About the generalization of the holography = holomorphy ansatz to general analytic functions

The general ansatz works also for analytic functions with poles since (f₁=0,f₂=0) implies that the poles do not belong to the space-time surface. What is required is that the roots are not essential singularities. For rational functions R_i=P_i/Q_i the vanishing conditions reduce to those for the polynomials P_i.

The generalization Rieman zeta to polyzeta S_n(s₁,...,s_n) is s function of n complex variables (see this) and satisfies identities analogous to those satisfied by Riemann zeta. This generalization is extremely interesting from the point of view of physics of chaotic and quantum critical systems. Polyzeta S₄ with four complex arguments would define as its roots a 6-D analog of the twistor space of the space-time surface expected to have an infinite number of 6-D roots having interpretation as a generalization of zeros of Riemann zeta.

One could have f₁=S₄ so that its roots would correspond to 6-D zeros of polyzeta S₄(s₁,...,s₄) defining the counterparts of twistor surfaces! f₂=0 could define a map from the M⁴ twistor sphere S²₁ to CP₂ twistor sphere S²₂ characterized by a winding number or vice versa.

A further extremely nice feature is that the space-time surfaces form a number field in the sense that one can sum, multiply and divide the members of f_i and g_i of (f₁,f₂) and (g₁,g₂) elementwise. Also functional composition is possible. One could say that the space-time surface is a number. One can also consider polynomials and polynomials with prime order behave like multiplicative primes. It is also possible to identify prime polynomials with respect to functional composition (see this).

See the article TGD as it is towards end of 2024: part I or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Are supernovae induced by the transformation of M89 nuclei at the stellar surface to ordinary nuclei?

Supernovae as explosions of M₈₉ magnetic bubbles?

Could the explosions of the M₈₉ magnetic bubbles proposed to rise to the planets induce supernovae? The following vision suggests itself.

1. The flux tubes as M₈₉ super-nuclei split to ordinary M₁₀₇ nuclei and produce ordinary nuclear matter and liberate energy. This transition would give an additional contribution to the nuclear matter outside stars compensating for the missing contribution due to the missing ordinary nuclear matter inside stars.
2. The decay of giant M₈₉ nuclei defined by the monopole flux tubes would also create nuclei heavier than Fe, which are not produced in the stellar cores.
3. The pressure pulse created in this way leads to the formation of supernovae and blackhole-like objects? Various giant stars would be the outcome of these kinds of explosions of the M₈₉ surface layer?

One can check whether this hypothesis might make sense in the case of supernovae. I attach here a piece of text from the Wikipedia article about supernovae (see this) almost as such.

1. A supernova occurs during the last evolutionary stages of a massive star, or when a white dwarf is triggered into runaway nuclear fusion. The original object, progenitor, either collapses to a neutron star or black hole, or is completely destroyed to form a diffuse nebula. The peak optical luminosity of a supernova can be comparable to that of an entire galaxy before fading over several weeks or months.
2. Theoretical studies indicate that most supernovae are triggered by one of two basic mechanisms: the sudden re-ignition of nuclear fusion in a white dwarf, or the sudden gravitational collapse of a massive star's core.
3. In the re-ignition of a white dwarf, the object's temperature is raised enough to trigger runaway nuclear fusion, completely disrupting the star. Possible causes are an accumulation of material from a binary companion through accretion, or by a stellar merger.
4. In the case of a massive star's sudden implosion, the core of a massive star will undergo sudden collapse once it is unable to produce sufficient energy from fusion to counteract the star's own gravity, which must happen once the star begins fusing iron, but may happen during an earlier stage of metal fusion.
5. Supernovae can expel several solar masses of material at speeds up to several percent of the speed of light. This drives an expanding shock wave into the surrounding interstellar medium, sweeping up an expanding shell of gas and dust observed as a supernova remnant. Supernovae are a major source of elements in the interstellar medium from oxygen to rubidium. The expanding shock waves of supernovae can trigger the formation of new stars. Supernovae are a major source of cosmic rays. They might also produce gravitational waves.

