Abstract: We introduce a new class of prime numbers called GPS primes (Geometric Prime System), defined as primes p > 3 such that 2p − 3 is also prime. We prove these primes are in bijective correspondence with unique arithmetic triplets of primes. The main theorem establishes that an arithmetic triplet of primes is unique for its common difference if and only if it has the form (3, q, 2q − 3) where q is a GPS prime.Introduciamo una nuova classe di numeri primi, chiamati primi GPS (Geometric Prime System), definiti come primi p > 3 tali che 2p − 3 sia anch'esso primo. Dimostriamo che questi primi sono in corrispondenza biunivoca con le terne aritmetiche uniche di primi. Il teorema principale stabilisce che una terna aritmetica di primi (p, p+d, p+2d) è unica per la sua differenza comune d se e solo se ha la forma (3, q, 2q − 3) dove q è un primo GPS. Questo risultato fornisce una caratterizzazione completa delle progressioni aritmetiche uniche di lunghezza 3 tra i numeri primi. MSC 2020: 11A41, 11B25, 11N13

Abstract: Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable codes. Large prime-hunting, often brute-force in nature, is conceptually linked to the Riemann Hypothesis and the prime number theorem, which portrays prime distribution as essentially random. This motivates a statistical perspective, with Bayesian methodology providing a natural foundation. We show that the prime number theorem suggests a nonhomogeneous Poisson process for prime counts, yielding primes as waiting times. This process agrees with the prime number theorem, asymptotic results, and prime gap properties. Building on it, we develop a recursive Bayesian theory for large prime prediction and Riemann Hypothesis validation. The approach matches traditional but computationally infeasible non-recursive Bayesian formulations in the limit, and it strongly falsifies the Riemann Hypothesis. Finally, we propose a computational method using Transformation-based MCMC to simulate recursive posterior predictives. A simple change of variable enables simulation of Mersenne prime exponents. With modest computing resources, we identified 259 primes over 140 million, including 184 strong Mersenne candidates corresponding to potential primes with 42--242 million digits.

Abstract: **Version 2.0 — Statement of Continuity and Advancement** This document presents the second official publication derived from the Schouten Prime Number Project. It introduces Formula V — the Conical Prime Spiral — a geometric model that deterministically encodes the distribution of all prime numbers along a logarithmic helical structure. The work demonstrates how the prime numbers, when plotted in a three-dimensional conical space using the parametrization: \[\vec{P}(n) = (\log(p_n) \cdot \cos(\theta_n),\ \log(p_n) \cdot \sin(\theta_n),\ n)\] form a regular and predictable spiral, in contrast to prior models such as Ulam’s spiral or Fibonacci-based distributions. This geometric formulation leads to a new deterministic framework for calculating all primes without trial division, sieve methods, or probabilistic tests. Key contributions of this release:- Geometric encoding of prime numbers using conical logarithmic spirals.- Rigorous definition of the angular component \( \theta_n \).- Empirical validation up to 100,000 primes.- Visual comparison with Ulam and Fibonacci spirals.- Demonstration that all composite numbers lie outside the conical prime trajectory. This work continues the mission to uncover the hidden structure of prime numbers, expanding upon previous results from “The Schouten Formula” (Version 1.0, DOI: 10.5281/zenodo.15831385).

