What's new inPhysics as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2017 
Questions inspired by quantum associationsAssociations have (or seem to have) different meaning depending on whether one is talking about cognition or mathematics. In mathematics the associations correspond to different bracketings of mathematical expressions involving symbols denoting mathematical objects and operations between them. The meaning of the expression  in the case that it has meaning  depends on the bracketing of the expression. For instance, one has a(b+c)≠ (ab)+c , that is ab+ac≠ ab+c). Note that one can change the order of bracket and operation but not that of bracket and object. For ordinary product and sum of real numbers one has associativity: a(bc)=(ab)c and a+(b+c) = (a+b)+c. Most algebraic operations such as group product are associative. Associativity of product holds true for reals, complex numbers, and quaternions but not for octonions and this would be fundamental in both classical and quantum TGD. The building of different associations means different groupings of n objects. This can be done recursively. Divide first the objects to two groups, divide these tow groups to to two groups each, and continue until you jave division of 3 objects to two groups  that is abc divided into (ab)c or a(bc). Numbers 3 and 2 are clearly the magic numbers. This inspires several speculative questions related to the twistorial construction of scattering amplitudes as associative singlets, the general structure of quantum entanglement, quantum measurement cascade as formation of association, the associative structure of manysheeted spacetime as a kind of linguistic structure, spin glass as a strongly associative system, and even the tendency of social structures to form associations leading from a fully democratic paradise to cliques of cliques of ... .

From amplituhedron to associahedronLubos has a nice article (see this) explaining the proposal represented in the newest article by Nima ArkaniHamed, Yuntao Bai, Song He, Gongwang Yan (see this). Amplituhedron is generalized to a purely combinatorial notion of associahedron and shown to make sense also in string theory context (particular bracketing). The hope is that the generalization of amplituhedron to associahedron allows to compute also the contributions of nonplanar diagrams to the scattering amplitudes  at least in N=4 SYM. Also the proposal is made that color corresponds to something less trivial than ChanPaton factors. The remaining problem is that 4D conformal invariance requires massless particles and TGD allows to overcome this problem by using a generalization of the notion of twistor: masslessness is realized in 8D sense and particles massless in 8D sense can be massive in 4D sense. In TGD nonassociativity at the level of arguments of scattering amplitude corresponds to that for octonions: one can assign to spacetime surfaces octonionic polynomials and induce arithmetic operations for spacetime surface from those for polymials (or even rational or analytic functions). I have already earlier demonstrated that associahedron and construction of scattering amplitudes by summing over different permutations and associations of external particles (spacetime surfaces). Therefore the notion of associahedron makes sense also in TGD framework and summation reduces to "integration" over the faces of associahedron. TGD thus provides a concrete interpretation for the associations and permutations at the level of spacetime geometry. In TGD framework the description of color and fourmomentum is unified at the level and the notion of twistor generalizes: one has twistors in 8D spacetime instead of twistors in 4D spacetime so ChanPaton factors are replaced with something nontrivial. 1. Associahedrons and scattering amplitudes The following describes briefly the basic idea between associahedrons. 1.1 Permutations and associations One starts from a noncommutative and nonassociative algebra with product (in TGD framework this algebra is formed by octonionic polynomials with real coefficients defining spacetime surfaces as the zero loci of their real or imaginary parts in quaternionic sense. One can indeed multiply spacetime surface by multiplying corresponding polynomials! Also sum is possible. If one allows rational functions also division becomes possible. All permutations of the product of n elements are in principle different. This is due to noncommutativity. All associations for a given ordering obtained by scattering bracket pairs in the product are also different in general. In the simplest case one has either a(bc) or (ab)c and these 2 give different outcomes. These primitive associations are building bricks of general associations: for instance, abc does not have welldefined meaning in nonassociative case. If the product contains n factors, one can proceed recursively to build all associations allowed by it. Decompose the n factors to groups of m and nm factors. Continue by decomposing these two groups to two groups and repeat until you have have groups consisting of 1 or two elements. You get a large number of associations and you can write a computer code computing recursively the number N(n) of associations for n letters. Two examples help to understand. For n=3 letters one obviously has N(3)= 2. For n=4 one has N(4)=5: decompose abcd to (abc)d and a(bcd) and (ab)(cd) and then the 3 letter groups to two groups: this gives 2+2+1 =5 associations and associahedron in 3D space has therefore 5 faces. 