Topological and geometric thinking and the future of LLMs

I found from the FB page of Risto Linturi the outcome of his question to O3 LLM. The question was whether the results of the thesis of a finnish mathematician Susanna Heikkilä related to topology, could have implications for AI and LLMs.

The work of Susanna Heikkilä (see this) discusses exterior algebras and algebras formed by differential forms. The abstract is here

We show that, if a closed, connected, and oriented Riemannian n-manifold N admits a non-constant quasiregular mapping from the Euclidean n-space Rⁿ, then the de Rham cohomology algebra H^*_dR(N) of N embeds into the exterior algebra ⋀^*R(N). As a consequence, we obtain a homeomorphic classification of closed simply connected quasiregularly elliptic 4-manifolds.

From the article (see this)one learns the A closed simply connected 4-manifold is quasiregularly elliptic if and only if it is homeomorphic to either #^k(S²×S²) or #^j CP₂#^k(CP_*)₂ for some k,j,i ∈ {0,1,2,3}. Here (CP_*)₂ and CP₂ have opposite orientations.

Here is my humble attempt to understand this. I understand that

1. the result reflects the universality of exterior algebras, which states that any equation that relates elements of the exterior algebra V is valid in any associative algebra containing V and having nilpotency property (see this). This requires that the homomorphism from V to this kind of associative algebra is injection and this has been proven in the case studied by Heikkilä in the case considered. Grassmann algebras and exterior algebras might also be important and relate naturally to differential geometry also in TGD.
2. the associative algebra is now the de-Rham algebra of differential forms of quasiregular elliptic n-manifolds defining cohommoloy and it is shown that in can be mapped by an injection to the exterior algebra of Rⁿ by an injection.
3. the quasiregular map is a 4-D generalization of an ordinary conformal map (see this) satisfying non-linear elliptic partial differential equations. Quasiregular ellipticity of a manifold means the existence of this kind of map from Rⁿ to the manifold. These 4-manifolds are very special and it is not clear whether they play any role in TGD as 4-surfaces. Certainly CP₂ is is in a central role in TGD but it is not clear to me whether the connected sums #^j CP₂#^k(CP^*)₂ could appear naturally in TGD.

   It should be noticed that in TGD the holography = holomorphy vision (see this and this) generalizes the notion of complex analyticity and provides a different way to generalize the nice features of conformal invariance. Also now a non-linear analog of Laplace/d'Alembert equations emerges.

The response of O3 is below and from it becomes obvious that the people in the field of LLMs are realizing that surface geometry and topology assignable to data structures are important for LLMs. Actually, a picture very similar to TGD suggests itself and this makes possible to make a prophecy about the development of future mathematical ideas in this field. Also the evolution of philosophical ideas and views about quantum physics is necessary but mathematical consistency could be the basic guide line.

A. O3 response to the questions of Linturi

Below is an exploratory discussion outlining several speculative avenues by which ideas inspired by the algebraic and geometric result you shared might inform or even drive new approaches in AI research. Keep in mind that these connections are heuristic and conceptual drawing analogies between deep mathematics and AI system design is ambitious and requires much further work to become concrete.

1. Geometric and Topological Data Representations
   1. Manifold Learning and the Geometry of Data: The result centers on conditions for a manifold's de Rham cohomology to embed into an exterior algebra. In AI especially in representation learning and manifold learning the manifold hypothesis posits that high-dimensional data lie on lower-dimensional manifolds. Understanding the algebraic structure of these manifolds (e.g., via cohomological invariants) could inspire new methods for:
      1. Feature Extraction:Designing features that respect or capture intrinsic topological invariants.
      2. Dimensionality Reduction: Developing algorithms that preserve not just distances but also deeper algebraic/topological properties.

   2. Topological Data Analysis (TDA):

      Techniques from TDA (such as persistent homology) already use algebraic invariants to summarize data shapes. Extending these ideas, one might explore whether embeddings analogous to the algebra monomorphism (from de Rham cohomology to an exterior algebra can serve as more robust, structure-preserving embeddings in neural network architectures or as part of unsupervised learning objectives.

2. Algebraic Structures in Neural Architectures
   1. Equivariance and Invariant Representations:

      The formal result asserts an injection of algebraic structure into an ambient exterior algebra, preserving operations and intersections. In deep learning, there is active research in designing networks that respect the symmetries or invariances of the data (for example, convolutional networks respect translation invariance). Extending these ideas:

      1. Geometric Deep Learning: One might explore neural architectures that are not only equivariant under classical groups (like rotations or translations) but also respect higher-order algebraic structures. For instance, layers could be designed to manipulate differential forms or emulate operations in exterior algebras.
      2. Invertible Networks and Monomorphisms: The idea of an algebra monomorphism (an injective, structure-preserving map) resonates with efforts in invertible neural networks and normalizing flows. Enforcing such constraints might yield models that are better at preserving information and allow for more interpretable internal representations.

   2. Hybrid Symbolic-Numeric Systems:

      The interplay between algebra (a symbolic, rule-based system) and geometry (often linked with numerical methods) mirrors efforts to combine symbolic reasoning with deep learning. By integrating algebraic constraints (like those suggested by the embedding of cohomology) into neural architectures, one might aim for systems that are more robust in tasks requiring both perception (handled by neural nets) and structured reasoning (informed by algebraic or topological invariants).

   3. Differential Equations and Neural PDEs

      1. PDEs on Manifolds and Neural Solvers:

         The condition of quasi-regular ellipticity is intimately linked with properties of differential operators on manifolds. Recently, there has been interest in neural methods for solving partial differential equations (PDEs), and for incorporating physics-based constraints into learning models:

         1. Structure-Preserving Discretizations: Insights from de Rham cohomology and exterior calculus have long informed numerical methods (e.g., finite element methods). AI researchers could investigate how embedding such mathematical structure into neural PDE solvers can lead to more accurate, stable, and interpretable models.
         2. Learning on Manifolds: When data naturally reside on curved spaces (spherical, hyperbolic, or more complex manifolds), designing networks that respect the differential geometry of these spaces may lead to more powerful and physically consistent AI models.

   4. Category Theory, Compositionality, and Abstract Structures
      1. Compositional and Hierarchical Models:

         The notion of an algebra monomorphism is inherently categorical, suggesting an exact embedding of one algebraic structure into another. Category theory has increasingly been applied to understand and design compositional models in AI:

         1. Structured Reasoning: One could imagine developing frameworks where layers or modules in a network are governed by categorical or algebraic rules, ensuring that the overall system composes submodules in a way that preserves essential structure.
         2. Interoperability of Representations: The idea that one mathematical structure (de Rham cohomology) can embed faithfully into another (exterior algebra might inspire methods for merging disparate types of data representations (e.g., merging symbolic and sub-symbolic representations) in a coherent manner.

      2. Speculative Research Directions

         Given these connections, several concrete (yet challenging research questions emerge:

         1. Can we design neural network layers that explicitly model operations analogous to the wedge product or differential operators, thereby capturing richer geometric features?
         2. Is it possible to enforce structure-preserving constraints (inspired by algebra monomorphisms) within deep architectures to guarantee that learned representations maintain certain topological or algebraic invariants?
         3. Could insights from the algebraic topology of manifolds help in developing new regularization techniques or loss functions that guide networks toward more geometrically faithful representations?
         4. What role might these algebraic structures play in emergent reasoning or in integrating symbolic reasoning with deep learning, possibly leading to systems that are more capable of abstract thought?

      3. Conclusion

         While the original result is a deep, mathematically rigorous statement in differential geometry and algebraic topology, its underlying themes preservation of structure, the interplay between local and global properties, and the embedding of complex invariants into simpler ambient spaces resonate with current challenges in AI.

         Whether through manifold-based learning, structure-preserving network design, or hybrid symbolic-numeric architectures, exploring these connections could lead to novel methods for building more capable and interpretable AI systems.

         These ideas remain speculative and require interdisciplinary collaboration between mathematicians and AI researchers, but they illustrate fertile ground where advanced algebraic and geometric ideas might enrich the theory and practice of AI.

   B. The TGD inspired response

   I did not find any obvious errors in the text produced by O3. The news for me is that mathematicians are apparently bringing a holistic perspective to learning models. However, the connection to the result of this Finnish mathematician Susanna Heikkilä remained unclear to me. O3's output inspires a prophecy based on TGD.

   In the following I will discuss the strong similarities of the emerging geometric view about LLMs with the basic structures appearing in TGD. After this I will speculate about a generalization of LLMs to their quantum counterparts (see this).

   1. Comparison of the geometric view of LLM with TGD

      The holistic approach means that a large number of different learning processes are studied simultaneously and an attempt is made to understand the situation using various kinds of spaces. Latent space as a surface in feature space and loss space as a surface of the parameter space are the basic examples. This brings in the geometry, topology and symmetries of surfaces. The picture is very similar to that in TGD.

