Some questions related to the maps g defining cognitive hierarchies realized as space-time surfaces

Rational maps g and possibly also their inverses which would be central in the realization of cognition and reflective hierarchies. These ideas are however far from their final form and in the following I try to imagine and exclude various alternatives. Some new results emerge.

1. Quantum realization of concepts as superpositions in the set of space-time surfaces defining the classical concept is more natural than the classical realization.
2. The roots of g_ºf correspond to classical non-determinism and would naturally correspond to generalized p-adicity and could also explain the p-adic length scale hypothesis.
3. The inverses g^-1 of the rational maps g correspond to algebraic functions unless g is analog of Möbius transformation. g^-1_ºf preserves the number of roots for f and decreases it for the iterate of g and in this case reduced complexity and negentropy. In the framework of TGD inspired theory of consciousness, this raises the question whether the quantum correlates for good and evil deeds as SFRs could correspond to maps of type g increasing algebraic complexity, information and quantum coherence and g^-1 possibly reducing them.

1. What could happen in the transition f→ g_ºf?

The proposal is that in SSFR the transition f→ g_ºf takes place. The number of roots becomes n-fold if g is a rational function of form P/Q. What could this transition mean physically? One can consider two options.

1.1 The option allowing quantum realization of concept

The nm roots (poles and zeros) for g _ºf , where f as m roots would be alternative outcomes of SSFR of which only a single outcome, or possible quantum superposition of the outcomes would be selected. What is so nice is that the classical non-determinism crucial for the TGD view of consciousness would follow automatically from the holography= holomorphy hypothesis without any additional assumptions.

Conservation laws conform with this view. All the alternative Bohr orbits would have the same classical conserved charges. The quantum superposition of the roots would represent a particular quantum realization of a concept and f→ g_ºf would mean a refinement of the quantum concept defined by f.

The hypothesis that the classical non-determinism correspond to the p-adic non-determinism would transform to a statement that different Bohr orbits associated g^_ºk define analogs for the sequences of k pinary digits if there are p outcomes for g_ºf. A possible interpretation would be in terms of a k-digit pinary digit sequence in powers of p. The largest integer would correspond to n=2^k for g^_ºk. The generalization of the notion of the notion of p-adic numbers for which p is replaced by a functional prime g and based on the generalization of Witt polynomials is suggestive. It remains unclear whether this could allow us to understand the generalization of the p-adic length scale hypothesis stating that a large prime p∼ p^k can be assigned to this set of Bohr orbits.

1.2 The option allowing a classical realization of concept

The union of nm space-time surfaces, where n is the degree of g and m is the number of roots of f, is generated in the step f→ g_ºf. The set of the nm space-time surfaces would give a classical realization of a concept as a set. Does this make sense? The first grave objection is that there is no continuous time evolution between f and g_ºf multiplying the number of space-time surfaces by n. Second objection relates to the conservation laws which seem to be violated. The third objection is that classical non-determinism is lost. It seems that this objection cannot be circumvented.

One can try to imagine ways to overcome the first two objections.

Option I: ZEO interpreted in the “eastern” sense in principle allows the creation of n space-time surfaces from each of the m space-time surfaces assignable with f. This is because the total classical charges of the zero energy states as sums of those for states at the boundaries of CD vanish. Zero energy state would be analogous to a quantum fluctuation.

Option II: In standard ontology, the classical realization of the concept as union of space-time surfaces defining its instances is possible only in a situation in which space-time surfaces are vacua or nearly vacua. Could this kind of surface serve as a template for the non-vacuum physical systems?

Cell replication, which would correspond to n=2 for g, was motivated by the consideration of both options, at least half-seriously. The instantaneous replication of the space-time surface representing the cell does not look sensible since the generation of biomatter requires a feed of metabolics and metabolic energy. Could a replicated field body serve as a kind of template for the formation of a final state involving two cells generated in f→ g_ºf? Could the replication occur at the level of the field body, proposed to control the biological body?

For Option II, conservation laws pose a problem for replication. In ZEO the classical charges of the space-time nm surfaces should be those associated with the passive boundary of CD and therefore same as those for f.

1. Could the space-time surfaces be special in the sense that the classical charges vanish? The vanishing of classical conserved charges is not possible unless the classical action reduces to Kähler action allowing vacuum extremals. The finite size of CD indeed allows by Uncertainty Principle a slight violation of the classical conservation laws assignable to the Poincare invariance (see this). This cannot be excluded and the original proposal (see this and this) indeed was that Kähler action defines the classical action by its unique property of having huge classical non-determinism defining the 4-D analog of spin-glass degeneracy (see this/sg} which could play a key role in biology.

If one assigns to M⁴ the analog of the Kähler structure (see this), this argument weakens since the induced M⁴ and CP₂ Kähler forms must vanish for the vacuum extremals. However, for a given Hamilton-Jacobi structure defining the M⁴ Kähler form, there exist space-time surfaces of this kind. They are Cartesian products of Lagrangian 2-manifolds of M⁴ and CP₂ defining vacuum string world sheets.

