# Physics as a Generalized Number Theory

Note: Newest contributions are at the top!

 Year 2017

### Questions inspired by quantum associations

Associations 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 many-sheeted space-time 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 ... .

1. In standard twistor approach 3-gluon amplitude is the fundamental building brick of twistor amplitudes constructed from on-shell-amplitudes with complex momenta recursively. Also in TGD proposal this holds true. This would naturally follow from the fact that associations can be reduced recursively to those of 3 objects. 2- and 3-vertex would correspond to a fundamental associations. The association defined 2-particle pairing (both associated particles having either positive or negative helicities for twistor amplitudes) and 3-vertex would have universal structure although the states would be in general decompose to associations.
2. Consider first the space-time picture about scattering (see this). CD defines interaction region for scattering amplitudes. External particles entering or leaving CD correspond to associative space-time surfaces in the sense that the tangent space or normal space for these space-time surfaces is associative. This gives rise to M8-H correspondence.

These surfaces correspond to zero loci for the imaginary parts (in quaternionic sense) for octonionic polynomial with coefficients, which are real in octonionic sense. The product of ∏iPi) of polynomials with same octonion structure satisfying IM(Pi)=0 has also vanishing imaginary part and space-time surface corresponds to a disjoint union of surfaces associated with factors so that these states can be said to be non-interacting.

Neither the choice of quaternion structure nor the choice of the direction of time axis assignable to the octonionic real unit need be same for external particles: if it is the particles correspond to same external particle. This requires that one treats the space of external particles (4-surfaces) as a Cartesian product of of single particle 4-surfaces as in ordinary scattering theory.

Space-time surfaces inside CD are non-associative in the sense that the neither normal nor tangent space is associative: M8-M4× CP2 correspondence fails and space-time surfaces inside CD must be constructed by applying boundary conditions defining preferred extremals. Now the real part of RE(∏iPi) in quaternionic sense vanishes: there is genuine interaction even when the incoming particles correspond to the same octonion structure since one does not have union of surfaces with vanishing RE(Pi). This follows from s rather trivial observation holding true already for complex numbers: imaginary part of zw vanishes if it vanishes for z and w but this does not hold true for the real part. If octonionic structures are different, the interaction is present irrespective of whether one assumes RE(∏iPi)=0 or IM(∏iPi)=0. RE(∏iPi)=0 is favoured since for IM(∏iPi)=0 one would obtain solutions for which IM(Pi)=0 would vanish for the i:th particle: the scattering dynamics would select i:th particle as non-interacting one.

3. The proposal is that the entire scattering amplitude defined by the zero energy state - is associative, perhaps in the projective sense meaning that the amplitudes related to different associations relate by a phase factor (recall that complexified octonions are considered), which could be even octonionic. This would be achieved by summing over all possible associations.
4. Quantum classical correspondence (QCC) suggests that in ZEO the zero energy states - that is scattering amplitudes determined by the classically non-associative dynamics inside CD - form a representation for the non-associative product of space-time surfaces defined by the condition RE(∏iPi)=0. Could the scattering amplitude be constructed from products of octonion valued single particle amplitudes. This kind of condition would pose strong constraints on the theory. Could the scattering amplitudes associated with different associations be octonionic - may be differing by octonion-valued phase factors - and could only their sum be real in octonionic sense (recall that complexified octonions involving imaginary unit i commuting with the octonionic imaginary units are considered)?
One can look the situation also from the point of view of positive and negative energy states defining zero energy states as they pairs.
1. The formation of association as subset is like formation of bound state of bound states of ... . Could each external line of zero energy state have the structure of association? Could also the internal entanglement associated with a given external line be characterized in terms of association.

Could the so called monogamy theorem stating that only two-particle entanglement can be maximal correspond to the decomposing of n=3 association to one- and two-particle associations? If quantum entanglement is behind associations in cognitive sense, the cognitive meaning of association could reduce to its mathematical meaning.

An interesting question relates to the notion of identical particle: are the many-particle states of identical particles invariant under associations or do they transform by phase factor under association. Does a generalization of braid statistics make sense?

2. In ZEO based quantum measurement theory the cascade of quantum measurements proceeds from long to short scales and at each step decomposes a given system to two subsystems. The cascade stops when the reduction of entanglement is impossible: this is the case if the entanglement probabilities belong to an extension of extension of rationals characterizing the extension in question. This cascade is nothing but a formation of an association! Since only the state at the second boundary of CD changes, the natural interpretation is that state function reduction mean a selection of association in 3-D sense.
3. The division of n objects to groups has also social meaning: all social groups tend to divide into cliques spoiling the dream about full democracy. Only a group with 2 members - Romeo and Julia or Adam and Eve - can be a full democracy in practice. Already in a group of 3 members 2 members tend to form a clique leaving the third member outside. Jules and Catherine, Jim and Catherine, or maybe Jules and Jim! Only a paradise allows a full democracy in which non-associativity holds true. In ZEO it would be realized only at the quantum critical external lines of scattering diagram and quantum criticality means instability. Quantum superposition of all associations could realize this democracy in 4-D sense.
A further perspective is provided by many-sheeted space-time providing classical correlate for quantum dynamics.
1. Many-sheeted space-time means that physical states have a hierarchical structure - just like associations do. Could the formation of association (AB) correspond basically to a formation of flux tube bond between A and B to give AB and serve as space-time correlate for (negentropic) entanglement. Could ((AB)C) would correspond to (AB) and (C) "topologically condensed" to a larger surface. If so, the hierarchical structure of many-sheeted space-time would represent associations and also the basic structures of language.
2. Spin glass is a system characterized by so called frustrations. Spin glass as a thermodynamical system has a very large number of minima of free energy and one has fractal energy landscape with valleys inside valleys. Typically there is a competition between different pairings (associations) of the basic building bricks of the system.

Could spin glass be describable in terms of associations? The modelling of spin glass leads to the introduction of ultrametric topology characterizing the natural distance function for the free energy landscape. Interestingly, p-adic topologies are ultrametric. In TGD framework I have considered the possibility that space-time is like 4-D spin glass: this idea was originally inspired by the huge vacuum degeneracy of Kähler action. The twistor lift of TGD breaks this degeneracy but 4-D spin glass idea could still be relevant.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.

### From amplituhedron to associahedron

Lubos has a nice article (see this) explaining the proposal represented in the newest article by Nima Arkani-Hamed, 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 non-planar diagrams to the scattering amplitudes - at least in N=4 SYM. Also the proposal is made that color corresponds to something less trivial than Chan-Paton factors.

The remaining problem is that 4-D 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 8-D sense and particles massless in 8-D sense can be massive in 4-D sense.

In TGD non-associativity at the level of arguments of scattering amplitude corresponds to that for octonions: one can assign to space-time surfaces octonionic polynomials and induce arithmetic operations for space-time 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 (space-time 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 space-time geometry.

In TGD framework the description of color and four-momentum is unified at the level and the notion of twistor generalizes: one has twistors in 8-D space-time instead of twistors in 4-D space-time so Chan-Paton factors are replaced with something non-trivial.

1. Associahedrons and scattering amplitudes

The following describes briefly the basic idea between associahedrons.

1.1 Permutations and associations

One starts from a non-commutative and non-associative algebra with product (in TGD framework this algebra is formed by octonionic polynomials with real coefficients defining space-time surfaces as the zero loci of their real or imaginary parts in quaternionic sense. One can indeed multiply space-time 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 non-commutativity. 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 well-defined meaning in non-associative 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 n-m 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 3-D 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 n-2-dimensional space with faces represented by the associations.

Associahedron is constructed by using the condition that adjacent faces (now 2-D polygons) intersecting along common face (now 1-D 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 non-planar 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 8-D analogs of twistors (see this).

1. By M8-H duality (H=M4× CP2) the scattering are assignable to complexified 4-surfaces in complexified M8. Complexified M8 is obtained by adding imaginary unit i commutating with octonionic units Ik, k=1,,..,7. Real space-time surfaces are obtained as restrictions to a Minkowskian subspace complexified M8 in which the complexified metric reduces to real valued 8-D Minkowski metric. This allows to define notions like Kähler structure in Minkowskian signature and the notion of Wick rotations ceases to be ad hoc concept. Without complexification one does not obtain algebraic geometry allowing to reduces the dynamics defined by partial differential equations for preferred extremals in H to purely algebraic conditions in M8. This means huge simplifications but the simplicity is lost at the QFT-GRT limit when many-sheeted space-time is replaced with slightly curved piece of M4.
2. The real 4-surface is determined by a vanishing condition for the real or imaginary part of octonionic polynomial with RE(P) and and IM(P) defined by the composition of octonion to two quaternions: o= RE(o)+ I4IM(o), where I4 is octonionic unit orthogonal to a quaternionic sub-space and RE(o) and IM(o) are quaternions. The coefficients of the polynomials are assumed to be real. The products of octonionic polynomials are also octonionic polynomials (this holds for also for general power series with real coefficients (no dependence on Ik). The product is not however neither commutative nor associative without additional conditions. Permutations and their associations define different space-time surfaces. The exchange of particles changes space-time surface. Even associations do it. Both non-commutativity and non-associativity have a geometric meaning at the level of space-time geometry!
3. For space-time surfaces representing external particles associativity is assumed to hold true: this in fact guarantees M8-H correspondence for them! For interaction regions associativity does not hold true but the field equations and preferred extremal property allow to construct the counterpart of space-time surface in H from the boundary data at the boundaries of CD fixing the ends of space-time surface.

Associativity poses quantization conditions on the coefficients of the polynomial determining it. The conditions are interpreted in terms of quantum criticality. In the interaction region identified naturally as causal diamond (CD), associativity does not hold true. For instance, if external particles as space-time surfaces correspond to vanishing of RE(Pi) for polynomials representing particles labelled by i, the interaction region (CD) could correspond to the vanishing of IM(Pi) and associativity would fail. At the level of H associativity and criticality corresponds to minimal surface property so that quantum criticality corresponds to universal free particle dynamics having no dependence on coupling constants.

4. Scattering amplitudes must be commutative and associative with respect to their arguments which are now external particles represented by polynomials Pi This requires that scattering amplitude is sum over amplitudes assignable to 4-surfaces obtained by allowing all permutations and all associations of a given permutation. Associations can be described combinatorially by the associahedron!

Remark:. In quantum theory associative statistics allowing associations to be represented by phase factors can be considered (this would be associative analog of Fermi statistics). Even a generalization of braid statistics can be considered.

Yangian variants of various symmetries are in central piece also in TGD although supersymmetries are realized in different manner and generalized to super-conformal symmetries: these include generalization of super-conformal symmetries by replacing 2-D surfaces with light-like 3-surfaces, supersymplectic symmetries and dynamical Kac-Moody symmetries serving as remnants of these symmetries after supersymplectic gauge conditions characterizing preferred extremals are applied, and Kac-Moody symmetries associated with the isometries of \$H\$ . The representation of Yangian symmetries as diffeomorphisms of the associahedron respecting positivity constraint encourages to think that associahedron is a useful auxiliary tool also in TGD. 3. Is color something more than Chan-Paton factors?

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 non-trivial dynamics involved. This is indeed the case in TGD framework. Also color quantum numbers are twistorialized in terms of the twistor space of CP2, and one performs a twistorialization at the level of M8 and M4× CP2. At the level of M8 momenta and color quantum numbers correspond to associative 8-momenta. Massless particles are now massless in 8-D sense but can be massive in 4-D 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 S4 (and generalized twistor space of M4 and CP2 have Kähler structure playing a crucial role in the twistorialization of TGD. To sum up, all roads lead to Rome. Everyone is well-come to Rome!

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 M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.

### Some layman considerations related to fundamentals of mathematics

I 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.