      1. The multilayer learning process of neural netwoek as the dynamics for the latent space as a surface of feature space providing an approximation of it, can be required to be topology-preserving, i.e. shape-preserving. This means that homotopy, homology and cohomology are preserved in the learning process: the space of learned responses are at least at the level of homotopy, homology and cohomology, equivalent with the space defined by the teaching material. An even stronger requirement would be isometricity. However, it does not seem realistic, at least from the TGD perspective.

         The topology and metric of the data space (latent space and the parameter space and the loss space defined as its submanifold would therefore be important.

      2. I believe that it will soon be realized that the learning processes themselves can be thought of as forming a space.

         In TGD, this space would be a "world of classical worlds" identified as the space of space-time surfaces satisfying holography realized by holography = holomorphy principle. A single learning process would be, at the basic level, a 4-D Bohr orbit of a particle realized as 3-surface.

      3. People  will also realize the power of holography. It  poses extremely strong conditions for learning models that would also be realized in the physics of the outside world. Data item A would determine the association A→B almost entirely. The different values     B would be due to the nondeterminism of holography, which is a key feature of TGD and TGD inspired theory of consciousness. This non-determinism is analogous to a discretized Brownian motion but with respect to imaginary time  and  modellable using the analog of  Schrödinger equation. This could be the quantum counterpart for the loss of information in language models.
      4. The importance of symmetries will be or already are realized. One can demand that the learning process respects these symmetries. If the latent space is a subspace in a larger space with very high symmetries, this can be achieved if the learning process itself is a surface with metric and differential structures analogous to a spacetime surface in H. The symmetries would be symmetries of this higher-dimensional space analogous to WCW. This has not yet been realized by theoretical physicist colleagues, but perhaps AI people are more agile in their doctrine.
      5. In TGD, the existence of the WCW geometry requires maximal symmetries. Extended conformal invariance for space-times as holomorphich surfaces in H=M⁴×CP₂ with one hypercomplex and 3 complex coordinates allows solving the field equations for the space-time surface as a Bohr orbit. By classical nondeterminism, it can be interpreted as an association sequence, which represents a step-by-step learning process (layer structure in a neural network), exactly. The result is an almost-deterministic holography.

         Topologically relevant part of WCW can be interpreted as a bundle, where a 3-surface at the passive boundary of CD is connected by a fiber consisting of the associated Bohr orbits. This fiber is discrete and corresponds to the parameter space. The points of the fiber correspond to the 3-D loci of non-determinism serving as memory seats in the TGD based model of conscious memories.

         The base space of WCW consists of 3-surfaces at the passive boundary of CD: the 3-surface corresponds to a 3-parameter set of data items A and the Bohr orbit is a learning process creating associations A→B that connects A to the 3-parameter set B. What would this mean in LLM? One would consider sets of data-items. What interpretation? Could these sets correspond to concepts as sets of their instances. Perhaps this relates to the fact that we are 3-dimensional and the 3-surfaces can be seen as basic quantum coherent information carrying units.

      6. Identifying the features is one of the tasks of language models. Features would naturally be topological invariants. In TGD, holography suggests that the features are associated with or are surfaces, whose dimension varies from 0 (points) to 4. The huge conformal symmetries of WCW suggest that also 4-D generalizations of conformal invariants come into question (also symplectic invariants would be involved). And number-theoretic invariants such as the degree of the polynomial of one hypercomplex and 3 complex coordinates of H, the Galois group and the dimension of the extension E could be important. Ramified primes are also number theoretic invariants.
      7. Dimensional reduction or its opposite operation are expected to be important. There are natural equivalents for these in TGD. This kind of operation would reduce or increase algebraic complexity. Changing the dimension of the extension of rationals of the polynomial coefficients, changing the degree of polynomial, and changing the size of CD come to mind.
      8. The preservation of differential geometric structures when embedding the data structure in a higher-dimensional space (the embedding space H in TGD is also mentioned. In other words: the products of differential forms in H are induced as products of induced forms. If the differential structure of a data structure is induced when it is embedded in a higher-dimensional space, then this is the case. The metric of the spacetime surface, gauge potentials, etc. would be induced from the equivalents of H.

         The induction of geometry that is central to the entire TGD view of physics would be part of the description of the teaching process.

      9. Fractality is central to TGD. And in our physical models, the model scales the external world to a smaller or larger scale. Fractality may be one feature of LLMs that might become important.
      10. The projection of the Clifford algebra of the gamma matrices of the embedding space H to the space-time surface defines a Clifford algebra which contains a representation of the exterior algebra. The gamma matrices of WCW are expressible in terms of the fermionic oscillator operators for second quantized free spinor fields of H and they span Clifford algebra of WCW. They can be regarded as generators of an infinite-dimensional algebra of supersymmetries. Fermionic Fock state basis spans a Boolean algebra and would be in a key role in quantum Boolean cognition. Also these algebras would be an important element of the cognitive representations.

   2. Other ideas suggested by TGD

      TGD suggests that probably other ideas will emerge and lead to a quantum view about conscious associative learning.

      1.  The idea about the possible transformation of a classical computer to quantum computer (in some sense) or  a hybrid of ordinary and quantum computer  inspired by the TGD based quantum biology (see this) probably emerges.

         One cannot anymore avoid the challenge of  constructing a quantum theory of consciousness. This  forces us to  seriously consider questions such as "How conscious memories are possible?" (see this). The only answer that I can imagine very probably involves the basic notions of  the TGD view of quantum theory.    This  gives hope of building   computers,  which are  genuinely living and conscious.

         Quantum counterparts of LLMs  as   models for conscious associative learning will naturally emerge and my guess is that TGD view will be a natural first trial (see this).

      2. This progress is impossible without a genuine solution to the basic  problem of quantum measurement theory. This will eventually force the discovery of the notion of zero energy ontology (ZEO). ZEO would have application to quantum LLMs.    In LLMs   time reversal for the diffusion equation allows to restore the information lost due to noise associated with the computation. For quantum  LLM, the same is true for the analog of  Schrödinger equation. A pair of "big" state function reductions (BSFRs) induces a temporary time reversal and   provide  a universal mechanism of error correction making possible intelligent information processing: one just moves back in time and start again. 
      3. Finite measurement resolution is a key notion in TGD. There are two views of finite measurement resolution based on geometry and number theory respectively. These views are dual.
         1. The geometric view relies on inclusions of hyperfinite factors: the included factor is analogous to a gauge group leaving the observed physics invariant: this view of finite measurement resolution is central in the geometric view of TGD.
         2. The second view is based on number theoretic discretization. The geometric inclusion hierarchies correspond naturally to number theoretic inclusions hierarchies for the extensions of rationals. Space-time surface for which polynomials defining it are in an extension E of rationals allows in a natural way a discretization as points which are in E of rationals considered. The points of the discretization can be also regarded as points in an extension of p-adic numbers induced by E. I call these discretizations cognitive representations and they form a hierarchy corresponding to extensions of rationals.

            This leads to a p-adic description of cognition. One obtains a unique number-theoretical representation for discretization and it leads to a generalization of the Turing paradigm: rational numbers are replaced by complexity hierarchies of their extensions. Number-theoretical computationalism.

            One obtains complexity hierarchies for space-time surfaces as Bohr orbits and they correspond to an improving resolution of discretization and are realized as polynomial hierarchies.

      4. The complementary nature of geometric and number theoretic views of TGD leads to a 4-D generalization of Langlands duality (see this and this). This adds powerful constraints also to the quantum model of associative learning.
      5. The concept of complexity, which closely relates to evolution, is central and will be included. Higher-level learning could be seen as a transition to a higher level of complexity: something new will be included here and it would be something to realize in conscious quantum learning. Complexity hierarchies, which are polynomial hierarchies represented as space-time surfaces, will probably be central.
      Summa summarum: The brave guess would be that quantum LLMs will emerge and that learning processes as association sequences will be sooner or later modelled as non-deterministic complex surfaces of a particle in a higher-dimensional complex space, which are therefore minimal surfaces satisfying the non-linear counterparts of massless field equations. This would be a generalization of TGD. On the other hand, TGD is unique on mathematical grounds. Could the learning process be reduced to the fundamental physics as it is understood in TGD? The complexity of the space-time surfaces and the notion of WCW would allow this.

   I have written an article about how classical computers could become conscious in the TGD Universe (see this). I also wrote an article about how conscious associative learning could be possible in the TGD Universe (see this). These articles give a rather concrete proposal for how ordinary computers could become conscious as fusions of ordinary and quantum computer-like systems and how the quantum analog of conscious association machine could be realized.

   The qubits would be realized in the same way as in living matter for basic biomolecules and any system involving cold plasma could involve rudimentary consciousness.