These facts suggest that both in the case of white dwarfs and massive stars, the transformation of M₈₉ nuclei to ordinary nuclei triggers the supernova by creating a powerful pressure pulse towards the core of the star.

In the case of a supernova, the mass thrown out is measured using solar mass M_Sun as a unit. For the explosions producing planets, the mass M_E of the Earth is the natural mass unit. Can one understand this?

1. In the case of the Sun The magnetic bubble consists of M₈₉ monopole flux tubes forming a mass of about .005M_Sun. The baryons produced in the transition make mass of about 3ME at most and would compensate for the missing nuclear mass inside the star. A good guess is that the model for the solar M₈₉ bubble generalizes as such so that the fraction of M₈₉ mass scales like (R_star/R_Sun)².
2. For blue giants (see this ), the masses are in the range 10 -300 M_Sun and the radii vary in the range 10 -100 R_E as the table of the Wikipedia article shows. The amount of ordinary baryons produced would be in the range 10²-10⁴M_E at most and considerably smaller than M_Sun∼ 10⁶M_E.
3. In accordance with the expectations, the explosion should also throw out a considerable amount of ordinary nuclear matter. The huge inward directed pressure pulse produced by the transformation of the M₈₉ layer to M₁₀₇ nuclear matter would produce as a reaction a strong inward pulse and this in turn would induce an outward pulse throwing the ordinary nuclear matter out.
4. In the case of white dwarf the inward directed pressure pulse could heat the core and re-ignite a runaway nuclear fusion inducing a total disruption of the white dwarf. In the case of a massive star this could induce a gravitational collapse of the core leading to a blackhole-like object or a neutron star.

To sum up, the TGD based model would solve the problem due to the missing nuclear mass and provide a missing link to the model of supernova. The decay of the giant M₈₉ nuclei to ordinary nuclei would also explain the origin of the nuclei heavier than Fe.

See the article Some solar mysteries or the chapter with the same title.

A mechanism of photosynthesis which does not involve biomolecules

Standard biology teaches us that photosynthesis is needed to produce oxygen, which is the basic prerequisite of life. Besides complex biological apparatus this requires photons, which provide the needed energy. At the bottom of the ocean there is very dark and this might form a bottleneck for the evolution of life. Now it has been found that at the bottom of ocean mineral deposits known as polymetallic nodules can generate oxygen in absence of photons (see this). They contain combinations of cobalt, copper, lithium, and manganese and the size of the nodule can be that of a human hand. The initiation of electrolysis splitting water to hydrogen and oxygen needs only 1.5 eV voltage in seawater. This means that one has a battery. It was found that the nodules involve voltage as high as .95 eV.

The nodules could make possible electrolysis and splitting of water. They could make it possible to overcome the hen and egg problem due the fact that a complex biomolecular apparatus is needed for photosynthesis but this apparatus cannot exist in primordial biology.

In the TGD Universe, multicellular life would have evolved in underground oceans and bursted to the surface in the Cambrian explosion for about 450 million years ago (see for instance this), which in the TGD Universe was caused by the expansion of the Earth radius by a factor 2 in a rather short period of time. TGD indeed predicts that the cosmic expansion of astrophysical objects occurs as short bursts. This explains why the astrophysical objects comove in expansion but do not expand themselves .

A heavy objection against this vision is that there are no photons in underground oceans so that photosynthesis is not possible. I have proposed that the light arriving as dark photons - ordinary photons but with a large value of effective Planck constant h_eff - from the Earth's core (the temperature is nearly the same as in the solar corona) could have provided the metabolic energy. Also solar photons arriving as dark photons along monopole flux tubes could have provided the energy.

It seems that also the polymetallic nodules could generate photons and make possible the splitting of water. What could be the mechanism making this possible? It must be added that also electrolysis, thought to represent ancient physics, is not a well-understood phenomenon. Remarkably, "cold fusion" was discovered in electrolytes (for the TGD view see this and this) . The voltages used in electrolysis are in eV range and in atomic physics length scales they correspond to ridiculously weak electric fields. How can they cause the ionization essential for electrolysis?