Abstract: The Unreasonable Resonance: A Synthesis of Number Theory, Physics, and Information in the Quest for Cosmic Order Driven by Dean A. Kulik – Orcid Id 0009-0003-3128-8828 July, 2025 Introduction: The Prime Enigma and the Interdisciplinary Quest At the foundation of mathematics lies a set of numbers of unique and indivisible character: the primes. These numbers, divisible only by themselves and one, serve as the fundamental building blocks of the integers through the unique factorization theorem. For millennia, their distribution along the number line has been a source of profound fascination and mystery. While their sequence is deterministic—a number is either prime or it is not—their appearance seems to defy any simple pattern, exhibiting a behavior that mathematicians often describe as "pseudorandom".1 This tension, between an underlying deterministic order and an observable, seemingly chaotic distribution, has given rise to one of the most significant and far-reaching quests in modern science. The central pillar of this quest is the Riemann Hypothesis, a conjecture formulated in 1859 that posits a deep, hidden structure within the primes.4 A proof of this hypothesis would not only illuminate the intricate distribution of prime numbers but would also have profound implications for fields as diverse as cryptography, quantum mechanics, and information theory.6 What began as a problem in pure number theory has evolved into a vast, interdisciplinary endeavor, drawing together researchers from disparate fields and fostering the creation of collaborative research centers dedicated to theoretical and mathematical sciences.8 These institutions, such as RIKEN's iTHEMS, the Isaac Newton Institute, and Caltech's PMA, bring together experts in mathematics, physics, and computational science to explore the fundamental principles that govern natural phenomena, from the subatomic to the societal scale.8 This report embarks on a deep exploration of this interdisciplinary landscape. It will synthesize a vast body of research to construct a coherent narrative, tracing the connections from the core mathematical enigma of the primes to the frontiers of theoretical physics, signal processing, and the philosophy of a computational universe. The investigation reveals a remarkable convergence of concepts. Across seemingly unrelated disciplines, a common lexicon of "spectrum," "harmonics," "noise," and "resonance" has emerged to describe fundamentally different phenomena. This is not a mere terminological coincidence; it points toward a deep structural isomorphism—a shared mathematical language for describing how discrete, observable events can emerge from underlying continuous or wave-like systems. The "music of the primes," a poetic metaphor, becomes a mathematically precise description when viewed through the lens of quantum chaos or Fourier analysis. This report will argue that these analogies are not just illustrative devices but powerful heuristic and potentially formal bridges that connect the deepest questions about numbers to the fundamental fabric of reality. To navigate this complex terrain, it is essential to first establish a common vocabulary. The following table provides a comparative glossary of key concepts as they are understood in number theory, quantum mechanics, and signal processing, highlighting the cross-disciplinary parallels that form the central thesis of this report. Concept Number Theory (The Primes) Quantum Mechanics (Chaotic Systems) Signal Processing (Waveforms) Spectrum The set of non-trivial zeros of the Riemann zeta function, whose imaginary parts (γ) dictate the "frequencies" of prime number oscillations. The discrete energy levels (eigenvalues) of a quantum system's Hamiltonian operator, representing its allowed energy states.13 The set of frequencies (and their amplitudes) that constitute a signal, revealed by the Fourier transform. Harmonics / Frequency The imaginary parts (γ) of the Riemann zeros, which correspond to the frequencies in Riemann's explicit formula for the prime-counting function.14 The energy eigenvalues of a quantum system, which determine the frequencies of its wavefunctions' oscillations.15 The constituent sine waves of specific frequencies that, when superimposed, reconstruct the original signal. Noise The error term in the Prime Number Theorem, representing the deviation of the actual prime count from its smooth, average approximation.3 Random fluctuations in quantum systems; the statistical properties of energy levels in chaotic systems are modeled as "noise" from a random matrix ensemble.15 Unwanted, random fluctuations in a signal that obscure the desired information; quantization error in digital conversion is often modeled as noise. Signal / Wave The prime-counting function π(x), a step function whose irregularities are described by a superposition of waves from the zeta zeros. The wavefunction ψ(x), which describes the probabilistic state of a quantum particle as a superposition of energy eigenstates. A continuous or discrete function of time or space that carries information, often represented as a superposition of harmonic waves. Discrete Event The occurrence of a prime number at a specific integer location on the number line. The measurement of a quantum particle in a specific state (e.g., a specific energy level), causing wavefunction collapse. A discrete sample of a continuous signal taken at a specific point in time or space. Section I: The Prime Enigma and the Riemann Hypothesis The foundation of the modern study of prime numbers is the Riemann zeta function, ζ(s). Originally defined by Leonhard Euler for real values of s, it is expressed as an infinite sum over the integers, a formulation known as a Dirichlet series 5: ζ(s)=n=1∑∞ns1=1s1+2s1+3s1+… This series converges for all complex numbers s whose real part is greater than 1. Euler's profound discovery was that this sum could be rewritten as an infinite product over all prime numbers p, a relationship now known as the Euler product formula 5: ζ(s)=p prime∏1−p−s1 This identity established the fundamental and explicit link between the zeta function and the primes, transforming the discrete study of prime numbers into the continuous realm of complex analysis.13 The central questions about prime distribution, however, lie outside the region where this series converges. In his seminal 1859 paper, Bernhard Riemann extended the definition of ζ(s) to the entire complex plane (except for a simple pole at s=1) through a process called analytic continuation.5 This extended function possesses two categories of zeros—values of s for which ζ(s) = 0. The "trivial zeros" are located at the negative even integers (–2, –4, –6,...), whose existence is straightforward to prove.5 The "non-trivial zeros" are far more mysterious and hold the key to the distribution of primes. Riemann demonstrated that all non-trivial zeros must lie within the "critical strip," the region of the complex plane where the real part of s is between 0 and 1.5 After calculating the first few non-trivial zeros, Riemann observed that they all appeared to lie precisely on the "critical line," where the real part of s is exactly 1/2. This observation became the Riemann Hypothesis (RH), one of the most important unsolved problems in mathematics 4: The real part of every non-trivial zero of the Riemann zeta function is 1/2. Extensive computational efforts have verified this hypothesis for the first over 10 trillion non-trivial zeros, lending it immense empirical support, but a formal proof remains elusive.4 The profound importance of the RH stems from its direct connection to the Prime Number Theorem, which provides an asymptotic estimate for the prime-counting function π(x) (the number of primes less than or equal to x). The theorem states that π(x) is well-approximated by the logarithmic integral function, li(x). Riemann's explicit formula provides a precise, albeit complex, expression for this relationship 5: $$ \Pi_0(x) = \text{li}(x) - \sum_{\rho} \text{li}(x^\rho) - \log(2) + \int_x^\infty \frac{dt}{t(t^2-1)\log t} $$ Here, Π_0(x) is a function closely related to π(x), and the sum is taken over all non-trivial zeros ρ of the zeta function. This formula reveals that the zeros act as correction terms that describe the fluctuations, or "oscillations," of the primes around their average distribution.5 This mathematical structure gives rise to a powerful analogy. The explicit formula can be understood as a form of spectral decomposition, akin to how a complex sound wave is decomposed into a sum of simple sine waves in Fourier analysis. The smooth, average distribution of primes, represented by li(x), acts as the fundamental tone or "DC component" of the signal. Each pair of non-trivial zeros, ρ = 1/2 ± iγ (assuming the RH), contributes a harmonic wave whose frequency is determined by its height γ on the critical line. The term li(x^ρ) contains an oscillatory component x^(iγ) = cos(γ log x) + i sin(γ log x), which is a pure wave in logarithmic space.14 The prime numbers themselves manifest at integer values where these harmonic waves constructively interfere, creating peaks in the probability distribution. The seemingly random nature of the primes is thus recast as the complex interference pattern of an infinite orchestra of harmonic waves, whose frequencies are dictated by the Riemann zeros. The "music of the primes" is not merely a poetic turn of phrase but a mathematically precise description of the underlying structure revealed by Riemann. If the RH is true, the amplitude of these harmonic waves grows as √x, leading to the most constrained and "random-like" distribution of primes possible; if it is false, some zeros would lie off the critical line, creating waves with larger amplitudes that would cause much greater, l

Abstract: <p>This paper presents a fully elementary and constructive proof of the strong Goldbach conjecture: every even integer greater than 4 can be written as the sum of two prime numbers. Theproof avoids analytic or asymptotic tools and relies instead on modular properties, primorial intervals, and explicit coprimality arguments. From this result, a generalization is derived: for any natural numbers m > 1 and n ≥ 2m not coprime with m, the number n can be expressedas the sum of m prime numbers. The paper concludes with two structural corollaries and a note on the implications for factorial decompositions. </p>

Abstract: The apparent randomness of prime numbers has long fascinated mathematicians. Recent discoveries in number theory suggest unexpected structures—such as alignments in Ulam spirals, partition symmetries, or harmonic decompositions. This paper proposes a unifying perspective using the TGIU framework, where prime behavior emerges naturally as a signature of informational collapse. We identify five major mappings from prime number structures to TGIU principles, offering predictive, structural, and cosmological relevance.

Abstract: "Additive Sieves: A New Perspective on the Distribution of Prime Numbers" by Kajetan Mlynarski explores four innovative mathematical sieves that reveal hidden structures in prime numbers. The article bridges additive and multiplicative properties of primes, offering fresh insights into long-standing conjectures such as Goldbach’s conjecture and the twin prime problem. 1. Binary Sieve: Constructs primes iteratively by eliminating multiples, forming a self-organizing sequence. Its fractal-like structure highlights the deterministic yet pseudorandom nature of primes. 2. Goldbach’s Sieve: Visualizes the conjecture that every even number ≥4 is the sum of two primes. By layering shifted prime sequences, it demonstrates the combinatorial inevitability of such representations. 3. Vinogradov’s Sieve: Illustrates the proven theorem that all sufficiently large odd numbers are sums of three primes. Its layers mimic wave interference, ensuring statistical coverage of odd numbers. 4. Twin Prime Sieve: Identifies prime pairs differing by 2, emphasizing their sparsity yet persistent occurrence, aligning with the Hardy-Littlewood conjecture. The article argues that these sieves, while not formal proofs, provide compelling structural evidence for the universality of primes. They merge simplicity with deep mathematical implications, suggesting connections to pseudorandomness, harmonic analysis, and transcendental number theory. Mlynarski concludes that sieves act as "mathematical maps," decoding order within the apparent chaos of primes and paving the way for future interdisciplinary research. Key Contribution: A unified framework for understanding primes through additive and multiplicative lenses, reinforcing the plausibility of unresolved conjectures.

