Summary of the Riemann Hypothesis and the Siegel Zero Problem under the FNT Paradigm

Author: Zhang Suji, Luoyang, Henan

I. Core Positioning of FNT and Prime Fractions

Fractal Number Theory (FNT), relying on the FCFG fractal continued-fraction geometric system, establishes a completely new topological analysis paradigm for prime distribution and the structure of complex zeros. Unlike traditional analytic number theory, which relies on quantitative research methods such as series approximation, inequality estimates, and sieve computations, FNT proceeds from the intrinsic scale structure of numbers, topological nesting rules, and manifold tension conservation, thereby achieving a structural and source-level qualitative reconstruction of core problems related to primes.

In the FNT system, the prime fraction is the only fundamental scale unit for characterizing primes.

For all primes p, define the prime fraction:

\rho_p=\frac{1}{p}

Its globally strictly fixed range is:

\boldsymbol{\rho_p \in (0,\;0.5]}

This interval is an innate topological constraint inherent in primes and the underlying law of the entire prime distribution system. It does not depend on analytic continuation or function fitting; it is a purely structural geometric axiom.

II. Dual Core Role of the Range of Prime Fractions

The scale boundary (0,0.5] of prime fractions uniformly governs the core structure of two top-level problems in analytic number theory:

1. Constraining the Source Position of the Critical Line of the Riemann Hypothesis

The maximum scale upper bound of prime fractions is 0.5, which is the limiting boundary of all prime scale primitives.

In the FNT complex topological manifold, the essence of the zeros of the \zeta function is the global interference cancellation of infinitely many prime-fraction operators.

Only at the symmetry axis \mathrm{Re}(s)=0.5, corresponding to the scale upper bound, can the global prime-fraction nesting system achieve perfect conjugate symmetry, scale balance, and phase cancellation.

It is thus proved that:

The Riemann critical line is equal to the maximum topological scale boundary of prime fractions and is an inevitable geometric result of the scale structure of primes.

2. Completely Ruling Out the Possibility of Siegel Zeros

Prime fractions satisfy the core topological axiom:

\forall p,\;\frac1p>0

Prime fractions always retain topological thickness and can neither be reduced to zero nor vanish.

If a Siegel zero near \sigma\to1 existed, it would mean that in a region with almost no deep nesting of prime fractions and extremely flattened topological tension, all positive-scale operators are forcibly superposed to zero. Such behavior would directly violate the tension conservation of the FNT manifold and cause topological tearing; it is a state impossible to establish at the geometric level.

It is thus proved from the source that:

Anomalous zeros near 1 have no topological condition for existence, and Siegel zeros do not exist.

III. Paradigm Comparison Summary

Previous research has always worked within the analytic framework to carry out quantitative restrictions, interval compression, and distance estimates, and can only impose boundary constraints on zero behavior.

This paper, relying on the FNT system of the Luoyang School of Mathematics, uses the same set of prime-fraction topological rules and the same fixed range to complete, at once, the topological reconstruction and construction of a proof framework for two core problems:

· Using the scale upper bound to lock the unique position of the Riemann critical line

· Using permanently positive thickness to annihilate the possibility of the existence of Siegel zeros

IV. Final Conclusion of the School

All deep laws of prime distribution are not numerical results of analytic operations, but the external manifestation of the innate topological scale \boldsymbol{(0,\;0.5]} of prime fractions.

The FNT paradigm unifies the scale of primes, the structure of complex manifolds, and the topological balance of zeros, and completes a structural breakthrough that traditional analytic number theory cannot achieve.

— Luoyang School of Mathematics, Zhang Suhang