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[16] ai.viXra.org:2510.0078 [pdf] submitted on 2025-10-31 16:11:08
Authors: Christos Thessalonikios
Comments: 8 Pages.
his work introduces a framework for exploring prime numbers through oscillatory fractal structures. Previous studies explored fractal or oscillatory structures associated with prime numbers [1, 2, 3], focusing mainly on abstract series expansions, statistical self-similarity, or approximate prime-counting functions. Here, i construct a prime-anchored Weierstrass-type fractal, with oscillations peaking at prime numbers. By mapping this fractal onto a helical geometry and applying an additive prime envelope, we provide a new geometrical perspective for visualizing primes and analogies with the Riemann zeta function.
Category: Number Theory
[15] ai.viXra.org:2510.0062 [pdf] replaced on 2025-10-29 20:41:16
Authors: Minkwon Chung
Comments: 3 Pages.
This paper establishes an analytic equivalence for the Riemann Hypothesis (RH) throughthe analysis of the exact-sieve error E(x, z) = Φ(x, z) − x · Q p≤z(1 − 1/p). Here, Φ(x, z) =#{1 ≤ n ≤ x : gcd(n, P(z)) = 1} and P(z) =Q p≤z p. A Schwartz test function ψ on alogarithmic scale defines the smoothed error term Eψ(X, z) = R R E(eu, z)ψ(u−X) du. Thisfunction admits a Mellin spectral representationEψ(X, z) = 1 2πi Z (c) esX b ψ(s)Gz(s) ds,where Gz(s) = ζ(s)Ez(s)/s − Ez(1)/(s − 1) and Ez(s) = Q p≤z(1 − p−s). The subtractionremoves the pole at s = 1, so Gz(s) is holomorphic on ℜs > 0; in particular, the nontrivial zeros of ζ(s) are not poles of Gz(s). The core result is a smoothed spectral equivalence: under RH, |Eψ(log x, z)| ≪ψ,ϵ zα(log z)βx1/2+ϵ; conversely, assuming this bound (and a necessary non-vanishing condition on Ez(ρ)) yields a contradiction via a smoothed explicit formula based on ζu2032/ζ [1], thereby proving RH. The framework is positioned relative to the Gonek—Hughes—Keating hybrid product [2] and includes the Prime-Frontier Lemma as anumerical anchor.
Category: Number Theory
[14] ai.viXra.org:2510.0053 [pdf] submitted on 2025-10-23 00:19:44
Authors: J. W. McGreevy
Comments: 21 Pages.
This manuscript presents Geometric-Representation Quantum Field Theory (GRQFT), a functorial framework that derives the Standard Model and General Relativity from arithmetic invariants via the Langlands program. Starting from the Monster group's moonshine module $V^atural$, GRQFT constructs the explicit pathway:[GalRep(mu_4) to AutInd(V^atural) to Twist(BQF) to DispMap(p^mu p_mu = m^2)]Key results include: (1) solution to Hilbert's 12th problem via $mu_4$ i-cycle preperiodic orbits; (2) Grothendieck's 'etale topos realized as the Higgs complex scalar doublet $Phi$; (3) Birch-Swinnerton-Dyer values $L'(E,1)_p approx 0.4472$ yielding the Higgs VEV $v_p = 246$ GeV; (4) Riemann zeta zeros as propagator interference patterns reproducing Planck CMB data; (5) entropy kernel $S_p to g_{muu_p}$ generating Einstein field equations. All predictions match experiment to sub-percent precision.
Category: Number Theory
[13] ai.viXra.org:2510.0047 [pdf] submitted on 2025-10-21 16:01:18
Authors: Monte Richardson
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We describe and present a symbolic framework to analyze the Collatz conjecture by encoding itsoperations and dynamics into deterministic motifs. Odd transformations, (3x+1), are represented by O. Ek represents maximal divisions by powers of 2. With this grammar, we are able to structurally analyze Collatz trajectories without computation or number theory. Five core results are proved:u2022 Non-trivial cycles are a structural impossibility. u2022 Descent from motifs dominates descent. u2022 All numeric trajectories map to a distinct symbolic path. u2022 Convergence is guaranteed u2022 Inverse motifs demonstrated that every positive integer is reachable from 1.Overall, these results create a symbolic resolution to the Collatz process. Over Z+, the grammar is lossless, complete, and reversible. Additionally, the grammar offers a universal and deterministic framework for understanding Collatz dynamics.