1. Gödel's problematics is shown to have a topological analog in real topology, which however disappears in p-adic topology which raises the question whether the replacement of the arithmetics of natural numbers with that of p-adic integers could allow to avoid Gödel's problematics.
2. Number theory looks from the point of view of TGD more fundamental than set theory and inspires the question whether the notion of algebraic number could emerge naturally from TGD. There are two ways to understand the emergence of algebraic numbers: the hierarchy of infinite primes in which ordinary primes are starting point and the arithmetics of Hilbert spaces with tensor product and direct sum replacing the usual arithmetic operations. Extensions of rationals give also rise to cognitive variants of n-D spaces.
3. The notion of empty set looks artificial from the point of view of physicist and a possible cure is to take arithmetics as a model. Natural numbers would be analogous to nonempty sets and integers would correspond to pairs of sets (A,B), A⊂ B or B⊂ A with equivalence (A,B)== (A∪ C,B∪ C). Empty set would correspond to pairs (A,A). In quantum context the generalization of the notion of being member of set a∈ A suggests a generalization: being an element in set would generalize to being single particle state which in general is de-localized to the set. Subsets would correspond to many-particle states. The basic operation would be addition or removal of element represented in terms of oscillator operator. The order of elements of set does not matter: this would generalize to bosonic and fermionic many particle states and even braid statistics can be considered. In bosonic case one can have multiple points - kind of Bose-Einstein condensate.
4. One can also start from finite-D Hilbert space and identify set as the collection of labels for the states. In infinite-D case there are two cases corresponding to separable and non-separable Hilbert spaces. The condition that the norm of the state is finite without infinite normalization constants forces selection of de-localized discrete basis in the case of a continuous set like reals. This inspires the question whether the axiom of choice should be given up. One possibility is that one can have only states localized to finite or at least discrete set of points which correspond points with coordinates in an extension of rationals.
1. Geometric analog for Gödel's problematics

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 p-adic 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 p-adic 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?

1. If one replaces open sets with their closures (the closure of open set includes also its boundary, which does not belong to the open set) and closed complements of open sets, the analog of Boolean algebra would consist of closed sets. Closure of an open set and the closure of its open complement - stament and its negation - share the common boundary. Statement and its negation would be simultaneously true at the boundary. This strange situation reminds of Russell's paradox but in geometric form.
2. If one replaces the closed complements of open sets with their open interiors, one has only open sets. Now the sphere would represent statement about which one cannot say whether it is true or false. This would look like Gödelian sentence but represented geometrically.

This leads to an already familiar conclusion: p-adic topology is natural for the geometric correlates of cognition, in particular Boolean cognition. Real topology is natural for the geometric correlates of sensory experience.

3. Gödelian problematics is encountered already for arithmetics of natural numbers although naturals have no boundary in the discrete topology. Discrete topology does not however allow well-ordering of natural numbers crucial for the definition of natural number. In the induced real topology one can order them and can speak of boundaries of subsets of naturals. The ordering of natural numbers by size reflects the ordering of reals: it is very difficult to think about discrete without implicitly bringing in the continuum.

For p-adic integers the induced topology is p-adic. Is Gödelian problematics is absent in p-adic Boolean logic in which set and its complement are both open and closed. If this view is correct, p-adic integers might replace naturals in the axiomatics of arithmetics. The new element would be that most p-adic integers are of infinite size in real sense. One has a natural division of them to cognitively representable ones finite also in real sense and non-representable ones infinite in real sense. Note however that rationals have periodic pinary expansion and can be represented as pairs of finite natural numbers.

In algebraic geometry Zariski topology in which closed sets correspond to algebraic surfaces of various dimensions, is natural. Open sets correspond to their complements and are of same dimension as the imbedding space. Also now one encounters asymmetry. Could one say that algebraic surfaces characterize "representable" (="geometrically provable"?) statements as elements of Boolean algebra and their complements the non-representable ones? 4-D space-time (as possibly associative/co-associative ) algebraic variety in 8-D octonionic space would be example of representable statement. Finite unions and intersections of algebraic surfaces would form the set of representable statements. This new-to-me notion of representability is somehow analogous to provability or demonstrability.

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?

1. One should construct rationals first. Suppose one starts from the notion of finite prime as something God-given. At the first step one constructs infinite primes as analogs for many-particle states in super-symmetric arithmetic quantum field theory. Ordinary primes label states of fermions and bosons. Infinite primes as the analogs of free many-particle states correspond to rationals in a natural manner.
2. One obtains also analogs of bound states which are mappable to irreducible polynomials, whose roots define algebraic numbers. This would give hierarchy of algebraic extensions of rationals. At higher levels of the hierarchy one obtains also analogs of prime polynomials with number of variables larger than 1. One might say that algebraic geometry has quantal representation. This might be very relevant for the physical representability of basic mathematical structures.

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 lower-dimensional 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!

Finite-dimensional 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 m-n. This identification would hold true also at the level of states. Hilbert spaces with negative dimension would correspond to pairs with (m-n)<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 Stern-Brocot (S-B) tree (see this and this) containing given rational number once. S-B 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 S-B 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 non-pediodic partial fraction and could correspond to non-periodic paths so that also they could have Hilbert spaces counterparts.

2.3 How to obtain the analogs higher-D spaces?

Algebraic extensions of rationals allow cognitive realization of spaces with arbitrary dimension identified as algebraic dimension of extension of rationals.

1. One can obtain n-dimensional spaces (in algebraic sense) with integer valued coordinates from n-D extensions of rationals. Now the n-tuples defining numbers of extension and differing by permutations are not equivalent so that one obtains n-D space rather than n-D space divided by permutation group Sn. This is enough at the level of cognitive representations and could explain why we are able to imagine spaces of arbitrary dimension although we cannot represent them cognitively.
2. One obtains also Galois group and orbits of set A of points of extension under Galois group G as G(A). One obtains also discrete coset spaces G/H and alike. These do not have any direct analog in the set theory. The hierarchy of Galois groups would bring in discrete group theory automatically. The basic machinery of quantum theory emerges elegantly from number theoretic vision.
3. In octonionic approach to quantum TGD one obtains also hierarchy of extensions of rationals since space-time surface correspond zero loci for RE or IM for octonionic polynomials obtained by algebraic continuation from real polynomials with coeffficients in extension of rationals (see this).

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.

1. From the point of quantum physics set theory and the notion of number based on set theory look somewhat artificial constructs. Nonempty set is a natural concept but empty set and set having empty set as element used as basic building brick in the construction of natural numbers looks weird to me.
2. From TGD point of view it would seem that number theory plus some basic pieces of quantum theory might be more fundamental than set theory. Could set theory emerge as a classical correlate for quantum number theory already considered and could quantal set theory make sense?
3.1 Quantum set theory

What quantum set theory could mean? Suppose that number theory-quantum theory connection really works. What about set theory? Or perhaps its quantum counterpart having ordinary set theory as a classical correlate?

1. A purely quantal input to the notion of set would be replacement of points delocalized states in the set. A generic single particle quantum state as analog of element of set would not be localized to a single element of set. The condition that the state has finite norm implies in the case of continuous set like reals that one cannot have completely localized states. This would give quantal limitation to the axiom of choice. One can have any discrete basis of state functions in the set but one cannot pick up just one point since this state would have infinite norm.

The idea about allowing only say rationals is not needed since there is infinite number of different choices of basis. Finite measurement resolution is however unvoidable. An alternative option is restriction of the domains of wave functions to a discrete set of points. This set can be chosen in very many manners and points with coordinates in extension of rationals are very natural and would define cognitive representation.

2. One can construct also the analogs of subsets as many-particle states. The basic operation would be addition/removal of a particle from quantum state represented by the action of creation/annihilation operator.

Bosonic states would be invariant under permutations of single particle states just like set is the equivalence class for a collection of elements (a1,...,an) such that any two permutations are equivalent. Quantum set theory would however bring in something new: the possibility of fermionic statistics. Permutation would change the state by phase factor -1. One would have fermionic and bosonic sets. For bosonic sets one could have multiple elements ("Bose-Einstein condensation"): in the theory of surfaces this could allow multiple copies of the same surface. Even braid statistics is possible. The phase factor in permutation could be complex. Even non-commutative statistics can be considered.

Many particle states formed from particles, which are not identical are also possible and now the different particle types can be ordered. On obtains n-ples decomposing to ordered K-ple of ni-ples, which are consist of identical particles and are quantum sets. One could talk about K-sets as a generalization of set as analogs of classical sets with K-colored elements. Group theory would enter into the picture via permutation groups and braid groups would bring in braid statistics. Braids strands would have K colors.

3.2 How to obtain classical set theory?

How could one obtain classical set theory?

1. Many-particle states represented algebraically are detected in lab as sets: this is quantum classical correspondence. This remains to me one of the really mysterious looking aspects in the interpretation of quantum field theory. For some reason it is usually not mentioned at all in popularizations. The reason is probably that popularization deals typically with wave mechanics but not quantum field theory unless it is about Higgs mechanism, which is the weakest part of quantum field theory!
2. From the point of quantum theory empty set would correspond to vacuum. It is not observable as such. Could the situation change in the presence of second state representing the environment? Could the fundamental sets be always non-empty and correspond to states with non-vanishing particle number. Natural numbers would correspond to eigenvalues of an observable telling the cardinality of set. Could representable sets be like natural numbers?
3. Usually integers are identified as pairs of natural numbers (m,n) such that integer corresponds to m-n. Could the set theoretic analog of integer be a pair (A,B) of sets such that A is subset of B or vice versa? Note that this does not allow pairs with disjoint members. (A,A) would correspond to empty set. This would give rise to sets (A,B) and their "antisets" (B,A) as analogs of positive and negative integers.

One can argue that antisets are not physically realizable. Sets and antisets would have as analogs two quantizations in which the roles of oscillator operators and their hermitian conjugates are changed. The operators annihilating the ground state are called annilation operators. Only either of these realization is possible but not both simultaneously.

In ZEO one can ask whether these two options correspond to positive and negative energy parts of zero energy states or to the states with state function reduction at either boundary of CD identified as correlates for conscious entities with opposite arrows of geometric time (generalized Zeno effect).

4. The cardinality of set, the number of elements in the set, could correspond to eigenvalue of observable measuring particle number. Many-particle states consisting of bosons or fermions would be analogs for sets since the ordering does not matter. Also braid statistics would be possible.

What about cardinality as a p-adic integer? In p-adic context one can assign to integer m, integer -m as m× (p-1)× (1+p+p2+...). This is infinite as real integer but finite as p-adic integer. Could one say that the antiset of m-element as analog of negative integer has cardinality -m= m(p-1)(1+p+p2+..). This number does not have cognitive representation since it is not finite as real number but is cognizable.

One could argue that negative numbers are cognizable but not cognitively representable as cardinality of set? This representation must be distinguished from cognitive representations as a point of imbedding space with coordinates in extension of rationals. Could one say that antisets and empty set as its own antiset can be cognized but cannot be cognitively represented?

Nasty mathematician would ask whether I can really start from Hilbert space of state functions and deduce from this the underlying set. The elements of set itself should emerge from this as analogs of completely localized single particle states labelled by points of set. In the case of finite-dimensional Hilbert space this is trivial. The number of points in the set would be equal to the dimension of Hilbert space. In the case of infinite-D Hilbert space the set would have infinite number of points.

Here one has two views about infinite set. One has both separable (infinite-D in discrete sense: particle in box with discrete momentum spectrum) and non-separable (infinite-D 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 bra-ket 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 hyper-finite factors of type II1 as basic algebraic objects in TGD based quantum theory as opposed to wave mechanics (type I) and quantum field theory (type III). They are infinite-D but allow excellent approximation as finite-D objects.

This picture could relate to the notion of non-commutative 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 M4× CP2 whereas at the level of M8 algebraic geometry might be enough. The notion of perfectoid as an extension of p-adic numbers Qp allowing all p:th roots of p-adic 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 infinite-dimensionality of perfectoid is in conflict with the vision about finiteness of cognition. For other p-adic number fields Qq, q≠ p the extension containing p:th roots of p would be however finite-dimensional even in the case of perfectoid. Furthermore, one has an entire hierarchy of almost-perfectoids allowing powers of pm:th roots of p-adic 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 almost-perfectoids 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 M8-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 p-adic space-time sheets (correlates for sensory and cognitive). One allows extensions of p-adics 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 (21/2) as the diameter of unit square). Stern-Brocot tree associated with partial fractions indeed allows to identify rationals as finite paths connecting the root of S-B 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 non-periodic paths. A very close analogy with chaos theory suggests itself.

### Super-number 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 light-cone boundary: this has interpretation in terms of propagation of signals with light-velocity 8-D sense. The emergence of fermions however requires super-octonions as super variant of number field. Rather surprisingly, it turns out that super-number theory makes perfect sense. One can define the inverse of super-number if its ordinary part is non-vanishing and also the notion of primeness makes sense and construct explicitly the super-primes 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.

### Is the quantum leakage between different signatures of the real sectors of the complexified M8 possible?

Complexified octonions have led to a dramatic progress in the understanding of TGD. One cannot however avoid a radical question about fundamentals.

