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Increase of the dimension of extension of rationals as the emergence of a reflective level of consciousness?
in TGD framework the hierarchy of extensions of rationals defines a hierarchy of adeles and evolutionary hierarchy. What could the interpretation for the events in which the dimension of the extension of rationals increases? Galois extension is extensions of an extension with relative Galois group Gal(rel)= Gal(new)/Gal(old). Here Gal(old) is a normal subgroup of Gal(new). A highly attractive possibility is that evolutionary sequences quite generally (not only in biology) correspond to this kind of sequences of Galois extensions. The relative Galois groups in the sequence would be analogous to conserved genes, and genes could indeed correspond to Galois groups (see this). To my best understanding this corresponds to a situation in which the new polynomial Pm+n defining the new extension is a polynomial Pm having as argument the old polynomial Pn(x): Pm+n(x)=Pm(Pn(x)).
What about the interpretation at the level of conscious experience? A possible interpretation is that the quantum jump leading to an extension of an extension corresponds to an emergence of a reflective level of consciousness giving rise to a conscious experience about experience. The abstraction level of the system becomes higher as is natural since number theoretic evolution as an increase of algebraic complexity is in question.
This picture could have a counterpart also in terms of the hierarchy of inclusions of hyperfinite factors of type II1 (HFFs). The included factor M and including factor N would correspond to extensions of rationals labelled by Galois groups Gal(M) and Gal(N) having Gal(M)⊂ Gal(M) as normal subgroup so that the factor group Gal(N)/Gal(M) would be the relative Galois group for the larger extension as extension of the smaller extension. I have indeed proposed (see this) that the inclusions for which included and including factor consist of operators which are invariant under discrete subgroup of SU(2) generalizes so that all Galois groups are possible. One would have Galois confinement analogous to color confinement: the operators generating physical states could have Galois quantum numbers but the physical states would be Galois singlets.
See the chapter Could Genetic Code Be Understood Number Theoretically? or the article Does M8-H duality reduce classical TGD to octonionic algebraic geometry?.
The question about possible variations of Geff (see this) led again to the old observation that sub-groups of Galois group could be analogous to conserved genes in that they could be conserved in number theoretic evolution. In small variations such as variation of Galois subgroup as analogs of genes would change G only a little bit. For instance, the dimension of Galois subgroup would change slightly. There are also big variations of G in which new sub-group can emerge.
The analogy between subgoups of Galois groups and genes goes also in other direction. I have proposed long time ago that genes (or maybe even DNA codons) could be labelled by heff/h=n . This would mean that genes (or even codons) are labelled by a Galois group of Galois extension (see this) of rationals with dimension n defining the number of sheets of space-time surface as covering space. This could give a concrete dynamical and geometric meaning for the notin of gene and it might be possible some day to understand why given gene correlates with particular function. This is of course one of the big problems of biology.
One should have some kind of procedure giving rise to hierarchies of Galois groups assignable to genes. One would also like to assign to letter, codon and gene and extension of rationals and its Galois group. The natural starting point would be a sequence of so called intermediate Galois extensions EH leading from rationals or some extension K of rationals to the final extension E. Galois extension has the property that if a polynomial with coefficients in K has single root in E, also other roots are in E meaning that the polynomial with coefficients K factorizes into a product of linear polynomials. For Galois extensions the defining polynomials are irreducible so that they do not reduce to a product of polynomials.
Any sub-group H⊂ Gal(E/K)) leaves the intermediate extension EH invariant in element-wise manner as a sub-field of E (see this). Any subgroup H⊂ Gal(E/K)) defines an intermediate extension EH and subgroup H1⊂ H2⊂... define a hierarchy of extensions EH1>EH2>EH3... with decreasing dimension. The subgroups H are normal - in other words Gal(E) leaves them invariant and Gal(E)/H is group. The order |H| is the dimension of E as an extension of EH. This is a highly non-trivial piece of information. The dimension of E factorizes to a product ∏i |Hi| of dimensions for a sequence of groups Hi.
Could a sequence of DNA letters/codons somehow define a sequence of extensions? Could one assign to a given letter/codon a definite group Hi so that a sequence of letters/codons would correspond a product of some kind for these groups or should one be satisfied only with the assignment of a standard kind of extension to a letter/codon?