1. In the Pollack effect (see this), the irradiation of water in the presence of the gel phase generates a voltage, and therefore produces a battery. This battery also makes possible electrolysis and the splitting of water producing oxygen. Pollack effect is not understood in the framework of standard chemistry.
2. The TGD explanation is that in the Pollack effect one fourth of the protons of water are transformed to dark photons and kicked to monopole flux tubes. This creates a negatively charged region called exclusion zone (EZ). This would generate a charge separation giving rise to the voltage. Photons would provide the needed energy to transform ordinary protons to dark protons with a larger value of h_eff and therefore larger energy.

   It has become gradually clear that what matters is energy. Therefore the Pollack effect can be realized in several ways. In particular, the formation of molecules as bound states of atoms can provide the needed energy: no photons would be needed (see this and this ).

   In particular, the reverse Pollack effect, that is dropping of dark protons from the monopole flux tubes back to ordinary protons, is also possible and would liberate ordinary photons needed in the splitting of water. This could also provide the photonic energy needed in photosynthesis and could provide a temporary storage of metabolic energy needed in photosynthesis and in the storage of energy to ATP (see this).

3. If this can happen in the nodules, the photosynthesis could have evolved in underground oceans via the fusion of atoms to molecules and completely without external light source.

See for instance the article Expanding Earth Hypothesis and Pre-Cambrian Earth or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Some solar mysteries

This article was inspired by the article "Is the Sun a Black Hole?" by Nassim Haramein. The article describes a collection of various anomalies related to the physics of the Sun, which I have also considered from the TGD point of view. The most important anomalies are the gamma ray anomalies and the missing nuclear matter of about 1500 Earth masses. The idea that the Sun could contain a blackhole led in the TGD framework to a refinement of the earlier model for blackhole-like objects (BHs) as maximally dense flux tube spaghettis predicting also their mass spectrum in terms of Mersenne primes and their Gaussian counterparts.

It however turned out that the TGD based model for the missing nuclear matter assigns the gamma ray anomalies to a magnetic bubble as a layer covering the surface of the Sun and consisting of closed monopole flux tube loops running in North-South direction and carrying M₈₉ nucleons with a mass which is 512 times the mass of the ordinary nucleon. This structure could be seen as a 2-D surface variant of the TGD counterpart of blackhole and under very natural assumptions its mass is the missing 1500 Earth masses of ordinary nuclear matter. This model conforms with the earlier model of the sunspot activity related to the reversal of the solar magnetic field. It also explains the gamma ray anomaly below 35 GeV.

A possible explanation for the TeV anomaly is in terms of M₇₉ nuclei generated in the TGD counterpart for the formation of quark gluon plasma, which in the TGD Universe would generate M₈₉ hadrons from M₁₀₇ hadrons. Now M₇₉ nuclei would be generated from M₈₉ hadrons in a process analogous to high energy nuclear collision, which would correspond to the collision of the M₈₉ flux tubes, whose distance would be larger than 2 Compton lengths of M₈₉ nucleons.

The model leads also to a proposal for the generation of the inner planets and Mars via explosion of the outer layer of the Sun consisting of M_k nucleons caused by the transformation of M_k nucleons to M₁₀₇ nucleons. M₈₉ would give the inner planets and cores of the outer planets, which would have got their gas envelopes by gravitational condensation.

See the article Some solar mysteries or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Galaxy without stars

Galaxy without stars and containing only hydrogen gas is the newest strange finding of astronomers (see this). The proposed explanation is that the galaxy-like structure is so young that the formation of stars has not yet begun.

The hydrogen galaxy might be also seen as a support for the TGD based view of the formation of galaxies and stars. The basic objects would be cosmic strings (actually 4-D objects as surfaces in M^4xCP_2 having 2-D M^4 projection) dominating the primordial cosmology. Cosmic strings would carry energy as analog of dark energy and would give rise to the TGD counterpart of galactic dark matter predicting the flat velocity spectrum of distance stars around the galaxy. Cosmic strings are unstable against thickening producing flux tube tangles. The reduction of string tension in the thickening liberates energy giving rise to the visible galactic matter, in particular stars. This process would be the TGD counterpart of inflation and produce galaxies and stars. Quasars would be formed first.

