Abstract: The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have a real part of $\frac{1}{2}$. As a pivotal conjecture in pure mathematics, it remains unproven and is equivalent to various statements, including one by Nicolas in 1983 asserting that the hypothesis holds if and only if $\prod_{p \leq x} \frac{p}{p - 1} > e^{\gamma} \cdot \log \theta(x)$ for all $x \geq 2$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, and $\log$ is the natural logarithm. Defining $N_n = 2 \cdot \ldots \cdot p_n$ as the $n$-th primorial, the product of the first $n$ primes, we employ Nicolas' criterion to prove that there exists a prime $p_k > 10^8$ and a prime $p_{k'}$ such that $\theta(p_{k'}) \leq \theta(p_k)^2$ and $p_k^{1.907} \ll p_{k'} < p_k^2$, where $p_k^{1.907} \ll p_{k'}$ implies $p_{k'}$ is significantly larger than $p_k^{1.907}$. This existence leads to $\frac{N_k}{\varphi(N_k)} \leq e^{\gamma} \cdot \log \log N_k$, contradicting Nicolas' condition and confirming the falsity of the Riemann Hypothesis. This result decisively refutes the conjecture, enhancing our insight into prime distribution and the behavior of the zeta function's zeros through analytic number theory.

Abstract: In this paper, we will give a new proof of a known result of the mean square of Riemann zeta-function

Abstract: The author sharpens a result of Jia and Liu (2000), showing that for sufficiently large $x$, the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{51}{53}-\varepsilon}$. This gives a solution with $\gamma = \frac{2}{53}$ to the Exercise 5.1 in Harman's book.

Abstract: The author sharpens the result of Rivat and Wu (2000), showing that for sufficiently large n, there are infinitely many primes of the form [n^c] for 1 < c < 211/178.

Abstract: In this paper, we prove that for every prime number $7 \leqslant p<100$, there are infinitely many practical numbers $q$ such that both $q^p$ and $q^p +2$ are practical numbers. This refine a result involving twin practical numbers of Wang and Sun. We also state a conjecture that the above theorem holds for all positive integers.

Abstract: The author prove that there are infinitely many primes of the form [n^c] for $1<c< \frac{919}{775}$. Using the theory of exponent pairs, the author also show that there are infinitely many almost primes of the form [n^c] with some larger c.

Abstract: Let \( K \) be an algebraically closed field of characteristic zero. The generalized Weyl algebra \( A_{n,f} \) is defined by generators \( x_1, x_2, \dots, x_n, y_1, \dots, y_n, z_1, \dots, z_n \) subject to certain commutation relations and additional structure determined by a collection of functions \( f = (f_1, \dots, f_n) \). We focus on the structure of left and right ideals in \( A_{n,f}(K) \), particularly proving that every left or right ideal can be generated by two elements. The proof is based on showing that if a left ideal can be generated by three elements, it can be reduced to two elements by applying the Noetherian property of the ring and an iterative reduction process. This result complements the simplicity of \( A_{n,f}(K) \), as established in prior work.

Abstract: In this paper, we introduce the 2-geometric mean and explore its connections with the spectral geometric mean and the Wasserstein mean for positive definite matrices. Additionally, we revisit and establish several inequalities for these means concerning the near order relation.

Abstract: This paper proposes the Law of Symbolic Dynamics, a new theoretical framework within Symbolic Field Theory (SFT) that explains the emergence of irreducible structures, such as prime numbers, Fibonacci numbers, and square-free integers, through symbolic interference in a dynamic compression field. Unlike traditional methods of prime number identification, which rely on sieving or divisibility testing, this model interprets primes as emergent points in a symbolic field, driven by the dynamics of symbolic curvature, force, mass, momentum, and energy. Using computational models of symbolic curvature based on Euler’s totient function, we show that symbolic collapse zones align with prime numbers with over 98.6\% accuracy, offering a deterministic approach to prime prediction. This work introduces the \textit{Orbital Collapse Law}, which allows for the stepwise prediction of primes without external verification or sieving, marking a transition from probabilistic number theory to a generative model based on geometric principles. Finally, we discuss the broader applicability of this framework to other irreducible structures and propose future directions for extending Symbolic Field Theory beyond number theory to domains such as language, perception, and music.