1.2 Geometric representation of association as face of associahedron Associations of n letters can be represented geometrically as so called Stasheff polytope (see this). The idea is that each association of n letters corresponds to a face of polytope in n2dimensional space with faces represented by the associations. Associahedron is constructed by using the condition that adjacent faces (now 2D polygons) intersecting along common face (now 1D edges). The number of edges of the face codes for the structure particular association. Neighboring faces are obtained by doing minimal change which means replacement of some (ab)c with a(bc) appearing in the association as a building bricks or vice versa. This means that the changes are carried out at the root level. 1.3 How does this relate to particle physics? In scattering amplitude letters correspond to external particles. Scattering amplitude must be invariant under permutations and associations of the external particles. In particular, this means that one sums over all associations by assigning an amplitude to each association. Geometrically this means that one "integrates" over the boundary of associahedron by assigning to each face an amplitude. This leads to the notion of associahedron generalizing that of amplituhedron. Personally I find it difficult to believe that the mere combinatorial structure leading to associahedron would fix the theory completely. It is however clear that it poses very strong conditions on the structure of scattering amplitudes. Especially so if the scattering amplitudes are defined in terms of "volumes" of the polyhedrons involved so that the scattering amplitude has singularities at the faces of associahedron. An important constraint on the scattering amplitudes is the realization of the Yangian generalization of conformal symmetries of Minkowski space. The representation of the scattering amplitudes utilizing moduli spaces (projective spaces of various dimensions) and associahedron indeed allows Yangian symmetries as diffeomorphisms of associahedron respecting the positivity constraint. The hope is that the generalization of amplituhedron to associahedron allows to generalize the construction of scattering amplitudes to include also the contribution of nonplanar diagrams of at N=4 SYM in QFT framework. 2. Associations and permutations in TGD framework Also in the number theoretical vision about quantum TGD one encounters associativity constraints leading to the notion of associahedron. This is closely related to the generalization of twistor approach to TGD forcing to introduce 8D analogs of twistors (see this).
Nima et al talk also about color structure of the scattering amplitudes usually regarded as trivial. It is claimed that this is actually not the case and that there is nontrivial dynamics involved. This is indeed the case in TGD framework. Also color quantum numbers are twistorialized in terms of the twistor space of CP_{2}, and one performs a twistorialization at the level of M^{8} and M^{4}× CP_{2}. At the level of M^{8} momenta and color quantum numbers correspond to associative 8momenta. Massless particles are now massless in 8D sense but can be massive in 4D sense. This solves one of the basic difficulty of the ordinary twistor approach. A further bonus is that the choice of the imbedding space H becomes unique: only the twistor spaces of S^{4} (and generalized twistor space of M^{4} and CP_{2} have Kähler structure playing a crucial role in the twistorialization of TGD. To sum up, all roads lead to Rome. Everyone is wellcome to Rome! See the articles Does M 8 − H duality reduce classical TGD to octonionic algebraic geometry? and From amplituhedron to associahedron. Addition: Marni Lee Sheppard wrote a thesis in which the notion of associahedron appeared. I remember discussions in some net group. Her motivations came from category theory. Marni had bad luck. Big boys rarely remember who proposed the idea first if she/he is not a name. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.

Some layman considerations related to fundamentals of mathematicsI am not a mathematician and therefore should refrain from consideration of anything related to fundamentals of mathematics. In the discussions with Santeri Satama I could not avoid the temptation to break this rule. I however feel that I must confess my sins and in the following I will do this.
Goedel's problematics involves statements which cannot be proved to be true or false or are simultaneously true and false. This problematics has also a purely geometric analog in terms of set theoretic representation of Boolean algebras when real topology is used but not when padic topology is used. The natural idea is that Boolean algebra is realized in terms of open sets such that the negation of statement corresponds to the complement of the set. In padic topologies open sets are simultaneously also closed and there are no boundaries: this makes them and  more generally Stone spaces  ideal for realizing Boolean algebra set theoretically. In real topology the complement of open set is closed and therefore not open and one has a problem. Could one circumvent the problem somehow?
2. Number theory from quantum theory Could one define or at least represent the notion of number using the notions of quantum physics? A natural starting point is hierarchy of extensions of rationals defining hierarchy of adeles. Could one obtain rationals and their extensions from simplest possible quantum theory in which one just constructs many particle states by adding or removing particles using creation and annihilation operators? 2.1 How to obtain rationals and their extensions? Rationals and their extensions are fundamental in TGD. Can one have quantal construction for them?