One can however consider a situation in which there is only hydrogen gas but no cosmic strings. If the hydrogen "galaxy" has this interpretation, the standard view of the formation of galaxies as gravitational condensation could be wrong. Galaxy formation would proceed from short to long length scales rather than vice versa.

See the article About the recent TGD based view concerning cosmology and astrophysics or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this. .

New understanding about the energetics of muscle contraction

The FB post of Robert Stonjek told about a popular article in Phys Org (see this) about the modelling of unexpected findings related to muscle contraction (see the Nature article). The article is very interesting from the point of view of TGD inspired quantum biology (see for instance this).

Muscle contraction requires energy. From the article one learns that the contraction is not actually well-understood. The interesting finding is that the rate of muscle contraction correlates with the rate of water flow through the muscle. As if the water flow would provide the energy needed by the contraction. How? This is not actually well-understood. This is only one example of the many failures of naive reductionism in recent biology.

TGD suggests a very general new physics mechanism for how a biosystem can gain metabolic energy.

1. One can start from biocatalysis, whose extremely rapid rate is a complete mystery in the framework of standard biochemistry. The energy wall which reactants must overcome makes the reactions extremely slow. A general mechanism of energy liberation allowing us to get over the wall, should exist. The reactants should also find each other in the molecular crowd.
2. The first problem is that one does not understand how reactants find each other. The magnetic monopole flux tubes, carrying phases of ordinary matter with effective Planck constant h_eff>h behaving like dark matter, make the living system a fractal network with molecules, cells, etc at the nodes. The U-shaped flux tubes acting as tentacles allow the reactant molecules to find each other: a resonance occur when the U-shaped flux tubes touching each other have same magnetic value of magnetic field and same thickness, a cyclotron resonance occurs, they reconnect to form a pair of flux tubes connecting the molecules. Molecules have found each other.
3. At the next step h_eff decreases and the connecting flux tube pair shortens. This liberates energy since the length of the flux tube pair increases with h_eff. Quite generally the increase of h_eff requires energy feed, and in biosystems this means metabolic energy feed. The liberated energy makes it possible to overcome the energy barrier making the reaction slow.
4. This mechanism applied to the monopole flux tubes associated with water clusters and bioactive molecules is a basic mechanism of the immune system and allows the organism to find bioactive molecules which do not belong to the system normally. Cyclotron frequency spectrum of the biomolecule serves as the fingerprint of the molecule. This is also the basic mechanism of water memory.

In muscle contraction, the flow of water involving these contracting flux tubes would liberate the energy needed by contraction and the process would be very fast. The water flowing through the muscle is a fuel carrying energy at its monopole flux tubs with h_eff>h. The energy is used and water becomes ordinary. The rate of the flow correlates with the rate of contraction and with the rate of the needed metabolic energy feed.

The interesting question is whether this mechanism reduces to the usual ATP-ADP mechanism in some sense or whether ATP-ADP mechanism is a special case of this mechanism

See for instance the article TGD view about water memory and the notion of morphogenetic field.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Do local Galois group and ramified primes make sense as general coordinate invariant notions?

In TGD, space-time surface can be regarded as a 4-D root for a pair P₁,P₂ of polynomials of generalized complex coordinates of H=M⁴× CP₂ (of of the coordinates is generalized complex coordinates varying along light-like curves). Each pair gives rise to a 6-D surface proposed to be identifiable as analog of twistor space and their intersection defines space-time surface as a common base of these twistor spaces as S².

One can also think of the space-time surface X⁴ as a base space of a twistor surface X⁶ in the product T(M⁴)× T(CP₂) of the twistor spaces of M⁴ and H. One can represent X⁴ as a section of this twistor space as a root of a single polynomial P. The number roots of a polynomial does not depend on the point chosen. One considers polynomials with rational coefficients but also analytic functions can be considered and general coordinate invariance would suggest that they should be allowed.