Abstract: This study introduces a comprehensive framework—Symbolic Field Theory (SFT)—for modeling the emergence and recurrence of irreducible mathematical structures, including prime numbers, square-free integers, Fibonacci and Lucas sequences, Mersenne primes, and other symbolic attractors. Building on prior work that established symbolic curvature collapse as a generative field geometry, we extend the analysis to over 10 irreducibility types using symbolic projection functions and curvature metrics applied across 30,000 natural numbers. Using statistical, logistic, and recurrence-based analysis, we find that symbolic field projections reliably separate irreducible types from the background distribution with high accuracy. Emergence convergence scores and symbolic recurrence rules achieve precision rates above 95\% for several irreducibles, with prime prediction reaching perfect classification under logistic modeling. This empirical geometry of collapse zones is supported by statistically significant $t$-tests, correlation matrices, and recurrence trace rules, offering a scalable method for identifying the generative structure behind symbolic constants. The results validate symbolic field collapse as a universal recurrence geometry and lay the foundation for a predictive science of irreducibility grounded in symbolic wave dynamics.

Abstract: This paper proposes the Law of Emergence, a theoretical principle within Symbolic Field Theory that aims to explain the appearance of irreducible structures—such asprime numbers, Fibonacci terms, and square-free integers—through symbolic interference. Rather than assuming irreducibles arise from isolated axioms or randomness, we explore the hypothesis that these structures emerge from the interaction of multiple symbolic fields operating over a shared domain. Using computational models of symbolic curvature (via Miller’s Law) and collapse zone detection, we observe consistent patterns of enrichment, wherein irreducibles tend to cluster at points of constructive symbolic interaction. While not definitive, these findings provide empirical support for the Law of Emergence as a generative framework, suggesting new directions for modeling pattern formation across mathematical and cognitive domains. We conclude by outlining the theoretical and experimental limitations, emphasizing that this work represents an early-stage contribution toward a unifying model of symbolic emergence. In a further extension of this work, we derive and empirically validate a symbolic recurrence rule—termed the Orbital Collapse Law —which predicts next prime from the previous with over 98.6% accuracy using only symbolic curvature collapse. This recurrence operates without sieving or verification, offering the first curvature-based generative law for prime emergence and marking a critical transition from descriptive alignment to predictive irreducibility within the Symbolic Field framework.

Abstract: This paper introduces a symbolic dynamical field model for the emergence of irreducible mathematical structures, such as prime numbers, from projection-induced curvature in symbolic space. By defining a symbolic collapse field \( S(x, t) \) governed by a second-order partial differential equation, \[ \frac{\partial^2 S}{\partial t^2} = \alpha \frac{\partial^2 S}{\partial x^2} - \beta \frac{\partial S}{\partial t} + \gamma S, \] we simulate the evolution of symbolic gradients and collapse potentials over discrete symbolic elements. Empirical fitting across curvature fields—derived from logarithmic, root-based, and modular projections—reveals stable attractor dynamics and recurrence consistent with symbolic emergence patterns. The field parameters \( \alpha = -0.31 \), \( \beta = 0.75 \), and \( \gamma = -0.046 \) correspond to localized symbolic contraction, damping, and dissipation, respectively. We visualize symbolic attractor trajectories, collapse zones, and bifurcation behavior, demonstrating that symbolic emergence follows predictable field dynamics. This work lays the foundation for a general symbolic physics framework based on curvature-induced collapse, with implications across number theory, complexity, and cognitive structure.

Abstract: In this study, the authors investigate the Collatz conjecture using a frequency-based iterative approach, demonstrating that all natural numbers ultimately converge to a reduced value with an increasing frequency rate, eventually leading to a cyclic loop. Furthermore, the authors present an argument suggesting that the probability of discovering cycles distinct from the known 4-2-1 loop is asymptotically close to zero. The findings of this research offer new insights into the fundamental properties of the Collatz process, potentially addressing several open questions related to the conjecture. In particular, the study explores the mathematical significance of the coefficients 3 and 2 in the transformation rules 3x + 1 and x/2, providing an explanation for their role in governing the conjecture’s behavior.

Abstract: Symbolic Field Theory (SFT) proposes that irreducible mathematical structures—such as prime numbers—emerge as a deterministic consequence of symbolic compression dynamics within discrete curvature fields. These fields are defined by projection functions ψ(x), which map integers into symbolic space, and a curvature operator κ(x), which quantifies local symbolic deviation between a number and its projection. Traditional formulations of Miller’s Law posit that emergence occurs at local minima of κ(x), known as collapse valleys, where symbolic tension is minimized and recursive compression is maximal. However, new analysis reveals a complementary pattern: irreducibles also tend to appear at curvature maxima—regions of symbolic folding and excitation. These inflection points, identified through Laplacian resonance, suggest a dual-collapse geometry where both valleys and peaks act as loci of structural emergence. Primes are found to align preferentially with both types of symbolic curvature extremum, indicating that irreducibility may arise from recursive folding dynamics—not from randomness or singular attraction. To evaluate this dual-collapse hypothesis, we compute symbolic curvature fields over the first million natural numbers using hybrid projection functions that combine modular symmetry and factor complexity. Local minima and maxima of the curvature field are identified and tested for prime enrichment using a Monte Carlo null model. Results show that both curvature valleys and peaks are statistically enriched with primes, with combined enrichment ratios exceeding 3.3 and Z-scores over 400. These findings confirm that symbolic curvature—whether compressive or excitatory—organizes the emergence of irreducibles in number space. By revealing that primes emerge not only from collapse valleys but also from recursive peaks, this work generalizes Miller’s Law into a bidirectional collapse geometry. SFT thus provides the first empirically grounded, curvature-driven framework for understanding irreducible structure as a product of symbolic field dynamics.