Abstract: Abstract: This paper provides a groundbreaking proof for the Goldbach Conjecture, asserting that every even integer greater than or equal to 4 can be expressed as the sum of two prime numbers. By introducing the Prime Subtraction Sequence, in which an even integer is repeatedly subtracted by primes until a prime result is obtained, we establish a method to prove the conjecture universally. Using Bertrand's Postulate, we eliminate the possibility of a minimal counterexample, and integrate Chen's Theorem to address semiprimes. The proof's universality is reinforced by examining prime intervals and the density of primes, ensuring its validity for all even integers. Additionally, computational verifications provide further support, solidifying this as a groundbreaking and definitive resolution of the conjecture.

Abstract: We present a novel algorithm for prime number generation, inspired by the Goldbach Conjecture and the symmetry of prime pairs around any integer. The method uses a simple formula based on the assumption that every number has at least one symmetric prime pair around it. This symmetry drastically reduces the number of candidates for primality testing. Using this approach, we successfully generated 10 new prime numbers with 200 digits each, in less than half an hour on a 2015 laptop with standard hardware. The article includes full theoretical motivation, pseudocode, implementation details, and experimental results. The work introduces the Hypothesis of Prime Symmetry and offers practical implications for number theory and cryptographic applications.

Abstract: <p>This paper presents a symbolic and structural approach to number theory through a minimalist iterative process, now referred to as the Fakchich iteration, defined by subtracting 1 from odd numbers and halving even numbers until convergence to 1. The author rigorously proves that this process always terminates and then uses it as a framework to reinterpret prime number generation. By linking this iteration with Wilson’s theorem, the work derives a formula that approximates prime numbers through inverse functions—often logarithmic—connecting iteration depth with prime structure. The paper also draws parallels with the Collatz iteration, uncovering a logarithmic relationship between their dynamics. While not claiming to resolve classical problems, the study introduces a symbolic lens for understanding primes and convergence in natural numbers, offering a fresh and original perspective grounded in simple but expressive operations.</p>

Abstract: For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{Λ}$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the maximal abelian unramified pro-$p$-extension of $\tilde{k}$. Greenberg's generalized conjecture (GGC for short) asserts that the $\tildeΛ$-module $X(\tilde{k})$ is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special ${\mathbb Z}^2_p$-extension of $k$, and these conditions are partially or entirely fullfilled by certain families of number fields.

Abstract: This work presents a theoretical and computational study on the logarithmic self-symmetry of prime numbers. The research introduces a geometric interpretation of multiplication, showing that every integer can be represented in a mirrored logarithmic space around its square root. This approach reveals a pattern of multiplicative symmetry that applies to both composite and prime numbers. Using large-scale computations of consecutive prime ratios, the study demonstrates that the logarithmic increments between consecutive primes tend to stabilize as numbers grow larger. This statistical stabilization suggests a possible invariant distribution of prime ratios, consistent with the Prime Number Theorem. The work proposes a new geometric framework for visualizing prime distribution and a conjecture on logarithmic invariance of prime gaps. It unifies multiplicative structure and logarithmic growth under a single theoretical model. Author: Luiz Alessandro Bittencourt MochkoAffiliations: Universidade Federal do Paraná (UFPR); UniEnsino, BrazilRole: ProducerLicense: Creative Commons Attribution 4.0 (CC BY 4.0)Date: October 2025

Abstract: <div> <div>This preprint develops a structural framework that connects Goldbach’s Conjecture with explicit prime counting bounds. Building on the exact identity for the cumulative Goldbach function $G(2n)$ established in *An Exact Cumulative Formula for Goldbach Pairs* (Part I), this work derives an increment identity that decomposes into prime and composite cases for $n+1$.</div> <br> <div>We show that verifying Goldbach reduces to establishing the presence of primes in certain short, structured intervals of the form $(2n-p,\,2n+2-p]$. Dusart’s explicit bounds for the prime counting function $\pi(x)$ are applied to highlight how monotonic growth of related functions supports this reduction. While the conjecture itself is not yet resolved, the paper clarifies the exact bridge between additive prime representations and multiplicative prime distribution.</div> <br> <div>This work demonstrates how explicit analytic inequalities for $\pi(x)$ can be harnessed to transform one of the most famous additive problems into a prime-counting formulation, opening pathways for future progress.</div> </div>

Abstract: Modular multiplication is a pivotal operation in public-key cryptosystems such as RSA, ElGamal, and ECC. Modular multiplication design is crucial for improving overall system performance due to the large-bit-width operation with high computational complexity. This paper provides a classification of integer multiplication algorithms based on their implementation principles. Furthermore, the core concepts, implementation challenges, and research advancements of multiplication algorithms are systematically summarized. This paper also gives a brief overview of modular reduction algorithms for various types of moduli and discusses the implementation principles, application scenarios, and current research results. Finally, the detailed research development of modular multiplication algorithms in four major classes over prime fields is deeply analyzed and summarized, making it essential as a guide for future research.