Category: Number Theory
[12] ai.viXra.org:2510.0045 [pdf] replaced on 2025-11-06 04:24:54
Authors: Sizwe Tshabalala
Comments: 30 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
In this paper I introduce a branch of math called Nembelo; from it I introduce a novel function I call the An() function. This function adds number to itself allowing us to study numbers in ways that enables us to have insight into number's behavior. The numbers studied are natural n, primarily 1. To answer the question on the nature of number, for so far the debate has been mostly philosophical and anecdotal, strange enough, not mathematical. This is because it has been taken as granted that math itself cannot answer this question but with the An() function; adding n to itself — we have a way of figuring out what numbers on themselves have to say. As we follow the behavior of n under An(n), it will appear that math has a say on ontology, metaphysics and theology. Theology, because we can naturally derive the concept of GOD from An(1) and not impute it as Gödelu201fs ontological proof. As many fields have been claimed for math & science in past times, e.g., weather prediction, so are these metaphysical fields. However the ambition of the math extends to the unification of math, science, biology, sociology, for which the current theses is not adequate to argue for. Nevertheless, the base from which one can stand is provided, i.e., evidence mapping mathematical conjecture into physical properties of the universe we observe. Keywords: kucwenga, Platonism, conjecture, behavior of n, return to itself, domain.
Category: Number Theory
[11] ai.viXra.org:2510.0043 [pdf] submitted on 2025-10-19 23:00:24
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript, the eighth in the series on Geometric-Representation Quantum Field Theory (GRQFT), reframes the functorial pathway from arithmetic invariants to physical laws through a kernel-based arithmetic quantum mechanics (AQM) framework, emphasizing the Rydberg-Ritz Combination Principle. We demonstrate how pre-periodic points in rational maps, linked to the Convolution Kernel, serve as class field generators for Hilbert’s 12th problem, extending Kronecker’s Jugendtraum to real quadratic fields. The Third Quadratic Kernel, derived from the Runge-Lenz vector (RLV) and binary quadratic forms (BQFs), embeds SO(4) symmetries and ties to dispersion invariants via the Dispersion Kernel. The Entropy Kernel formalizes thermodynamic derivations inspired by Planck, bounding scales with Spec(Z) primes and Monster supersingular primes. U(1) gauge symmetry, with field strength tensors ( F_{muu} ), and the BSD Matrix Kernel physicalize torsion-to-curvature transitions, culminating in the capstone BSD conjecture. The i-cycle bundle’s 4-torsion, modeled through kernel interactions, connects to 4-vectors via quadratic dispersion. This kernel-centric approach strengthens GRQFT’s resolution of pre-Planck dynamics, with implications for quantum gravity and class field theory.
Category: Number Theory
[10] ai.viXra.org:2510.0037 [pdf] submitted on 2025-10-14 21:07:16
Authors: J. W. McGreevy
Comments: 7 Pages.
This manuscript, the seventh in the series on Geometric-Representation Quantum Field Theory (GRQFT), formalizes the functorial pathway from arithmetic structures to physical dispersion relations using category theory. We define key categories including Galois representations (GalRep), quadratic fields (QuadField), automorphic forms (AutForm), and physical dispersion relations (PhysDisp). A chain of functors is described: automorphic induction (AutInd) from GalRep to AutForm, the i-cycle twist (Twist$_i$) from real to imaginary quadratic fields, and the dispersion mapping (DispMap) from AutForm to PhysDisp. The categorical product resolves the hyperbolic frontier of Hilbert's 12th problem, generating class fields as limits and pullbacks. This framework relates to class field theory, providing a physical-geometric resolution to extension problems via quadratic unification and dispersion kernels.
Category: Number Theory
[9] ai.viXra.org:2510.0035 [pdf] submitted on 2025-10-14 20:25:19
Authors: Chandhru Srinivasan
Comments: 7 Pages.
This work presents empirical evidence for a novel phenomenon concerning the sums of prime factors of odd semiprimes. For a semiprime N=p⋅q, where p,q are odd primes, this work investigates the modular residue properties of the sum s=p+q modulo a dynamically chosen modulus m. We conjecture that s (mod"u2009u2009" m) lies within a small subset R_m⊂Z_m whose size grows sublinearly relative to m. By combining multiple such modular constraints via the Chinese Remainder Theorem (CRT), the candidate space for s shrinks dramatically and help us to reconstruct the prime factors, leading to potential improvements in semiprime factorization algorithms.