Could one generalize the notion of the Galois group so that one could speak of a Galois group acting on 4-surface X⁴ permuting its sheets as roots of the polynomial? Could one speak of a local Galois group with local groups Gal(x) assigned with each point x of the space-time surface. Could one have a general coordinate invariant definition for the generalized Galois group, perhaps working even when one considers analytic functions f₁,f₂ instead of polynomials. Also a general coordinate invariant definition of ramified primes identifiable as p-adic primes defining the p-adic length scales would be desirable.

The required view of the Galois group would be nearer to the original view of Galois group as permutations of the roots of a polynomial whereas the standard definition identifies it as a group acting as an automorphism in the extension of the base number field induced by the roots of the polynomial and leaving the base number field. The local variant of the ordinary Galois group would be defined for the points of X⁴ algebraic values of X⁴ coordinates and would be trivial for most points. Something different is needed.

In the TGD framework, a geometric realization for the action of the Galois group permutings space-time regions as roots of a polynomial equation is natural and the localization of the Galois group is natural. I have earlier considered a realization as a discrete subgroup of a braid group which is a covering group of the permutation group. The twistor approach leads to an elegant realization as discrete permutations of the roots of the polynomial as values of the S² complex coordinate of the analog of twistor bundle realized as a 6-surface in the product of twistor spaces of M⁴ and CP₂. This realization makes sense also for the P₁,P₂ option.

The natural idea is that the Galois group acts as conformal transformations or even isometries of the twistor sphere S². The isometry option leads to a connection with the McKay correspondence. Only the Galois groups appearing in the hierarchy finite subgroups of rotation groups appearing in the hierarchy of Jones inclusions of hyper-finite factors of type II₁ are realized as isometries and only the isometry group of the cube is a full permutation group. For the conformal transformations the situation is different. In any case, Galois groups representable as isometries of S² are expected to be physically very special so that the earlier intuitions seems to be correct.

General coordinate invariance allows any coordinates for the space-time surface X⁴ as the base space of X⁶ as the analog of twistor bundle and the complex coordinate z of S² is determined apart from linear holomorphies z → az+b, which do not affect the ramimifed primes as factors of the discriminant defined by the product of the root differences.

See the article TGD as it is towards end of 2024: part I or a chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The mystery of the magnetic field of the Moon

In Bighthink there was an interesting story telling about the strange finding related to the faces of the Moon. The finding is that the faces of the Moon are very different. Moon and Earth are in rotational resonance meaning that the we see always the same face of the Moon. In 1959 the first spacecraft flew around the Moon and it was found that the two sides of the Moon are very different.

The near side is heavily cratered and the lighter areas are in general more cratered that the dark areas known as maria. Craters have a fractal structure: craters within craters. Dark areas have different decomposition. At the far side there are relatively few dark maria and the dark side is thoroughly cratered and "rays" appear to radiate out from them.

The "obvious" explanation for the difference between the two sides is that there is a massive bombardment by heavy towards the far side whereas Earth has shielded the near side. This explanation fails quantitatively: the number of collisions at the near side should be only 1 per cent smaller at the far side. The far side is about 30 per cent more heavily cratered than the near side. There is no explanation for the size and abundance difference of the maria.

The article discusses the explanation in terms of Theia hypothesis stating that Moon was formed as a debris resulting from a collision of Mars size planet with Earth. If the Earth was very hot, certain elements would have been depleted from the surface of the Moon and chemical gradients would have changed its chemical decomposition. The very strong tidal forces when the Moon and Earth were near to each other would have led to a tidal locking. If the near side has thinner crust, Maria could be understood as resulting from molten lava flows into great basins and lowlands of the near side. If the maria solidified much later than the highlands one can understand why the number of craters is much lower. The impact did not leave any scars. The hot Earth near the Moon also explain the difference in crustal thickness.