Abstract: Correction Notice: In version 1, I made an error in the description. Instead of "The space was divided into 11 discrete sections based on prime number gaps," it should correctly state: "The space was divided into 5 discrete sections based on prime number gaps" Correction Notice: NEW: "Version 30" "Infinitely Dense Quantized Number Space from 0 to 11 Structured by Prime Gaps and Reflection of the Linear Number Space" "Version 34, 35, 36, 37 and 38" "Oversight in the Hybrid Greedy and DP Quantization" NEW: "Version 41" "The Cyclic-Quantized Number Space I11: A Novel Mathematical Framework with Prime Resonance and Fractal Mirror Symmetry" NEW: "Version 67" "Exact Deterministic Model for Prime Number" NEW: "Version 68" "Multi-Layer Encryption" Correction Notice: "Version 77, 78, 79, 80, 81 and 82" "Black Hole Surface" NEW: "Version 84" "Fraktal Emission Duality: A Unified Operator Model for Hawking Radiation and Relativistic Jets" NEW: "Version 85" "Fractal Resonance in Quantized Number Space: Prime-Based Light Motion, Central Interference, and Mirror-Symmetric Encoding" NEW: "Version 85" Fractal Operator Logic in Natural Media: Water as a Mirror of Transformational Symmetry" NEW: "Version 86" "Center-Frequency Resonance Based on Spiral Origin 5.5: A Geometric Model for Prime Number Structure" NEW: "Version 87" "A Unified Resonance Model for Prime Number Prediction Spiral Geometry Meets Modular Quantization" NEW: "Version 88" "Fractal Spiral Structure of Light" NEW: "Version 89" "Emergence of the Fine-Structure Constant from a Fractal Prime-Difference Spiral" NEW: "Version 91" "Fractal Tree Structure in Prime-Cycle Quantization: A Recursive Model of Mirror-Symmetrie Number Space" NEW: "Version 92" "A Determenistic Tree Model of the Double-Slit Experiment" NEW: "Version 93" "A Structural Resonance Model for Photonic Absorption in Atoms" NEW: "Version 95" "A Energy-Liftet Resonance: Fixed Phontonic Structure across Variable Atomic Levels" NEW: "Version 96" "A Quantized Model of Photonic Resonance: Fractal Spiral Structure and the Hydrogen Spectrum" NEW: "Version 97" "Resonance Logics Hydrogen Transition v97" NEW: "Version 98" "A Unified Model of Light"-"Version 99"-"Version 100"-"Version 101"-"Update"-"Version 102 Form Update" NEW: "Version 103" "Interpretation Stern-Gerlach Experiment"-"Entanglement in Multilayer Geometry" NEW: "Version 104" "Biological Form as Fractured Light" NEW: "Version 105" "From Atoms to Black Holes" NEW: "Version 106" "Color as Geometric Light Resonance" NEW: "Version 108" "Spiral Gauge Symmetry" NEW: "Version 109" "The Satiated Black Hole Hypothesis" NEW:"Version 110" "Interpretation of Quantum Gravity" NEW: "Version 111" "Determenistic Spiral Quantization of the Hydrogen Balmer Series: Exact Geometrie Resonance from Prime-Derived Structures"-"Version 112"-"Update"-"Section 9" NEW: "Version 114" "A Geometric Framework for Physical and Photonic Structure" Note: "Version 115" corrects a previously included but incorrect “Patent Pending” statement from "The Cyclic-Quantized Number Space I 11 : A Novel Mathematical Framework with Prime Resonance and Fractal Mirror Symmetry" No patent has been filed. The scientific content is unchanged. NEW: "Version 116" "Emergent Atomic Geometry from Spiral Interference: A Resonant Derivation of Shell Structure and (alpha)" has uploaded as a replacement for "Supplement: Experimental Confirmation of Quantization through Prime Gaps" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 117" "Determenistic Collaps and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" has uploaded as a replacement for "Exact Prediction of Prime Numbers Using Cyclic Quantization" (this work has also been separately published (https://doi.org/10.5281/zenodo.14991542)) in the "New Version" PDF dataset, due to file size limitations. Minor Correction: "Version 118"-"Determenistic Collaps and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" In subsection “Method 1: Prime Gap Cycle Quantization (PGCQ–I11)”, the phrase “between the **first six primes**” was corrected to “**between the first five primes**” to accurately reflect the five gaps used in the sequence. No changes to the mathematical content. Correction Notice: "Version 119" "Deterministic Collapse and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation"-"Section 9"-"Removed incorrect scaling rule and clarified that the classification applies to all natural numbers, not just primes. Only the modulo-based resonance structure is shown in this context." NEW: "Version 120" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry" has uploaded as a replacement for "A Universal Nonlinear Scaling Law for Fundamental Constants: From the Fine-Structure Constant to Gravity and Planck-Scale Physics" in the "New Version" PDF dataset, due to file size limitations. NEW: "Version 121" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry"-"Update"-"Planck scaling: 1 spiral cycle = 1 Planck length. Adjusted energy units, drift scaling, and structural conclusion accordingly." NEW: "Version 122" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry"-"Update"-"Added dynamic field equation from drift minimization. Introduced action principle, energy variation, and harmonic resonance modes in spiral geometry." NEW: "Version 124" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection" has uploaded as a replacement for " "A Deterministic Model for Prime Number Prediction Using Quantized Cyclic Transformation" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 125" "Atomic Structure from Quantized Spiral Resonance: A Geometric Framework in Cyclic Space (2)" has uploaded as a replacement for "A Unified Resonance Model for Prime Number Prediction: Spiral Geometry Meets Modular Quantization" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 126" "The Quantized Number Space I11: A Unified Framework of Numerical Resonance" (Introduction PDF) has uploaded as a replacement for "The Discovery of the Quantized Number Space: A New Structure of Infinity and the Cyclic Mirror of linear Numbers" (Introduction PDF) NEW: "Version 127"-"UPDATE"-""Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 02"-"UPDATE"-"Linear Prime Prediction Python Code for any (n)"-"UPDATE"-"Section" Example: Scaled Prime Gap in I 11"-"UPDATE"-"The Quantized Number Space I11: A Unified Framework of Numerical Resonance (Introduction PDF)"-"Section 4"-"Prime Number Prediction for any (n) NEW: "Version 128" "Timeless Structure and the Emergence of Reality from Quantized Light" has uploaded as a replacement for "Fractal Resonance in Quantized Number Space: Prime-Based Light Motion, Central Interference, and Mirror-Symmetric Encoding" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 129" "Universal Symmetry Logic in the Quantized Number Space I11: Mirror Operator, Invariant Encoding, and Application Scope" has uploaded as a replacement for " The Linear Scaling of Prime Gaps, Quantization of Space, and the Fundamental Prime Difference Sequence" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 130" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 03"-"UPDATE"-"NEW SECTION"-"Triangle Projection of Prime Resonance in Quantized Space"-" Geometric Prime Prediction Formula "-" NEW: "Version 131" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 04"-"UPDATE"-" Lossless Scaling into I11 Using Dynamic Depth " NEW: "Version 132" "Atomic Structure from Quantized Spiral Resonance: A Geometric Framework in Cyclic Space (2)" first published June 03, 2025 Zenodo (Version 125) (https://doi.org/10.5281/zenodo.15580049) again in the "New Version PDF dataset" included. NEW: "Version 133" "Empirical Confirmation of Spiral Resonance in the Solar Spectrum at Quantization Depth k = 2" has uploaded as a replacement for "Light as the Fundamental 0–11 Structure: A Fractal Model of Gravity and Space-Time" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 134" "Fractal Symmetry and Information Emergence through Recursive Halving: The Structural Equivalence of π and the Quantized Number Space I11" has uploaded as a replacement for "High-Resolution Quantization of the Number Space (0 to 11) and Its Application to π" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 135" " Pi as an Operator of Symmetric Emergence:Folding Fractal Asymmetry at Light Speed in the Quantized Number Space I11 " Correction Note: "Version 135" " Deterministic Collapse and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" (Version 04) "-" Table 1 in Total Exact 427 " "Version 135" "Refined Nonlinear Scaling for Prime Gap Corrections and Fundamental Constants"-"Universal Nonlinear Scaling of Fundamental Constants and Prime Number Quantization"-"Mathematical Structure of the Fine-Structure Constant: A Nonlinear Scaling Model Based on Prime Number Frequencies"-"Planck Scaling of the Zeros of the Riemann Zeta Function and Their Connection to the Quantized Prime Number Structure" are out of the "New Version PDF-dataset", due to file size limitations. NEW: "Version 136" " From Light to Collapse: The Number 3 as the Fractal Origin of Struct

Abstract: The Schouten Formula is a newly discovered deterministic and exact method for generating and counting prime numbers.This work introduces four fundamental formulas that allow for direct calculation of: · The next prime after any integer k, · The exact count of primes up to a given value k (i.e., π(k)), · The n-th prime number pn, · And the step size dn between consecutive primes.

