- Both hierarchy of Planck constant and preferred p-adic primes are now understood number theoretically.

- 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.

- 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).

- 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 h_{eff}/h=n as a factor of the order of Galois group
of extension of rationals.

- 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 x
^{n}+y^{n}=z^{n}has no rational points for n>2, raises the hope that the resolution scale could emerge spontaneously.

- 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 8
^{th}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.

- But how does quantum criticality relate to number theory and adelic physics? h
_{eff}/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 h

_{eff}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.

- 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 x

^{n}+ a_{n-1}x^{n-1}+...+a_{0}. 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= ∏ P

_{i}^{ei}, where e_{i}is the ramification index. If e_{i}>1 is true for some i,*ramification*occurs. P_{i}: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.

- 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= ∏ P
_{i}^{e(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.

- 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 P_{i}:s (product of prime factors of P in L). These ideals represent the analogs of product of preferred primes P of K and primes P_{i}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?

- Ramified p-adic prime P=P
_{i}^{e}would be replaced with its e:th root P_{i}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 #{P_{i}}<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: L_{p}∝ p^{1/2}L_{1}→ p^{1/2e}L_{1}? The only physical option is that the p-adic temperature for P would be scaled down T_{p}=1/n → 1/ne for its e:th root (for fermions serving as fundamental particles in TGD one actually has T_{p}=1). Could the lower temperature state be more stable and select the preferred primes as maximimally ramified ones? What about general ramified primes?

- 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.

- 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.

- 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.

- Only the expressions for the scatterings amplitudes should should satisfy NTU. This does not require that the functional integral satisfies NTU.

- Since the Gaussian and metric determinants cancel in WCW Kähler metric the contributions form maxima are proportional to action exponentials exp(S
_{k}) divided by the ∑_{k}exp(S_{k}). Loops vanish by quantum criticality.

- 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 α
_{K}. These contributions are normalized by the vacuum amplitude.

It is enough to require NTU for X

_{i}=exp(S_{i})/∑_{k}exp(S_{k}). This requires that S_{k}-S_{l}has form q_{1}+q_{2}iπ + q_{3}log(n). The condition brings in mind homology theory without boundary operation defined by the difference S_{k}-S_{l}. NTU for both S_{k}and exp(S_{k}) would only values of general form S_{k}=q_{1}+q_{2}iπ + q_{3}log(n) for S_{k}and this looks quite too strong a condition.

- 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.

For a summary of earlier postings see Latest progress in TGD.

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