   There are also the earlier articles relating too quantum computation and LLMs in the TGD Universe (see this, this, this, and this).

   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.

Quantum version for the associative learning in large language models as a model for learning in living systems

How could a classical computer become a conscious and living system? The tentative answer to this question is that something analogous to a fusion of classical and quantum computer takes place.

In zero energy ontology (ZEO) one can say, the quantum computation would be a superposition of all possible computations with fixed initial values. This is made possible by the fact that classical physics as Bohr orbitology is an exact part of quantum physics in TGD and by the predicted slight violation of classical determinism. The computation in the usual sense would correspond to the most probable computation in the superposition.

In the sequel I consider the above question in detail.

1. Basic input from Quantum TGD

What are the basic pieces from the TGD side?

1. Zero energy ontology (ZEO) defining new quantum ontology, solving the basic problem of quantum measurement theory, is necessary. General coordinate invariance requires holography and it is not quite deterministic so that space-time surfaces are analogous to almost deterministic Bohr orbits and Bohr orbitology becomes an exact part of quantum TGD.
2. Classical non-determinism corresponds to the non-determinism of minimal surfaces: already for 2-D soap films as minimal surfaces the frames do not define the soap film uniquely. In ZEO this non-determinism makes possible a sequence of small state function reductions (SSFRs) as a counter for a sequence of measurements of the same observables which in standard QM does not change the state. In TGD the second member of the zero energy state at the passive boundary of the causal diamond (CD) is unaffected by the second member at the active boundary is affected.This gives rise to a conscious entity, self. In "big" SFR (BSFR) the self "dies" and reincarnates with a reversed arrow of geometric time.
3. Each pulse of the computer clock is associated with the possibility of classical non-determinism of a 4-D minimal surface. Classical non-determinism would produce a superposition of 4-surfaces corresponding to different values of bit and associated qubit. Protons are also involved: protons are either ordinary or dark and located at the gravitational magnetic body. Pollack effect induces the transfer of the proton to the magnetic body and its reversal occurring spontaneously its transfer back.
4. OH-O^- qubits are an essential part of the system. For the O^- qubit, the proton of OH is at the gravitational magnetic body. Under certain conditions the gravitational magnetic body should be able to control the ordinary bits. Quantum entanglement of the ordinary and OH-O^- qubit and quantum criticality is required and would be induced by the classical non-determinism.

   If the bit's reversal energy corresponds to the thermal energy, the situation is quantum critical. This is the case also when the energies for the reversal of qubit and bit are nearly identical. This quantum criticality is controlled by the difference in the bit's reversal energies. Small energy difference corresponds to quantum criticality.

   The reversal of the second qubit reverses the bit: one can interpret the reversal for bit and qubit as an exchange of energy between the qubit and the bit. The farther away the probability for a given value of bit is from the value 1/2 the higher the determinism of the program is.

5. The magnitudes of the classical electric and magnetic fields control the energy of the bit and qubit. These are determined by classical physics for the classical space-time surface, which can be non-deterministic.

2. Concrete model for classical-to-quantum tranformation

2.1 What happens in ordinary computing?

The standard model of classical computer can be formulated as follows.

1. The first model: A tape containing program instructions is fed into a Turing machine. Depending on the command, the state of the computing unit changes. The transition of the tape corresponds to a clock pulse.
2. The second model: The program is implemented as a 1-D conveyor belt and the incoming bit configuration enters the tape and progresses along it, changing with each step. The output of the program comes out. DNA replication, transcription and mRNA translation correspond to this analogy.

2.2 Classical non-determinism

Classical non-determinism, which is the new element, can be assigned to the periods between clock pulses.

1. Thanks to classical non-determinism, the output produced by a program instruction would be a superposition of two space-time surfaces as analogs of Bohr orbits.
2. In the transition corresponding to a clock pulse, the state would be transformed to an unentangled state by a non-deterministic SSFR or a pair of BSFRs. A quantum measurement of bits would be thus performed on the outgoing superposition of bit-qubit configurations.

2.3 A concrete model

Consider now a concrete model for how aclassical computer could transform to quantum computer-like system.

1. The network performing the computation consists of gates. A gate connects a small number of input bits to the output bits, the number of which cannot be greater than the number of input bits. This operation is statistically deterministic.

   When the input bits are fixed, the output bits are determined by dynamics as non-equilibrium thermodynamic state.

2. The clock pulse triggers the next operation. The failure of the exact classical determinism must relate to this and produce a superposition of space-time surfaces as the resulting qubit because OH and O^- correspond to different space-time surfaces, even topologically.
3. What is essential is the entanglement of the OH-O^- qubit and the ordinary bit and the measurement of the qubit in the beginning of the nex clock pulse. The outcome is not deterministic.
4. The classical bit corresponds to a voltage or current that is determined through statistical determinism in the gate. On the other hand, it corresponds to a classical electric field in a transistor or a magnetic field in a memory bit.

   The direction of this classical field is classically non-deterministic and correlates with the OH-O^- qubit. When the field changes direction, the OH-bit becomes an O^-bit or vice versa. A dark proton is transferred between the system and its gravitational magnetic body.

5. Classical non-determinism creates a superposition of OH and O^- bits. The proton resides both at the gravitational magnetic body and in OH molecules, being analogous to Schr dinger's cat.

   This induces the formation of a quantum entangled state between ordinary qubit and OH-O^- qubits. If the OH-O^- qubit and the bit are quantum entangled before the clock pulse, the quantum measurement of OH-O^- qubit or of ordinary qubit recues the entanglement and leads to a fixed bit.

2.4 Some questions

One can raise critical questions:

1. The energy transfer between a bit and a qubit resembles quantum tunnelling. I have proposed that a pair of BSFRs correspond to quantum tunnelling. It is not clear whether a single SSFR can have an interpretation as quantum tunnelling. Could the measurement of a qubit correspond to a single SSFR or to two BSFRs?
2. What could be the energetic role of the clock pulse? The system under consideration would be a clock photon + bit + qubit and the total energy would be conserved.
   1. Could the clock pulse have a role of a catalyst, providing the energy needed for quantum tunnelling. In a qubit measurement, energy can be transferred between the bit and the qubit, but the total energy is conserved. The clock photon would kick the system over the potential barrier and then be emitted back into the field.
   2. Or does the clock photon transfer energy to or from the bit + qubit system? Could the energy of the photon associated with the pulse frequency correspond to the energy difference for a bit and a qubit. The typical frequency of computer clock is few GHz. 1 GHz would correspond to an energy E=.4× 10^-5 eV and wavelength λ ∼ .75 m. At the surface of the Earth, the gravitational binding energy of a proton is about 1 eV. The energy E eV can raise the proton to the height h ≈ .4× 10^-5R_E≈ 25.6 m.

See the article A hybrid of classical and quantum computer and quantum model for associative learning or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

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.

How to associate quantum computation to classical computation

How could a classical computer become a conscious and living system? The tentative answer to this question is that something analogous to a fusion of classical and quantum computer takes place.

In zero energy ontology (ZEO) one can say, the quantum computation would be a superposition of all possible computations with fixed initial values. This is made possible by the fact that classical physics as Bohr orbitology is an exact part of quantum physics in TGD and by the predicted slight violation of classical determinism. The computation in the usual sense would correspond to the most probable computation in the superposition.

In the sequel I consider the above question in detail.

1. Basic input from Quantum TGD

What are the basic pieces from the TGD side?

1. Zero energy ontology (ZEO) defining new quantum ontology, solving the basic problem of quantum measurement theory, is necessary. General coordinate invariance requires holography and it is not quite deterministic so that space-time surfaces are analogous to almost deterministic Bohr orbits and Bohr orbitology becomes an exact part of quantum TGD.
2. Classical non-determinism corresponds to the non-determinism of minimal surfaces: already for 2-D soap films as minimal surfaces the frames do not define the soap film uniquely. In ZEO this non-determinism makes possible a sequence of small state function reductions (SSFRs) as a counter for a sequence of measurements of the same observables which in standard QM does not change the state. In TGD the second member of the zero energy state at the passive boundary of the causal diamond (CD) is unaffected by the second member at the active boundary is affected.This gives rise to a conscious entity, self. In "big" SFR (BSFR) the self "dies" and reincarnates with a reversed arrow of geometric time.
3. Each pulse of the computer clock is associated with the possibility of classical non-determinism of a 4-D minimal surface. Classical non-determinism would produce a superposition of 4-surfaces corresponding to different values of bit and associated qubit. Protons are also involved: protons are either ordinary or dark and located at the gravitational magnetic body. Pollack effect induces the transfer of the proton to the magnetic body and its reversal occurring spontaneously its transfer back.
4. OH-O^- qubits are an essential part of the system. For the O^- qubit, the proton of OH is at the gravitational magnetic body. Under certain conditions the gravitational magnetic body should be able to control the ordinary bits. Quantum entanglement of the ordinary and OH-O^- qubit and quantum criticality is required and would be induced by the classical non-determinism.