Irreducible polynomials define Galois extensions and one should understand what happens to an irreducible polynomial of an extension EH in a further extension to E. The degree of EH increases by a factor, which is dimension of E/EH and also the dimension of H. Is there a standard manner to construct irreducible extensions of this kind?
There was an interesting popular article in Quanta Magazine titled "Oxygen and Stem Cells May Have Reshaped Early Complex Animals".
The article dicusses the work of geobiologist Emma Hammarlund and tumor biologist Sven Påhlman: their interdisciplinary hypothesis is published as article in Nature with title "Refined control of cell stemness allowed animal evolution in the oxic realm".
Here is the abstract of their article.
Animal diversification on Earth has long been presumed to be associated with the increasing extent of oxic niches. Here, we challenge that view. We start with the fact that hypoxia (<1-3 per cent O2) maintains cellular immaturity (stemness), whereas adult stem cells continuously—and paradoxically—regenerate animal tissue in oxygenated settings. Novel insights from tumour biology illuminate how cell stemness nevertheless can be achieved through the action of oxygen-sensing transcription factors in oxygenated, regenerating tissue. We suggest that these hypoxia-inducible transcription factors provided animals with unprecedented control over cell stemness that allowed them to cope with fluctuating oxygen concentrations. Thus, a refinement of the cellular hypoxia-response machinery enabled cell stemness at oxic conditions and, then, animals to evolve into the oxic realm. This view on the onset of animal diversification is consistent with geological evidence and provides a new perspective on the challenges and evolution of multicellular life.
The proposal of Hammarlund and Påhlman
Cambrian explosion during which highly advanced lifeforms suddenly emerged - prolifiration and diversification of animal life are the terms used about this - is one of the mysteries of biology. For most of its 4.5-billion-year history, Earth has sustained life — but that life was largely limited to microbial organisms: bacteria, plankton, algae. For about 540 million years ago did larger, more complex species are assumed to dominate the oceans, but within just a few tens of millions of years (very short time on the evolutionary timescale), the planet had filled up with all kinds of animals. The fossil record from that period shows the beginnings of almost all modern animal lineages: animals with shells and animals with spines, animals that swam and animals that burrowed, animals that could hunt and animals that could defend themselves from predators. Also many lineages that disappeared were present as one learns from the book of Stephen Jay Gould describing in detail the Burgess Shale finding that revolutionized the picture about evolutionary biology and remains still a puzzle.
The belief is that the environment became considerable more oxic - that is contained oxygen - and lifeforms had to cope with this change. Before the change the animals in seas (believed to exist!) were anaerobic. The shifting to aerobic respiration was however an enormous metabolic advantange since the effectiveness of metabolic energy gain become roughly 20-fold. Increased metabolic feed in turn made possible the emergence of complexity during Cambrian period.
2. TGD view
My own proposal is that life evolved in underground oceans and entered to the surface of Earth in Cambrian explosion when oceans were formed at the surface of Eearth from cracks formed when Earth expanded rapidly in geological time scale. Before the explosion Earth did not have oceans and continents and was like Mars nowadays: even its radius was that of Mars. This picture follows from TGD based variant of Expanding Earth hypothesis. See this and this .
The habitat changed in the rapid expansion of Earth from hypoxic to oxic and the emergence of the hypothetical HIF-1 transcription factor would have been forced by this evolutionary pressure and made it possible for the lifeforms to adapt oxygen based metabolism. This would have led to a rapid evolution of animals and emergence of vertebrates. One can of course think that oxygenation developed already in the underground oceans as cracks caused in the crust by the expansion of Earth began to develop and provided oxygen. The alternative - not so plausible sounding - option is that the highly developed organisms developed underground slowly and only bursted to the surface of Earth in the explosion.
What can one say about the emergence of animal life in TGD framework?
For some time ago I represented what I called reckless speculations about higher level variants of genetic code (see this for the updated version of the original article). The speculations turned out to be not only reckless but to contain besides an unrealistic working hypothesis for p-adic length scale of dark DNA also a numerical error in the estimate of dark nuclear excitation energy scale leading to a wrong track.
