Do numbers exist independently of human invention?
It has been a matter of debate, rather a philosophical more than a scientific. Geometry and arithmetic are deeply rooted in human culture, but it's difficult to assert their roles in biology.
In the What is Life? essay, written by Erwin Schrodinger, a concept of negative entropy is introduced. It has been acknowledged by Francis Crick and James Watson as an influential work in their discovery of DNA.
We have introduced a theoretical formulation of the scale-space information tunnelling1, describing the dynamics of synergistic waves coupling, in scale. Binding information quanta dynamically, via coordinate transformations, express multidimensional information with 5-dimensional decomposition.
Reversely, the quantum code outlines evolving patterns of the multidimensional information, in the scale-space. It is an active research area of quantum information, computing, analysis, and control.
In the theory of stochastic resonance synergies, we propose a genotype information processing2. A cascade of quadrupoles computation has been applied in computational physics and neuroscience, aiming to understand their distributions, in the scale-space.
The prime numbers distribution and the scale-space wave function
The structure of the distribution of the prime numbers has attracted attention beyond the original conjecture in Riemann's hypothesis. The implications of such studies are reaching in mathematics, physics, and biology.
The scale-space wave function preserves information about the distribution of the prime numbers. It follows from the theory of stochastic resonance synergies that the prime numbers are the zeros of the scale-space wave function.
Reflecting on infinity in triplets: is 3-6-9 the "key" to the universe, as stated by Nikola Tesla?
Large scale networked structures are usually considered methodologically separate from the smallest because of negligible effects it has on its behaviour.
In this study we bind clusters of positive and negative integer numbers within the quadrupole information qantas. Starting from the 6·k +/- 1 scale distribution of prime numbers, the quadrupole binding structures extend up to 3 triplets, 9·6·k, in a cascade.
Taken in chunks of 9·6·k numbers, this approach scales up the computation of the prime number, checking for all possible 9 discordance from the 6·k +/- 1 prime gap. A closed-loop structure clusters the integer numbers with positive and negative values of the Green function. The zeros of the scale-space wave function binds the opposite clusters at the prime numbers. It is also a polynomial decomposition of prime numbers, in scale.
Towards solving the Riemann's hypothesis, in contribution to Schrodinger's "what is life" question
Mathematically, the proof of the Riemann hypothesis is narrowed to the distribution of the prime gaps for triplets, in the scale-space domain. In the closed-loop structure, there are 9 possible cases to check if the Riemann zeta function takes zero at the positive prime numbers. Alternatively, this approach finds a counterexample with the quadrupole decomposition algorithm.
The results of this study, in our view, support the roles of mirror symmetries in computational physics, as well as, in informatics.
1 Jovovic, M., Hierarchical scale quantization and coding of motion information in image sequences, Informacione Tehnologije VI, Zabljak, 2002.
2 Jovovic, M., Stochastic Resonance Synergetics – Quantum Information Theory for Multidimensional Scaling, Journal of Quantum Information Science, 5/2:47-57, 2015.