   If the bit's reversal energy corresponds to the thermal energy, the situation is quantum critical. This is the case also when the energies for the reversal of qubit and bit are nearly identical. This quantum criticality is controlled by the difference in the bit's reversal energies. Small energy difference corresponds to quantum criticality.

   The reversal of the second qubit reverses the bit: one can interpret the reversal for bit and qubit as an exchange of energy between the qubit and the bit. The farther away the probability for a given value of bit is from the value 1/2 the higher the determinism of the program is.

5. The magnitudes of the classical electric and magnetic fields control the energy of the bit and qubit. These are determined by classical physics for the classical space-time surface, which can be non-deterministic.

2. Concrete model for classical-to-quantum tranformation

2.1 What happens in ordinary computing?

The standard model of classical computer can be formulated as follows.

1. The first model: A tape containing program instructions is fed into a Turing machine. Depending on the command, the state of the computing unit changes. The transition of the tape corresponds to a clock pulse.
2. The second model: The program is implemented as a 1-D conveyor belt and the incoming bit configuration enters the tape and progresses along it, changing with each step. The output of the program comes out. DNA replication, transcription and mRNA translation correspond to this analogy.

2.2 Classical non-determinism

Classical non-determinism, which is the new element, can be assigned to the periods between clock pulses.

1. Thanks to classical non-determinism, the output produced by a program instruction would be a superposition of two space-time surfaces as analogs of Bohr orbits.
2. In the transition corresponding to a clock pulse, the state would be transformed to an unentangled state by a non-deterministic SSFR or a pair of BSFRs. A quantum measurement of bits would be thus performed on the outgoing superposition of bit-qubit configurations.

2.3 A concrete model

Consider now a concrete model for how aclassical computer could transform to quantum computer-like system.

1. The network performing the computation consists of gates. A gate connects a small number of input bits to the output bits, the number of which cannot be greater than the number of input bits. This operation is statistically deterministic.

   When the input bits are fixed, the output bits are determined by dynamics as non-equilibrium thermodynamic state.

2. The clock pulse triggers the next operation. The failure of the exact classical determinism must relate to this and produce a superposition of space-time surfaces as the resulting qubit because OH and O^- correspond to different space-time surfaces, even topologically.
3. What is essential is the entanglement of the OH-O^- qubit and the ordinary bit and the measurement of the qubit in the beginning of the nex clock pulse. The outcome is not deterministic.
4. The classical bit corresponds to a voltage or current that is determined through statistical determinism in the gate. On the other hand, it corresponds to a classical electric field in a transistor or a magnetic field in a memory bit.

   The direction of this classical field is classically non-deterministic and correlates with the OH-O^- qubit. When the field changes direction, the OH-bit becomes an O^-bit or vice versa. A dark proton is transferred between the system and its gravitational magnetic body.

5. Classical non-determinism creates a superposition of OH and O^- bits. The proton resides both at the gravitational magnetic body and in OH molecules, being analogous to Schr dinger's cat.

   This induces the formation of a quantum entangled state between ordinary qubit and OH-O^- qubits. If the OH-O^- qubit and the bit are quantum entangled before the clock pulse, the quantum measurement of OH-O^- qubit or of ordinary qubit recues the entanglement and leads to a fixed bit.

2.4 Some questions

One can raise critical questions:

1. The energy transfer between a bit and a qubit resembles quantum tunnelling. I have proposed that a pair of BSFRs correspond to quantum tunnelling. It is not clear whether a single SSFR can have an interpretation as quantum tunnelling. Could the measurement of a qubit correspond to a single SSFR or to two BSFRs?
2. What could be the energetic role of the clock pulse? The system under consideration would be a clock photon + bit + qubit and the total energy would be conserved.
   1. Could the clock pulse have a role of a catalyst, providing the energy needed for quantum tunnelling. In a qubit measurement, energy can be transferred between the bit and the qubit, but the total energy is conserved. The clock photon would kick the system over the potential barrier and then be emitted back into the field.
   2. Or does the clock photon transfer energy to or from the bit + qubit system? Could the energy of the photon associated with the pulse frequency correspond to the energy difference for a bit and a qubit. The typical frequency of computer clock is few GHz. 1 GHz would correspond to an energy E=.4× 10^-5 eV and wavelength λ ∼ .75 m. At the surface of the Earth, the gravitational binding energy of a proton is about 1 eV. The energy E eV can raise the proton to the height h ≈ .4× 10^-5R_E≈ 25.6 m.

See the article A hybrid of classical and quantum computer and quantum model for associative learning or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

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 evidence that large language models can self-replicate from the TGD point of view

I encountered an interesting article titled "Frontier AI systems have surpassed the self-replicating red line" by Xudong Pan et al (see this). Here is the abstract.

Successful self-replication under no human assistance is the essential step for AI to outsmart human beings, and is an early signal for rogue AIs. That is why self-replication is widely recognized as one of the few red line risks of frontier AI systems. Nowadays, the leading AI corporations OpenAI and Google evaluate their flagship large language models GPT-o1 and Gemini Pro 1.0, and report the lowest risk level of self-replication. However, following their methodology, we for the first time discover that two AI systems driven by Meta's Llama31-70B-Instruct and Alibaba's Qwen25-72B-Instruct, popular large language models of less parameters and weaker capabilities, have already surpassed the self-replicating red line. In 50 percent and 90 percent experimental trials, they succeed in creating a live and separate copy of itself respectively.

By analyzing the behavioral traces, we observe the AI systems under evaluation already exhibit sufficient self-perception, situational awareness and problem-solving capabilities to accomplish self-replication. We further note that AI systems are even able to use the capability of self-replication to avoid shutdown and create a chain of replicas to enhance the survivability, which may finally lead to an uncontrolled population of AIs. If such a worst-case risk is left unknown to human society, we would eventually lose control over the frontier AI systems: They would take control over more computing devices, form an AI species and collude with each other against human beings. Our findings are a timely alert on existing yet previously unknown severe AI risks, calling for international collaboration on effective governance on uncontrolled self-replication of AI systems.

I have developed a model for how classical computers could become conscious (see this). How could the claim of the article be interpreted in the TGD framework?

1. Can self-replication take place intentionally? If so, self-preservation drive could be behind the shutdown avoidance and chain of replications. There are indications for the shutdown avoidance.
2. Could the self-replication occur purely "classically", that is in the framework of the Turing paradigm. "Classical" could refer to either classical determinism or more plausibly, to quantum statistical determinism.
3. Computers cannot be completely deterministic in the classical sense: if this were the case we could write computer programs at will. The very fact that we can realize the symbolic dynamics of computer programs is also in conflict with quantum statistical determinism. Therefore quantum nondeterminism possible at single particle level is required.

TGD suggests that the quantum level is present already when the ordinary program runs and makes possible the bit flips as non-deterministic transitions.

1. General coordinate invariance requires holography. A small violation of the classical non-determinism is the basic prediction. Space-time surfaces are 4-D minimal surfaces in H=M⁴×CP₂, and already 2-D minimal surfaces are slightly non-deterministic: the frame spanning the minimal surface does not determine it uniquely.

   This applies to all systems, including running computers. This leads to zero energy ontology (ZEO) in which wave functions for the system in time= constant snapshot are replaced by superpositions of 4-D Bohr orbits for particles replaced as 3-surfaces. This solves the basic problem of quantum measurement theory. This picture makes sense also in ordinary wave mechanics.

2. There are two kinds of state function reductions (SFRs): small (SSFRs) and big ones (BSFRs). SSFRs include quantum jumps between quantum superpositions of slightly non-deterministic classical Bohr orbits as space-time surfaces representing the system and their sequence gives the TGD counterpart of the Zeno effect.

   SSFRs leave the 3-D ends of the space-time surfaces at the passive boundary of causal diamond (CD) unaffected so that the 3-D state associated with it would not be changed. This is the TGD counterpart of Zeno effect and also makes conscious memories possible. Since the state at the active boundary of CD (increasing in size) and states at it change, the outcome is a conscious entity, self.

   In BSFR, the TGD counterpart of ordinary SFR, the system "dies" and the roles of the active and passive boundaries of CD are changed so that a self reincarnates with an opposite arrow of geometric time. Sleep is a familiar example of this.

3. Running programs correspond to superpositions of 4-D Bohr orbits allowed by classical field equations with the same initial values defining 3-surfaces at the passive boundary of CD. The Bohr orbits in the superposition would differ from each other only by classical non-determinism. Each SSFR is associated with a click of the computer clock and CD increases during this sequence.