TGD suggests a different explanation consistent with the Theia hypothesis. TGD predicts that cosmic expansion consists of a sequence of rapid expansions. This explains why the astrophysical objects participate in cosmic expansion but do not seem to expand themselves. The prediction is that astrophysical objects have experienced expansions. The latest expansion would have occurred .5 billion years ago and increased the radius of Earth by a factor 2. These epansion can be also explosions throwing away a layer of matter. Sun would created planets in this kind of explosions by the gravitational condensation of the resulting spherical layers to form the planet. Also Moon could have emerged in an explosion of Earth throwing out a thin expanding spherical layer. This would explains why the composition of Moon is similar to that of Earth.

The hypothesis resembles the Theia hypothesis. The hypothesis however suggests that the Moon should consist of a material originating from both Theia and Earth. The compositions of Earth and Moon are however similar. Why Theia and Earth would have had similar compositions?

This spherical layer was unstable against gravitational condensation to form the Moon. If the condensation was such that there was no radial mixing, the layer's inner side remained towards the Earth. This together with the tidal locking could allow to understand the differences between the near and far sides of the Moon. The chemical composition of the near side would correspond to that in the Earth's interior at certain depth h. One can estimate the thickness h of the layer as h= R_M^3/R_E² ≈ R_E/48 from R_M≈ R_E/4. This gives h≈ 130 km. The temperature of the recent Earth at this depth is around 1000 K (see this). At the time of the formation of Moon, the temperature could have been considerably higher, and it could have been in molten magma state.

Orbital locking would rely on the same mechanism as in Theia model. The half-molten state would have favored the development of the locking. The far side would represent the very early Earth affected by the meteoric bombardment or some other mechanism creating the craters.

Another mysterious observation is that Moon has apparentely turned itself inside out! The proposed mechanism indeed explains this. See the blog post.

See the article Moon is mysterious or the chapter Magnetic Bubbles in TGD Universe: Part I.

TGD as it is towards end of 2024: part II

This article is the second part of the article trying to give a rough overall view about Topological Geometrodynamics (TGD) as it is towards the end of 2024. Various views about TGD and their relationship are discussed at the general level. In the first part of the article the geometric and number theoretic visions of TGD were discussed.

In the first part of the article the two visions of TGD: physics as geometry and physics as number theory were discussed. The second part is devoted to the details of M⁸-H duality relating these two visions, to zero energy ontology (ZEO), and to a general view about scattering amplitudes.

Classical physics is coded either by the space-time surfaces of H or by 4-surfaces of M⁸ with Euclidean signature having associative normal space, which is metrically M⁴. M⁸-H duality as the analog of momentum-position duality relates geometric and number theoretic views. The pre-image of causal diamond cd, identified as the intersection of oppositely directed light-cones, at the level of M⁸ is a pair of half-light-cones. M⁸-H duality maps the points of cognitive representations as momenta of fermions with fixed mass m in M⁸ to hyperboloids of CD\subset H with light-cone proper time a= h_eff/m.

Holography can be realized in terms of 3-D data in both cases. In H the holographic dynamics is determined by generalized holomorphy leading to an explicit general expression for the preferred extremals, which are analogs of Bohr orbits for particles interpreted as 3-surfaces. At the level of M⁸ the dynamics is determined by associativity of the normal space.

Zero energy ontology (ZEO) emerges from the holography and means that instead of 3-surfaces as counterparts of particles their 4-D Bohr orbits, which are not completely deterministic, are the basic dynamical entities. Quantum states would be superpositions of these and this leads to a solution of the basic problem of the quantum measurement theory. It also leads also to a generalization of quantum measurement theory predicting that in the TGD counterpart of the ordinary state function reduction, the arrow of time changes.

A rather detailed connection with the number theoretic vision predicting a hierarchy of Planck constants labelling phases of the ordinary matter behaving like dark matter and ramified primes associated with polynomials determining space-time regions as labels of p-adic length scales. There has been progress also in the understanding of the scattering amplitudes and it is now possible to identify particle creation vertices as singularities of minimal surfaces associated with the partonic orbits and fermion lines at them. Also a connection with exotic smooth structures identifiable as the standard smooth structure with defects identified as vertices emerges.