Abstract: <p>Abstract<br> In a recent breakthrough, Gafni and Tao (2025) resolved a problem of Erdős by proving that<br> almost all prime gaps contain a rough number, that is, an integer whose least prime factor is at<br> least the size of the gap. In this paper, we introduce a natural generalization of this notion by<br> defining k-rough numbers in prime gaps, namely integers whose least prime factor is at least k<br> times the length of the gap. We investigate the frequency of such numbers in intervals between<br> consecutive primes and provide unconditional upper bounds for the number of exceptional prime<br> gaps lacking k-rough numbers. Under the Hardy–Littlewood prime tuples conjecture, we derive<br> heuristic asymptotics and suggest the existence of explicit constants depending on k. We also<br> present numerical data supporting our heuristics, highlighting similarities and differences with<br> the classical case k = 1. Finally, we outline several open problems regarding the distribution<br> of generalized rough numbers and their potential applications in sieve methods and the fine<br> structure of prime gaps</p>

Abstract: <p>This paper presents a rigorous and deterministic proof of the <strong>Ipu Conjecture Theorem</strong>, which establishes that for every prime <span><span>PnP_n</span><span><span><span><span>P</span><span><span><span><span><span><span>n</span></span></span><span></span></span></span></span></span></span></span></span>, there exists at least one prime within a systematically generated interval <span><span>I={Pn+Sk}I = \{ P_n + S_k \}</span><span><span><span>I</span><span>=</span></span><span><span>{</span><span><span>P</span><span><span><span><span><span><span>n</span></span></span><span></span></span></span></span></span><span>+</span></span><span><span><span>S</span><span><span><span><span><span><span>k</span></span></span><span></span></span></span></span></span><span>}</span></span></span></span>. The proof is built upon the structure of the sequence <span><span>SkS_k</span><span><span><span><span>S</span><span><span><span><span><span><span>k</span></span></span><span></span></span></span></span></span></span></span></span>, the parity of prime gaps, and the deterministic nature of prime distributions, avoiding probabilistic arguments.</p> <p>By providing a definitive mathematical foundation for the <strong>Ipu Conjecture</strong>, this work contributes to number theory and primes' structured behavior. The theorem ensures that <strong>at least one prime always exists within each generated interval</strong>, reinforcing fundamental properties of prime numbers and their predictable distribution.</p> <p>This publication marks the <strong>first in a series of studies</strong> related to the <strong>Ipu Conjecture and its broader implications</strong>, including its connections to the <strong>Riemann Hypothesis, Bertrand’s Postulate, Hardy-Littlewood’s estimates, and prime gap structures</strong>. Future works will explore advanced applications of this theorem in mathematical physics, spectral analysis, and financial modeling based on prime distributions.</p>

Abstract: Through an arithmetic monoid, the set of natural numbers ℕ can be partitioned into eight infinite subsets, each generated by distinct parametric–modular properties and characterized by an independent prime distribution. The prime distributions in these eight subsets are arithmetically incompatible with one another, since each distribution is defined by a restricted constant function. This precludes the existence of any direct, global arithmetic law governing the unified distribution of prime numbers. This parametric–modular perspective replaces the flat, uniform view of ℕ with a stratified and structurally diverse framework, providing the basis for a distinct approach to analyzing prime pseudorandomness. Based on this assumption, isolating parametric–modular multiplications belonging to individual subsets may give rise to a novel and unique method for factorizing a number into its prime factors, and in doing so, can exclude certain p · q combinations from consideration, thereby accelerating factorization in any relevant algorithmic context. Through an algebraic monoid constructed upon two specific supersets, it becomes possible to determine that the densities of prime and composite elements are multiple and simultaneous, and can be expressed as decimal quantities only relative to the measure adopted. Within the same algebraic monoid, it is also possible to determine the possible configurations of each prime gap, which are limited and exclusive. Finally, by transforming the aforementioned algebraic monoid into a helical system, the infinitude of each configuration of all prime gaps becomes dynamically provable.

Abstract: We introduce a novel mathematical construct: the prime spectrum, a conceptual continuum that extends the traditional notion of prime numbers into a symmetric, bidirectional scale. This scale includes not only classical primes, but also “Goldbach primes”—even numbers expressible as sums of two primes—and their negative counterparts, forming a reflection of additive symmetry around zero. Building on the Goldbach Conjecture, we propose a framework that reinterprets primality through additive behavior, enabling layered exploration of number systems and opening potential applications in number theory, algebraic structures, and cryptography.