2.2 Arithmetics of Hilbert spaces The notions of prime and divisibility and even basic arithmetics emerge also from the tensor product and direct sum for Hilbert spaces. Hilbert spaces with prime dimension do not decompose to tensor products of lowerdimensional Hilbert spaces. One can even perform a formal generalization of the dimension of Hilbert space so that it becomes rational and even algebraic number. For some years ago I indeed played with this thought but at that time I did not have in mind reduction of number theory to the arithemetics of Hilbert spaces. If this really makes sense, numbers could be replaced by Hilbert spaces with product and sum identified as tensor product and direct sum! Finitedimensional Hilbert space represent the analogs of natural numbers. The analogs of integers could be defined as pairs (m,n) of Hilbert spaces with spaces (m,n) and (m+r,n+r) identified (this space would have dimension mn. This identification would hold true also at the level of states. Hilbert spaces with negative dimension would correspond to pairs with (mn)<0: the canonical representives for m and m would be (m,0) and (0,m). Rationals can be defined as pairs (m,n) of Hilbert spaces with pairs (m,n) and (km,kn) identified. These identifications would give rise to kind of gauge conditions and canonical representatives for m and 1/m are (m,1) and (1,m). What about Hilbert spaces for which the dimension is algebraic number? Algebraic numbers allow a description in terms of partial fractions and SternBrocot (SB) tree (see this and this) containing given rational number once. SB tree allows to see information about algebraic numbers as constructible by using an algorithm with finite number of steps, which is allowed if one accepts abstraction as basic aspect of cognition. Algebraic number could be seen as a periodic partial fraction defining an infinite path in SB tree. Each node along this path would correspond to a rational having Hilbert space analog. Hilbert space with algebraic dimension would correspond to this kind of path in the space of Hilbert spaces with rational dimension. Transcendentals allow identification as nonpediodic partial fraction and could correspond to nonperiodic paths so that also they could have Hilbert spaces counterparts. 2.3 How to obtain the analogs higherD spaces? Algebraic extensions of rationals allow cognitive realization of spaces with arbitrary dimension identified as algebraic dimension of extension of rationals.
3. Could quantum set theory make sense? In the following my view point is that of quantum physicist fascinated by number theory and willing to reduce set theory to what could be called called quantum set theory. It would follow from physics as generalised number theory (adelic physics) and have ordinary set theory as classical correlate.
What quantum set theory could mean? Suppose that number theoryquantum theory connection really works. What about set theory? Or perhaps its quantum counterpart having ordinary set theory as a classical correlate?
How could one obtain classical set theory?
Here one has two views about infinite set. One has both separable (infiniteD in discrete sense: particle in box with discrete momentum spectrum) and nonseparable (infiniteD in real sense: free particle with continuous momentum spectrum) Hilbert spaces. In the latter case the completely localized single particle states would be represented by delta functions divided by infinite normalization factors. They are routinely used in Dirac's braket formalism but problems emerge in quantum field theory. A possible solution is that one weakens the axiom of choice and accepts that only discrete points set (possibly finite) are cognitively representable and one has wave functions localized to discrete set of points. A stronger assumption is that these points have coordinates in extension of rationals so that one obtains number theoretical universality and adeles. This is TGD view and conforms also with the identification of hyperfinite factors of type II_{1} as basic algebraic objects in TGD based quantum theory as opposed to wave mechanics (type I) and quantum field theory (type III). They are infiniteD but allow excellent approximation as finiteD objects. This picture could relate to the notion of noncommutative geometry, where set emerges as spectrum of algebra: the points of spectrum label the ideals of the integer elements of algebra. See the chapter Unified Number Theoretical Vision or the article Some layman considerations related to fundamentals of mathematics.

Could the precursors of perfectoids emerge in TGD?The work of Peter Stolze based on the notion of perfectoid has raised a lot of interest in the community of algebraic geometers. One application of the notion relates to the attempt to generalize algebraic geometry by replacing polynomials with analytic functions satisfying suitable restrictions. Also in TGD this kind of generalization might be needed at the level of M^{4}× CP_{2} whereas at the level of M^{8} algebraic geometry might be enough. The notion of perfectoid as an extension of padic numbers Q_{p} allowing all p:th roots of padic prime p is central and provides a powerful technical tool when combined with its dual, which is function field with characteristic p. Could perfectoids have a role in TGD? The infinitedimensionality of perfectoid is in conflict with the vision about finiteness of cognition. For other padic number fields Q_{q}, q≠ p the extension containing p:th roots of p would be however finitedimensional even in the case of perfectoid. Furthermore, one has an entire hierarchy of almostperfectoids allowing powers of p^{m}:th roots of padic numbers. The larger the value of m, the larger the number of points in the extension of rationals used, and the larger the number of points in cognitive representations consisting of points with coordinates in the extension of rationals. The emergence of almostperfectoids could be seen in the adelic physics framework as an outcome of evolution forcing the emergence of increasingly complex extensions of rationals. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article Could the precursors of perfectoids emerge in TGD?.