1. The basic structure at M8 side consists of complexified octonions. The metric tensor for the complexified inner product for complexified octonions (no complex conjugation with respect to i for the vectors in the inner product) can be taken to have any signature (ε1,...,ε8), εi=+/- 1. By allowing some coordinates to be real and some coordinates imaginary one obtains effectively any signature from say purely Euclidian signature. What matters is that the restriction of complexified metric to the allowed sub-space is real. These sub-spaces are linear Lagrangian manifolds for Kähler form representing the commuting imaginary unit i. There is analogy with wave mechanics. Why M8 -actually M4 - should be so special real section? Why not some other signature?
2. The first observation is that the CP2 point labelling tangent space is independent of the signature so that the problem reduces to the question why M4 rather than some other signature (ε1,..,ε4). The intersection of real subspaces with different signatures and same origin (t,r)=0 is the common sub-space with the same signature. For instance, for (1,-1,-1,-1) and (-1,-1,-1,-1) this subspace is 3-D t=0 plane sharing with CD the lower tips of CD. For (-1,1,1,1) and (1,1,1,1) the situation is same. For (1,-1,-1,-1) and (1,1,-1,-1) z=0 holds in the intersection having as common with the lower boundary of CD the boundary of 3-D light-cone. One obtains in a similar manner boundaries of 2-D and 1-D light-cones as intersections.
3. What about CDs in various signatures? For a fully Euclidian signature the counterparts for the interiors of CDs reduce to 4-D intervals t∈ [0,T] and their exteriors and thus the space-time varieties representing incoming particles reduce to pairs of points (t,r)=(0,0) and (t,r)= (T,0): it does not make sense to speak about external particles. For other signatures the external particles correspond to 4-D surfaces and dynamics makes sense. The CDs associated with the real sectors intersect at boundaries of lower dimensional CDs: these lower-dimensional boundaries are analogous to subspaces of Big Bang (BB) and Big Crunch (BC).
4. I have not found any good argument for selecting M4=M1,3 as a unique signature. Should one allow also other real sections? Could the quantum numbers be transferred between sectors of different signature at BB and BC? The counterpart of Lorentz group acting as a symmetry group depends on signature and would change in the transfer. Conservation laws should be satisfied in this kind of process if it is possible. For instance, in the leakage from M4=M1,3 to Mi,j, say M2,2, the intersection would be M1,2. Momentum components for which signature changes, should vanish if this is true. Angular momentum quantization axis normal to the plane is defined by two axis with the same signature. If the signatures of these axes are preserved, angular momentum projection in this direction should be conserved. The amplitude for the transfer would involve integral over either boundary component of the lower-dimensional CD.

Final question: Could the leakage between signatures be detected as disappearance of matter for CDs in elementary particle scales or lab scales?

See the article Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

### What the properties of octonionic product can tell about fundamental physics?

In developing the view about M8-H duality reducing physics to algebraic geometry for complexified octonions at the level of M8, I became aware of trivial looking but amazingly profound observation about the basic arithmetics of complex, quaternion, and octonion number fields.

1. Imaginary part for the product z1z2 of complex numbers is

Im(z1z2)= Im(z1)Re(z2)+Re(z1)Im(z2)

and linear in Im(z1) and Im(z2).

2. Real part Re(z1z2)= Re(z1)Re(z2)-Im(z1)Im(z2).

is not linear in real parts:

This generalizes to the product of octonions with Re and Im replaced by RE and IM in the decomposition to two quaternions: o= RE(o)+J IM(o), J is octonion imaginary unit not belonging to quaternionic subspace.

This extremely simple observation turns out to contain amazingly deep physics.

1. Space-time surfaces can be identified as IM(P)= loci or RE(P)=0 loci. When one takes product of two polynomials P1P2 the IM(P1P2)=0 locus as space-time surface is just the union of IM(p1)=0 locus and IM(P2) locus. No interaction: free particles as space-time surfaces! This picture generalizes also to rational functions R=P1/P2 and an their zero and infty loci.
2. For RE(P1P2)=0 the situation changes. One does not obtain union of RE(P1)=0 and RE(P2) space-time surfaces. There is interaction and most naturally this interaction generates wormhole contacts connecting the space-time surfaces (sheet) carrying fermions at the throats of the wormhole contact!

The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers!

### Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications to TGD?

Riemann-Roch theorem (RR) is a central piece of algebraic geometry. Atyiah-Singer 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 Yang-Mills theories have applications in gauge theories and string models.

The advent of octonionic approach to the dynamics of space-time surfaces inspired by M8-H duality (see this and this) gives hopes that dynamics at the level of complexified octonionic M8 could reduce to algebraic equations plus criticality conditions guaranteeing associativity for space-time surfaces representing external particles, in interaction region commutativity and associativity would be broken. The complexification of octonionic M8 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 light-like 3-surfaces and curves (string boundaries) as generalized poles and zeros with topological (but not metric) dimension one unit higher than in Euclidian signature.

Atyiah-Singer 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 M8-H duality suggesting very simple counting for the numbers of solutions at M8 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 sub-algebras with conformal weights coming as n-multiples for those for the entire algebras.

### About enumerative algebraic geometry in TGD framework

I 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 M-theory 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.

1. One might want to know the number of points of sub-variety belonging to the number field defining the coefficients of the polynomials. This problem is very relevant in M8 formulation of TGD, where these points are carriers of sparticles. In TGD based vision about cognition they define cognitive representations as points of space-time surface, whose M8 coordinates can be thought of as belonging to both real number field and to extensions of various p-adic number fields induced by the extension of rationals. If these cognitive representations define the vertices of analogs of twistor Grassmann diagrams in which sparticle lines meet, one would have number theoretically universal adelic formulation of scattering amplitudes and a deep connection between fundamental physics and cognition.
2. Second kind of problem involves a set algebraic surfaces represented as zero loci for polynomials - lines and circles in the simplest situations. One must find the number of algebraic surfaces intersecting or touching the surfaces in this set. Here the notion of incidence is central. Point can be incident on line or two lines (being their intersection), line on plane, etc.. This leads to the notion of Grassmannians and flag-manifolds. In twistor Grassmannian approach algebraic geometry of Grassmannians play key role. Also in twistor Grassmannian approach to TGD algebraic geometry of Grassmannians play a key role and some aspects of this approach are discussed.
3. In string models the notion of brane leads to what might be called quantum variant of algebraic geometry in which the usual rules of algebraic geometry do not apply as such. Gromow-Witten invariants provide an example of quantum invariants allowing sharper classification of algebraic and symplectic geometries. In TGD framework M8-H duality suggests that the construction of scattering amplitudes at level of M8 reduces to a super-space analog of algebraic geometry for complexified octonions. Candidates for TGD analogs of branes emerge naturally and G-W invariants could have applications also in TGD.
In the sequel I will summarize the understanding of novice about enumerative algebraic geometry and discuss possible TGD applications. This material can be also found in earlier articles but it seemed appropriate to collect the material about enumerative algebraic geometry to a separate article.

### Super-octonions, super-twistors and twistorial construction of scattering amplitudes

For 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 "super-octonionic" 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 space-time 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 space-time surface with M8 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.

1. The surprise that RE(P)=0 and IM(P)=0 conditions have as singular solutions light-cone interior and its complement and 6-spheres S6(tn) with radii tn given by the roots of the real P(t), whose octonionic extension defines the space-time variety X4. The intersections X2= X4∩ S6(tn) are tentatively identified as partonic 2-varieties defining topological interaction vertices. S6 and therefore also X2 are doubly critical, S6 is also singular surface.

The idea about the reduction of zero energy states to discrete cognitive representations suggests that interaction vertices at partonic varieties X2 are associated with the discrete set of intersection points of the sparticle lines at light-like orbits of partonic 2-surfaces belonging to extension of rationals.

2. CDs and therefore also ZEO emerge naturally. For CDs with different origins the products of polynomials fail to commute and associate unless the CDs have tips along real (time) axis. The first option is that all CDs under observation satisfy this condition. Second option allows general CDs.

The proposal is that the product ∏ Pi of polynomials associated with CDs with tips along real axis the condition IM(∏ Pi)=0 reduces to IM(Pi)=0 and criticality conditions guaranteeing associativity and provides a description of the external particles. Inside these CDs RE(∏ Pi)=0 does not reduce to RE(∏ Pi)=0, which automatically gives rise to geometric interactions. For general CDs the situation is more complex.

3. The possibility of super-octonionic geometry raises the hope that the twistorial construction of scattering amplitudes in N=4 SUSY generalizes to TGD in rather straightforward manner to a purely geometric construction. Functional integral over WCW would reduce to summations over polynomials with coefficients in extension of rationals and criticality conditions on the coefficients could make the summation well-defined by bringing in finite measurement resolution.

If scattering diagrams are associated with discrete cognitive representations, one obtains a generalization of twistor formalism involving polygons. Super-octonions as counterparts of super gauge potentials are well-defined if octonionic 8-momenta are quaternionic. Indeed, Grassmannians have quaternionic counterparts but not octonionic ones. There are good hopes that the twistor Grassmann approach to N=4 SUSY generalizes. The core part in the calculation of the scattering diagram would reduce to the construction of octonionic 4-varieties and identifying the points belonging to the appropriate extension of rationals.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

### Geometrization of fermions using super version of the octonionic algebraic geometry

Could 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 M8 using the variant of the twistor approach discussed earlier.

The idea about super-geometry 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 M8-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 super-space might look like.

1. What makes octonions so nice is the octonionic triality. One has three 8-D representations: vector representation 8V, spinor representation 8s and its conjugate 8*s. The tensor products of two representations gives the third representation in the triplet. This is the completely unique feature of dimension 8 and makes octonionic physics so fascinating an option. The octonionic triality is central also in super-string models but in a different manner since of starts from 10-D situation and ends up with effectively 8-D situation for physical states.
2. One can define super octonion as os= o +θ1 + θ2. Here o is bosonic octonionic coordinate. θi= θki Ek, where Ek are octonionic units, is Grassmann valued octonion in 8s satisfying the usual anti-commutations and θ2 transforms as 8*s. (I have already earlier considered as natural candidates for spinors in octonionic M8).

The first interpretation is that θ1 and θ2 correspond to objects with opposite fermion numbers. If this is not the case, one could perhaps define the conjugate of super-coordinate as o*s=o* +θ*1 + θ*2. This looks however ugly.

3. What could be the physical interpretation? One should obtain particles and antiparticles naturally as also separately conserved baryon and lepton numbers (I have also considered the identification of hadrons in terms of anyonic bound states of leptons with fractional charges).

Quarks and leptons have different coupling to the induced Kähler form at the level of H. It seems impossible to understand this at the level of M8, where the dynamics is purely algebraic and contains no gauge couplings.

The difference between quarks and leptons is that they allow color partial waves with triality t=+/- 1 and triality t=0. Color partial waves correspond to wave functions in the moduli space CP2 for M40 ⊃ M20. Could the distinction between quarks and leptons emerge at the level of this moduli space rather than at the fundamental octonionic level? There would be no need for gauge couplings to distinguish between quarks and leptons at the level of M8. All couplings would follow from the criticality conditions guaranteeing 4-D associativity for external particles (on mass shell states would be critical).

If so, one would have only the super octonions os= θ1+ θ2 =θ*1 and θ1 and θ2 =θ*1 would correspond to fermions and antifermions with no differentiation to quarks or leptons. Fermion number conservation would be coded by the Grassmann algebra.

One can imagine also other options but they have their problems. Therefore this option will be considered in the sequel.

2. Super version of octonionic algebraic geometry

Instead of super-fields one would have a super variant of octonionic algebraic geometry.

1. Super polynomials make still sense and reduce to a sum of octonionic polynomials Pklθ1kθ2l, where the integers k and l would be tentatively identified as fermion numbers.

One would clearly have an upper bound for k and l for given CD. Therefore these many-fermion states must correspond to fundamental particles rather than many-fermion Fock states. One would obtain bosons with non-vanishing fermion numbers if the proposed identification is correct. Octonionic algebraic geometry for single CD would describe only fundamental particles or states with bounded fermion numbers. Fundamental particles would be indeed fundamental also geometrically.