The wrong working hypothesis was the assumption that ordinary DNA, RNA, etc correspond to same p-adic length scale as their dark variants. Simple argument shows that the dark scales must result via radial scaling of the typically linear structures such as DNA, RNA, etc and also 2-D structures such as membranes and microtubules giving rise to 2-D lattice like realizations of genetic code generalizing the ordinary 1-D realizations.
Also new improved picture conforms with the vision that dark realizations of genetic code at various p-adic length scales serve as controllers of the ordinary biochemistry, which is kind of shadow dynamics. Replication, certainly one of the most mysterious feats of living matter, would reduce to the replication at the level of dark DNA in various p-adic length scales involved. This would be a huge simplification.
A hierarchy of dark nuclear physics with hierarchy of n= heff/h=n coming as certain powers of two so that the corresponding length scales correspond to p-adic length scales is an attractive idea. I have speculated with this idea already earlier.
Consider first the general ideas.
In the sequel I discuss in more quantitative level the idea that cell membrane and neuronal membrane realize analogs of genes as 2-D structures.
2.1 The p-adic length scales associated with the dark variants of 2-D structures?
Consider next the p-adic length scales associated with the structures considered.
Lipids would be analogs of ordinary RNA codons and dark RNA codons would code for them: this would predict 64 different lipids in cell membrane. Single dark RNA would correspond to the size scale of single lipid given by L(143)=2L(141)=.625 nm. The dark nuclear physics would correspond to k=149. The number N of parallel dark RNA strands would be roughly the circumference of the axonal lipid layer divided by the size of single lipid about L(143)=.625 nm given by N∼ 2π × Le(167)/Le(143) = π × 224 ∼ 5× 106.
2.2. Thermodynamical constraints
Could this totally irresponsible speculation about p-adic hierarchy of dark nuclear physics and genetic codes survive thermodynamical constraints?
Remark: In TGD view about information processing in brain active linear neuron groups relate to verbal cognition and 2-D neuronal groups relate to the geometric cognition associated with the decomposition of perceptive field to objects. At cellular level DNA and cell membrane could perhaps be seen as counterparts for these structures. In TGD framework neuronal membrane is proposed to be a constructor of sensory representations communicated to the magnetic body (MB) using generalized Josephson radiation whereas motor control by MB has been assumed to take place via DNA.
3. Microtubules as quantum critical systems
Also microtubules (see this) are 2-D structures having a strong resemblance with the lipid layers of cell membrane. Could a higher level representation of genetic code similar to the one proposed for lipid layers make sense for them. Also now one can imagine that the microtubular surface is accompanied by its dark variant realizing 2-D genes with scaled up size. The p-adic prime should correspond to k>151 so that higher level realization of genetic code would be in question. In the case of axons a possible identification for the dark scale would be as the radius of the axonal membrane.
This kind of map would give rise to a kind of abstraction about what happens at the level of axonal membrane integrating out un-necessary details. This abstraction is natural since microtubules would indeed correspond to a higher level of cognitive hierarchy. Roughly N=216 lipids would contribute to the information received by single tubulin. Could nerve pulse patterns can induce braiding of the flux tubes in this scale?
See the chapter About the Correspondence of Dark Nuclear Genetic Code and Ordinary Genetic Code or the article with the same title, or the shorter article About dark variants of DNA, RNA, and amino-acids .
To make progress one must construct a concrete model for the dark nuclei. The recent picture relies strongly on various anomalies to which TGD provides a solution. The TGD inspired model for "cold fusion leads to the notion of dark nuclear physics - actually hierarchy of them labelle by the values of heff/h=n and corresponding p-adic length scales. Second basic idea is that cylindrical variants of EZs discovered by Pollack (see this) give rise to the dark counterparts of DNA, RNA, and amino-acids as dark proton sequences. tRNAs would be analogs of tritium and 3He. Pollack effect serves as a strong constraint for the model. Also the effects of ELF em fields on vertebrate brain combined with the rather recent finding about clustering of RNA II polymerase molecules exhibiting Comorosan effect provide valuable constraints on the model (see this) . The outcome of the arguments is that single strand of DNA, mRNA, tRNA and amino-acids most naturally correspond to k=149 and double stranded DNA to k=151.
Remark: The following argumentation is kind of Sherlock-Holmes-ing using all possible hints as constraints to select between imagined options rather than glorious march from axioms to theorems and thus not science in the usual sense.