What does cognitive representability really mean?I had a debate with Santeri Satama about the notion of number leading to the question about what cognitive representability of number could mean. This inspired writing of an articling discussing the notion of cognitive representability. Numbers in the extensions of rationals are assumed to be cognitively representable in terms of points common to real and various padic spacetime sheets (correlates for sensory and cognitive). One allows extensions of padics induced by extension of rationals in question and the hierarchy of adeles defined by them. One can however argue that algebraic numbers do not allow finite representation as do rational numbers. A weaker condition is that the coding of information about algorithm producing the cognitively representable number contains a finite amount of information although it might take an infinite time to run the algorithm (say containing infinite loops). Furthermore, cognitive representations in TGD sense are also sensory representations allowing to represent algebraic numbers geometrically (2^{1/2}) as the diameter of unit square). SternBrocot tree associated with partial fractions indeed allows to identify rationals as finite paths connecting the root of SB tree to the rational in question. Algebraic numbers can be identified as infinite periodic paths so that finite amount of information specifies the path. Transcendental numbers would correspond to infinite nonperiodic paths. A very close analogy with chaos theory suggests itself. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article What does cognitive representability really mean? 
Supernumber fields: does physics emerge from the notion of number?A proposal that all physics emerges from the notion of number field is made. The first guess for the number field in question would be complexified octonions for which inverse exists except at complexified lightcone boundary: this has interpretation in terms of propagation of signals with lightvelocity 8D sense. The emergence of fermions however requires superoctonions as super variant of number field. Rather surprisingly, it turns out that supernumber theory makes perfect sense. One can define the inverse of supernumber if its ordinary part is nonvanishing and also the notion of primeness makes sense and construct explicitly the superprimes associated with ordinary primes. The prediction of new number piece of theory can be argued to be a strong support for the integrity of TGD. See the the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article Supernumber fields: does physics emerge from the notion of number?.

Is the quantum leakage between different signatures of the real sectors of the complexified M^{8} possible?Complexified octonions have led to a dramatic progress in the understanding of TGD. One cannot however avoid a radical question about fundamentals.

What the properties of octonionic product can tell about fundamental physics?In developing the view about M^{8}H duality reducing physics to algebraic geometry for complexified octonions at the level of M^{8}, I became aware of trivial looking but amazingly profound observation about the basic arithmetics of complex, quaternion, and octonion number fields.
This extremely simple observation turns out to contain amazingly deep physics.
The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers! See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article Do RiemannRoch theorem and AtyiahSinger index theorem have applications in TGD?. 
Do RiemannRoch theorem and AtyiahSinger index theorem have applications to TGD?RiemannRoch theorem (RR) is a central piece of algebraic geometry. AtyiahSinger index theorem is one of its generalizations relating the solution spectrum of partial differential equations and topological data. For instance, characteristic classes classifying bundles associated with YangMills theories have applications in gauge theories and string models. The advent of octonionic approach to the dynamics of spacetime surfaces inspired by M^{8}H duality (see this and this) gives hopes that dynamics at the level of complexified octonionic M^{8} could reduce to algebraic equations plus criticality conditions guaranteeing associativity for spacetime surfaces representing external particles, in interaction region commutativity and associativity would be broken. The complexification of octonionic M^{8} replacing norm in flat space metric with its complexification would unify various signatures for flat space metric and allow to overcome the problems due to Minkowskian signature. Wick rotation would not be a mere calculational trick. For these reasons time might be ripe for applications of possibly existing generalization of RR to TGD framework. In the following I summarize my admittedly unprofessional understanding of RR discussing the generalization of RR for complex algebraic surfaces having real dimension 4: this is obviously interesting from TGD point of view. I will also consider the possible interpretation of RR in TGD framework. One interesting idea is possible identification of lightlike 3surfaces and curves (string boundaries) as generalized poles and zeros with topological (but not metric) dimension one unit higher than in Euclidian signature. AtyiahSinger index theorem (AS) is one of the generalizations of RR and has shown its power in gauge field theories and string models as a method to deduce the dimensions of various moduli spaces for the solutions of field equations. A natural question is whether AS could be useful in TGD and whether the predictions of AS at H side could be consistent with M^{8}H duality suggesting very simple counting for the numbers of solutions at M^{8} side as coefficient combinations of polynomials in given extension of rationals satisfying criticality conditions. One can also ask whether the hierarchy of degrees n for octonion polynomials could correspond to the fractal hierarchy of generalized conformal subalgebras with conformal weights coming as nmultiples for those for the entire algebras. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article Do RiemannRoch theorem and AtyiahSinger index theorem have applications in TGD?. 