Abstract: Let $\mathcal{X}$ be a p-adic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.

Abstract: The search for a deterministic law governing prime number emergence has challenged mathematicians for centuries. In this study, we present a novel approach rooted in Symbolic Field Theory (SFT), which models irreducible numbers as emergent structures within a multidimensional symbolic curvature field. Using four key projection functions—Euler’s totient function, the Möbius function, the divisor count function, and the prime sum function—we define symbolic curvature, force, mass, and momentum to capture the structural dynamics underlying prime and non-prime numbers. By analyzing the collapse behavior of these projections across the integers, we identify emergent collapse zones that predict prime positions with high accuracy. A symbolic regression model, trained on multidimensional collapse scores, demonstrates over 97\% accuracy in discriminating primes from non-primes. This paper introduces the field-invariant collapse equation, which generalizes symbolic curvature across multiple arithmetic functions, offering a unified framework for understanding and predicting irreducible emergence in number theory. The method provides a new perspective on the deterministic dynamics governing the distribution of primes and other irreducible numbers.

Abstract: The Collatz Conjecture, a long-standing open problem in number theory, asserts that every positive integer sequence generated by the Collatz function eventually reaches the 4-2-1 cycle. This paper presents a deterministic proof by modeling the Collatz dynamics using a 17-state finite state machine (FSM) derived from a structured partition of the integers. This FSM comprises a two-state precursor stage (for multiples of 3), a 12-state transient core (for other numbers outside the cycle), and a 3-state terminal stage (representing the 4-2-1 cycle). We analyze the deterministic transitions within this FSM and prove that every state in the precursor and core stages has a finite path leading inevitably to the terminal cycle stage, guaranteeing convergence for all starting integers. Our approach resolves the conjecture through deterministic finite-state analysis, demonstrating the inevitable collapse of any Collatz sequence into the unique 4-2-1 attractor.

Abstract: The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zeta function lie on the critical line ℜ(s) = 12, a conjecture long held central to the understanding of prime distribution. A recent preprint challenges RH by identifying heat instability in the de Bruijn–Newman evolution equation, arguing that irregularities in prime gaps introduce an unbounded forcing term that displaces zeros off the critical line. In this response, we present an alternative yet reinforcing theoretical framework grounded in symbolic collapse geometry—a field-based theory of recursive structure emergence. We demonstrate that prime-gap irregularity and heat instability naturally follow from symbolic curvature tension, a compression-driven field phenomenon that governs recursive emergence across number space. Within this model, the Riemann zeros are not statically harmonic, but rather dynamically emergent from local minima of symbolic field curvature. This reinterpretation offers a unifying explanation for the observed instability in RH, subsuming it under a broader geometric law of recursive symbolic tension, suggesting a shift in perspective from symmetry preservation to symbolic emergence dynamics.

Abstract: This paper proposes a novel unified framework connecting fractal geometry with abstract algebraic structures and analytical methods. I develop a formalism that characterizes self-similar structures through the lens of group actions, measure theory, and operator algebras. My approach bridges previously disparate mathematical traditions, establishing formal connections between fractal dimension, algebraic invariants, and spectral properties of operators. I introduce several new theoretical constructs, including fractal homology groups, measure-preserving group actions on fractals, and spectral decomposition methods for self-similar operators. I demonstrate applications of this framework to physical systems with multiscale dynamics and biological pattern formation. The unification of algebraic and analytical perspectives offers new insights into the fundamental nature of selfsimilarity and creates opportunities for cross-disciplinary approaches to complex systems.

Abstract: Diophantine equations or, more generally, polynomial equations whose solutions are restricted to integers, have captivated mathematicians from the era of Diophantus to modern researchers. In this review, we trace the historical evolution of solution methods from early classical techniques (such as the Euclidean algorithm and the method of infinite descent) to modern approaches involving geometry and computational methods. We survey recent advances, discuss important open problems (including exponential Diophantine equations), and examine the intersection of theory and applications in areas such as cryptography. Our exposition draws on three recent pieces of literature (2020–2025), two works from 2010–2025, and several foundational classical texts.