1. Why one must have k=151 for dark DNA
Concerning the identification of the size scale of dark DNA one can consider several options. It however turns out that the p-adic length scale assignable to dark DNA is most naturally k=151 corresponding to the thickness 10 nm of DNA coil. The hypothesis that the integer k labelling p-adic length scale is prime is attractive working hypothesis leaving very few options under consideration. The options k=137 and k=149 are excluded since the pairing of dark DNA and ordinary DNA would not be possible without the coiling of ordinary RNA around dark DNA. This leaves only options for which k≥ 149 for prime values of k.
For prime values of k the options k<149 are not possible for dark DNA since ordinary DNA should coil around dark DNA. There is also second objection against k<149 from energetics inspiring the hypothesis DNA corresponds to k=151.
Also RNA and amino-acids should have dark variants and one should understand their role. Suppose that the integer k characterizing the p-adic length scale is prime. The vision about RNA era preceding DNA era suggests that RNA accompanying dark RNA is at lower level in the evolution, and hence the value of heff is smaller for dark RNA than for dark DNA. Also the p-adic length scale for RNA would be shorter.
The time scales τ equal 5, 10, and 20 seconds appear in the clustering of RNA II polymerase proteins and Mediator proteins (see this and the previous posting). What is intriguing that so called Comorosan effect involves time scale of 5 seconds and its multiples claimed by Comorosan long time ago to be universal time scales in biology. The origin of these time scales has remained more or less a mystery although I have considered several TGD inspired explanations for this time scale is based on the notion of gravitational Planck constant (see this).
One can consider several starting point ideas, which need not be mutually exclusive.
To sum up, the model suggests that the idealization of flux tubes as kind of universal Josephson junctions. The model is consistent with bio-photon hypothesis. The constraints on hgr= GMDm/v0 are consistent with the earlier views and allows to assign Comorosan time scale 5 seconds to proton and nerve pulse time scale to electron as Josephson time scales. This inspires the question whether the dynamics of bio-catalysis and nerve pulse generation be seen as scaled variants of each other at quantum level? This would not be surprising if MB controls the dynamics. The earlier assumption that Bend=0.2 Gauss is minimal value for Bend must be replaced with the assumption that it is maximal value of Bend.
I received a link to a highly interesting popular article telling about the work of Ibrahim Cisse at MIT and colleagues (see this): at this time about clustering of proteins in the transcription of RNA. Similar clustering has been observed already earlier and interpreted as a phase separation Similar clustering has been observed already earlier and interpreted as a phase separation (see this). Now this interpretation is not proposed by experiments but experimenters say that it is quite possible but they cannot prove it.
I have already earlier discussed the coalescence of proteins into droplets as this kind of process in TGD framework. The basic TGD based ideas is that proteins - and biomolecules in general - are connected by flux tubes characterized by the value of Planck constant heff=n× h0 for the dark particles at the flux tube. The higher the value of n is the larger the energy of given state. For instance, the binding energies of atoms decrease like 1/n2. Therefore the formation of the molecular cluster liberates energy usable as metabolic energy.
TGD view about the findings
Gene control switches - such as RNA II polymerases in the DNA transcription to RNA - are found to form clusters called super-enhancers. Also so called Mediator proteins form clusters. In both cases the number of members is in the range 200-400. The clusters are stable but individual molecules spend very brief time in them. Clusers have average lifetime of 5.1±.4 seconds.
Why the clustering should take place? Why large number of these proteins are present although single one would be enough in the standard picture. In TGD framework one can imagine several explanations. One can imagine at least following reasons.
A FB discussion inspired by the recent findings of NASA suggesting the presence of life under the surface of Mars raised the question whether the TGD based Expanding Earth model is consistent with plate tectonics and with the motivating claim of Adams that the continents fit together nicely to cover the entire surface of Earth if its radius were one half of the recent radius. The outcome was what one might call Platonic plate tectonics.
There was a very interesting popular article in Spacedaily with title "Scientists crack how primordial life on Earth might have replicated itself" (see this). The research paper "Ribozyme-catalysed RNA synthesis using triplet building blocks" is here.