Category: Number Theory
[8] ai.viXra.org:2510.0027 [pdf] submitted on 2025-10-11 22:44:53
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript, the sixth in the Foundations of Geometric-Representation Quantum Field Theory (GRQFT) series, advances the quadratic unification framework through a categorical approach using the Yoneda lemma. We establish a rigorous, explicit functorial mapping from the Runge-Lenz Vector (RLV) and Johnson-Lippmann Operator (JLO) conic quadratics---defined by the conservation law $A^2$ (quadratic in position and momentum)---to binary quadratic forms (BQFs) over the integers, mediated by the group law on the elliptic curve $E : y^2 = x^3 - x$ with complex multiplication (CM) by $mathbb{Z}[i]$ and 4-torsion $E[4]$. This Yoneda unification connects arithmetic structures to physical metrics via quadratic dispersion relations, leveraging the Laplace transform to bridge $p$-adic valuations and spacetime geometry. The spectral action $operatorname{Tr}(f(D/Lambda))$ embeds this into energy-momentum tensors, deriving gravity and the Standard Model (SM) from moonshine via the $i$-cycle bundle and mod-4 primes, with ties to Grothendieck's Weil proofs.
Category: Number Theory
[7] ai.viXra.org:2510.0020 [pdf] submitted on 2025-10-09 17:34:03
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the fifth in the series on Geometric-Representation Quantum Field Theory (GRQFT), consolidates the quadratic unification framework as a rigorous mathematical description of quantized energy/momentum oscillations in the pre-Planck spacetime lump. We demonstrate that GRQFT describes the lump as a torsion-dominated epoch, with curvature emerging as an internal symmetry via the Einstein-Cartan formalism. Gravity is associated with p-adic attraction in the arithmetic base, where p-adic norms minimize shared p-factors, echoing gravitational pull. The i-cycle bundle, ramified prime 2, and mod-4 primes are explicitly connected as the arithmetic mechanism for symmetry breaking and metric emergence. Kronecker's complex multiplication extensions are incorporated, showing how abelian extensions from roots of unity generate the torsion structures. Dispersion relations are derived quadratically, unifying relativistic physics with arithmetic structures.
Category: Number Theory
[6] ai.viXra.org:2510.0019 [pdf] submitted on 2025-10-08 18:16:49
Authors: Gongshan Liu
Comments: 21 Pages.
We report two independent but deeply interconnected discoveries in the statisticalbehavior of Riemann zeta function zeros. First, near Primorial values (products ofthe first k primes), the distribution of zero spacings exhibits systematic deviationsfrom Random Matrix Theory predictions, characterized by variance anomalies (ratio1.72, p < 10−10) and precise log-normal fits (KS p = 0.08), completely departingfrom GUE distributions (p < 10−27). Second, the derivative amplitude H(ρ) = |ζu2032(ρ)|correlates with minimal neighbor spacing via Spearman ρ ≈ 1/√2, with this correlationstrengthening by 23.5% within Primorial windows.These findings reveal a three-layer arithmetic structure in prime distribution: theclassical Riemann ζ function (zero locations), a derivative operator (error weights),and a Primorial modulation operator (local enhancement). We propose a unified modulation function G(t, P) that quantitatively predicts H-value suppression (34%), spacing variance increase (72%), and correlation enhancement (24%) at Primorial points,achieving agreement with observations within 5% error.This framework provides a new perspective on prime distribution and demonstratesthe potential of human-AI collaboration in mathematical discovery.
Category: Number Theory
[5] ai.viXra.org:2510.0017 [pdf] submitted on 2025-10-08 03:13:31
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the fourth in the series on Geometric-Representation Quantum Field Theory (GRQFT), extends the functorial pathway outlined in previous installments by focusing on the emergence of diffeomorphism invariance and the evolution of metrics from the arithmetic base to those in general relativity (GR) and Einstein-Cartan theory. Building on the arithmetic-geometric foundations over the ``field with one element'' F$_1$, the i-cycle bundle, elliptic torsion, Runge-Lenz vector (RLV)/Johnson-Lippmann operator (JLO) algebra, and binary quadratic forms (BQFs), we demonstrate how the Néron-Tate height pairing and p-adic valuations define an arithmetic ``metric'' that evolves into the spacetime metric $g_{muu}$. Diffeomorphism covariance emerges from the invariance under modular transformations in the moduli space and Galois actions, ensuring coordinate-independent laws. Torsion in Einstein-Cartan arises from ramification asymmetries, sourced by spinors via the i-cycle twists. This provides a unified derivation of gravitational structures from arithmetic vacua, with explicit mappings and consistency checks.