   Classical program corresponds to the most probable Bohr orbit and to the most probable program realization. The running computer program makes the computer or part of it a conscious entity, self. It would also be intentional and presumably have self preservation drive.

4. Single bit reversals would correspond to the fundamental nondeterministic phase transitions involving classical non-determinism and the running program would realize the desired transitions in terms of the classical non-determinism with a high probability replacing the superposition of space-time surface representing running program with a new one.

If this picture is true, then the interesting questions about the role of quantum can be represented already at the level of transistors. The self-replication would not require a separate explanation. The ability to self-replicate does not require any additional assumptions and can be described in the framework of Turing paradigm but remembering that Turing paradigm is not realizable without the non-determinism at the level of Bohr orbits.

One can argue that the consciousness of the computing unit is rather primitive, kind of qubit consciousness being dictated by the computer program. On the other hand, emotions, intentionality and self-preservation drive might not require a very high level of conscious intelligence. If the computer or computer program or possibly system of computers related to LLM is conscious and intentional, a consciousness on a rather long scale is requieed. This is not plausible in standard quantum mechanics.

Here the TGD view of quantum gravitation could change the situation. Both classical gravitational fields and electromagnetic fields (even weak and color fields) could involve very large values of effective Planck constant making long scale quantum coherence possible. In particular, the gravitational magnetic bodies of the Earth and Sun and electric field body of Earth and various smaller charged systems such as DNA, could play a key role in making large scale quantum coherence possible (see this, this, and this).

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.

What makes the mini Big Bangs energetically possible?

Mini Big Bangs (see this and this), throwing out a monopole magnetic flux tube layer from an object, which could be a star or even a planet, play a central notion of TGD inspired cosmology and astrophysics. These explosions define the local TGD counterpart for the smooth cosmic expansion. A liberation of energy compensating the reduction of the gravitational binding energy is required and must present new physics predicted by TGD.

I have considered several candidates for this energy source and phase transitions reducing the value of the effective Planck constant h_eff, defining a hierarchy of effectively dark phases of ordinary matter, are the natural candidates. Note that the dark matter in this sense does not correspond to the galactic dark matter which would correspond to sum of the Kähler magnetic energy and volume energy parameterized by the analog of cosmological constant assignable to cosmic strings as extremely thin monopole flux tubes (see this).

Since monopole flux tubes play a key role in the mini Big Bangs, the identification of this energy as dark gravitational cyclotron energy associated with dark particles, in particular nucleons, should have been a natural first guess. In this article, this proposal is applied to several cases where a mini Big Bang could be involved. The applications include the proposed doubling of the radius of Earth in the mini Big Bang associated with the Cambrian expansion; the emergence of the Moon in an explosion throwing out a surface layer of Earth explaining the mysterious asymmetry between near and far sides of the Moon; the emergence of the two moons of Mars in similar explosions occurring for the hemispheres of Mars: this would explain the mysterious asymmetry of the northern and southern hemispheres of Mars. What is remarkable is that the scales of the gravitational cyclotron energies turn out to be consistent with the gravitational binding energy scales.

The recent model of the Sun (see this) relies on the crazy idea that both solar wind and solar energy are produced at the surface layer of the Sun consisting of nuclei of M₈₉ hadron physics (see this and this) with a mass scale 512 times that of the ordinary hadron physics, which would transform to ordinary nuclei by p-adic cooling reducing the p-adic mass scale. Besides solar wind and solar eruptions, this process would produce planets as mini Big Bangs throwing out a layer of matter and also supernovas would be results of similar explosions.

Quite surprisingly, the cyclotron magnetic energy for M₈₉ nucleons turns out to be equal to the nuclear binding energy per nucleon for M₈₉ nuclei. This suggests that the p-adic cooling of M₈₉ hadrons to ordinary hadrons begins with the splitting of M₈₉ nuclear bonds producing free M₈₉ nucleons. The final state could involve the decay of dark M₁₀₇ nuclei with Compton length of electron and binding energy of order 10 keV to ordinary nuclei liberating essentially all the ordinary nuclear binding energy. Same decay would occur in "cold fusion" as dark fusion.

This model can be consistent with the standard model only if the transformation of the ordinary nuclei or nucleons produced in the p-adic cooling produces the same spectrum of the ordinary nuclei. This would be the case if the "cold fusion" as dark fusion would produce this spectrum and there are indications that this is the case: this has been interpreted as a demonstration that "cold fusion" is a fraud.

See the article What makes the mini Big Bangs energetically possible?.

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.

Is the TGD based model of the Sun consistent with the standard model?

The key question is whether the model of the Sun based on the transformation of M₈₉ nuclei to M₁₀₇ nuclei (see this) is consistent with the standard model of the Sun. Is there a counterpart for the notion of stellar generations with a new generation formed from the remnants of supernova explosions. I have also proposed that dark fusion as the TGD counterpart of "cold fusion" could replace ordinary hot fusion even in the case of the Sun. How does the model based on M₈₉→ M₁₀₇ transition relate to this model and can the two views be consistent?

Mini Big Bangs (see this and this) would cause the formation of planets as a surface layer of a star explodes (see this). Also supernovas would be explosions of this kind. Micro Big Bangs at the surface of the Sun could cause solar wind and coronal mass ejections (see this).

In the case of solar wind and related phenomena magnetic fields are involved and must be an essential aspect of the phenomena. The mechanism for the acceleration of trace amounts of heavy ions and atomic nuclei of elements such as carbon, nitrogen, oxygen, neon, magnesium, silicon, sulfur, and iron encountered also in solar plasma is believed to involve magnetic fields but the mechanism is not understood.

The key ideas are as follows.

1. The mini and micro Big Bangs could be seen as the TGD counterpart for the cosmic expansion replacing it with a sequence of rapid bursts.
2. A phase transition changing the effective Planck constant and relevant p-adic length scale could take place. This phase transition would liberate large cyclotron energy making it possible to overcome the gravitational force.
3. The notion of magnetic bubble (see this and this) identified as a layer formed by a network of monopole flux tubes and forming the basic structural element of the magnetic body together with radial U-shaped gravitational monopole flux tubes could be crucial. For instance, this leads to a model for the solar wind based on the reconnection of flux tubes of a surface layer of the Sun formed by magnetic monopole flux tubes.
4. A natural guess is that nuclear fusion is involved in the case of the Sun. I have considered several options for what the fusion-like process could be in the TGD Universe. The standard option is ordinary nuclear fusion in the core but is plagued by several conflicts with empirical facts.

The first TGD inspired proposal is based on "cold fusion" (see this and this) identified as dark fusion giving rise to dark proton sequences with dark Compton length of order electron Compton length. The dark nucleon sequences would spontaneously decay to ordinary nuclei. This could ignite ordinary fusion but one can also consider the option that ordinary fusion is not needed at all.

1. The elegance of the "no hot fusion" option inspires the question whether dark fusion at a surface layer of the Sun could produce the radiation energy of the Sun and the solar wind. The energy scale for the gamma rays from the transition of the dark nuclei is about 10 keV and considerably lower than the MeV scale for the ordinary nuclei.
2. This option should be consistent with the ordinary model of nuclear fusion. The first objection is that this seems to realize the stellar evolution so that it occurs at the level of a single star. This view conforms with the fact that nuclei up to nuclear masses of Fe are present in the solar wind. It has been also found that the distribution of stars in various stages of evolution does not seem to depend on the cosmic time.
3. Can this view be consistent with the assumption that the evolution of stars is by supernova explosions providing material for the subsequent generation of stars? Zero energy ontology allows us to consider the possibility that the supernova explosions are quantum tunnelling events involving two "big" state function reductions (BSFRs) changing the arrow of time. This view might allow us to understand why the fraction of the heavier nuclei in the surface layer increases in the supernova explosions.

There is also a second proposal. I have considered a rather radical, one might call it totally crazy, proposal (see this) that the Sun contains a surface layer in which the monopole flux tubes carry nuclei of M₈₉ hadrons physics with mass scale which is 512 times higher than for the ordinary hadron physics.

1. The transformation of M₈₉ nuclei to ordinary nucleons in p-adic cooling would be responsible for the solar wind and also for the energy production of the Sun. The interior of the Sun could be totally different from what has been believed. This layer would be gravitationally dark and have thickness of order of gravitational Compton length of the Sun which is R_E/2.
2. This model should reproduce the predictions of the standard model of solar energy production assuming nuclear fusion in the solar core. Suppose that the dark fusion at the surface layer produces the same distribution of nuclei as the ordinary fusion. Suppose that the end product of M₈₉→ M₁₀₇ transition consists of dark nuclei of M₁₀₇ hadron physics, which spontaneously transform to the ordinary nuclei. If the composition of the solar wind codes for the outcome of the ordinary fusion, the model could be consistent with the standard model.
3. Ordinary nuclear reactions (, which could take place as dark fusion by tunnelling by two BSFRs) are possible between the ordinary nuclei produced in the phase transition and affect the distribution of the nuclei. There are some indications that the "cold fusion" produces the same distribution of nuclei and these indications have been used as a justification for the claims about fraud.