2. I have already earlier considered the question whether the partonic 2-surfaces can carry also many-fermion states or not, and adopted the working hypothesis that fermion numbers is not larger than 1 for given wormhole throat, possibly for purely dynamical reasons. This picture however looks too limited. The many fermion states might not however propagate as ordinary particles (the proposal has been that their propagator pole corresponds to higher power of p2).
3. The result looks somewhat disappointing at first. It would seem that the states with high fermion numbers must be described in terms of Cartesian products just like in condensed matter physics with interactions described by the proposed braney mechanism in which intersection of space-time surfaces with S6 giving analogs of partonic 2-surfaces are involved.
4. One can also now define space-time varieties as zero loci via the conditions RE(Ps)(os)=0 or IM(Ps)(os)=0. One obtains a collection of 4-surfaces as zero loci of Pkl. One would have a correlation with between fermion content and algebraic geometry of the space-time surface unlike in the ordinary super-space approach, where the notion of the geometry remains rather formal and there is no natural coupling between fermionic content and classical geometry. At the level of H this comes from quantum classical correspondence (QCC) stating that the classical Noether charges are equal to eigenvalues of fermionic Noether charges.

There are several questions about quantum numbers.

1. Could octonionic super geometry code for quantum numbers of the particle states? It seems that super-octonionic polynomials multiplied by octonionic multi-spinors inside single CD can code only for the electroweak quantum numbers of fundamental particles besides their fermion and anti-fermion numbers.

As already suggested, color corresponds to partial waves in CP2 serving as moduli space for M40⊃ M20 and quarks and leptons have different trialities. Also four-momentum and angular momentum are naturally assigned with the translational degrees for the tip of CD assignable with the fundamental particle.

Remark: There is a funny accident that deserves to be noticed. Octonionic spinor decomposes to 1⊕ 1⊕ 3 ⊕3* under SU(3)⊂ G2. Could it be that 1⊕ 1 corresponding to real unit and preferred imaginary unit assignable to M20 correspond to color wave functions in CP2 transforming like leptons and 3+3* corresponds to wave functions transforming like quarks and antiquarks? Unfortunately, one cannot understand electroweak quantum numbers in this framework. There would be uncertainty principle allowing to measure either of these quantum numbers but not both.

2. What about twistors in this framework? M4× CP1 as twistor space with CP1 coding for the choice of M20⊂ M40 allows projection to the usual twistor space CP3. Twistor wave functions describing spin elegantly would correspond to wave functions in the twistor space and one expects that the notion of super-twistor is well-defined also now. The 6-D twistor space SU(3)/U(2)× U(1) of CP2 would code besides the choice of M40⊃ M20 also quantization axis for color hypercharge and isospin.
3. What about the sphere S6 serving as the moduli space for the choices of M8+? Should one have wave functions in S6 or can one restrict the consideration to single M8+? As found, one obtains S6 also as the zero locus of Im(P)=0 for some radii identifiable as values tn of time coordinates given as roots of P(t). This would be crucial for the braney description of interactions between space-time surfaces associated with different CDs.
4. Could scattering amplitudes be computed at the level of M8?

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 M8. Something rather similar to N=4 twistor Grassmann approach suggests itself.

1. In ZEO picture one would consider the situation in which the passive boundary of CD and members of state pairs at it appearing in zero energy state remain fixed during the sequence of state function reductions inducing stepwise drift of the active boundary of CD and change of states at it by unitary U-matrix at each step following by a localization in the moduli space for the positions of the active boundary.
2. At the active boundary one would obtain quantum superposition of states corresponding to different octonionic geometries for the outgoing particles. Instead of functional integral one would have sum over discrete points of WCW. WCW coordinates would be the coefficients of polynomial P in the extension of rationals. This would give undefined result without additional constraints since rationals are a dense set of reals.

Criticality however serves as a constraint on the coefficients of the polynomials and is expected to realize finite measurement resolution, and hopefully give a well defined finite result in the summation. Criticality for the outgoing states would realize purely number theoretically the cutoff due to finite measurement resolution and would be absolutely essential for the finiteness and well-definedness of the theory.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

### Emergence of Zero Energy Ontology and Causal Diamonds from octonionic algebraic surface dynamics

Octonionic 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 (8-D) and quaternionic (4-D) context. Light-like boundary of CD as also light-cone emerge naturally as zeros of octonionic polynomials. This does not yet give CDs and ZEO: one should have intersection of future and past directed light-cones. The intuitive picture is that one has a hierarchy of CDs and that also the space-time surfaces inside different CDs an interact. It turns out that CDs and thus also ZEO emerge naturally both at the level of M8 and M4 .

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)I4, 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 4-D space-time surfaces as zero loci. Appendix shows explicitly for P(o)=o2 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)=o2 having double root at origin.

1. Consider first the octonionic polynomials P(o) satisfying P(0)=0 restricted to the light-like boundary δ M8+ assignable to 8-D CD, where the octonionic norm of o vanishes.
1. P(o) reduces along each light-ray of δ M8+ to the same real valued polynomial P(t) of a real variable t apart from a multiplicative unit E= (1+in)/2 satisfying E2=E. Here n is purely octonion-imaginary unit vector defining the direction of the light-ray.

IM(P)=0 corresponds to quaterniocity. If the E4 (M8= M4× E4) projection is vanishing, there is no additional condition. 4-D light-cones M4+/- are obtained as solutions of IM(P)=0. Note that M4+/- can correspond to any quaternionic subspace.

If the light-like ray has a non-vanishing projection to E4, one must have P(t)=0. The solutions form a collection of 6-spheres labelled by the roots tn of P(t)=0. 6-spheres are not associative.

2. RE(PE)=0 corresponding to co-quaternionicity leads to P(t)=0 always and gives a collection of 6-spheres.
2. Suppose now that P(t) is shifted to P1(t)=P(t)-c, c a real number. Also now M4+/- is obtained as solutions to IM(P)=0. For RE(P)=0 one obtains two conditions P(t)=0 and P(t-c)=0. The common roots define a subset of 6-spheres which for special values of c is not empty.
The above discussion was limited to δ M8+ and light-likeness of its points played a central role. What about the interior of 8-D CD?
1. The natural expectation is that in the interior of CD one obtains a 4-D variety X4. For IM(P)=0 the outcome would be union of X4 with M4+ and the set of 6-spheres for IM(P)=0. 4-D variety would intersect M4+ in a discrete set of points and the 6-spheres along 2-D varieties X2. The higher the degree of P, the larger the number of 6-spheres and these 2-varieties.
2. For RE(P)=0 X4 would intersect the union of 6-spheres along 2-D varieties. What comes in mind that these 2-varieties correspond in H to partonic 2-surfaces defining light-like 3-surfaces at which the induced metric is degenerate.
3. One can consider also the situation in the complement of 8-D CD which corresponds to the complement of 4-D CD. One expects that RE(P)=0 condition is replaced with IM(P)=0 condition in the complement and RE(P)= IM(P)=0 holds true at the boundary of 4-D CD.
6-spheres and 4-D empty light-cones are special solutions of the conditions and clearly analogs of branes. Should one make the (reluctant-to-me) conclusion that they might be relevant for TGD at the level of M8.
1. Could M4+ (or CDs as 4-D objects) and 6-spheres integrate the space-time varieties inside different 4-D CDs to single connected structure with space-time varieties glued to the 6-spheres along 2-surfaces X2 perhaps identifiable as pre-images of partonic 2-surfaces and maybe string world sheets? Could the interactions between space-time varieties X4i assignable with different CDs be describable by regarding 6-spheres as bridges between X4i having only a discrete set of common points. Could one say that X2i interact via the 6-sphere somehow. Note however that 6-spheres are not dynamical.
2. One can also have Poincare transforms of 8-D CDs. Could the description of their interactions involve 4-D intersections of corresponding 6-spheres?
3. 6-spheres in IM(P)=0 case do not have image under M8-H correspondence. This does not seem to be possible for RE(P)=0 either: it is not possible to map the 2-D normal space to a unique CP2 point since there is 2-D continuum of quaternionic sub-spaces containing it.
2. Some general observations about CDs

CD defines the basic geometric object in ZEO. It is good to list some basic features of CDS, which appear as both 4-D and 8-D variants.

1. There are both 4-D and 8-D CDs defined as intersections of future and past directed light-cones with tips at say origin 0 at real point T at quaternionic or octonionic time axis. CDs can be contained inside each other. CDs form a fractal hierarchy with CDs within CDs: one can add smaller CDs with given CD in all possible manners and repeat the process for the sub-CDs. One can also allow overlapping CDs and one can ask whether CDs define the analog of covering of O so that one would have something analogous to a manifold.
2. The boundaries of two CDs (both 4-D and 8-D) can intersect along light-like ray. For 4-D CD the image of this ray in H is light-like ray in M4 at boundary of CD. For 8-D CD the image is in general curved line and the question is whether the light-like curves representing fermion orbits at the orbits of partonic 2-surfaces could be images of these lines.
3. The 3-surfaces at the boundaries of the two 4-D CDs are expected to have a discrete intersection since 4 + 4 conditions must be satisfied (say RE(Pik))=0 for i=1,2, k=1,4. Along line octonionic coordinate reduces effectively to real coordinate since one has E2=E for E=(1+in)/2, n octonionic unit. The origins of CDs are shifted by a light-like vector kE so that the light-like coordinates differ by a shift: t2= t1-k. Therefore one has common zero for real polynomials RE(P1k(t)) and RE(P2k(t-k)).

Are these intersection points somehow special physically? Could they correspond to the ends of fermionic lines? Could it happen that the intersection is 1-D in some special cases? The example of o2 suggest that this might be the case. Does 1-D intersection of 3-surfaces at boundaries of 8-D CDs make possible interaction between space-time surfaces assignable to separate CDs as suggested by the proposed TGD based twistorial construction of scattering amplitudes?

4. Both tips of CD define naturally an origin of quaternionic coordinates for D=4 and the origin of octonionic coordinates for D=8. Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be along the real line (time axis) connecting the tips of CD. Only the translations in this specified direction are symmetries preserving the commutativity and associativity of the polynomial algebra.
5. One expects that also Lorentz boosts of 4-D CDs are relevant. Lorentz boosts leave second boundary of CD invariant and Lorentz transforms the other one. Same applies to 8-D CDs. Lorentz boosts define non-equivalent octonionic and quaternionic structures and it seems that one assume moduli spaces of them.
One can of course ask whether the still somewhat ad hoc notion of CD general enough. Should one generalize it to the analog of the polygonal diagram with light-like geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the light-like boundaries assignable to CDs but leaves open the question whether more complex structures with light-like boundaries are possible. How do the space-time surfaces associated with different quaternionic structures of M8 and with different positions of tips of CD interact?

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 light-like boundaries of CDs? In Minkowskian case the complex norm qqci 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 3-D boundary of light-cone 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 light-rays of O indeed demonstrated that both 8-D land 4-D light-cones and their complements emerge naturally, and that the M4 projections of these light-cones and even of their boundaries are 4-D future - or past directed light-cones. What one should understand is how CDs as their intersections, and therefore ZEO, emerge.

1. One manner to obtain CDs naturally is that the polynomials are sums P(t)= ∑k Pk(o) of products of form Pk(o) =P1,k(o)P2,k(o-T), where T is real octonion defining the time coordinate. Single product of this kind gives two disjoint 4-varieties inside future and past directed light-cones M4+(0) and M4-(T) for either RE(P)=0 (or IM(P)=0) condition. The complements of these cones correspond to IM(P)=0 (or RE(P)=0) condition.
2. If one has nontrivial sum over the products, one obtains a connected 4-variety due the interaction terms. One has also as special solutions M4+/- and the 6-spheres associated with the zeros P(t) or equivalently P1(t1)== P(t), t1=T-t vanishing at the upper tip of CD. The causal diamond M4+(0)∩ M4-(T) belongs to the intersection.

Remark: Also the union M4-(0)∪ M4+(T) past and future directed light-cones belongs to the intersection but the latter is not considered in the proposed physical interpretation.

3. The time values defined by the roots tn of P(t) define a sequence of 6-spheres intersecting 4-D CD along 3-balls at times tn. These time slices of CD must be physically somehow special. Space-time variety intersects 6-spheres along 2-varieties X2n at times tn. The varieties X2n are perhaps identifiable as 2-D interaction vertices, pre-images of corresponding vertices in H at which the light-like orbits of partonic 2-surfaces arriving from the opposite boundaries of CD meet.

The expectation is that in H one as generalized Feynman diagram with interaction vertices at times tn. The higher the evolutionary level in algebraic sense is, the higher the degree of the polynomial P(t), the number of tn, and more complex the algebraic numbers tn. P(t) would be coded by the values of interaction times tn. If their number is measurable, it would provide important information about the extension of rationals defining the evolutionary level. One can also hope of measuring tn with some accuracy! Octonionic dynamics would solve the roots of a polynomial! This would give a direct connection with adelic physics.