About enumerative algebraic geometry in TGD frameworkI wrote a brief summary about basic ideas of enumerative algebraic geometry and proposals for applications to TGD. Here is a short abstract of the article summarizing the new results. String models and Mtheory involve both algebraic and symplectic enumerative geometry. Also in adelic TGD enumerative algebraic geometry emerges. This article gives a brief summary about the basic ideas involved and suggests some applications to TGD.
See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article About enumerative algebraic geometry in TGD framework. 
Superoctonions, supertwistors and twistorial construction of scattering amplitudesFor last month or so I have been writing an article about reduction of quantum classical TGD to octonionic algebraic geometry. During writing process "octonionic" has been replaced with "superoctonionic" and it has turned out that the formalism of twistor Grassmannian approach could generalize to TGD. The twistorial scattering diagrams have interpretation as cognitive representations with vertices assignable to points of spacetime surface in appropriate extension of rationals . In accordance with the vision that classical theory is exact part of quantum theory, the core part of the calculations of scattering amplitudes would reduce to the determination of zero loci for real and imaginary parts of octonionic polynomials and finding of the points of spacetime surface with M^{8} coordinates in extension of rationals. At fundamental level the physics would reduce to number theory. A proposal for the description of interactions is discussed in the article. Here is a brief summary.

Geometrization of fermions using super version of the octonionic algebraic geometryCould the octonionic level provide an elegant description of fermions in terms of super variant of octonionic algebraic geometry? Could one even construct scattering amplitudes at the level of M^{8} using the variant of the twistor approach discussed earlier. The idea about supergeometry is of course very different from the idea that fermionic statistics is realized in terms of the spinor structure of "world of classical worlds" (WCW) but M^{8}H duality could however map these ideas and also number theoretic and geometric vision to each other. The angel of geometry and the devil of algebra could be dual to each other. 1. Octonionic superspace Consider now what super version of the octonionic superspace might look like.
Instead of superfields one would have a super variant of octonionic algebraic geometry.
There are several questions about quantum numbers.
It would be extremely nice if the scattering amplitudes could be computed at the octonionic level by using a generalization of twistor approach in ZEO finding a nice justification at the level of M^{8}. Something rather similar to N=4 twistor Grassmann approach suggests itself.

Emergence of Zero Energy Ontology and Causal Diamonds from octonionic algebraic surface dynamicsOctonionic polynomials provide a promising approach to the understanding of zero energy ontology (ZEO) and causal diamonds (CDs) defined as intersections of future and past directed lightcones: CD makes sense both in octonionic (8D) and quaternionic (4D) context. Lightlike boundary of CD as also lightcone emerge naturally as zeros of octonionic polynomials. This does not yet give CDs and ZEO: one should have intersection of future and past directed lightcones. The intuitive picture is that one has a hierarchy of CDs and that also the spacetime surfaces inside different CDs an interact. It turns out that CDs and thus also ZEO emerge naturally both at the level of M^{8} and M^{4} . Remark: In the sequel RE(o) and IM(o) refer to real and imaginary parts of octonions in quaternionic sense: one has o= RE(o)+IM(o)I_{4}, where RE(o) and IM(o) are quaternions. 1. General view about solutions to RE(P)=0 and IM(P)=0 conditions The first challenge is to understand at general level the nature of RE(P)=0 and IM(P)=0 conditions expected to give 4D spacetime surfaces as zero loci. Appendix shows explicitly for P(o)=o^{2} that Minkowski signature gives rise to unexpected phenomena. In the following these phenomena are shown to be completely general but not quite what one obtains for P(o)=o^{2} having double root at origin.
CD defines the basic geometric object in ZEO. It is good to list some basic features of CDS, which appear as both 4D and 8D variants.
3. The emergence of CDs CDs are a key notion of zero energy ontology (ZEO). Could the emergence of CDs be understood in terms of singularities of octonion polynomials located at the lightlike boundaries of CDs? In Minkowskian case the complex norm qq^{c}_{i} is present in P (^{c} is conjugation changing the sign of quaternionic unit but not that of the commuting imaginary unit i). Could this allow to blow up the singular point to a 3D boundary of lightcone and allow to understand the emergence of causal diamonds (CDs) crucial in ZEO. The study of the special properties for zero loci of general polynomial P(o) at lightrays of O indeed demonstrated that both 8D land 4D lightcones and their complements emerge naturally, and that the M^{4} projections of these lightcones and even of their boundaries are 4D future  or past directed lightcones. What one should understand is how CDs as their intersections, and therefore ZEO, emerge.