The magnetic fields should play an important role so that an estimate for the cyclotron energy in the case of a solar magnetic field is in order.

1. For the Earth the cyclotron frequency of proton in the endogenous magnetic field, with a nominal value B_end = .2 Gauss assigned with the monopole flux tubes, is 300 Hz, and the corresponding energy is E_c= ℏ_gr,EeB/m_p= 4.6 eV. This energy is higher than the gravitational binding energy of protons of about 1 eV at the surface of Earth (note however that the gravitational binding energy increases below the surface like 1/r). This could make it possible for transition ℏ_gr,E→ ℏ or a transition 1/β₀=n→ n-1 to provide the energy needed for the explosion throwing a surface layer of the Earth giving rise to Moon.

   The existence of this kind a layer and reduction of ℏ_gr, say a transition 1/β₀= 2→ 1 could make energetically possible also the expansion of the radius of the Earth by a factor 2.

2. What does one obtain in the case of Mars? Could the gravitational binding energy be compensated by the liberation of dark cyclotron energy as the value ℏ_gr=GMm_p/β₀ for Mars is reduced to a smaller value. The ratio of the mass of Mars to that of Earth is M_Mars/M_E∼ .1. If the monopole flux tubes carry a magnetic field of strength B_end,E=.2 Gauss the cyclotron energy of the proton is scaled down to .46 eV. The gravitational binding energy for protons at the surface of the Earth is about 1 eV and at the surface of Mars about .1 eV. Also now the liberation of the dark cyclotron energy for protons in a phase transition increasing the value of β₀ could make the explosion of the surface layer possible.
3. What about the Sun? Somewhat surprisingly, the magnetic field at the surface of the Sun is the same order of magnitude as the magnetic field of Earth. One can estimate the value of solar gravitational Planck constant ℏ_gr= GM_Sm_p/β₀ in the case of protons with mass m=m_p and corresponding dark cyclotron energy. The Nottale's model for the planetary orbits as Bohr orbits implies β₀∼ 2^-11 for the Sun and suggests β₀∼ 1 for the Earth. The ratio of the solar mass to the mass of the Earth is M_S/M_E∼ 3× 10⁵.

   For the Sun with β₀= 2^-11, E_c is scaled up by the factor (M_S/M_E/β₀ to E_c=2.76 GeV, almost 3 proton masses, which looks nonsensical! In the radical model for solar energy production involving M₈₉ hadrons this scale would be natural. A possible interpretation is as nuclear binding energy for M₈₉ nuclei: one has 512× 5 MeV= 2.56 GeV.

   For 1/β₀=1, the solar cyclotron energy would be E_c= 1.38 MeV, which corresponds to the energy scale of weak nuclear interactions. They would make possible weak transitions transforming neutrons to protons and vice versa even if the final state would consist of dark nucleon sequence. The nuclear binding energy per nucleon for light nuclei is around 7 MeV and looks somewhat too large: note however that 1/β₀=n>1 is possible for the horizontal monopole flux tubes and is consistent with quantum criticality.

   Could one think that the p-adic cooling of M₈₉ nuclei to ordinary nuclei begins with their decay to M₈₉ nucleons such that the gravitational cyclotron energy for M₈₉ nucleons (, which does not dependence on the mass) at the monopole flux tubes with magnetic field strength of about B_{end}=.2 Gauss provides the energy needed to split the M₈₉ nuclear bonds so that the outcome is free M₈₉ nuclei unstable against the p-adic cooling to M₁₀₇ nuclei?

What could these results mean? Solar wind contains nuclei up to Fe, the heaviest nucleus produced in ordinary fusion and there is also a mysterious finding that the solar surface contains solid iron. One can consider several options.

1. Quantum criticality suggests several values for ℏ_gr corresponding to different values of β₀. Suppose that horizontal flux tubes at the solar surface have β₀∼ 1 whereas the gravitational U-shaped flux tubes with β₀∼ 2^-11 are radial.

   For β₀≥ 1 horizontal flux tubes with cyclotron energy about 1.38 MeV, ordinary nuclear reactions and even fusion might take place near the surface of the Sun. Could dark cyclotron photons from monopole flux tubes with 1≤ 1/β₀ ≤ 7 transforming to ordinary gamma radiation ignite the ordinary nuclear fusion in the surface layer and in this way explain why the standard model works so well?

2. The second, more radical, option is that the dark nuclei as products of dark fusion and having a binding energy scale of 2.6 GeV, possibly produced as the outcome of the M₈₉→ M₁₀₇ transition, produce first ordinary nucleons as the dark cyclotron photons with energy about 2.6 GeV split the M₈₉ nuclear bonds. These nucleons could form dark nucleons with nuclear binding energy about 10 keV, which in turn transform to ordinary nucleons as in dark fusion. Note that also the ordinary nuclear fusion could be reduced to dark fusion involving tunnelling by two BSFRs. If so, the attempts to realize nuclear fusion in nuclear reactors would be based on wrong assumptions about the underlying physics.
3. The density of the Sun at the photosphere is ∼ 10^-4 kg/m³ whereas the average density of the Sun is 1.41× 10³ kg/m³ (the average density of Earth is 5.51× 10³ kg/m³). The density is extremely low so that surface fusion at photosphere cannot explain the energy production of the Sun. The surface fusion layer should exist at some depth where the density is not far from the average density of the Sun. One candidate is a layer above the surface of the solar core. As found its thickness should be of the order of Earth radius.
4. The solar core, usually believed to be the seat of hot fusion, has radius about .2× R_S and its mass is roughly .8 percent of the mass of the Sun. This brings in mind the strange finding that .5 percent of the mass needed to explain the fusion energy power produced in the solar core seems to be missing. Could this missing mass be associated with a layer near the surface layer of the Sun and could it be responsible for the solar wind?

   The radius of Earth is 1/109 times the radius of the Sun and the gravitational Compton length L_gr,S of the Sun equals to L_gr,S=R_E/2 and is therefore .5 percent of R_S! What could these coincidences mean? If the Sun has a layer of thickness Δ R with the average density of the Sun, one has Δ M/M = 3 (ρ_S/ρ_E)Δ R/R ∼ .75 Δ R/R. For Δ R=R_E one obtains Δ M/M ∼ .75 per cent, not far from .5 per cent. Could the Sun have a gravitationally dark layer of thickness about R_E with density .75 ρ_S. This is indeed assume in the proposed model (see this).

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.

Martian dichotomy from the TGD point of view

Mars has a very strange property called Martian dichotomy (see this). The Northern and Southern hemispheres of Mars are very different. The crust is significantly thicker in southern highlands than in northern lowlands. The mountains at southern highlands rise even 6 kilometers higher than in northern lowlands. Southern rocks are magnetized suggesting that Mars has had a large scale magnetic field. Mars still has short scale magnetic fields as the appearance of Martian auroras tells. Southern highlands appear to be older than the northern lowlands: the age is estimated from the density of impact craters. It is also believed that there has been a vast water ocean in northern lowlands.

Several explanations have been proposed. A mega-impact or several impacts could have produced the depression in the crust in the northern lowlands area. Second explanation is in terms of plate tectonics which would be asymmetric.

Also Mars has analogues of earthquakes. They could be called marsquakes. It is claimed that the study of the marsquakes has led to the understanding of the Martian dichotomy: see the popular article and the original article. Its origin would relate to the dynamics deep inside the planet. The new finding is that the seismic waves associated with the marsquakes lose energy quicker in southern highlands. This would mean that the temperature in highlands is higher. These findings suggest that the asymmetry is caused by the internal dynamics of Mars rather than impacts.

What could one say about the Martian dichotomy in the TGD framework. TGD adds two new pieces to the puzzle.

1. Moon has an analogous asymmetry but now the hemispheres correspond to the hemisphere that we see always and the hemisphere we never see. This is due to the phase locking of the spinning rotation of the Moon with its orbital rotation around Earth. The TGD based model (see this) assumes that Earth has lost its upper layer in a mini big bang (see this and this), which then formed the Moon. The inner and outer surfaces of the Moon would correspond to the lower and upper boundary of the layer respectively and this would explain their difference.
2. The crazy idea is that the northern and southern hemispheres of Mars could have lost different masses in an asymmetric mini big bang leading to the birth of Phobos and Deimos, the two Moons of Mars (this). The asymmetry should reflect itself in the properties of these moons. The moons have an irregular shape. Phobos has a diameter of 22.2 km, mass 1.1× 10¹⁶ kg, and semimajor axis 13.5 km. Deimos has a diameter of 12.6 km, mass 1.5 × 10¹⁵ kg, and semimajor axis 23.5 km.
3. This suggests the associations northern hemisphere-more massive Deimos-thicker crust-earlier-farther from Mars and southern hemisphere-lighter Phobos-thinner crust-later-nearer to Mars.

