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FTNILO: Explicit Multivariate Function Inversion, Optimization and Counting, Cryptography Weakness and Riemann Hypothesis Solution Equation with Tensor Networks

Alejandro Mata Ali

TL;DR

The paper introduces Field Tensor Network Integral Logical Operator (FTNILO), a continuous-variable extension of the MeLoCoToN framework, to derive explicit equations for inverting and optimizing multivariate functions and for counting zeros of functions such as the Riemann zeta function. By tensorizing logical circuits into Field Tensor Networks and using delta-function constraints, FTNILO converts multivariate problems into tractable, explicit equations that yield the solution vector, the number of solutions, and, in principle, properties like zero locations. The work recovers MeLoCoToN in the discrete limit, establishes mathematical implications such as universal inversion for injective/bijective functions, and discusses cryptographic weaknesses implied by explicit inversion. Applied to the Riemann hypothesis, the framework produces explicit FTNILO equations whose delta-consistency properties could, in principle, certify or refute the hypothesis, illustrating a novel bridge between tensor networks, continuous function inversion, and foundational problems in number theory.

Abstract

In this paper, we present a new formalism, the Field Tensor Network Integral Logical Operator (FTNILO), to obtain the explicit equation that returns the minimum, maximum, and zeros of a multivariable injective function, and an algorithm for non-injective ones. This method extends the MeLoCoToN algorithm for inversion and optimization problems with continuous variables, by using Field Tensor Networks. The fundamentals of the method are the conversion of the problem of minimization of $N$ continuous variables into a problem of maximization of a dependent functional of a single variable. It can also be adapted to determine other properties, such as the zeros of any function. For this purpose, we use an extension of the imaginary time evolution, the new method of continuous signals, and partial or total integration, depending on the case. In addition, we show a direct way to recover both the tensor networks and the MeLoCoToN from this formalism. We show some examples of application, such as the Riemann hypothesis resolution. We provide an explicit integral equation that gives the solution of the Riemann hypothesis, being that if it results in a zero value, it is correct; otherwise, it is wrong. This algorithm requires no deep mathematical knowledge and is based on simple mathematical properties.

FTNILO: Explicit Multivariate Function Inversion, Optimization and Counting, Cryptography Weakness and Riemann Hypothesis Solution Equation with Tensor Networks

TL;DR

The paper introduces Field Tensor Network Integral Logical Operator (FTNILO), a continuous-variable extension of the MeLoCoToN framework, to derive explicit equations for inverting and optimizing multivariate functions and for counting zeros of functions such as the Riemann zeta function. By tensorizing logical circuits into Field Tensor Networks and using delta-function constraints, FTNILO converts multivariate problems into tractable, explicit equations that yield the solution vector, the number of solutions, and, in principle, properties like zero locations. The work recovers MeLoCoToN in the discrete limit, establishes mathematical implications such as universal inversion for injective/bijective functions, and discusses cryptographic weaknesses implied by explicit inversion. Applied to the Riemann hypothesis, the framework produces explicit FTNILO equations whose delta-consistency properties could, in principle, certify or refute the hypothesis, illustrating a novel bridge between tensor networks, continuous function inversion, and foundational problems in number theory.

Abstract

In this paper, we present a new formalism, the Field Tensor Network Integral Logical Operator (FTNILO), to obtain the explicit equation that returns the minimum, maximum, and zeros of a multivariable injective function, and an algorithm for non-injective ones. This method extends the MeLoCoToN algorithm for inversion and optimization problems with continuous variables, by using Field Tensor Networks. The fundamentals of the method are the conversion of the problem of minimization of continuous variables into a problem of maximization of a dependent functional of a single variable. It can also be adapted to determine other properties, such as the zeros of any function. For this purpose, we use an extension of the imaginary time evolution, the new method of continuous signals, and partial or total integration, depending on the case. In addition, we show a direct way to recover both the tensor networks and the MeLoCoToN from this formalism. We show some examples of application, such as the Riemann hypothesis resolution. We provide an explicit integral equation that gives the solution of the Riemann hypothesis, being that if it results in a zero value, it is correct; otherwise, it is wrong. This algorithm requires no deep mathematical knowledge and is based on simple mathematical properties.
Paper Structure (28 sections, 18 theorems, 110 equations, 15 figures)

This paper contains 28 sections, 18 theorems, 110 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Field Tensor Network Integral Logical Operator (FTNILO)
  3. Logical Circuits
  4. Inversion Problem
  5. Computing the zeros from the sum of a sequence
  6. Optimization Problem
  7. Optimizing a quadratic function with linear chain interaction to one neighbor
  8. Optimizing the cos-sin function
  9. Optimizing a quadratic function with linear chain interaction to one neighbor with the extra restriction $\sum_{i=0}^{n-1}a_ix_i<W$.
  10. Circuit Field Tensorization
  11. Inversion Problem
  12. Optimization Problem
  13. Solution extraction
  14. Inversion Problem
  15. Optimization Problem
  16. ...and 13 more sections

Key Result

Theorem 2.1

$$ Given a vector-valued function $f:\mathcal{X}\rightarrow \mathcal{Y}$, with $\mathcal{X}\subseteq \mathbb{R}^n$ and $\mathcal{Y}\subseteq \mathbb{R}^m$, and a set of values $\vec{Y}\in \mathcal{Y}$, if $\exists! \vec{X}\in\mathcal{X} \backslash \ \vec{Y}=f(\vec{X})$, then there always exists an being $\Omega_j$ the FTN associated to the $j$-th variable determination and $\Phi(\vec{x},\vec{y}

Figures (15)

  • Figure 1: Tensor Network with four tensors.
  • Figure 2: Logical circuits for a) Computing the zeros from the sum of a sequence, b) Optimizing a quadratic function with linear chain interaction to one neighbor, c) Optimizing the cos-sin function.
  • Figure 3: Logical Circuit for Optimizing a quadratic function with linear chain interaction to one neighbor with the restriction $\sum_k^{n-1}a_kx_k<10$.
  • Figure 4: Tensorized circuits for a) Computing the zeros from the sum of a sequence, b) Optimizing a quadratic function with linear chain interaction to one neighbor, c) Optimizing a quadratic function with linear chain interaction to one neighbor with the restriction $\sum_k^{n-1}a_kx_k<10$.
  • Figure 5: Iteration process to determine first, second and third variable values in the inversion problem for \ref{['eq: first inversion function']}.
  • ...and 10 more figures

Theorems & Definitions (22)

  • Theorem 2.1: General inversion equation for one solution
  • Theorem 2.2: Checker of solutions
  • Theorem 2.3: Infinite number of general inversion equation for one solution
  • Theorem 2.4: General optimization equation with no degeneration
  • Theorem 2.5: General optimization equation with no degeneration out of the limit
  • Theorem 2.6: Infinite number of general optimization equations with no degeneration
  • Theorem 2.8: Number of solutions equation of inversion problems
  • Theorem 2.9: Protocol of obtaining the solution of inversion problems with degeneration
  • Theorem 2.10: Protocol of obtaining all the solutions of inversion problems with degeneration
  • Theorem 4.1: Injective function inverse
  • ...and 12 more