4. How could the spacetime varieties associated with different CDs interact? The interaction of spacetime surfaces inside given CD is welldefined. Sitation is not so clear for different CDs for which the choice of octonionic coordinate origin is in general different and polynomial bases for different CDs do not commute nor associate. The intuitive expectation is that 4D/8D CDs can be located everywhere in M^{4}/M^{8}. The polynomials with different origins neither commute nor are associative. Their sum is a polynomial whose coefficients are not real. How could one avoid losing the extremely beautiful associative and commutative algebra? It seems that one cannot form their products and sums and must form the Cartesian product of M^{8}:s with different origins and formulate the interaction at M^{8} level in this framework. Note that CayleyDickson hierarchy does not seem to be relevant since the dimension are powers of 2 rather than multiples of 8. Should one give up associativity and allow products (but not sums since one should give up the assumption that the coefficients of polynomials are real) of polynomials associated with different CDs as an analog for the formation of free manyparticle states. One can still have separate vanishing of the polynomials in separate CDs but how could one describe their interaction? If one does not give up associativity and commutativity, how can one describe the interactions between spacetime surfaces inside different CDs at the level of M^{8}?
All big pieces of quantum TGD are now tightly interlinked.

M^{8}H duality: summary and future prospectsIn the following I give a brief summary about what has been done. I concentrate on M^{8}H duality since the most significant results are achieved here. It is fair to say that the new view answers the following a long list of open questions.

Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?I have used last month to develop a detailed vision about M^{8}H duality and now I dare to speak about genuine breakthrough. I attach below the abstract of the resulting article. TGD leads to several proposals for the exact solution of field equations defining spacetime surfaces as preferred extremals of twistor lift of Kähler action. So called M^{8}H duality is one of these approaches. The beauty of M^{8}H duality is that it could reduce classical TGD to algebraic geometry and would immediately provide deep insights to cognitive representation identified as sets of rational points of these surfaces. In the sequel I shall consider the following topics.
Octonions and quaternions are 20 year old part of TGD: one of the three threads in physics as generalized number theory vision. Second vision is quantum physics as geometry of WCW. The question has been how to fuse geometric and number theory visions. Algebraic geometry woul do it since it is both geometry and algebra and it has been also part of TGD but only now I realized how to get acceess to its enormous power. Even the proposal discussed now about the algebra of octonionic polynomials with real coefficients was made about two decades ago but only now I managed to formulate it in detail. Here the general wisdom gained from adelic physics helped enormously. I dare say that classical TGD at the most fundamental level is solved exactly. From the point of pure mathematics the generalization of complex analyticity and linear Cauchy Riemann conditions to multilinear variants for quaternions, octonions and even for the entire hierarchy of algebras obtained by CayleyDickson construction is a real breakthrough. Consider only the enormous importance of complex analyticity in mathematics and physics, including string models. I do not believe that this generalization has been discovered: otherwise it would be key part of mathematical physics. Quaternionic and octonionic analyticities will certainly mean huge evolution in mathematics. I had never ended to these discoveries without TGD: TGD forced them. At these moments I feel deep sadness when knowing that the communication of these results to collegues is impossible in practice. This stupid professional arrogance is something which I find very difficult to accept even after 4 decades. I feel that when society pays a monthly salary for a person for being a scientists, he should feel that his duty is to be keenly aware what is happening in his field. When some idiot proudly tells that he reads only prestigious journals, I get really angry. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part II.

Generalization of Riemann zeta to Dedekind zeta and adelic physics?A further insight to adelic physics comes from the possible physical interpretation of the Lfunctions appearing also in Langlands program (see this. The most important Lfunction would be generalization of Riemann zeta to extension of rationals. I have proposed several roles for ζ, which would be the simplest Lfunction assignable to rational primes, and for its zeros.