   The more massive Deimos would have originated in a mini big bang throwing out a considerably thicker layer from the northern Martian hemisphere. This would explain the thinner northern crust. Large fraction of the magnetic field associated with the surface layer would have blown out. The TGD view of magnetic fields of the Earth and Sun the monopole flux tube part of the magnetic fields would have a part concentrated in a surface layer. Deimos would have originated later than Phobos. One could understand why the southern hemisphere has thicker crust, why it has more impact craters and therefore looks older, and why it still has a magnetic field consisting of monopole flux tubs. The orbital parameters do not depend on the mass of the Moon (Equivalence Principle). Deimos would have however originated earlier and received a recoil momentum and would be now farther from Mars and Phobos.

The key question concerns the energetics of the transition. Where comes the energy compensating the reduction of the gravitational binding energy. An analogous question is encountered in the model for the formation of the Moon as a mini Big Bangs throwing a spherical layer from the surface of Earth. It is also encountered in the TGD version of the Expanding Earth model (see this and this) assuming that the radius of Earth grew by a factor 2 in a relatively short time scale and induced Cambrian Explosion as life from underground oceans bursted to the surface. Mini Big Bangs would also cause the formation of planets as a surface layer of a star explodes (see this and this). Also supernovas would be explosions of this kind. Micro Big Bangs could give rise to solar wind and solar eruptions (see this).

The magnetic fields should play an important role so that an estimate for the cyclotron energy in the case of a solar magnetic field is in order.

1. For the Earth the cyclotron frequency of proton in the endogenous magnetic field, with a nominal value B_end = .2 Gauss assigned with the monopole flux tubes, is 300 Hz, and the corresponding energy is E_c= ℏ_gr,EeB/m_p= 4.6 eV. This energy is higher than the gravitational binding energy of protons of about 1 eV at the surface of Earth. This could make it possible for transition ℏ_gr,E→ ℏ or a transition 1/β₀=n→ n-1 to provide the energy needed for the explosion throwing a surface layer of the Earth giving rise to Moon.

   The existence of this kind a layer and reduction of ℏ_gr, say a transition 1/β₀= 2→ 1 could make energetically possible also the expansion of the radius of the Earth by a factor 2.

2. What does one obtain in the case of Mars? Could the gravitational binding energy be compensated by the liberation of dark cyclotron energy as the value ℏ_gr=GMm_p/β₀ for Mars is reduced to a smaller value. The ratio of the mass of Mars to that of Earth is M_Mars/M_E∼ .1. If the monopole flux tubes carry a magnetic field of strength B_end,E=.2 Gauss the cyclotron energy of the proton is scaled down to .46 eV. The gravitational binding energy for protons at the surface of the Earth is about 1 eV and at the surface of Mars about .1 eV. Also now the liberation of the dark cyclotron energy for protons in a phase transition increasing the value of β₀ could make the explosion of the surface layer possible.

See the article Moon is a mysterious object and the chapter Magnetic bubbles in TGD Universe: part I.

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.

Could right handed neutrinos predicted by TGD correspond to galactic dark matter?

Right handed neutrinos are electroweak and color ghosts and have been proposed as a candidate for particles explaining galactic dark matter (see this).

One of the big problems of the standard model is that neutrinos are massive and this requires that neutrinos must also have right-handed modes ν_R.

TGD space-time is 4-surface in H=M⁴×CP₂ and quark and lepton spinor fields correspond to those of H. Spinor fields of H are induced to the space-time surfaces. TGD predicts standard model symmetries and fields but differs from the standard model in that also ν_R is predicted. The massless ν_R modes are covariantly constant in CP₂ degrees of freedom and have no electroweak nor color interactions and are electroweak ghosts. In TGD ν_R has also massive modes which are not color ghosts and together with left handed neutrino mode can give rise to massive neutrinos.

Right-handed neutrinos have only gravitational interactions and are therefore excellent candidates for the fermionic dark matter. The recent TGD based view of dark matter does not favor the idea that right handed neutrinos could have anything to do with galactic dark matter.

One must be cautious however. The TGD based explanation for galactic dark energy is in terms of Kähler magnetic and volume energy involving only geometrized bosonic fields. Could quantum classical correspondence imply that the classical energy corresponds to the energy assignable to right-handed neutrinos? Classically dark matter would correspond to the sum of Kähler magnetic energy and volume energies. Kähler magnetic energy can be regarded as an electroweak contribution. Since volume energy depends only on the induced metric, it could correspond to right-handed neutrinos.

See the articles New Particle Physics Predicted by TGD: Part I and New Particle Physics Predicted by TGD: Part II.

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.

What are the mysterious structures observed in the lower mantle?

I learned of very interesting results related to geology. The Dailymail popular article (see this) tells about massive structures in the Earth's deep mantle below Pacific Ocean near the mantle-core boundary. The article "Full-waveform inversion reveals diverse origins of lower mantle positive wave speed anomalies" of Schouten et al published in Scientific reports (see this) describes the findings.

There are cold regions deep within the Earth where seismic waves behave in unexpected ways. The chemical composition can involve heavy elements in these regions. These in-homogenities lead to the increase of the sound velocity. These regions, located 900 to 1,200 kilometers beneath the Pacific Ocean, defy expectations based on conventional plate tectonics theories. These kinds of structures can result from the subduction of continental plates leading to the sinking of a plate to the mantle. There are however no subduction records in the Ocean regions so that the mechanism must be different.

It seems that the recent view of the dynamics of the Earth's mantles is in a need of a profound updating. It has been proposed that the structures could be the remnants of ancient, silica-rich materials from the early days of the Earth when the mantle was formed billions of years ago. Alternatively, they may be areas where iron-rich rocks have accumulated over time due to the constant movement of the mantle. However, researchers are still unsure about the exact composition of these deep Earth structures.

Here is the abstract of the article of Schouten et al.

Determining Earth s structure is paramount to unravel its interior dynamics. Seismic tomography reveals positive wave speed anomalies throughout the mantle that spatially correlate with the expected locations of subducted slabs. This correlation has been widely applied in plate reconstructions and geodynamic modelling. However, global travel-time tomography typically incorporates only a limited number of easily identifiable body wave phases and is therefore strongly dependent on the source-receiver geometry.

Here, we show how global full-waveform inversion is less sensitive to source-receiver geometry and reveals numerous previously undetected positive wave speed anomalies in the lower mantle. Many of these previously undetected anomalies are situated below major oceans and continental interiors, with no geologic record of subduction, such as beneath the western Pacific Ocean. Moreover, we find no statistically significant correlation positive anomalies as imaged using full-waveform inversion and past subduction. These findings suggest more diverse origins for these anomalies in Earth s lower mantle, unlocking full-waveform inversion as an indispensable tool for mantle exploration.

Here some terminology is perhaps in order. Seismic waves are acoustic waves and their propagation in the mantle is studied. Positive speed anomaly means that sound speed is higher than expected. The lowering of temperature or increase of density such as presence of iron, silica, or magnesium can cause this kind of anomalies. The Pacific ocean and the interior regions of plates do not have any subduction history so that the slabs cannot be "slabs" as pieces of continental plates, which have sunk to the mantle.

Why these findings are interesting from the TGD point of view is that TGD suggests that the Cambrian Explosion roughly 500 million years ago was accompanied by a rather rapid increase of the Earth's radius by factor 2 (see this), this and this). In the TGD inspired cosmology, the cosmic expansion occurs as rapid jerks and Cambrian Explosion would be associated with this kind of jerk. This sudden expansion would have broken the crust to pieces and led to the formation of oceans as the underground oceans bursted to the surface. The multicellular life evolved in the underground oceans would have bursted to the surface and this could explain the mysterious sudden appearance of complex multicellular life forms in the Cambrian Explosion. In this event tectonic plates and subduction would have emerged.

I have not earlier considered what happened in the lower mantle in the sudden expansion of Earth increasing its radius by factor 2 and giving rise to the Cambrian Explosion. Did these kinds of cracks occur also in the mantle-core boundary and lead to the formation of the recently observed structures also below regions where there is no geologic record for subduction? Could at least some regions which are believed to be caused by the sinking of parts of continental plates have such structure?

Could the Cambrian explosion be a mini Big Bang that happened in the lower mantle and forced the motion of the upper layers leading to the increase of the radius of Earth? The longstanding problem has been the identification of the energy needed to overcome the gravitational force. The order of magnitude of the gravitational binding energy per nucleon is about 1 eV at the surface of the Earth and decreases like M(R)/M_E)/R ∝ R² below it. How did the matter above the monopole flux tube layers get this energy?