Remark: Could corresponding construction for higher algebras obtained by Cayley-Dickson construction solve the "roots" of polynomials with larger number of variables? Or could Cartesian product of octonionic spaces perhaps needed to describe interactions of CDs with arbitrary positions of tips lead to this?

4. Above I have considered only the interiors of light-cones. Also their complements are possible. The natural possibility is that varieties with RE(P)=0 and IM(P)=0 are glued at the boundary of CD, where RE(P)=IM(P)=0 is satisfied. The complement should contain the external (free) particles, and the natural expectation is that in this region the associativity/co-associativity conditions can be satisfied.
5. The 4-varieties representing external particles would be glued at boundaries of CD to the interacting non-associative solution in the complement of CD. The interaction terms should be non-vanishing only inside CD so that in the exterior one would have just product P(o)=P1,k0(o)P2,k0(o-T) giving rise to a disjoint union of associative varieties representing external particles. In the interior one could have interaction terms proportional to say t2(T-t)2 vanishing at the boundaries of CD in accordance with the idea that the interactions are switched one slowly. These terms would spoil the associativity.
Remark: One can also consider sums of the products ∏k Pk(o-Tk) of n polynomials and this gives a sequence CDs intersecting at their tips. It seems that something else is required to make the picture physical.

4. How could the space-time varieties associated with different CDs interact?

The interaction of space-time surfaces inside given CD is well-defined. 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 4-D/8-D CDs can be located everywhere in M4/M8. 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 M8:s with different origins and formulate the interaction at M8 level in this framework. Note that Cayley-Dickson 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 many-particle 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 space-time surfaces inside different CDs at the level of M8?

1. Could the intersection of space-time varieties with zero loci for RE(Pi) and IM(Pi) define the loci of interaction. As already found, the 6-D spheres S6 with radii tn given by the zeros of P(t) are universal and have interpretation as t=tn snapshots of 7-D spherical light front.

The 2-D intersections X2 of 4-D space-time variety X4 with S6 would define natural candidates for the intersections and might allow interpretation as pre-images of partonic 2-surfaces. X2 would be the contact of X4 with S6 associated with second 8-D CD. Together with SH this gives hopes about an elegant description of interactions in terms of connected space-time varieties.

2. The following picture is suggestive. Consider two space-time varieties X4i, i=1,2 associated with CDs with different origins and connected by a connected sum contact, which at the level of H corresponds to a wormhole contact connecting space-time sheets with different octonionic coordinates. The partonic 2-varieties X2i= X4i∩ S6i are labelled by time values t=ti,ni.

Assume that there is tube-like 3-surface X31,2 connecting X21 and X22. The union X21∩ X22 of partonic 2-surfaces must be homologically trivial in order to define a boundary of 3-surface X31,2. The surfaces X2i must therefore have opposite homology charges. X31,2 would be pre-image of a wormhole contact connecting different space-time sheets to which the CDs are assigned.

The 6-spheres S6i intersect along 4-D surface X41,2= S61∩ S62 in M8. One should have X31,2⊂ X41,2 and X31,2 should be non-critical but associative and therefore 3-D. This surface should allow a realization as a zero locus of RE(P1,2(u)) or IM(P1,2(u)) and belong to X41,2. One would not have manifold-topology. Rather, one could speak of two 4-D branes X4i (3-branes) connected by a 3-D brane X31,2 (2-brane). Two 2 parallel 4-planes joint by a 1-D curve is the lower-dimensional analogy. The interaction would be instantaneous inside X4i.

3. The polynomials associated with different 8-D CDs do not commute nor associate. Should one allow their products so that one would still effectively have a Cartesian product of commutative and associative algebras?

Or should one introduce Cartesian powers of O and CD:s inside these powers to describe the interaction? This would be analogous to what one does in condensed matter physics. What seems clear is that M8-H correspondence should map all the factors of (M8)n to the same M4× CP2 by a kind of diagonal projection.

4. Partonic 2-surfaces define wormhole throats and appear in pairs if they carry monopole charges. Could one think that the above mentioned 2-surfaces are intersections of X1i with Ski+1 for the pair of space-time sheets assignable to different CDs? Could the image in H of the structure formed by {X21,X22, S61, S62} under M8-H correspondences be wormhole contact.
5. Summary

All big pieces of quantum TGD are now tightly interlinked.

1. The notion of causal diamond (CD) and therefore also ZEO can be now regarded as a consequence of the number theoretic vision and M8-H correspondence, which is also understood physically.
2. The hierarchy of algebraic extensions of rationals defining evolutionary hierarchy corresponds to the hierarchy of octonionic polynomials.
3. Associative varieties for which the dynamics is critical are mapped to minimal surfaces with universal dynamics without any dependence on coupling constants as predicted by twistor lift of TGD. The 3-D associative boundaries of non-associative 4-varieties are mapped to initial values of space-time surfaces inside CDs for which there is coupling between Kähler action and volume term.
4. Free many particle states as algebraic 4-varieties correspond to product polynomials in the complement of CD and are associative. Inside CD the addition of interaction terms vanishing at its boundaries spoils associativity and makes these varieties connected.
5. The basic building bricks of topological scattering diagrams identified as space-time surfaces having as vertices partonic 2-surfaces emerge from the special features of the octonionic algebraic geometry predicting sequence of 3-balls as intersections of hyperplanes t= tn with CD. One can say that octonionic dynamics solves roots of the polynomial P(t) whose octonionic extension defines space-time surfaces as zero loci. Furthermore, the generic prediction is the existence of 6-spheres inside octonionic CDs having 2-D partonic 2-variety as intersection with space-time surface inside CD and interpreted as a vertex of generalized scattering diagram.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

### M8-H duality: summary and future prospects

In the following I give a brief summary about what has been done. I concentrate on M8-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.

1. When M8-H correspondence is true (to be honest, this question emerged during this work!)? What are the explicit formulas expressing associativity of the tangent space or normal space of the 4-surface?

The key element is the formulation in terms of complexified M8 identified in terms of octonions and restriction to M8. One loses the number field property but for polynomials ring property is enough. The level surfaces for real and imaginary parts of octonionic polynomials with real coefficients define 4-D surfaces in the generic case.

Associativity condition is an additional condition reducing the dimension of the space-time surface unless some components of RE(P) or IM(P) are critical meaning that also their gradients vanish. This conforms with the quantum criticality of TGD and provides a concrete first principle realization for it.

2. How this picture corresponds to twistor lift? The twistor lift of Kähler action (dimensionally reduced Kähler action in twistor space of space-time surface) one obtains two kinds of space-time regions. The regions, which are minimal surfaces and obey dynamics having no dependence on coupling constants, correspond naturally to the critical regions in M8 and H.

There are also regions in which one does not have extremal property for both Kähler action and volume term and the dynamics depends on coupling constant at the level of H. These regions are associative only at their 3-D ends at boundaries of CD and at partonic orbits, and the associativity conditions at these 3-surfaces force the initial values to satisfy the conditions guaranteeing preferred extremal property. The non-associative space-time regions are assigned with the interiors of CDs. . The particle orbit like space-time surfaces entering to CD are critical and correspond to external particles.

3. The surprise was that M4⊂ M8 is naturally co-associative. If associativity holds true also at the level of H, M4 ⊂ H must be associative. This is possible if M8-H duality maps tangent space in M8 to normal space in H and vice versa.
4. The connection to the realization of the preferred extremal property in terms of gauge conditions of subalgebra of SSA is highly suggestive. Octonionic polynomials critical at the boundaries of space-time surfaces would determine by M8-H correspondence the solution to the gauge conditions and thus initial values and by holography the space-time surfaces in H.
5. A beautiful connection between algebraic geometry and particle physics emerges. Free many-particle states as disjoint critical 4-surfaces can be described by products of corresponding polynomials satisfying criticality conditions. These particles enter into CD , and the non-associative and non-critical portions of the space-time surface inside CD describe the interactions. One can define the notion of interaction polynomial as a term added to the product of polynomials. It can vanish at the boundary of CD and forces the 4-surface to be connected inside CD. It also spoils associativity: interactions are switched on. For bound states the coefficients of interaction polynomial are such that one obtains a bound state as associative space-time surface.
6. This picture generalizes to the level of quaternions. One can speak about 2-surfaces of space-time surface with commutative or co-commutative tangent space. Also these 2-surfaces would be critical. In the generic case commutativity/co-commutativity allows only 1-D curves.

At partonic orbits defining boundaries between Minkowskian and Euclidian space-time regions inside CD the string world sheets degenerate to the 1-D orbits of point like particles at their boundaries. This conforms with the twistorial description of scattering amplitudes in terms of point like fermions.

For critical space-time surfaces representing incoming states string world sheets are possible as commutative/co-commutative surfaces (as also partonic 2-surfaces) and serve as correlates for (long range) entaglement) assignable also to macroscopically quantum coherent system (heff/h=n hierarchy implied by adelic physics).

7. The octonionic polynomials with real coefficients form a commutative and associative algebra allowing besides algebraic operations function composition. Space-time surfaces therefore form an algebra and WCW has algebra structure. This could be true for the entire hierarchy of Cayley-Dickson algebras, and one would have a highly non-trivial generalization of the conformal invariance and Cauchy-Riemann conditions to their n-linear counterparts at the n:th level of hierarchy with n=1,2,3,.. for complex numbers, quaternions, octonions,... One can even wonder whether TGD generalizes to this entire hierarchy!
What mathematical challenges one must meet?
1. One should prove more rigorously that criticality is possible without the reduction of dimension of the space-time surface.
2. One must demonstrate that SSA conditions can be true for the images of the associative regions (with 3-D or 4-D). This would obviously pose strong conditions on the values of coupling constants at the level of H.
1. Does associativity hold true in H for minimal surface extremals obeying universal critical dynamics? As found, the study of the known extremals supports this view.
2. Could one construct the scattering amplitudes at the level of M8? Here the possible problems are caused by the exponents of action (Kähler action and volume term) at H side. Twistorial construction however leads to a proposal that the exponents actually cancel. This happens if the scattering amplitude can be thought as an analog of Gaussian path integral around single extremum of action and conforms with the integrability of the theory. In fact, nothing prevents from defining zero energy states in this manner! If this holds true then it might be possible to construct scattering amplitudes at the level of M8.
3. What about coupling constants? Coupling constants make themselves visible at H side both via the vanishing conditions for Noether charges in sub-algebra of SSA and via the values of the non-vanishing Noether charges. M8-H correspondence determining the 3-D boundaries of interaction regions within CDs suggests that these couplings must emerge from the level M8 via the criticality conditions posing conditions on the coefficients of the octonionic polynomials coding for interactions.

Could all coupling constant emerge from the criticality conditions at the level of M8? The ratio of R2/lP2 of CP2 scale and Planck length appears at H level. Also this parameter should emerge from M8-H correspondence and thus from criticality at M8 level. Physics would reduce to a generalization of the catastrophe theory of Rene Thom!

4. There are questions related to ZEO. Is the notion of CD general enough or should one generalize it to the analog of the polygonal diagram with light-like geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the light-like boundaries assignable to CDs but leaves open the question whether more complex structures with light-like boundaries are possible. How the space-time surfaces associated with different quaternionic structures of M8 and with different positions of tips of CD interact?
5. Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be at real line (time axis) in the octonionic sense. All CDs cannot be located along this line. How do the varieties associated with octonionic polynomials with different origins interact? The polynomials with different origins neither commute nor are associative. 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 M8:s with different origins and formulate the interaction in this framework. Could Cayley-Dickson hierarchy be necessary to describe the interactions between different CDs (note however that the dimension are powers of 2) than multiples of 8?

Is the interaction nr well-defined only at the level of H inside CD to which these 4-D varieties arrive through the boundary of this CD? All CDs, whose tips are along light-like ray of CD boundary, share this ray. There is a common M20 shared by these CDs. Could M20 make possible the interaction. The CDs able to interact with given CD would have tips at the 3-D boundary of this CD and share common M20. These M20:s are labelled by the points of twistor sphere so that twistoriality seems to enter into the game in non-trivial manner also at the level of M8!