Are Preferred Extremals QuaternionAnalytic in Some Sense?A generalization of 2D conformal invariance to its 4D variant is strongly suggestive in TGD framework, and leads to the idea that for preferred extremals of action spacetime regions have (co)associative/(co)quaternionic tangent space or normal space. The notion of M^{8}H correspondence allows to formulate this idea more precisely. The beauty of this notion is that it does not depend on the signature of Minkowski space M^{4} representable as subspace of of complexified quaternions M^{4}_{c}, which in turn can be seen as subspace of complexified octonions M^{8}_{c}. The 4D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. This notion is however not so straightforward even in Euclidian signature, and the generalization to Minkowskian signature brings in further problems. The CauchyRiemannFuerter conditions make however sense also in Minkowskian quaternionic situation and the problem is whether they allow the physically expected solutions. One should also show that the possible generalization is consistent with (co)associativity. In this article these problems are considered. Also a comparison with Igor Frenkel's ideas about hierarchy of Lie algebras, loop, algebras and double look algebras and their quantum variants is made: it seems that TGD as a generalization of string models replacing string world sheets with spacetime surfaces gives rise to the analogs of double loop algebras and they quantum variants and Yangians. The straightforward generalization of double loop algebras seems to make sense only at the lightlike boundaries of causal diamonds and at lightlike orbits of partonic 2surfaces but that in the interior of spacetime surface the simple form of the conformal generators is not preserved. The twistor lift of TGD in turn corresponds nicely to the heuristic proposal of Frenkel for the realization of double loop algebras. See the article Are Preferred Extremals QuaternionAnalytic in Some Sense? or the chapter Unified Number Theoretical Vision. 
Philosophy of Adelic PhysicsThe padic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the padic ideas are inspired by TGD based view about physics. pAdic mass calculations relying on padic generalization of thermodynamics and supersymplectic and superconformal symmetries are summarized. Number theoretical existence constrains lead to highly nontrivial and successful physical predictions. The notion of canonical identification mapping padic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from padics to reals and vice versa. A view about padicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various padic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of padic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection. The existence of padic variants of definite integral, Fourier analysis, Hilbert space, and Riemann geometry is far from obvious and various constraints lead to the idea of number theoretic universality (NTU) and finite measurement resolution realized in terms of number theory. An attractive manner to overcome the problems in case of symmetric spaces relies on the replacement of angle variables and their hyperbolic analogs with their exponentials identified as roots of unity and roots of e existing in finitedimensional algebraic extension of padic numbers. Only group invariants  typically squares of distances and norms  are mapped by canonical identification from padic to real realm and various phases are mapped to themselves as number theoretically universal entities. Also the understanding of the correspondence between real and padic physics at various levels  spacetime level, imbedding space level, and level of "world of classical worlds" (WCW)  is a challenge. The gigantic isometry group of WCW and the maximal isometry group of imbedding space give hopes about a resolution of the problems. Strong form of holography (SH) allows a nonlocal correspondence between real and padic spacetime surfaces induced by algebraic continuation from common string world sheets and partonic 2surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and padic surfaces providing automatically finite "cognitive" resolution. he existence padic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it. I will also sum up the role of padic physics in TGD inspired theory of consciousness. Negentropic entanglement (NE) characterized by number theoretical entanglement negentropy (NEN) plays a key role. Negentropy Maximization Principle (NMP) forces the generation of NE. The interpretation is in terms of evolution as increase of negentropy resources. For details see the new chapter Philosophy of Adelic Physics. 
Why would primes near powers of two (or small primes) be important?The earlier What's New was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at spacetime and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely. Also new insights about how preferred padic primes identified as ramified primes of extension emerge. The picture suggests strong resemblance with the evolution of genetic code with conserved genes having ramified primes as their analogs. Category theoretic thinking in turn suggests that the positions of fermions at partonic 2surfaces correspond to singularities of the Galois covering so that the number of sheets of covering is not maximal and that the singularities has as their analogs what happens for ramified primes. pAdic length scale hypothesis states that physically preferred padic primes come as primes near prime powers of two and possibly also other small primes. Does this have some analog to complexity theory, period doubling, and with the superstability associated with period doublings? Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.
For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?. 
h_{eff}/h=n hypothesis and Galois groupThe previous What's New was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at spacetime and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely. I ended up to rather interesting information theoretic interpretation about the understanding of effective Planck constant assigned to flux tubes mediating as gravitational/electromagnetic/etc... interactions. The real surprise was that this leads to a proposal how monocellulars and multicellulars differ! The emergence of multicellulars would have meant emergence of systems with mass larger than critical mass making possible gravitational quantum coherence. Penrose's vision about the role of gravitation would be correct although OrchOR as such has little to do with reality! The natural hypothesis is that h_{eff}/h=n equals to the order of Galois group in the case that it gives the number of sheets of the covering assignable to the spacetime surfaces. The stronger hypothesis is that h_{eff}/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.
These arguments support the view that quantum information theory indeed closely relates not only to gravitation but also other interactions. Speculations revolving around blackhole, entropy, and holography, and emergence of space would be replaced with the number theoretic vision about cognition providing information theoretic interpretation of basic interactions in terms of entangled tensor networks (see this). Negentropic entanglement would have magnetic flux tubes (and fermionic strings at them) as topological correlates. The increase of the complexity of quantum states could occur by the "fusion" of Galois groups associated with various nodes of this network as macroscopic quantum states are formed. Galois groups and their representations would define the basic information theoretic concepts. The emergence of gravitational quantum coherence identified as the emergence of multicellulars would mean a major step in biological evolution. For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?. 