It is possible to replicate unfolded RNA strands in Lab by using enzymes known as ribozymes, which are RNA counterparts of enzymes, which are amino-adic sequences. In the presence of folding the replication is however impossible. Since ribozymes are in general folded, they cannot catalyze their own replication in this manner. The researchers however discovered that the replication using RNA triplets - genetic codons - as basic unit can be carried out in laboratory even for the folded RNA strands and with rather low error rate. Also the ribozyme involved can thus replicate. For units longer than 3 nucleotides the replication becomes prone to errors.
These findings are highly interesting in TGD framework. In TGD chemical realization of genetic code is not fundamental. Rather, dark matter level would provide the fundamental realizations of analogs of DNA, RNA, tRNA, and amino-acids as dark proton sequences giving rise to dark nuclei at magnetic flux tubes. Also ordinary nuclei correspond in TGD Universe to sequences of protons and neutrons forming string like entities assignable to magnetic flux tubes.
The basic unit representing DNA, RNA and tRNA codon and amino-acid would consist of 3 entangled dark protons. The essential aspect is that by entanglement the dark codons do not decompose to products of letters. This is like words of some languages, which do not allow decomposition to letters. This representation is holistic. As we learn to read and write, we learn the more analytic western view about words as letter sequences. Could the same hold true in evolution so that RNA triplets would have come first as entities pairing with dark RNA codons from from dark proton triplets as a whole? Later DNA codons would have emerged and paired with dark DNA codons. Now the coupling would have have been letter by letter in DNA replication and transcription to mRNA.
It is intriguing that tRNA consists of RNA triplets combined from amino-acids and analogs of mRNA triplets! The translation of mRNA to amino-acids having no 3-letter decomposition of course forces the holistic view but one can ask whether something deeper is involved. This might be the case. I have been wondering whether during RNA era RNA replicated using a prebiotic form of translational machinery, which replicated mRNA rather than translated RNA to protein formed from amino-acids (AAs).
Peter Gariaev and colleagues have applied the linguistic notions of synonymy and homonymy to genetic code. Also the notion of syhomy fusing these concepts is introduced. Homonymy is visible in mRNa-tRNA pairing and induced by the 1-to-many pairing of the third mRNA nucleotide with tRNA nucleotide. The homonymy in mRNA-AA (AA for amino-acid) pairing is also present albeit rare.
The codons for the standard code can be divided to two classes. For 32 codons the first two letters fix AA completely. For the remaining 32 codons this is not the case. There is however almost unbroken symmetry in that U and C resp. A and G code for the same AA. The breaking of this symmetry is minimal appearing only for 3 4-columns of the code table and present for A-G only. The deviations from the standard code as a rule break A-G or T-C symmetry or re-establish it.
The notion of homonymy is highly interesting from TGD point of view. TGD leads to two basic proposals for non-chemical realization of genetic code predicting the numbers of DNA codons coding for amino-acid (AA) rather successfully. The first proposal relies on TGD based view about dark matter as heff/h=n phases of ordinary matter and identifies counterparts of DNA, RNA, tNRA, and AAs as entangled dark proton triplets.
Second proposal emerged from the model of music-harmony based on fusion of icosahedral and tetrahedral geometries. Codons are represented as photon triplets (dark or ordinary) defining the allowed 3-chords of given harmony defined by Hamilton cycle at icosahedron extended to Hamilton cycle to the fusion of icosahedron with tetrahedron along common face. Photon triplets give rise to resonant coupling giving rise to physical pairing of biomolecule and its dark counterpart. Remarkably, there are 3 different realizations of tRNA in terms of 3-chords. There is large number of bio-harmonies corresponding to Hamiltonian cycles. Since music expresses and creates emotions, the proposal is that a realization of emotions at molecular level adding additional degrees of freedom not visible at the level of chemistry is in question. This might give rise to a context dependence of the code.
The proposal is that genetic code at dark level extends to a sequence DDNA → DmRNA → DtRNA → DAA of horizontal pairings analogous to projections is fundamental one. Codon-codon pairings are realized via dark photon triplet resonance and mRNA-AA pairing by resonant coupling to the sum fXYZ=f1+f2+f3 of 3-chord frequencies: the codons coding same AA would have frequencies fXYZ differing only by a multiple of octave. One might perhaps say that AA sequence defines melody and mRNA sequence the accompaniment.
