Comments inspired by the article Jenny Nielsen and Lucis Semita about provability of Riemann Hypothesis

Jenny Lorraine Nielsen and Lucis Semita have published and interesting and very speculative article (this)) related to the provability of Riemann Hypothesis (RH). Here is th abstract of the article.

We then prove the truth of RH in two logically independent ways. First, we introduce strong reflection principles, termed Bundle Cardinal Axioms, under which the Λ obstruction collapses and RH becomes provable. We show that these axioms are Π₀-1 conservative over ZFC, implying that such provability entails truth in the standard model of arithmetic.

Second, independently of any reflection axiom, we prove that if RH is Π₀-1 and ZFC-independent, then it must already be true in the standard model, by arithmetic witness reflection. Together these results constitute a proof that RH is independent of ZFC and true in N, with its truth derivable by two independent logical routes.

I am not metamathematician and many notions involved are unfamiliar to me so that I cannot say anything about the proposed proof of Riemann hypothesis (I have considered a possible proof based on physics inspired arguments (see this). There notions are however very interestingand I cannot resist the temptation to look for the TGD counterparts of some of these notions. I have already earlier consider Gödel’s theorems from the TGD point of view from the point of view holography = holomorphy vision providing an exact solution of the field equations for space-time surfaces irrespective of the action principle as long it is general coordinate invariant and expressible in terms of induced geometry (see this).

1. The idea about the space of representational paradigms is an interesting notion. The meaning of arithmetic and analytic spectral paradigms of number theory are intuitively clear in the case of RH. Primes and the zeros of zeta are related (here Google language model helps): statements about prime distribution correspond to statements about the distribution of zeros of zeta and one can formulate Riemann hypothesis as statements about these distributions. This brings in mind number theory and geometry related in the geometric Langlands duality.

Wikipedia informs that Lambda calculus relates to mathematical logic and describes computation as function abstraction and application using variable binding and substitution. Lambda irreducibility means that a further simplification is not possible. One interpretation could be that the proof of the theorem as a path in the space to allowed sentences is the shortest one possible.

Could the TGD counterpart of Lambda calculus code for the rules for the decomposition of pieces of the non-deterministic space-time surface? What can happen at the loci of non-determinism would be told by the rules of the calculus. Could the analog of Λ irreducibility mean that the representation as a space-time surface is the simplest possible one. Is this implied by the field equations solved by holography = holomorphy principle.

Round trip translation brings to mind holonomy in Riemann geometry. The “linear” in linear paradigm of number theory and “curved” in curved analytic spectral paradigm cannot however relate to this holonomy. I understand that there are maps between the languages of the representational paradigms, maybe one could call them morphisms. Morphisms as correspondences in the most general sense can be many-to-one and even 1-to-many.

The singularities involve 1→2 vertices for fermion pair creation as turning of fermion backwards in time and having in TGD description involving smooth exotic structures possible only in 4-D space-time and edges of X⁴. One can speak of Brownian motion of a 4-surface X⁴ in H with discontinuous changes of the direction at the 3-D vertices as loci of the classical non-determinism. The counterparts of Feynman diagrams for fermions involve only 2-vertices and this means huge simplification (see this).

Could the obstruction mean that one of the maps or both between different paradigms are not homomorphisms implying that they are not equivalent? If so, both paradigms are needed for a full description.

M⁸-H duality can be 1-to.very-many in M⁸→H direction in a different way. At peak-like singularities of Y⁸⊂M⁸ a single point is mapped to an infinite number of CP₂ points since the tangent space at the singularity is not unique. The set of CP₂ points is 3-dimensional for the singular points. CP₂ type extremals with Euclidean metric are the building blocks of elementary particles in a geometric sense.

How does this relate to the provability of a statement, say, RH? I do not have the competence to comment the Cardinal Axiom and the two proofs of RH.

Holography= holomorphy principle would define the axioms of TGD as a physical theory. The space of space-time surfaces (“world of classical worlds” (WCW)) together with WCW spinor fields as correlates for Boolean logic define quantum Platonia. WCW could be seen as the space of proofs for theorems represented geometrically. The possibility to assign elements of various number fields, also of function fields, corresponds to the Gödel numbering assigning to a proof of a theorem as a space-time surface a Gödel number.]

Gödel theorems force us to conclude that there is an infinite hierarchy of axiom systems. In TGD, this hierarchy is implied by a finite measurement resolution. The polynomials appearing in the equations defining X⁴ resp. Y⁴ have numbers in an extension of rationals as Taylor coefficients. The extensions form infinite inclusion hierarchies. Also analytic functions are possible. The extension of rationals defines a natural discretization characterizing also a given axiomatics. The higher the complexity of the algebraic extension, the more powerful the axioms system is and becomes optimal at the limit of algebraic numbers. Iterations of polynomials define inclusion hierarchies for the axiom system. If the coefficient field consists of real numbers, real numbers characterize the axiomatics.

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/2025/12/comments-inspired-by-article-jenny.html