This however allows interactions only between varieties with same M40. What about a more general picture in which unit octonions define 6-D sphere S6 of directions of 8-D light-rays and parameterize different quaternionic structures with fixed M20. Could the condition for interaction be that S6 coordinates are same so that M20 is shared. In this case light rays in 7-D light-cone would parameterize the allowed origins for octonionic polynomials for which the interaction of zero loci is possible. The images of these light-rays in H would be more complex. This could allow varieties with different M40 but common M20 to interact via the common light ray/M20. Somewhat similar picture involving preferred M20 for given connected part of twistor graph emerges from the construction of twistor amplitudes.

What would be the interaction at the intersection? Light-like ray naturally defines a string like object with fermions at its ends. Could this fermionic string be transferred between space-time surfaces in the intersecting CDs. Also a branching of a string like object between space-time varieties in different CDs intersecting along this ray would be possible. This would describe stringy reaction vertex with incoming strings in different CDs. It must be admitted that this unavoidably brings in mind branes and all the nasty things that I have said about them during years!

Or could the strange singularity behavior in Minkowski signature play a role in the interactions? In the generic situation the intersection consists of discrete points but as the study of o2 shows the surface RE(o2)=0 and IM(o2)=0 can have dimensions 5 and 6 and their intersection naively expected to consist of discrete points can be interior or exterior of light-cone. Could the zero loci of singular polynomials play a key role in the interaction and allow 4-varieties in M8 to interact by being glued to this higher than 4-D objects. 4-D space-time variety could have 2-D intersection with 6-D variety and the 6-D variety could allow the interactions to between two 4-D varieties by correlating them. Also this strongly brings in mind branes but looks less elegant that above proposal.

6. What is the connection with Yangian symmetry, whose generalization in TGD framework is highly suggestive?
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

### Does M8-H duality reduce classical TGD to octonionic algebraic geometry?

I have used last month to develop a detailed vision about M8-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 space-time surfaces as preferred extremals of twistor lift of Kähler action. So called M8-H duality is one of these approaches. The beauty of M8-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.

1. I will discuss basic notions of algebraic geometry such as algebraic variety, surface, and curve, rational point of variety central for TGD view about cognitive representation, elliptic curves and surfaces, and rational and potentially rational varieties. Also the notion of Zariski topology and Kodaira dimension are discussed briefly. I am not a mathematician and what hopefully saves me from horrible blunders is physical intuition developed during 4 decades of TGD.
2. It will be shown how M8-H duality could reduce TGD at fundamental level to algebraic geometry. Space-time surfaces in M8 would be algebraic surfaces identified as zero loci for imaginary part IM(P) or real part RE(P) of octonionic polynomial of complexified octonionic variable oc decomposing as oc= q1c+q2c I4 and projected to a Minkowskian sub-space M8 of complexified O. Single real valued polynomial of real variable with algebraic coefficients would determine space-time surface! As proposed already earlier, spacetime surfaces would form commutative and associative algebra with addition, product and functional composition.

One can interpret the products of polynomials as correlates for free many-particle states with interactions described by added interaction polynomial, which can vanish at boundaries of CDs thanks to the vanishing in Minkowski signature of the complexified norm qcqc* appearing in RE(P) or IM(P) caused by the quaternionic non-commutativity. This leads to the same picture as the view about preferred extremals reducing to minimal surfaces near boundaries of CD. Also zero zero energy ontology (ZEO) could emerge naturally from the failure of number field property for for quaternions at light-cone boundaries.

3. The fundamental challenge is to prove that the octonionic polynomials with real coefficients determine associative/quaternionic surfaces as the zero loci of their imaginary/real parts in quaternionic sense. Here the intuition comes from the idea that the octonionic polynomials map from octonionic space O to second octonionic space W. Real and imaginary parts in W are quaternionic/co-quaternionic. These planes correspond to surfaces in O defined by the vanishing of real/imaginary parts, and the natural guess is that they are quaternionic/co-quaternionic, that is associative/co-associative.

The hierarchy of notions involved is well-ordering for 1-D structures, commutativity for complex numbers, and associativity for quaternions. This suggests a generalization of Cauchy-Riemann conditions for complex analytic functions to quaternions and octonions. Cauchy Riemann conditions are linear and constant value manifolds are 1-D and thus well-ordered. Quaternionic polynomials with real coefficients define maps for which the 2-D spaces corresponding to vanishing of real/imaginary parts of the polynomial are complex/co-complex or equivalently commutative/co-commutative. Commutativity is expressed by conditions bilinear in partial derivatives. Octonionic polynomials with real coefficients define maps for which 4-D surfaces for which real/imaginary part are quaternionic/co-quaternionic, or equivalently associative/co-associative. The conditions are now 3-linear.

In fact, all algebras obtained by Cayley-Dickson construction adding imaginary units to octonionic algebra are power associative so that polynomials with real coefficients define an associative and commutative algebra. Hence octonion analyticity and M8-H correspondence could generalize.

4. It turns out that in the generic case associative surfaces are 3-D and are obtained by requiring that one of the coordinates RE(Y)i or IM(Y)i in the decomposition Yi=RE(Y)i +IM(Y)iI4 of the gradient of RE(P)= Y=0 with respect to the complex coordinates zik, k=1,2, of O vanishes that is critical as function of quaternionic components z1k or z2k associated with q1 and q2 in the decomposition o= q1+q2I4, call this component Xi. In the generic case this gives 3-D surface.

In this generic case M8-H duality can map only the 3-surfaces at the boundaries of CD and light-like partonic orbits to H, and only determines the boundary conditions of the dynamics in H determined by the twistor lift of Kähler action. M8-H duality would allow to solve the gauge conditions for SSA (vanishing of infinite number of Noether charges) explicitly.

One can also have criticality. 4-dimensionality can be achieved by posing conditions on the coefficients of the octonionic polynomial P so that the criticality conditions do not reduce the dimension: Xi would have possibly degenerate zero at space-time variety. This can allow 4-D associativity with at most 3 critical components Xi. Space-time surface would be analogous to a polynomial with a multiple root. The criticality of Xi conforms with the general vision about quantum criticality of TGD Universe and provides polynomials with universal dynamics of criticality. A generalization of Thom's catastrophe theory emerges. Criticality should be equivalent to the universal dynamics determined by the twistor lift of Kähler action in H in regions, where Kähler action and volume term decouple and dynamics does not depend on coupling constants.

One obtains two types of space-time surfaces. Critical and associative (co-associative) surfaces can be mapped by M8-H duality to preferred critical extremals for the twistor lift of Kähler action obeying universal dynamics with no dependence on coupling constants and due to the decoupling of Kähler action and volume term: these represent external particles. M8-H duality does not apply to non-associative (non-co-associative) space-time surfaces except at 3-D boundary surfaces. These regions correspond to interaction regions in which Kähler action and volume term couple and coupling constants make themselves visible in the dynamics. M8-H duality determines boundary conditions.

5. Cognitive representations are identified as sets of rational points for algebraic surfaces with "active" points containing fermion. The representations are discussed at both M8- and H level. Rational points would be now associated with 4-D algebraic varieties in 8-D space. General conjectures from algebraic geometry support the vision that these sets are concentrated at lower-dimensional algebraic varieties such as string world sheets and partonic 2-surfaces and their 3-D orbits, which can be also identified as singularities of these surfaces.
6. Some aspects related to homology charge (Kähler magnetic charge) and genus-generation correspondence are discussed. Both are central in the proposed model of elementary particles and it is interesting to see whether the picture is internally consistent and how algebraic surface property affects the situation. Also possible problems related to heff/h=n hierarchy realized in terms of n-fold coverings of space-time surfaces are discussed from this perspective.
In order to get more perspective I add an FB response relating to this.

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 Cayley-Dickson 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.

### Generalization of Riemann zeta to Dedekind zeta and adelic physics?

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

1. Riemann zeta itself could be identifiable as an analog of partition function for a system with energies given by logarithms of prime. In ZEO this function could be regarded as complex square root of thermodynamical partition function in accordance with the interpretation of quantum theory as complex square root of thermodynamics.
2. The zeros of zeta could define the conformal weights for the generators of super-symplectic algebra so that the number of generators would be infinite. The rough idea - certainly not correct as such except at the limit of infinitely large CD - is that corresponding functions would correspond to functions of radial light-like coordinate rM of light-cone boundary (boundary of causal diamond) of form (rM/r0)sn, sn=1/2+iy, sn would be radial conformal weight. Periodic boundary conditions for CD do not allow all possible zeros as conformal weights so that for given CD only finite subset corresponds to generators of supersymplectic algebra. Conformal confinement would hold true in the sense that the sum sn for physical states would be integer. Roots and their conjugates should appear as pairs in physical states.
3. On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the set formed by zeros of zeta consists of primes so that one could regard zeros as one-dimensional quasi-crystal. This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has piy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is motivated by number theoretical universality.
4. I have considered the possibility (see this) that the inverse of the electro-weak U(1) coupling constant for a gauge field assignable to the Kähler form of CP2 corresponds to poles of the fermionic zeta ζF(s)= ζ(s)/ζ(2s) coming from sn/2 (denominator) and pole at s=1 (numerator) zeros of zeta assignable to rational primes. Here one can consider scaling of argument of ζF(s). More general coupling constant evolutions could correspond to ζF(m(s)) where m(s)= (as+b)/(cs+d) is Möbius transformation performed for the argument mapping upper complex plane to itself so that a,b,c,d are real and also rational by number theoretical universality.
Suppose for a moment that more precise formulations of these physics inspired conjectures hold true and even that their generalization for extensions K/Q of rationals holds true. This would solve quite a portion of adelic physics! Not surprisingly, the generalization of zeta function was proposed already by Dedekind (see this).
1. The definition of Dedekind zeta function ζK relies on the product representation and analytic continuation allows to deduce ζK elsewhere. One has a product over prime ideals of K/Q of rationals with the factors 1/(1-p-s) associated with the ordinary primes in Riemann zeta replaced with the factors X(P) =1/(1-NK/Q(P)-s), where P is prime for the integers O(K) of extension and NK/Q(P) is the norm of P in the extension. In the region s>1 where the product converges, ζK is non-vanishing and s=1 is a pole of ζK. The functional identifies of ζ hold true for ζK as well. Riemann hypothesis is generalized for ζK.
2. It is possible to interpret ζK in terms of a physical picture. By the general results (see this) one NK/Q(P)= pr, r>0 integer. One can deduce for r a general expression. This implies that one can arrange in ζK all primes P for which the norm is power or given p in the same group. The prime ideals p of ordinary integers decompose to products of prime ideals P of the extension: one has p= ∏r=1g Prer, where er is so called ramification index. One can say that each factor of ζ decomposes to a product of factors associated with corresponding primes P with norm a power of p. In the language of physics, the particle state represented by p decomposes in an improved resolution to a product of many-particle states consisting of er particles in states Pr, very much like hadron decomposes to quarks.

The norms of NK/Q(Pr) = pdr satisfy the condition ∑r=1g dr er= n. Mathematician would say that the prime ideals of Q modulo p decompose in n-dimensional extension K to products of prime power ideals Prer and that Pr corresponds to a finite field G(p,dr) with algebraic dimension dr. The formula ∑r=1g dr er = n reflects the fact the dimension n of extension is same independent of p even when one has g<n and ramification occurs.

Physicist would say that the number of degrees of freedom is n and is preserved although one has only g<n different particle types with er particles having dr internal degrees of freedom. The factor replacing 1/(1-p-s) for the general prime p is given by ∏r=1g 1/(1-p-erdrs).

3. There are only finite number of ramified primes p having er>1 for some r and they correspond to primes dividing the so called discriminant D of the irreducible polynomial P defining the extension. D mod p obviously vanishes if D is divisible by p. For second order polynomials P=x2+bx+c equals to the familiar D=b2-4c and in this case the two roots indeed co-incide. For quadratic extensions with D= b2-4c>0 the ramified primes divide D.

Remark: Resultant R(P,Q) and discriminant D(P)= R(P,dP/dx) are elegant tools used by number theorists to study extensions of rationals defined by irreducible polynomials. From Wikipedia articles one finds an elegant proof for the facts that R(P,Q) is proportional to the product of differences of the roots of P and Q, and D to the product of squares for the differences of distinct roots. R(P,Q)=0 tells that two polynomials have a common root. D mod p=0 tells that polynomial and its derivative have a common root so that there is a degenerate root modulo p and the prime is indeed ramified. For modulo p reduction of P the vanishing of D(P) mod p follows if D is divisible by p. There exist clearly only a finite number of primes of this kind.