There is context dependence and homonymies already in DmRNA-DtRNA pairing and due the fact that DtRNA corresponds to a 2-harmony which is sub-harmony of 3-harmony and can be chosen in 3 different manners. The vertical pairings DDNA → DNA, DmRNA → mRNA, etc. also mediated by frequency couplings induce ordinary genetic code and horizontal pairings in DNA → mRNA → tRNA → AA. DAA → AA pairing dictates mRNA → AA pairing and mRNA → tRNA homonymy does not matter and actually makes the translation safer by increasing the number of tRNAs performing the same task.
The rather rare homonymies in DNA-AA pairing can be understood as accidental degeneracies. AA couples resonantly to the sum fXYZ=f1+f2+f2 of frequencies associated with codon XYZ and it can occur that the sum frequencies can be identical for two codons.
See the chapter Homonymy of the genetic code from TGD point of view or the article with the same title.
The idea about the realization of genetic code in terms of dark proton sequences giving rise to dark nuclei is one of the key ideas of TGD inspired quantum biology (see this). This vision was inspired by the totally unexpected observation that the states of three dark protons (or quarks) can be classified to 4 classes in which the number of states are same as those of DNA, RNA, tRNA, and amino-acids. Even more, it is possible to identify genetic code as a natural correspondence between the dark counterparts of DNA/RNA codons and dark amino-acids and the numbers of DNAs/RNAs coding given amino-acid are same as in the vertebrate code. What is new is that the dark codons do not reduce to ordered products of letters.
During years I have considered several alternatives for the representations of genetic code. For instance, one can consider the possibility that the letters of the genetic code correspond to the four spin-isospin states of nucleon or quark or for spin states of electron pair. Ordering of the letters as states is required and this is problematic from the point of view of tensor product unless the ordering reflects spatial ordering for the positions of particles representing the letters. One representation in terms of 3-chords formed by 3-photon states formed from dark photons emerges from the model of music harmony (see this). By octave equivalence the ordering of the notes is not needed.
The above observations inspire several speculative insights.
The model must satisfy stringent conditions.
The basic problem in the understanding of the prebiotic evolution is how DNA, RNA, amino-acids and tRNA and perhaps even cell membrane and microtubules . The individual nucleotides and amino-acids emerge without the help of enzymes or ribozymes but the mystery is how their polymers emerged. If the dark variants of these molecules served as templates for their generation one avoids this hen-and-egg problem. The problem how just the biomolecules were picked up from a huge variety of candidates allowed by chemistry could be solved by the resonance condition making possible metabolic energy transfer between biomolecules and dark nuclei.
Simple scaling argument shows that the assumption that ordinary genetic code corresponds to heff/h=n=218 and therefore to the p-adic length scale L(141)≈ .3 nm corresponding to the distance between DNA and RNA bases predicts that the scale of dark nuclear excitation energies is .5 eV, the nominal value of metabolic energy quantum. This extends and modifies the vision about how prebiotic evolution led via RNA era to the recent biology. Unidentified infrared bands (UIBs) from interstellar space identified in terms of transition energies of dark nuclear physics support this vision and one can compre it to PAH world hypothesis.
p-Adic length scale hypothesis and thermodynamical considerations lead to ask whether cell membrane and microtubules could correspond to 2-D analogs of RNA strands associated with dark RNA codons forming lattice like structures. Thermal constraints allow cell membrane of thickness about 5 nm as a realization of k=149 level with n= 222 in terms of lipids as analogs of RNA codons. Metabolic energy quantum is predicted to be .04 eV, which corresponds to membrane potential. The thickness of neuronal membrane in the range 8-10 nm and could correspond to k=151 and n=223 in accordance with the idea that it corresponds to higher level in the cellular evolution reflecting that of dark nuclear physics.
Also microtubules could correspond to k=151 realization for which metabolic energy quantum is .02 eV slightly below thermal energy at room temperature: this could relate to the inherent instability of microtubules. Also a proposal for how microtubules could realize genetic code with the 2 conformations of tubulin dimers and 32 charges associated with ATP and ADP accompanying the dimer thus realizing the analogs of 64 analogs of RNA codons is made.
See the chapter About the Correspondence of Dark Nuclear Genetic Code and Ordinary Genetic Code or the article with the same title.