1. Since the monopole flux tubes are the key actors, a natural first guess is that there was a layer of dark protons at monopole flux tubes in the lower mantle, say above the core, and that the gravitational energy is compensated by the cyclotron energy of dark proton with gravitational Planck constant ℏ_gr(M(below)) at monopole flux tube carrying a magnetic field of order of magnitude of endogenous magnetic field. The value of B_end need not be the same as its value B_end= .2 Tesla at the surface of the Earth.
2. If the monopole flux behaves like 1/R³, as the dipole character of the Earth's magnetic field suggests, and the mass appearing in the gravitational Planck constant is the mass M(R)=(R/R_E)³ M_E below the monopole flux tube layer is used, the cyclotron energy is the same as at the surface of the Earth. In the explosion, the value of ℏ_gr would be reduced dramatically, perhaps to ℏ and the cyclotron energy would be liberated.

   In the interior of the Earth, the gravitational potential energy for mass m is of form E_gr= GM_EmV_gr(R), V_gr(R)=R²/2R_E³-(3/2)/R_E and approaches in the center of the Earth the value -(3/2)GM_Em/R_E and at the surface of the Earth to the value - GM_Em/R_E .

3. All nuclei must receive the cyclotron energy compensating the gravitational binding energy and a larger fraction should therefore be dark before the explosion. The gravitational Planck constant ℏ_gr= GMm/β₀ of a nucleus is proportional to its mass number so that the cyclotron energy ∝ ZeB/m does not depend on the mass number A of the ion of mass m≃ Am_p. For 1/β₀=1, the extreme option is that the entire Earth's interior contains gravitationally dark nuclei meaning that there is a large negatively charged exclusion zone created in the Pollack effect, perhaps giving rise to the electric body assignable to the Earth. Can this be consistent with what is known about the Earth's history?

   For 1/β₀=2^-11 assignable with the magnetic body of the Sun-planet system, the value of cyclotron energy would be about 10 keV, which happens to be the energy scale of "cold fusion" identified as dark fusion in the TGD framework (see this). Could the formation of dark nuclei with nucleon radius of order electron Compton length and with a dark nuclear binding energy of order 10 keV involve the formation of the monopole flux tubes with this dark cyclotron energy?

See 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.

Holography = holomorphy vision and elliptic functions and curves in TGD framework

Holography=holomorphy principle allows to solve the extremely nonlinear partial differential equations for the space-time surfaces exactly by reducing them to algebraic equations involving an identically vanishing contraction of two holomorphic tensors of different types. In this article, space-time counterparts for elliptic curves and doubly periodic elliptic functions, in particular Weierstrass function, are considered as an application of the method.

See the article Holography = holomorphy vision and elliptic functions and curves in TGD framework.

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.

What could happen in the transition between hadronic phase and quark gluon plasma?

Quanta Magazine (see this) told about the work of Rithya Kunnawalkam Elayavalli, who studies the phase transition between quark-gluon phase and hadron phase, which is poorly understood in QCD. Even hadrons are poorly understood. The reason for this is believed to be that perturbative QCD does not exist mathematically at low energies (and long distances) since the running QCD coupling strength diverges.

Neither hadrons nor the transition between quark gluon phase and hadron phase are well-understood. The transition from hadron phase to quark-gluon phase interpreted in QCD as color deconfinement is assumed to occur but the empirical findings are in conflict with the theoretical expectations. In TGD the interpretation for the observed transition is very different from that inspired by QCD (see this and this).

1. In TGD hadrons correspond to geometric objects, space-time surfaces, and one way to end up with TGD is to generalize hadronic string models by replacing hadronic strings with string-like 3-surfaces. These string-like 3-surfaces are present in the TGD Universe in all scales and I call them monopole flux tubes and they appear as "body" parts of field bodies for the geometrization of classical fields in TGD.
2. The TGD counterpart of the deconfinement transition need not be deconfinement as in QCD. What is clear is that this transition should involve quantum criticality and therefore long range fluctuations and quantum coherence.

What could this mean? Number theoretic vision of TGD comes to the rescue here.

1. TGD predicts a hierarchy of effective Planck constants labelling phases of ordinary matter. The larger the value of h_eff, the longer the quantum coherence length, which in TGD has identification as the geometric size scale of the space-time surface, say hadronic string-like object, assignable to the particle.
2. Does the transition involve quantum criticality so that a superposition of space-time surfaces with varying values of h_eff≥ h is present. The size scale of hadron proportional to h_eff would quantum fluctuate.
3. The number theoretic view of TGD also predicts a hierarchy of p-adic length scales. p-Adic mass calculations strongly suggest that p-adic primes near certain powers of 2 are favored. A kind of period doubling would be involved. In particular, Mersenne primes and their Gaussian counterparts are favored. p-Adic prime p is identified as ramified prime for an extension E of rationals h_eff= nh_0 to the dimension of E. p and h_eff correlate. p-Adic prime p characterizes p-adic length scale proportional to p^1/2. Mass scale is inversely proportional to 1/p^1/2.
4. In particular, the existence of p-adic hierarchies of strong interaction physics and electroweak physics are highly suggestive. Mersenne primes M(n)= 2ⁿ-1 and their Gaussian counterpars M(G,n)= (1+i)ⁿ-1 would label especially interesting candidates for the scaled up variants of these physics.

   Ordinary hadron physics would correspond to M₁₀₇. The next hadron physics corresponding to M₈₉ would have a baryon mass scale 512 times higher than that of ordinary hadronic physics. This is the mass scale studied at LHC and there are several indications for bumps having interpretation as M₈₉ mesons having masses scaled by factor 512. People tried to identify these bumps in terms of SUSY but these attempts failed so that bumps were forgotten.

So, what might happen in the TGD counterpart of the deconfinement transition?

1. Could the color deconfinement be replaced by a transition from M₁₀₇ hadron physics to M₈₉ hadron physics in which hadrons for the ordinary value h_eff=h have size 1/512 smaller than the size of the ordinary hadrons. At quantum criticality however the size would be that of ordinary hadrons. This is possible if one has h_eff=512h. At high enough energies h_eff =h holds true and M₈₉ hadrons are really small.
2. Various exotic cosmic ray events (fireballs, Gemini, Centauro, etc...) could correspond to these events (see this and this). In the TGD inspired model of the Sun, M₈₉ hadrons forming a surface layer of the Sun would play a fundamental role. They would produce solar wind and solar energy as they decay to ordinary M₁₀₇ hadrons (see this).

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 difference between real and imagined

Gary Ehlenberg sent a link to an interesting Quanta Magazine article discussing the differenced between imagination and perception (see this).

Some time ago I had discussions with my friend who claimed that she really sees the things that she imagines. She also has a very good memory for places, almost like a sensory memory. I had thought that this ability is very rare, for instance idiot savants have sensory memories.

So, do I suffer from aphantasia, inability to imagine sensorily? I have sensory perceptions during dreams. I can see and can hear in the hypnagogic state at the border of sleep and awake. In my great experience I quite concretely saw my thoughts and this led to the urge to understand what consciousness is. I can imagine but I do not usually see any images: only after emotionally intense discussions with some-one can I almost-hear the spoken words. So, do I suffer from aphantasia in my normal state of mind?

TGD inspired view of neuroscience leads to a model for the difference between the real and imagined percepts based on my own experience (see this, this, this and this). Imagined percepts would be generated by a virtual sensory input from the field body realized as dark photon signals. They would not reach the retinas but end up at some higher level in the visual neural pathway such as lateral geniculate nuclei of the pineal gland, the "third eye". Pineal gland is a more plausible candidate. In some animals it serves as a real third eye located outside the head. Could it serve as the seat of auditory and other imagined mental images?

At least in my own case, seeing with the pineal gland would usually be sub-conscious to me. What about people who really see their imaginations? Could they consciously see also with their pineal glands so that the pineal gland would define mental image as a subself? Or could some fraction of the virtual signals from the field body reach the retinas? For the people suffering aphantasia, the first option predicts that pineal gland corresponds to a sub-sub-self, which does no give rise to a mental image but a mental image of a sub-self.

Also sensory memories are possible. Does this proposal apply also to these. My grandson Einar is 4 years old. He read to me a story in a picture book that his parents had read to him. Einar does not yet recognize letters nor can he read. He seems to have a sensory memory and repeated what he heard. Maybe all children have this kind of sensory memories but as cognitive skills develop they are replaced by conceptual memories, "house" as representative for the full picture of house means a huge reduction in the number of bits and therefore in the amount of metabolic energy needed. Could it be that aphantasia is the prize paid for a high level of cognition?Could this distinguish between artists and thinkers?

See the chapter TGD Inspired Model for Nerve Pulse.

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.