Most primes are unramified. If one has maximum number n of factors in the decomposition and er=1, maximum splitting of p occurs. The factor 1/(1-p-s) is replaced with its n:th power 1/(1-p-s)n. The geometric interpretation is that space-time sheet is replaced with n-fold covering and each sheet gives one factor in the power. It is also possible to have a situation in which no splitting occurs and one as er=1 for one prime Pr=p. The factor is in this case equal to 1/(1-p-ns).

From Wikipedia one learns that for Galois extensions L/K the ratio ζLK is so called Artin L-function of the regular representation (group algebra) of Galois group factorizing in terms of irreps of Gal(L/K) is holomorphic (no poles!) so that ζL must have also the zeros of ζK. This holds in the special case K=Q. Therefore extension of rationals can only bring new zeros but no new poles!
1. This result is quite far reaching if one accepts the hypothesis about super-symplectic conformal weights as zeros of ζK and the conjecture about coupling constant evolution. In the case of ζF,K this means new poles meaning new conformal weights due to increased complexity and a modification of the conjecture for the coupling constant evolution due to new primes in extension. The outcome looks physically sensible.
2. Quadratic field Q(m1/2) serves as example. Quite generally, the factorization of rational primes to the primes of extension corresponds to the factorization of the minimal polynomial for the generating element θ for the integers of extension and one has p= Piei, where ei is ramification index. The norm of p factorizes to the produce of norms of Piei.

Rational prime can either remain prime in which case x2-m does not factorize mod p, split when x2-m factorizes mod P, or ramify when it divides the discriminant of x2-m = 4m. From this it is clear that for unramfied primes the factors in ζ are replaced by either 1/(1-p-s)2 or 1/(1-p-2s)= 1/(1-p-s)(1+p-s). For a finite number of unramified primes one can have something different.

For Gaussian primes with m=-1 one has er=1 for p mod 4=3 and er=2 for p=~mod~4=1. zK therefore decomposes into two factors corresponding to primes p ~mod~4=3 and p mod 4=1. One can extract out Riemann zeta and the remaining factor

p mod 4=3 1/(1-p-s) × ∏p mod 4=1 1/(1+p-s)

should be holomorphic and without poles but having possibly additional zeros at critical line. That ζK should possess also the poles of ζ as poles looks therefore highly non-trivial.

### Are Preferred Extremals Quaternion-Analytic in Some Sense?

A generalization of 2-D conformal invariance to its 4-D variant is strongly suggestive in TGD framework, and leads to the idea that for preferred extremals of action space-time regions have (co-)associative/(co-)quaternionic tangent space or normal space. The notion of M8-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 M4 representable as sub-space of of complexified quaternions M4c, which in turn can be seen as sub-space of complexified octonions M8c.

The 4-D 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 Cauchy-Riemann-Fuerter 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 space-time 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 light-like boundaries of causal diamonds and at light-like orbits of partonic 2-surfaces but that in the interior of space-time 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 Quaternion-Analytic in Some Sense? or the chapter Unified Number Theoretical Vision.

The p-adic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the p-adic ideas are inspired by TGD based view about physics.

p-Adic mass calculations relying on p-adic generalization of thermodynamics and super-symplectic and super-conformal symmetries are summarized. Number theoretical existence constrains lead to highly non-trivial and successful physical predictions. The notion of canonical identification mapping p-adic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from p-adics to reals and vice versa.

A view about p-adicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various p-adic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of p-adic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection.

The existence of p-adic 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 finite-dimensional algebraic extension of p-adic numbers. Only group invariants - typically squares of distances and norms - are mapped by canonical identification from p-adic to real realm and various phases are mapped to themselves as number theoretically universal entities.

Also the understanding of the correspondence between real and p-adic physics at various levels - space-time 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 non-local correspondence between real and p-adic space-time surfaces induced by algebraic continuation from common string world sheets and partonic 2-surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and p-adic surfaces providing automatically finite "cognitive" resolution. he existence p-adic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it.

I will also sum up the role of p-adic 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 space-time 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 p-adic 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 2-surfaces 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.

p-Adic length scale hypothesis states that physically preferred p-adic 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 super-stability associated with period doublings?

Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.

1. Any rational prime p can be decomposes to a product of powers Pki of primes Pi of extension given by p= ∏i Piki, ∑ ki=n. If one has ki≠ 1 for some i, one has ramified prime. Prime p is Galois invariant but ramified prime decomposes to lower-dimensional orbits of Galois group formed by a subset of Piki with the same index ki . One might say that ramified primes are more structured and informative than un-ramified ones. This could mean also representative capacity.
2. Ramification has as its analog criticality leading to the degenerate roots of a polynomial or the lowering of the rank of the matrix defined by the second derivatives of potential function depending on parameters. The graph of potential function in the space defined by its arguments and parameters if n-sheeted singular covering of this space since the potential has several extrema for given parameters. At boundaries of the n-sheeted structure some sheets degenerate and the dimension is reduced locally . Cusp catastrophe with 3-sheets in catastrophe region is standard example about this.

Ramification also brings in mind super-stability of n-cycle for the iteration of functions meaning that the derivative of n:th iterate f(f(...)(x)== fn)(x) vanishes. Superstability occurs for the iteration of function f= ax+bx2 for a=0.

3. I have considered the possibility that that the n-sheeted coverings of the space-time surface are singular in that the sheet co-incide at the ends of space-time surface or at some partonic 2-surfaces. One can also consider the possibility that only some sheets or partonic 2-surfaces co-incide.

The extreme option is that the singularities occur only at the points representing fermions at partonic 2-surfaces. Fermions could in this case correspond to different ramified primes. The graph of w=z1/2 defining 2-fold covering of complex plane with singularity at origin gives an idea about what would be involved. This option looks the most attractive one and conforms with the idea that singularities of the coverings in general correspond to isolated points. It also conforms with the hypothesis that fermions are labelled by p-adic primes and the connection between ramifications and Galois singularities could justify this hypothesis.

4. Category theorists love structural similarities and might ask whether there might be a morphism mapping these singularities of the space-time surfaces as Galois coverings to the ramified primes so that sheets would correspond to primes of extension appearing in the decomposition of prime to primes of extension.

Could the singularities of the covering correspond to the ramification of primes of extension? Could this degeneracy for given extension be coded by a ramified prime? Could quantum criticality of TGD favour ramified primes and singularities at the locations of fermions at partonic 2-surfaces?

Could the fundamental fermions at the partonic surfaces be quite generally localize at the singularities of the covering space serving as markings for them? This also conforms with the assumption that fermions with standard value of Planck constants corresponds to 2-sheeted coverings.

5. What could the ramification for a point of cognitive representation mean algebraically? The covering orbit of point is obtained as orbit of Galois group. For maximal singularity the Galois orbit reduces to single point so that the point is rational. Maximally ramified fermions would be located at rational points of extension. For non-maximal ramifications the number of sheets would be reduced but there would be several of them and one can ask whether only maximally ramified primes are realized. Could this relate at the deeper level to the fact that only rational numbers can be represented in computers exactly.
6. Can one imagine a physical correlate for the singular points of the space-time sheets at the ends of the space-time surface? Quantum criticality as analogy of criticality associated with super-stable cycles in chaos theory could be in question. Could the fusion of the space-time sheets correspond to a phenomenon analogous to Bose-Einstein condensation? Most naturally the condensate would correspond to a fractionization of fermion number allowing to put n fermions to point with same M4 projection? The largest condensate would correspond to a maximal ramification p= Pin.
Why ramified primes would tend to be primes near powers of two or of small prime? The attempt to answer this question forces to ask what it means to be a survivor in number theoretical evolution. One can imagine two kinds of explanations.
1. Some extensions are winners in the number theoretic evolution, and also the ramified primes assignable to them. These extensions would be especially stable against further evolution representing analogs of evolutionary fossils. As proposed earlier, they could also allow exceptionally large cognitive representations that is large number of points of real space-time surface in extension.
2. Certain primes as ramified primes are winners in the sense the further extensions conserve the property of being ramified.
1. The first possibility is that further evolution could preserve these ramified primes and only add new ramified primes. The preferred primes would be like genes, which are conserved during biological evolution. What kind of extensions of existing extension preserve the already existing ramified primes. One could naively think that extension of an extension replaces Pi in the extension for Pi= Qikki so that the ramified primes would remain ramified primes.
2. Surviving ramified primes could be associated with a exceptionally large number of extensions and thus with their Galois groups. In other words, some primes would have strong tendency to ramify. They would be at criticality with respect to ramification. They would be critical in the sense that multiple roots appear.

Can one find any support for this purely TGD inspired conjecture from literature? I am not a number theorist so that I can only go to web and search and try to understand what I found. Web search led to a thesis (see this) studying Galois group with prescribed ramified primes.

The thesis contained the statement that not every finite group can appear as Galois group with prescribed ramification. The second statement was that as the number and size of ramified primes increases more Galois groups are possible for given pre-determined ramified primes. This would conform with the conjecture. The number and size of ramified primes would be a measure for complexity of the system, and both would increase with the size of the system.

3. Of course, both mechanisms could be involved.
Why ramified primes near powers of 2 would be winners? Do they correspond to ramified primes associated with especially many extension and are they conserved in evolution by subsequent extensions of Galois group. But why? This brings in mind the fact that n=2k-cycles becomes super-stable and thus critical at certain critical value of the control parameter. Note also that ramified primes are analogous to prime cycles in iteration. Analogy with the evolution of genome is also strongly suggestive.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

### heff/h=n hypothesis and Galois group

The 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 space-time 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 mono-cellulars 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 Orch-OR as such has little to do with reality!

The natural hypothesis is that heff/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 space-time surfaces. The stronger hypothesis is that heff/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.

1. The basic idea is that Mother Nature is theoretician friendly. As perturbation theory breaks down, the interaction strength expressible as a product of appropriate charges divided by Planck constant, is reduced in the phase transition hbar→ hbareff.
2. In the case of gravitation GMm→ = GMm (h/heff). Equivalence Principle is satisfied if one has hbareff=hbargr = GMm/v0, where v0 is parameter with dimensions of velocity and of the order of some rotation velocity associated with the system. If the masses move with relativistic velocities the interaction strength is proportional to the inner product of four-momenta and therefore to Lorentz boost factors for energies in the rest system of the entire system. In this case one must assume quantization of energies to satisfy the constraint or a compensating reduction of v0. Interactions strength becomes equal to β0= v0/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality.
3. The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z1Z2α→ Z1Z2α (h/ hem) and hbarem=Z1Z2α/&beta0 giving Universal interaction strength. In the case of color interactions the phase transition would lead to the emergence of hadron and it could be that inside hadrons the valence quark have heff/h=n>1. In this case one could consider a generalization in which the product of masses is replaced with the inner product of four-momenta. In this case quantization of energy at either or both ends is required. For astrophysical energies one would have effective energy continuum.
This hypothesis suggests the interpretation of heff/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n1 and n2, then the composite system would be n2-dimensional extension of n1-dimensional extension and have dimension n1× n2. This could be also true for the orders of Galois groups. This would be the case if Galois group of the entire system is free group generated by the G1 and G2. One just takes all products of elements of G1 and G2 and assumes that they commute to get G1× G2. Consider gravitation as example.
1. The order of Galois group should coincide with hbareff/hbar=n= hbargr/hbar= GMm/v0hbar. The transition occurs only if the value of hbargr/hbar is larger than one. One can say that the order of Galois group is proportional the product of masses using as unit Planck mass. Rather large extensions are involved and the number of sheets in the Galois covering is huge.

Note that it is difficult to say how larger Planck constants are actually involved since by gravitational binding the classical gravitational forces are additive and by Equivalence principle same potential is obtained as sum of potentials for splitting of masses into pieces. Also the gravitational Compton length λgr= GM/v0 for m does not depend on m at all so that all particles have same λgr= GM/v0 irrespective of mass (note that v0 is expressed using units with c=1).

The maximally incoherent situation would correspond to ordinary Planck constant and the usual view about gravitational interaction between particles. The extreme quantum coherence would mean that both M and m behave as single quantum unit. In many-sheeted space-time this could be understood in terms of a picture based on flux tubes. The interpretation for the degree of coherence is in terms of flux tube connections mediating gravitational flux.

2. hgr/h would be order of Galois group, and there is a temptation to associated with the product of masses the product n=n1n2 of the orders ni of Galois groups associated masses M and m. The order of Galois group for both masses would have as unit mP01/2, β0=v0/c, rather than Planck mass mP. For instance, the reduction of the Galois group of entire system to a product of Galois groups of parts would occur if Galois groups for M and m are cyclic groups with orders with have no common prime factors but not generally.