Holography= holomorphy principle, implying that Bohr orbits are minimal surfaces, seems to hold true for any classical action, which is general coordinate invariant and is determined by the induced geometry. For the Kähler action, the coefficient Λ of the volume term, defining the analog of cosmological constant, would vanish. Holography= holomorphy principle does not allow Cartesian products of Lagrangian 2-manifolds of M⁴ and CP₂. One could hope that their vacuum property could change the situation but this does not look an elegant option.

For the standard ontology, one can also consider another option. The classical action, and therefore the classical conserved charges, are for the twistor lift proportional to 1/α_K, where α_K is Kähler coupling strength. The conservation of charges would suggest α_K→ nα_K requiring h_eff→ h_eff/n in the n-fold multiplication. For h_eff=h this would require h→ h/n. This looks strange.

h need not however be the minimal value of h_eff and I have considered the possibility that one has h =n₀h₀ (see this), where n₀ corresponds to the ratio R²(CP₂)/l_P². CP₂ size scale would be given by Planck length l_P size scale but for h=n₀h₀ the size scale would be scaled up to R² ∼ n₀l_P², n₀ ∈ [10⁷,10⁸]. The estimate for n₀ is given by n₀=(7!)² having numbers 2,3,5,6,7 (primes 2,3,5,7) as factors (see this). R(CP₂) would naturally correspond to the M⁴ size of a wormhole throat. h could be reduced by a factor appearing in n₀ and there is some evidence for the reduction of h_eff by a small power of 2 (see this). This mechanism could work for a functional prime g characterized by prime p∈{2,3,5,7}. To classical realization of concept does not look realistic except possibly for Option I.

2. About the interpretation of the inverses of the maps g

What could be the interpretation of the inverse maps g^-1 for g=P/Q, assuming that they can occur? g^-1 is a multivalued algebraic function analogous to z^1/n. In f→ g^-1_ºf the roots r_n of f are mapped to g(r_n) so that their number does not increase. For the iterate of g, g^-1 means the reduction of the number of roots by 1/n. The complexity does not increase and can even decrease.

This is just the opposite for what occurs in f→ g_ºf. The increase of complexity is assigned with number theoretic evolution and NMP. Suppose for a moment that the inverses g^-1 are allowed. What could be their interpretation?

1. The sequence of the inverses g^-1 does not correspond to non-determinism and does not give rise to a refinement of either classical or quantum concept. There is no increase of complexity and it can be reduced for iterates.

Could the reduction of the cell to stem cell level as a reverse of cell differentiation, which occurs by cell replications, correspond at the level of the field body to a sequence of g^-1:s reducing the complexity. Could cancer correspond to this kind of process? This would conform with the interpretation in terms of the reduction of negentropy.

The first option is that the maps of type g^-1 are possible for both arrows of the geometric time. For the iterates of g, g^-1 destroys complexity and information and reduces the level of cognition in this case. g^-1 would obey anti-NMP in this case. Both maps g and g^-1 make possible a trial and error process. If an iterate of g is not involved, the roots r_n of h_ºf are mapped by g to roots g(r_n) and the number of roots is preserved. It is not clear whether the algebraic complexity is increased or reduced.

This suggests that NMP (see this) is not lost if both maps of type g and g^-1 are allowed? Furthermore, there is a lower bound for algebraic complexity but no upper bound so that it seems that NMP remains true even if maps of type g^-1 are allowed.

Any quantum theory of consciousness should be able to say something about the quantum correlates of ethics (see this). In TGD, one can assign the notion of good to state function reductions (SFRs) inducing the increase of quantum coherence occurring in a statistical sense in SFRs. It would correspond to the increase of algebraic complexity and would be accompanied by the increase of h_eff and the amount of potentially conscious information. Is evil something something analogous to a thermodynamic fluctuation reducing entropy or can one speak of an active evil? Could the notion of evil as something active be assigned with the occurrence of maps of type g^-1?

The maps of type g and g^-1 are reversals of each other and differ unless they act as symmetries analogous to M”obius transformations. Could they be assigned with SSFRs with opposite arrows of geometric time? If so, negentropy would not increase for both arrows of the geometric time and there would be a universal arrow of time analogous to that assumed in standard thermodynamics and defined by negentropy increase. If a universal arrow of time exists, it should somehow relate to the violation of time reflection symmetry T. To me this option does not look plausible.

If this is the case, the trial and error process allowed by ZEO and based on pairs of BSFRs would involve a map of type g^-1 induced by SSFRs whereas the second BSFR would correspond to a map of type g. The sequence of SSFRs after the first BSFR would preserve or even reduce complexity and would mean starting from a new state at the passive boundary (PB) of CD. If the first BSFR is followed by a sequence of SSFRs of type g, it in general leads to a more negentropic new initial state at PB. See the article A more detailed view about the TGD counterpart of Langlands correspondence 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.

Source: https://matpitka.blogspot.com/2025/04/some-questions-related-to-maps-g.html