What could be the role of complexity theory in TGD?Chaotic (or actually extremely complex and only apparently chaotic) systems seem to be the diametrical opposite of completely integrable systems about which TGD is a possible example. There is however also something common: in completely integrable classical systems all orbits are cyclic and in chaotic systems they form a dense set in the space of orbits. Furthermore, in chaotic systems the approach to chaos occurs via steps as a control parameter is changed. Same would take place in adelic TGD fusing the descriptions of matter and cognition. In TGD Universe the hierarchy of extensions of rationals inducing finitedimensional extension of padic number fields defines a hierarchy of adelic physics and provides a natural correlate for evolution. Galois groups and ramified primes appear as characterizers of the extensions. The sequences of Galois groups could characterize an evolution by phase transitions increasing the dimension of the extension associated with the coordinates of "world of classical worlds" (WCW) in turn inducing the extension used at spacetime and Hilbert space level. WCW decomposes to sectors characterized by Galois groups G_{3} of extensions associated with the 3surfaces at the ends of spacetime surface at boundaries of causal diamond (CD) and G_{4} characterizing the spacetime surface itself. G_{3} (G_{4}) acts on the discretization and induces a covering structure of the 3surface (spacetime surface). If the state function reduction to the opposite boundary of CD involves localization into a sector with fixed G_{3}, evolution is indeed mapped to a sequence of G_{3}s. Also the cognitive representation defined by the intersection of real and padic surfaces with coordinates of points in an extension of rationals evolve. The number of points in this representation becomes increasingly complex during evolution. Fermions at partonic 2surfaces connected by fermionic strings define a tensor network, which also evolves since the number of fermions can change. The points of spacetime surface invariant under nontrivial subgroup of Galois group define singularities of the covering, and the positions of fermions at partonic surfaces could correspond to these singularities  maybe even the maximal ones, in which case the singular points would be rational. There is a temptation to interpret the padic prime characterizing elementary particle as a ramified prime of extension having a decomposition similar to that of singularity so that category theoretic view suggests itself. One also ends up to ask how the number theoretic evolution could select preferred padic primes satisfying the padic length scale hypothesis as a survivors in number theoretic evolution, and ends up to a vision bringing strongly in mind the notion of conserved genes as analogy for conservation of ramified primes in extensions of extension. h_{eff}/h=n has natural interpretation as the order of Galois group of extension. The generalization of hbar_{gr}= GMm/v_{0}=hbar_{eff} hypothesis to other interactions is discussed in terms of number theoretic evolution as increase of G_{3}, and one ends up to surprisingly concrete vision for what might happen in the transition from prokaryotes to eukaryotes. For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD? of the article What could be the role of complexity theory in TGD?. 
Progress in adelic physicsThe preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note ideas related directly to adelic TGD are discussed.
Simple arguments lead to the identification of h_{eff}/h=n as a factor of the order of Galois group of extension of rationals.
The intuitive feeling is that the notion of preferred prime is something extremely deep and to me the deepest thing I know is number theory. Does one end up with preferred primes in number theory? This question brought to my mind the notion of ramification of primes (more precisely, of prime ideals of number field in its extension), which happens only for special primes in a given extension of number field, say rationals. Ramification is completely analogous to the degeneracy of some roots of polynomial and corresponds to criticality if the polynomial corresponds to criticality (catastrophe theory of Thom is one application). Could this be the mechanism assigning preferred prime(s) to a given elementary system, such as elementary particle? I have not considered their role earlier also their hierarchy is highly relevant in the number theoretical vision about TGD.
In TGD framework the extensions of rationals (see this) and padic number fields (see this) are unavoidable and interpreted as an evolutionary hierarchy physically and cosmological evolution would gradually proceed to more and more complex extensions. One can say that string world sheets and partonic 2surfaces with parameters of defining functions in increasingly complex extensions of prime emerge during evolution. Therefore ramifications and the preferred primes defined by them are unavoidable. For padic number fields the number of extensions is much smaller for instance for p>2 there are only 3 quadratic extensions. How could ramification relate to padic and adelic physics and could it explain preferred primes?
Number theoretical vision relies on NTU. In fermionic sector NTU is necessary: one cannot speak about real and padic fermions as separate entities and fermionic anticommutation relations are indeed number theoretically universal. What about NTU in case of functional integral? There are two opposite views.
See the chapter Unified Number Theoretic Vision or the article pAdicization and adelic physics. 