The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m<mP01/2. Planck mass is about 1.3 × 1019 proton masses and corresponds to a blob of water with size scale 10-4 meters - size scale of a large neuron so that only above these scale gravitational quantum coherence would be possible. For v0<1 it would seem that even in the case of large neurons one must have more than one neurons. Maybe pyramidal neurons could satisfy the mass constraint and would represent higher level of conscious as compared to other neurons and cells. The giant neurons discovered by the group led by Christof Koch in the brain of of mouse having axonal connections distributed over the entire brain might fulfil the constraint (see this).

3. It is difficult to avoid the idea that macroscopic quantum gravitational coherence for multicellular objects with mass at least that for the largest neurons could be involved with biology. Multicellular systems can have mass above this threshold for some critical cell number. This might explain the dramatic evolutionary step distinguishing between prokaryotes (mono-cellulars consisting of Archaea and bacteria including also cellular organelles and cells with sub-critical size) and eukaryotes (multi-cellulars).
4. I have proposed an explanation of the fountain effect appearing in super-fluidity and apparently defying the law of gravity. In this case m was assumed to be the mass of 4He atom in case of super-fluidity to explain fountain effect. The above arguments however allow to ask whether anything changes if one allows the blobs of superfluid to have masses coming as a multiple of mP01/2. One could check whether fountain effect is possible for super-fluid volumes with mass below mP01/2.
What about hem? In the case of super-conductivity the interpretation of hem/h as product of orders of Galois groups would allow to estimate the number N= Q/2e of Cooper pairs of a minimal blob of super-conducting matter from the condition that the order of its Galois group is larger than integer. The number N=Q/2e is such that one has 2N(α/β0)1/2=n. The condition is satisfied if one has α/β0=q2, with q=k/2l such that N is divisible by l. The number of Cooper pairs would be quantized as multiples of l. What is clear that em interaction would correspond to a lower level of cognitive consciousness and that the step to gravitation dominated cognition would be huge if the dark gravitational interaction with size of astrophysical systems is involved. Many-sheeted space-time allows this in principle.

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 multi-cellulars 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 finite-dimensional extension of p-adic 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 space-time and Hilbert space level. WCW decomposes to sectors characterized by Galois groups G3 of extensions associated with the 3-surfaces at the ends of space-time surface at boundaries of causal diamond (CD) and G4 characterizing the space-time surface itself. G3 (G4) acts on the discretization and induces a covering structure of the 3-surface (space-time surface). If the state function reduction to the opposite boundary of CD involves localization into a sector with fixed G3, evolution is indeed mapped to a sequence of G3s.

Also the cognitive representation defined by the intersection of real and p-adic 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 2-surfaces connected by fermionic strings define a tensor network, which also evolves since the number of fermions can change.

The points of space-time surface invariant under non-trivial 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 p-adic 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 p-adic primes satisfying the p-adic 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. heff/h=n has natural interpretation as the order of Galois group of extension. The generalization of hbargr= GMm/v0=hbareff hypothesis to other interactions is discussed in terms of number theoretic evolution as increase of G3, 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?.

The 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.

1. Both hierarchy of Planck constant and preferred p-adic primes are now understood number theoretically.
2. The realization of number theoretical universality (NTU) for functional integral seems like a formidable problem but the special features of functional integral makes the challenge tractable. NTU of functional integral is indeed suggested by the need to describe also cognition quantally.
3. Strong form of holography (SH) is now understood. 2-D surfaces (string world sheets and possibly partonic 2-surfaces) are not quite enough: also number theoretic discretization of space-time surface is required. This allows to understand the distinction between imagination in terms of p-adic space-time surfaces and reality in terms of real space-time surface. The number of imaginations is much larger than realities (p-adic pseudo-constants).
4. The localization of spinor modes to string world sheets can be understand only as effective: this resolves several interpretational problems. These spinors give all information needed to construct 4-D spinor modes. Also 2-D modified Dirac action and area action are enough to construct scattering amplitudes once number theoretic discretization of space-time surface responsible for dark matter is given. This means enormous simplification of the theory.
Galois group of number theoretic discretization and hierarchy of Planck constants

Simple arguments lead to the identification of heff/h=n as a factor of the order of Galois group of extension of rationals.

1. The strongest form of NTU would require that the allowed points of imbedding space belonging an extension of rationals are mapped as such to corresponding extensions of p-adic number fields (no canonical identification). At imbedding space level this correspondence would be extremely discontinuous. The "spines" of space-time surfaces would however contain only a subset of points of extension, and a natural resolution length scale could emerge and prevent the fluctuation. This could be also seen as a reason for why space-times surfaces must be 4-D. The fact that the curve xn+yn=zn has no rational points for n>2, raises the hope that the resolution scale could emerge spontaneously.
2. The notion of monadic geometry - discussed in detail here would realize this idea. Define first a number theoretic discretization of imbedding space in terms of points, whose coordinates in group theoretically preferred coordinate system belong to the extension of rationals considered. One can say that these algebraic points are in the intersection of reality and various p-adicities. Overlapping open sets assigned with this discretization define in the real sector a covering by open sets. In p-adic sector compact-open-topology allows to assign with each point 8th Cartesian power of algebraic extension of p-adic numbers. These compact open sets define analogs for the monads of Leibniz and p-adic variants of field equations make sense inside them.

The monadic manifold structure of H is induced to space-time surfaces containing discrete subset of points in the algebraic discretization with field equations defining a continuation to space-time surface in given number field, and unique only in finite measurement resolution. This approach would resolve the tension between continuity and symmetries in p-adic--real correspondence: isometry groups would be replaced by their sub-groups with parameters in extension of rationals considered and acting in the intersection of reality and p-adicities.

The Galois group of extension acts non-trivially on the "spines" of space-time surfaces. Hence the number theoretical symmetries act as physical symmetries and define the orbit of given space-time surface as a kind of covering space. The coverings assigned to the hierarchy of Planck constants would naturally correspond to Galois coverings and dark matter would represent number theoretical physics.

This would give rise to a kind of algebraic hierarchy of adelic 4-surfaces identifiable as evolutionary hierarchy: the higher the dimension of the extension, the higher the evolutionary level.

3. But how does quantum criticality relate to number theory and adelic physics? heff/h=n has been identified as the number of sheets of space-time surface identified as a covering space of some kind. Number theoretic discretization defining the "spine for a monadic space-time surface defines also a covering space with Galois group for an extension of rationals acting as covering group. Could n be identifiable as the order for a sub-group of Galois group?

If this is the case, the proposed rule for heff changing phase transitions stating that the reduction of n occurs to its factor would translate to spontaneous symmetry breaking for Galois group and spontaneous - symmetry breakings indeed accompany phase transitions.

Ramified primes as referred primes for a given extension

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.

1. Stating it very roughly (I hope that mathematicians tolerate this sloppy language of physicist): as one goes from number field K, say rationals Q, to its algebraic extension L, the original prime ideals in the so called integral closure over integers of K decompose to products of prime ideals of L (prime ideal is a more rigorous manner to express primeness). Note that the general ideal is analog of integer.

Integral closure for integers of number field K is defined as the set of elements of K, which are roots of some monic polynomial with coefficients, which are integers of K having the form xn+ an-1xn-1 +...+a0. The integral closures of both K and L are considered. For instance, integral closure of algebraic extension of K over K is the extension itself. The integral closure of complex numbers over ordinary integers is the set of algebraic numbers.

Prime ideals of K can be decomposed to products of prime ideals of L: P= ∏ Piei, where ei is the ramification index. If ei>1 is true for some i, ramification occurs. Pi:s in question are like co-inciding roots of polynomial, which for in thermodynamics and Thom's catastrophe theory corresponds to criticality. Ramification could therefore be a natural aspect of quantum criticality and ramified primes P are good candidates for preferred primes for a given extension of rationals. Note that the ramification make sense also for extensions of given extension of rationals.

2. A physical analogy for the decomposition of ideals to ideals of extension is provided by decomposition of hadrons to valence quarks. Elementary particles becomes composite of more elementary particles in the extension. The decomposition to these more elementary primes is of form P= ∏ Pie(i), the physical analog would be the number of elementary particles of type i in the state (see this). Unramified prime P would be analogous a state with e fermions. Maximally ramified prime would be analogous to Bose-Einstein condensate of e bosons. General ramified prime would be analogous to an e-particle state containing both fermions and condensed bosons. This is of course just a formal analogy.
3. There are two further basic notions related to ramification and characterizing it. Relative discriminant is the ideal divided by all ramified ideals in K (integer of K having no ramified prime factors) and relative different for P is the ideal of L divided by all ramified Pi:s (product of prime factors of P in L). These ideals represent the analogs of product of preferred primes P of K and primes Pi of L dividing them. These two integers ideals would characterize the ramification.
Ramified primes for preferred extensions as preferred p-adic primes?

In TGD framework the extensions of rationals (see this) and p-adic 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 2-surfaces 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 p-adic 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 p-adic and adelic physics and could it explain preferred primes?

1. Ramified p-adic prime P=Pie would be replaced with its e:th root Pi in p-adicization. Same would apply to general ramified primes. Each un-ramified prime of K is replaced with e=K:L primes of L and ramified primes P with #{Pi}<e primes of L: the increase of algebraic dimension is smaller. An interesting question relates to p-adic length scale. What happens to p-adic length scales. Is p-adic prime effectively replaced with e:th root of p-adic prime: Lp∝ p1/2L1 → p1/2eL1? The only physical option is that the p-adic temperature for P would be scaled down Tp=1/n → 1/ne for its e:th root (for fermions serving as fundamental particles in TGD one actually has Tp=1). Could the lower temperature state be more stable and select the preferred primes as maximimally ramified ones? What about general ramified primes?
2. This need not be the whole story. Some algebraic extensions would be more favored than others and p-adic view about realizable imaginations could be involved. p-Adic pseudo constants are expected to allow p-adic continuations of string world sheets and partonic 2-surfaces to 4-D preferred extremals with number theoretic discretization. For real continuations the situation is more difficult. For preferred extensions - and therefore for corresponding ramified primes - the number of real continuations - realizable imaginations - would be especially large.

The challenge would be to understand why primes near powers of 2 and possibly also of other small primes would be favored. Why for them the number of realizable imaginations would be especially large so that they would be winners in number theoretical fight for survival?

NTU for functional integral

Number theoretical vision relies on NTU. In fermionic sector NTU is necessary: one cannot speak about real and p-adic fermions as separate entities and fermionic anti-commutation relations are indeed number theoretically universal.

What about NTU in case of functional integral? There are two opposite views.

1. One can define p-adic variants of field equations without difficulties if preferred extremals are minimal surface extremals of Kähler action so that coupling constants do not appear in the solutions. If the extremal property is determined solely by the analyticity properties as it is for various conjectures, it makes sense independent of number field. Therefore there would be no need to continue the functional integral to p-adic sectors. This in accordance with the philosophy that thought cannot be put in scale. This would be also the option favored by pragmatist.
2. Consciousness theorist might argue that also cognition and imagination allow quantum description. The supersymmetry NTU should apply also to functional integral over WCW (more precisely, its sector defined by CD) involved with the definition of scattering amplitudes.
The general vision involves some crucial observations.
1. Only the expressions for the scatterings amplitudes should should satisfy NTU. This does not require that the functional integral satisfies NTU.
2. Since the Gaussian and metric determinants cancel in WCW Kähler metric the contributions form maxima are proportional to action exponentials exp(Sk) divided by the ∑k exp(Sk). Loops vanish by quantum criticality.
3. Scattering amplitudes can be defined as sums over the contributions from the maxima, which would have also stationary phase by the double extremal property made possible by the complex value of αK. These contributions are normalized by the vacuum amplitude.

It is enough to require NTU for Xi=exp(Si)/∑k exp(Sk). This requires that Sk-Sl has form q1+q2 iπ + q3log(n). The condition brings in mind homology theory without boundary operation defined by the difference Sk-Sl. NTU for both Sk and exp(Sk) would only values of general form Sk=q1+q2 iπ + q3log(n) for Sk and this looks quite too strong a condition.

4. If it is possible to express the 4-D exponentials as single 2-D exponential associated with union of string world sheets, vacuum functional disappears completely from consideration! There is only a sum over discretization with the same effective action and one obtains purely combinatorial expression.

See the chapter Unified Number Theoretic Vision or the article p-Adicization and adelic physics.