Abstract: The Unreasonable Resonance: A Synthesis of Number Theory, Physics, and Information in the Quest for Cosmic Order Driven by Dean A. Kulik – Orcid Id 0009-0003-3128-8828 July, 2025 Introduction: The Prime Enigma and the Interdisciplinary Quest At the foundation of mathematics lies a set of numbers of unique and indivisible character: the primes. These numbers, divisible only by themselves and one, serve as the fundamental building blocks of the integers through the unique factorization theorem. For millennia, their distribution along the number line has been a source of profound fascination and mystery. While their sequence is deterministic—a number is either prime or it is not—their appearance seems to defy any simple pattern, exhibiting a behavior that mathematicians often describe as "pseudorandom".1 This tension, between an underlying deterministic order and an observable, seemingly chaotic distribution, has given rise to one of the most significant and far-reaching quests in modern science. The central pillar of this quest is the Riemann Hypothesis, a conjecture formulated in 1859 that posits a deep, hidden structure within the primes.4 A proof of this hypothesis would not only illuminate the intricate distribution of prime numbers but would also have profound implications for fields as diverse as cryptography, quantum mechanics, and information theory.6 What began as a problem in pure number theory has evolved into a vast, interdisciplinary endeavor, drawing together researchers from disparate fields and fostering the creation of collaborative research centers dedicated to theoretical and mathematical sciences.8 These institutions, such as RIKEN's iTHEMS, the Isaac Newton Institute, and Caltech's PMA, bring together experts in mathematics, physics, and computational science to explore the fundamental principles that govern natural phenomena, from the subatomic to the societal scale.8 This report embarks on a deep exploration of this interdisciplinary landscape. It will synthesize a vast body of research to construct a coherent narrative, tracing the connections from the core mathematical enigma of the primes to the frontiers of theoretical physics, signal processing, and the philosophy of a computational universe. The investigation reveals a remarkable convergence of concepts. Across seemingly unrelated disciplines, a common lexicon of "spectrum," "harmonics," "noise," and "resonance" has emerged to describe fundamentally different phenomena. This is not a mere terminological coincidence; it points toward a deep structural isomorphism—a shared mathematical language for describing how discrete, observable events can emerge from underlying continuous or wave-like systems. The "music of the primes," a poetic metaphor, becomes a mathematically precise description when viewed through the lens of quantum chaos or Fourier analysis. This report will argue that these analogies are not just illustrative devices but powerful heuristic and potentially formal bridges that connect the deepest questions about numbers to the fundamental fabric of reality. To navigate this complex terrain, it is essential to first establish a common vocabulary. The following table provides a comparative glossary of key concepts as they are understood in number theory, quantum mechanics, and signal processing, highlighting the cross-disciplinary parallels that form the central thesis of this report. Concept Number Theory (The Primes) Quantum Mechanics (Chaotic Systems) Signal Processing (Waveforms) Spectrum The set of non-trivial zeros of the Riemann zeta function, whose imaginary parts (γ) dictate the "frequencies" of prime number oscillations. The discrete energy levels (eigenvalues) of a quantum system's Hamiltonian operator, representing its allowed energy states.13 The set of frequencies (and their amplitudes) that constitute a signal, revealed by the Fourier transform. Harmonics / Frequency The imaginary parts (γ) of the Riemann zeros, which correspond to the frequencies in Riemann's explicit formula for the prime-counting function.14 The energy eigenvalues of a quantum system, which determine the frequencies of its wavefunctions' oscillations.15 The constituent sine waves of specific frequencies that, when superimposed, reconstruct the original signal. Noise The error term in the Prime Number Theorem, representing the deviation of the actual prime count from its smooth, average approximation.3 Random fluctuations in quantum systems; the statistical properties of energy levels in chaotic systems are modeled as "noise" from a random matrix ensemble.15 Unwanted, random fluctuations in a signal that obscure the desired information; quantization error in digital conversion is often modeled as noise. Signal / Wave The prime-counting function π(x), a step function whose irregularities are described by a superposition of waves from the zeta zeros. The wavefunction ψ(x), which describes the probabilistic state of a quantum particle as a superposition of energy eigenstates. A continuous or discrete function of time or space that carries information, often represented as a superposition of harmonic waves. Discrete Event The occurrence of a prime number at a specific integer location on the number line. The measurement of a quantum particle in a specific state (e.g., a specific energy level), causing wavefunction collapse. A discrete sample of a continuous signal taken at a specific point in time or space. Section I: The Prime Enigma and the Riemann Hypothesis The foundation of the modern study of prime numbers is the Riemann zeta function, ζ(s). Originally defined by Leonhard Euler for real values of s, it is expressed as an infinite sum over the integers, a formulation known as a Dirichlet series 5: ζ(s)=n=1∑∞ns1=1s1+2s1+3s1+… This series converges for all complex numbers s whose real part is greater than 1. Euler's profound discovery was that this sum could be rewritten as an infinite product over all prime numbers p, a relationship now known as the Euler product formula 5: ζ(s)=p prime∏1−p−s1 This identity established the fundamental and explicit link between the zeta function and the primes, transforming the discrete study of prime numbers into the continuous realm of complex analysis.13 The central questions about prime distribution, however, lie outside the region where this series converges. In his seminal 1859 paper, Bernhard Riemann extended the definition of ζ(s) to the entire complex plane (except for a simple pole at s=1) through a process called analytic continuation.5 This extended function possesses two categories of zeros—values of s for which ζ(s) = 0. The "trivial zeros" are located at the negative even integers (–2, –4, –6,...), whose existence is straightforward to prove.5 The "non-trivial zeros" are far more mysterious and hold the key to the distribution of primes. Riemann demonstrated that all non-trivial zeros must lie within the "critical strip," the region of the complex plane where the real part of s is between 0 and 1.5 After calculating the first few non-trivial zeros, Riemann observed that they all appeared to lie precisely on the "critical line," where the real part of s is exactly 1/2. This observation became the Riemann Hypothesis (RH), one of the most important unsolved problems in mathematics 4: The real part of every non-trivial zero of the Riemann zeta function is 1/2. Extensive computational efforts have verified this hypothesis for the first over 10 trillion non-trivial zeros, lending it immense empirical support, but a formal proof remains elusive.4 The profound importance of the RH stems from its direct connection to the Prime Number Theorem, which provides an asymptotic estimate for the prime-counting function π(x) (the number of primes less than or equal to x). The theorem states that π(x) is well-approximated by the logarithmic integral function, li(x). Riemann's explicit formula provides a precise, albeit complex, expression for this relationship 5: $$ \Pi_0(x) = \text{li}(x) - \sum_{\rho} \text{li}(x^\rho) - \log(2) + \int_x^\infty \frac{dt}{t(t^2-1)\log t} $$ Here, Π_0(x) is a function closely related to π(x), and the sum is taken over all non-trivial zeros ρ of the zeta function. This formula reveals that the zeros act as correction terms that describe the fluctuations, or "oscillations," of the primes around their average distribution.5 This mathematical structure gives rise to a powerful analogy. The explicit formula can be understood as a form of spectral decomposition, akin to how a complex sound wave is decomposed into a sum of simple sine waves in Fourier analysis. The smooth, average distribution of primes, represented by li(x), acts as the fundamental tone or "DC component" of the signal. Each pair of non-trivial zeros, ρ = 1/2 ± iγ (assuming the RH), contributes a harmonic wave whose frequency is determined by its height γ on the critical line. The term li(x^ρ) contains an oscillatory component x^(iγ) = cos(γ log x) + i sin(γ log x), which is a pure wave in logarithmic space.14 The prime numbers themselves manifest at integer values where these harmonic waves constructively interfere, creating peaks in the probability distribution. The seemingly random nature of the primes is thus recast as the complex interference pattern of an infinite orchestra of harmonic waves, whose frequencies are dictated by the Riemann zeros. The "music of the primes" is not merely a poetic turn of phrase but a mathematically precise description of the underlying structure revealed by Riemann. If the RH is true, the amplitude of these harmonic waves grows as √x, leading to the most constrained and "random-like" distribution of primes possible; if it is false, some zeros would lie off the critical line, creating waves with larger amplitudes that would cause much greater, l