See the article TGD as it is towards end of 2024: part II or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

TGD as it is towards end of 2024: part I

This article is the first part of the article, which tries to give a rough overall view about Topological Geometrodynamics (TGD) as it is towards the end of 2024. Various views about TGD and their relationship are discussed at the general level.

1. The first view generalizes Einstein's program for the geometrization of physics. Space-time surfaces are 4-surfaces in H=M⁴× CP₂ and general coordinate invariance leads to their identification as preferred extremals of an action principle satisfying holography. This implies zero energy ontology (ZEO) allowing to solve the basic paradox of quantum measurement theory.
2. Holography = holomorphy principle makes it possible to construct the general solution of field equations in terms of generalized analytic functions. This leads to two different views of the construction of space-time surfaces in H, which seem to be mutually consistent.
3. The entire quantum physics is geometrized in terms of the notion of "world of classical worlds" (WCW), which by its infinite dimension has a unique K\"ahler geometry. Holography = holomorphy vision leads to an explicit general solution of field equations in terms of generalized holomorphy and has induced a dramatic progress in the understanding of TGD.

Second vision reduces physics to number theory.

1. Classical number fields (reals, complex numbers, quaternions, and octonions) are central as also p-adic number fields and extensions of rationals. Octonions with number theoretic norm RE(o²) is metrically Minkowski space, having an interpretation as an analog of momentum space M⁸ for particles identified as 3-surfaces of H, serving as the arena of number theoretical physics.
2. Classical physics is coded either by the space-time surfaces of H or by 4-surfaces of M⁸ with Euclidean signature having associative normal space, which is metrically M⁴. M⁸-H duality as analog of momentum-position duality relates these views. The pre-image of CD at the level of M⁸ is a pair of half-light-cones. M⁸-H duality maps the points of cognitive representations as momenta of fermions with fixed mass m in M⁸ to hyperboloids of CD\subset H with light-cone proper time a= h_eff/m.

   Holography can be realized in terms of 3-D data in both cases. In H the holographic dynamics is determined by generalized holomorphy leading to an explicit general expression for the preferred extremals, which are analogs of Bohr orbits for particles interpreted as 3-surfaces. At the level of M⁸ the dynamics is determined by associativity. The 4-D analog of holomorphy implies a deep analogy with analytic functions of complex variables for which holography means that analytic function can be constructed using the data associated with its poles and cuts. Cuts are replaced by fermion lines defining the boundaries of string world sheets as counterparts of cuts.

3. Number theoretical physics means also p-adicization and adelization. This is possible in the number theoretical discretization of both the space-time surface and WCW implying an evolutionary hierarchy in which effective Planck constant identifiable in terms of the dimension of algebraic extension of the base field appearing in the coefficients of polynomials is central.

This summary was motivated by a progress in several aspects of TGD.

1. The notion of causal diamond (CD), central to zero energy ontology (ZEO), emerges as a prediction at the level of H. The moduli space of CDs has emerged as a new notion.
2. Galois confinement at the level of M⁸ is understood at the level of momentum space and is found to be necessary. Galois confinement implies that fermion momenta in suitable units are algebraic integers but integers for Galois singlets just as in the ordinary quantization for a particle in a box replaced by CD. Galois confinement could provide a universal mechanism for the formation of all bound states.
3. There has been progress in the understanding of the quantum measurement theory based on ZEO. From the point of view of cognition BSFRs would be like heureka moments and the sequence of SSFRs could correspond to an analysis, possibly having the decay of 3-surface to smaller 3-surfaces as a correlate.

In the first part of the article the two visions of TGD: physics as geometry and physics as number theory are discussed. The second part is devoted to M⁸-H duality relating these two visions, to zero energy ontology (ZEO), and to a general view about scattering amplitudes.

See the article TGD as it is towards end of 2024: part I or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.