Fractal potential wells are an extension of classical quantum potential wells with fractal scale dependence; energy surfaces are modulated with wavy, self-similar structures, and the probability distribution of particles is shaped by multi-scale fractal motifs. This approach offers a wide field of application, ranging from atomic transitions at the micro-level to the energy flow around black holes at the macro-level.
Purpose: To reinterpret the probabilistic and non-local structure revealed by Bell’s theorem through quantum entanglement within the framework of Fractal Mechanics. This report extends the classical quantum interpretation with the concepts of fractal derivatives, energy flow, and multiscale resonance.
A synthesis summarizing the entire collapse, from the center-periphery inequality to the natural/artificial motif debate:
The fractal analysis of Surah Al-Fatiha shows that the recurring motifs (Rabb, Rahman, Malik, Din, Ibadah, Sirat al-Mustaqim) in the semantic and hermeneutic layers of the surah reflect each other in a scalable manner. This reveals that the text exhibits a “fractal” structure within itself, both linguistically and theologically.
Although Marxism offers a strong critique of capitalism, it has been heavily criticized for its uni-centric historical determinism, economic reductionism, and failures in practical applications. From the perspective of spiral-fractal logic, Marxism imposes a unilinear model of progress rather than building bridges across micro-meso-macro scales.
John Steinbeck stands out in American literature as a powerful author who treats the themes of “land, poverty, and the inner conflicts of man.” His works present individual dramas as fractal repetitions of the social structure: small family conflicts are miniature motifs of the larger social order.
Reading Leo Tolstoy’s literature through fractal analysis shows that the motifs of individual conscience, family, society, and history in his works expand as repeating patterns. Tolstoy’s texts exhibit a fractal expansion from small-scale moral questionings to large-scale historical and cosmic orders.
CONTENTS: Fractal Taylor Series Visual Fractal Taylor In this diagram, the fractal extension of the classical Taylor expansion is visualized, where derivative terms become scale-dependent through self-similar modulations. Fractal Laplace Transform Visual This graph shows the fractal extension of the classical Laplace transform: damping behavior on the amplitude axis and self-similar resonances on the frequency
Fractal series expansions are the redefined forms of classical Taylor, Maclaurin, and Fourier series using the principle of self-similarity. The aim here is to capture not only the local behavior of functions but also their fractal resonances that repeat at every scale.
Fractal Beginning Axiom System 1. Beginning Constant Axiom ∀𝑋 ∈ 𝒰, ∃! 𝐵(𝑋) Every system has a singular, unmultipliable, and irreducible beginning. 2. Reduction Axiom 𝑥/0 = 1 ⇒ 𝑥 ↦ 𝐵 Every mathematical expression is reduced to the beginning. 3. Fractal Evolution Axiom 𝐵(𝑋) ⇒ {𝑌1, 𝑌2, … , 𝑌∞} The beginning is singular,