally correspond to 4-surfaces Y⁴ with Euclidean signature and bound states would be formed by gluing together the tachyonic and non-tachyonics states to Feynman graph-like structures. Note that the on-mass-shell 2-spheres are in general different from those satisfying the conditions (Re(f), Im(f))=(0,0) proposed to define vertices for the generalized Feynman graphs.

   Note that the on mass shell 2-spheres are in general different from those satisfying the conditions (Re(f),Im(f))=(0,0) proposed to define vertices for the generalized Feynman graphs.

See the article Does M⁸-H duality reduce to local G₂ symmetry? 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.

How TGD avoids the catastrophe caused by observer-free Universe?

Gary Ehlenberger sent a link to a very intersting Quanta Magazine article titled "Cosmic Paradox Reveals the Awful Consequence of an Observer-Free Universe". The two first paragraphs of the article give a good view of the problem.

Tinkering at their desks with the mathematics of quantum space and time, physicists have discovered a puzzling conundrum. The arcane rules of quantum theory and gravity let them imagine many different kinds of universes in precise detail, enabling powerful thought experiments that in recent years have addressed long-standing mysteries swirling around black holes.

But when a group of researchers examined a universe intriguingly like our own in 2019, they found a paradox: The theoretical universe seemed to admit only a single possible state. It appeared so simple that its contents could be described without conveying even a single bit of data, not even a choice of a zero or a one. This result clashed with the fact that this type of universe should be capable of hosting black holes, stars, planets — and people. Yet all those rich details were nowhere to be seen.

To me this result is not terribly surprising. The paradox of heat death is a well-known intuitive way to state the problem. The outcome is the final catastrophe putting an end to the materialistic view. The catastrophe is also due to the sticking to too simple mathematics based on real numbers which have no internal anatomy, in accordance with materialism in which only magnitude matters. Universe is much much more complex. AdS/CFT brings in strings and extended particles but is not a realistic solution.

1. One part of the problem is the completely wrong view of space-time as a single smooth manifold such as AdS. In TGD, space-time is a union of an arbitrarily large number of 4-D Bohr orbits of 3-surfaces in H=M⁴×CP₂ obeying holography, which is almost deterministic: this is absolutely crucial for having fermion interactions for formally free fermions.

   Intersections of space-time surfaces as 2-D string world sheets give rise to the geometric aspect of interactions. H=M⁴×CP₂ explains symmetries and fields of the standard model and is mathematically the only possible choice: number theory-geometry duality at the level of physics and the existence of the twistor lift dictates this choice. The dynamics reduces to that of space-time surfaces. By general coordinate invariance, there are only four local degrees of freedom.

2. There is a fractal hierarchy of size scales defining the size scales of the space-time sheets. Holography = holomorphy principle is a crucial part of the solution and generalizes the holomorphy of string models and transforms extremely linear classical field equations to algebraic equations. One obtains minimal surfaces irrespective of classical action as long as it is general coordinate invariant and expressible in terms of the induced geometry (see this and (this). One also obtains direct counterparts of functional iteration hierarchies associated with 4-D analogs 2-D fractals like Mandelbrot fractal and Julia set.
3. Besides topology and algebraic geometry, also number theory brings in structure. Entire hierarchies of extensions of rationals emerge and Galois groups appear as symmetry groups. Galois confinement is an attractive dynamical principle forcing for instance the total momenta be rational valued. functional fields appear too.

   Classical number fields, reals, complex numbers, quaternions, octonions emerge as an essential part of dynamics via M⁸-H duality (see this). p-Adic number fields appear also and the corresponding functional fields appear too.

   Space-time surfaces represent numbers in both the classical sense and as elements of function fields see (this, this, this).

4. TGD also brings in consciousness and one gets rid of the curse of materialism (this). The Universe as space of space-time surfaces becomes Quantum Platonia consisting of classical and quantum states as mathematical objects and quantum jumps between these quantum states make quantum Platonia conscious entity (or a union of conscious entities), which learns and remembers and its complexity and level of consciousness unavoidably increases since the number space-time surfaces more complex than a given surface is infinitely larger than the number of those which are simpler.

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.