Abstract: Correction Notice: In version 1, I made an error in the description. Instead of "The space was divided into 11 discrete sections based on prime number gaps," it should correctly state: "The space was divided into 5 discrete sections based on prime number gaps" Correction Notice: NEW: "Version 30" "Infinitely Dense Quantized Number Space from 0 to 11 Structured by Prime Gaps and Reflection of the Linear Number Space" "Version 34, 35, 36, 37 and 38" "Oversight in the Hybrid Greedy and DP Quantization" NEW: "Version 41" "The Cyclic-Quantized Number Space I11: A Novel Mathematical Framework with Prime Resonance and Fractal Mirror Symmetry" NEW: "Version 67" "Exact Deterministic Model for Prime Number" NEW: "Version 68" "Multi-Layer Encryption" Correction Notice: "Version 77, 78, 79, 80, 81 and 82" "Black Hole Surface" NEW: "Version 84" "Fraktal Emission Duality: A Unified Operator Model for Hawking Radiation and Relativistic Jets" NEW: "Version 85" "Fractal Resonance in Quantized Number Space: Prime-Based Light Motion, Central Interference, and Mirror-Symmetric Encoding" NEW: "Version 85" Fractal Operator Logic in Natural Media: Water as a Mirror of Transformational Symmetry" NEW: "Version 86" "Center-Frequency Resonance Based on Spiral Origin 5.5: A Geometric Model for Prime Number Structure" NEW: "Version 87" "A Unified Resonance Model for Prime Number Prediction Spiral Geometry Meets Modular Quantization" NEW: "Version 88" "Fractal Spiral Structure of Light" NEW: "Version 89" "Emergence of the Fine-Structure Constant from a Fractal Prime-Difference Spiral" NEW: "Version 91" "Fractal Tree Structure in Prime-Cycle Quantization: A Recursive Model of Mirror-Symmetrie Number Space" NEW: "Version 92" "A Determenistic Tree Model of the Double-Slit Experiment" NEW: "Version 93" "A Structural Resonance Model for Photonic Absorption in Atoms" NEW: "Version 95" "A Energy-Liftet Resonance: Fixed Phontonic Structure across Variable Atomic Levels" NEW: "Version 96" "A Quantized Model of Photonic Resonance: Fractal Spiral Structure and the Hydrogen Spectrum" NEW: "Version 97" "Resonance Logics Hydrogen Transition v97" NEW: "Version 98" "A Unified Model of Light"-"Version 99"-"Version 100"-"Version 101"-"Update"-"Version 102 Form Update" NEW: "Version 103" "Interpretation Stern-Gerlach Experiment"-"Entanglement in Multilayer Geometry" NEW: "Version 104" "Biological Form as Fractured Light" NEW: "Version 105" "From Atoms to Black Holes" NEW: "Version 106" "Color as Geometric Light Resonance" NEW: "Version 108" "Spiral Gauge Symmetry" NEW: "Version 109" "The Satiated Black Hole Hypothesis" NEW:"Version 110" "Interpretation of Quantum Gravity" NEW: "Version 111" "Determenistic Spiral Quantization of the Hydrogen Balmer Series: Exact Geometrie Resonance from Prime-Derived Structures"-"Version 112"-"Update"-"Section 9" NEW: "Version 114" "A Geometric Framework for Physical and Photonic Structure" Note: "Version 115" corrects a previously included but incorrect “Patent Pending” statement from "The Cyclic-Quantized Number Space I 11 : A Novel Mathematical Framework with Prime Resonance and Fractal Mirror Symmetry" No patent has been filed. The scientific content is unchanged. NEW: "Version 116" "Emergent Atomic Geometry from Spiral Interference: A Resonant Derivation of Shell Structure and (alpha)" has uploaded as a replacement for "Supplement: Experimental Confirmation of Quantization through Prime Gaps" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 117" "Determenistic Collaps and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" has uploaded as a replacement for "Exact Prediction of Prime Numbers Using Cyclic Quantization" (this work has also been separately published (https://doi.org/10.5281/zenodo.14991542)) in the "New Version" PDF dataset, due to file size limitations. Minor Correction: "Version 118"-"Determenistic Collaps and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" In subsection “Method 1: Prime Gap Cycle Quantization (PGCQ–I11)”, the phrase “between the **first six primes**” was corrected to “**between the first five primes**” to accurately reflect the five gaps used in the sequence. No changes to the mathematical content. Correction Notice: "Version 119" "Deterministic Collapse and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation"-"Section 9"-"Removed incorrect scaling rule and clarified that the classification applies to all natural numbers, not just primes. Only the modulo-based resonance structure is shown in this context." NEW: "Version 120" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry" has uploaded as a replacement for "A Universal Nonlinear Scaling Law for Fundamental Constants: From the Fine-Structure Constant to Gravity and Planck-Scale Physics" in the "New Version" PDF dataset, due to file size limitations. NEW: "Version 121" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry"-"Update"-"Planck scaling: 1 spiral cycle = 1 Planck length. Adjusted energy units, drift scaling, and structural conclusion accordingly." NEW: "Version 122" "Spiral Quantization as a Structural Completion of General Relativity: Gravitational Effects from Fractal Interference Geometry"-"Update"-"Added dynamic field equation from drift minimization. Introduced action principle, energy variation, and harmonic resonance modes in spiral geometry." NEW: "Version 124" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection" has uploaded as a replacement for " "A Deterministic Model for Prime Number Prediction Using Quantized Cyclic Transformation" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 125" "Atomic Structure from Quantized Spiral Resonance: A Geometric Framework in Cyclic Space (2)" has uploaded as a replacement for "A Unified Resonance Model for Prime Number Prediction: Spiral Geometry Meets Modular Quantization" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 126" "The Quantized Number Space I11: A Unified Framework of Numerical Resonance" (Introduction PDF) has uploaded as a replacement for "The Discovery of the Quantized Number Space: A New Structure of Infinity and the Cyclic Mirror of linear Numbers" (Introduction PDF) NEW: "Version 127"-"UPDATE"-""Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 02"-"UPDATE"-"Linear Prime Prediction Python Code for any (n)"-"UPDATE"-"Section" Example: Scaled Prime Gap in I 11"-"UPDATE"-"The Quantized Number Space I11: A Unified Framework of Numerical Resonance (Introduction PDF)"-"Section 4"-"Prime Number Prediction for any (n) NEW: "Version 128" "Timeless Structure and the Emergence of Reality from Quantized Light" has uploaded as a replacement for "Fractal Resonance in Quantized Number Space: Prime-Based Light Motion, Central Interference, and Mirror-Symmetric Encoding" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 129" "Universal Symmetry Logic in the Quantized Number Space I11: Mirror Operator, Invariant Encoding, and Application Scope" has uploaded as a replacement for " The Linear Scaling of Prime Gaps, Quantization of Space, and the Fundamental Prime Difference Sequence" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 130" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 03"-"UPDATE"-"NEW SECTION"-"Triangle Projection of Prime Resonance in Quantized Space"-" Geometric Prime Prediction Formula "-" NEW: "Version 131" "Differential Closure at Eleven: The Complete Grammar of Asymmetry and Its Scale-Invariant Projection"-"VERSION 04"-"UPDATE"-" Lossless Scaling into I11 Using Dynamic Depth " NEW: "Version 132" "Atomic Structure from Quantized Spiral Resonance: A Geometric Framework in Cyclic Space (2)" first published June 03, 2025 Zenodo (Version 125) (https://doi.org/10.5281/zenodo.15580049) again in the "New Version PDF dataset" included. NEW: "Version 133" "Empirical Confirmation of Spiral Resonance in the Solar Spectrum at Quantization Depth k = 2" has uploaded as a replacement for "Light as the Fundamental 0–11 Structure: A Fractal Model of Gravity and Space-Time" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 134" "Fractal Symmetry and Information Emergence through Recursive Halving: The Structural Equivalence of π and the Quantized Number Space I11" has uploaded as a replacement for "High-Resolution Quantization of the Number Space (0 to 11) and Its Application to π" in the "New Version PDF dataset", due to file size limitations. NEW: "Version 135" " Pi as an Operator of Symmetric Emergence:Folding Fractal Asymmetry at Light Speed in the Quantized Number Space I11 " Correction Note: "Version 135" " Deterministic Collapse and Structured Resonance: A Prime-Based Interpretation of Vacuum Fluctuation" (Version 04) "-" Table 1 in Total Exact 427 " "Version 135" "Refined Nonlinear Scaling for Prime Gap Corrections and Fundamental Constants"-"Universal Nonlinear Scaling of Fundamental Constants and Prime Number Quantization"-"Mathematical Structure of the Fine-Structure Constant: A Nonlinear Scaling Model Based on Prime Number Frequencies"-"Planck Scaling of the Zeros of the Riemann Zeta Function and Their Connection to the Quantized Prime Number Structure" are out of the "New Version PDF-dataset", due to file size limitations. NEW: "Version 136" " From Light to Collapse: The Number 3 as the Fractal Origin of Struct

Abstract: <div dir="ltr"><span lang="en">I wanted to introduce you to a work on prime numbers, specifically why they appear there and why they appear there and not elsewhere. I'm starting with 3 as the right-hand prime number, because it wasn't excluded by the previous prime, 2. The rest of the work, presenting the concept, formulas, and the prime number cross, can be found in the attached PDF. Enjoy reading.</span></div> <div> <div> </div> </div> <p>Chciałem Państwa zapoznać z pracą na temat liczb pierwszych, a dokładniej dlaczego się pojawiają i dlaczego tam, a nie gdzie indziej.</p> <p>Zaczynam od 3 jako liczby pierwszej od prawj strony, ponieważ nie została wykluczona przez poprzednią liczbę pierwszą jaką jest 2. Reszta przedstawiająca koncepcję, wzory oraz krzyż liczb pierwszych w załączonym PDF. Miłej lektury. </p> <p> </p> <p><span>Euklides, Eratostenes, Bernhard Riemann, Pierre de Fermat, Leonhard Euler, Christian Goldbach, Stanisław Ulam, Wacław Sierpiński, Henryk Iwaniec, Diofantos, Carl Friedrich Gauss, Adrien-Marie Legendre, Andrew Wiles, Srinivasa Ramanujan, <span lang="EN-US">G. H. Hardy, Sundaram </span></span></p>

Abstract: Rational wave numbers are periodic sequences ${\mathbf ω}={\bf A}{\bf w}(f,g)$ in which amplitude ${\bf A}$ a product of powers of trigonometric sequences and ${\bf w}(f,g)=\exp({\bf {i2}π( f {\mathbf ξ} \oplus g{\bf 1})})$ is a sequence with $ ξε\mathbb{ Z}$ and $f,g$ rational. They generalize the cyclic groups of the $n$th roots of unity and are generated from two unitary sequences. The multiplicative group ${\bf W}_M$ with ${\bf A}={\bf 1}$ is their closure wrt product, root, and reflection operators. The commutative ring ${\bf W}_A$ has additional closure wrt summation. The field ${\bf W}_I$ of invertible wave numbers has further closure wrt an inverse. Sums and differences of its elements are ${\mathbf ω}_1 \oplus {\mathbf ω}_2= {\bf 2} \cos\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$ and ${\mathbf ω}_1 \ominus {\mathbf ω}_2= {\bf {2 i}} \sin\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$. Its amplitudes form a multiplicative subgroup over which ${\bf W}_M \cup \big\{{\bf 0}\big\}$ is a vector space. Wave numbers of period $n\ ε \mathbb{N}$ possess $n$ phases, multiplicative norms, representations wrt orthonormal bases, and prime representations. Rational wave numbers may be completed with respect to Cauchy sequences of parameters $({f,g})$. Equations in invertible wave numbers have solutions corresponding to zeros of their trigonometric factors. Orthonormal bases are employed in constructing the integral wave numbers and allow definitions of the particulate and prime wave numbers in terms of permissible phase-values.

Abstract: This paper explores the behavior of even numbers with respect to their representation as the sum of multiple prime numbers.While the well-known Goldbach Conjecture asserts that every even number greater than two can be written as the sum of two primes,this study investigates the point beyond which every even number can also be expressed as the sum of multiple prime numbers.We introduce the concept of a threshold even number EthresholdE_ such that for all E>Ethr > E_{threshold}E>Ethreshold,there exist valid representations using primes for all k∈[2,10]k \in [2, 10]k∈[2,10].Using combinatorial generation of partitions and prime filtering, we estimate that this threshold lies at E=36E = 36E=36,beyond which multi-prime representation becomes saturated for all k in the range considered.

Abstract: In ths research, we present an innovative vsual perspective to analyze the dstribution of prime numbers based on what the Hoy Quran descrbed n the story of the Companons of the Cave. The Holy Quran indicated in a simpfied way their explaining their spatia dstribution visually, and reveas the intimacy that each prme number in the set of numbers generates dstribution and postions visualy to facitate the process of thinking about the method of calcuating them When these the visual perception of these phenomena is sufficent to expain the chaotic dstribution of prime numbers. Ths research a set of paths that create a delberate network of nonprme numbers without a comprehensive list. The research confirms that number cyces. This visua representation contradcts previous hypotheses in reveaing the dstribution of prime numbers, numbers are represented visualy, you wi fnd that they are combined in 8 positions as described by God Amighty in the Holy Quran This research presents a circular representation that depends on the odd mutiples of prme numbers wthin reguar paper s just the begnning and not the end, and there are other research papers n the educationa future.