Category: Number Theory
[4] ai.viXra.org:2510.0013 [pdf] submitted on 2025-10-07 18:00:35
Authors: Berkouk Mohamed
Comments: 18 Pages. In French (Please fill the Submission Form in Englished!)
In 1742, Christian Goldbach wrote to Leonhard Euler proposing the following weak conjecture: Every odd number greater than 5 can be written as a sum of three prime numbers. Euler replied with the stronger version of the conjecture: Every even number greater than two can be written as a sum of two prime numbers. we started by sketching the demonstration of the weak Goldbach conjecture by living, I would not dwell on it, in fact I would bet against it"by putting a little order among the multiple choice of variables and their values, to arrive at a better distribution of the arcs in the circle including the rational numbers a/q, which define points Hardy-Littlewood circle method, on its application to the strong conjecture, A.Helfgott replied in a video-tube: "I think that without hope for the moment, if this were to be given up from my around which the major arcs are articulated and their contribution for the integral > 0. I opted for the Cantor coupling function which generally restores the pairs of integers in perfect order, then within the pairs, a second function called "selection" (f(n)=(n^2+n+4)/2) introduces a second order to choose q, the divisor of the rational in question, afterwards, I was surprised to find a third function which objectifies the decrease and the disappearance of their influence in the integral R'(n).
Category: Number Theory
[3] ai.viXra.org:2510.0012 [pdf] submitted on 2025-10-07 17:55:27
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the third in the series on Geometric-Representation Quantum Field Theory (GRQFT), extends the functorial pathway outlined in ``The Threefold Way: Derivation of the Standard Model's Three Generations from the Monster Group'' cite{McGreevy2025a} and ``Foundations of GRQFT: Elliptic Torsion, the $i$-Cycle Bundle, and Hidden Symmetries in the Threefold Way -- Part II'' cite{McGreevy2025b}. We rigorously derive a connection between the algebra of the Runge-Lenz vector (RLV) and Johnson-Lippmann operator (JLO) and binary quadratic forms (BQFs), grounded in the arithmetic geometry of elliptic curves with complex multiplication (CM). Starting from first principles, we map the quadratic relationships in RLV orbital equations to BQF discriminants and norms, embedding the so(4) Lie algebra into the space of forms via quaternion actions and lattice endomorphisms. This extension stabilizes the $i$-cycle bundle and torsion structures in the F$_1$ geometry base, providing a unified arithmetic origin for conserved quantities in classical and quantum mechanics. The role in defining geometry over the ``field with one element'' F$_1$ is emphasized, with BQFs classifying the absolute lattices that seed emergent symmetries. Consistency checks and implications for pre-Planck unification are discussed.
Category: Number Theory
[2] ai.viXra.org:2510.0007 [pdf] submitted on 2025-10-05 02:51:14
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript expands upon the Geometric-Representation Quantum Field Theory (GRQFT) framework introduced in the inaugural paper of this series, The Threefold Way: Derivation of the Standard Model's Three Generations from the Monster Group'' cite{McGreevy2025}. Building on the functorial pathway from the Riemann zeta function to the Monster group's McKay-Thompson series via automorphic induction and monstrous moonshine, we delve into the arithmetic-geometric base over the field with one element'' ($mathbb{F}_1$) geometry. Specifically, we rigorously derive the SO(4) hidden symmetry of the Kepler problem from the 4-torsion structure of the elliptic curve $E: y^2 = x^3 - x$ over $mathbb{Q}(i)$, ramified at the prime $p=2$. We introduce the $i$-cycle bundle as a principal $mu_4$-bundle encoding the complex multiplication (CM) action, and establish explicit mappings from the Rosati involution and N'eron-Tate height pairing to the Runge-Lenz vector (RLV) and Johnson-Lippmann operator (JLO). This provides a unified arithmetic origin for classical and quantum conserved quantities in GRQFT, linking torsion imbalances to eccentricity and unitarity. The exposition emphasizes mathematical rigor, with explicit computations and embeddings, positioning this as a foundational component for deriving Standard Model structures from arithmetic vacua.
Category: Number Theory
[1] ai.viXra.org:2510.0005 [pdf] submitted on 2025-10-03 19:37:37
Authors: Zirui Wang
Comments: 8 Pages.
This paper presents a complete resolution of the Erdős—Straus conjecture. We introduce a systematic approach termed the "Constraint-Construction Method," which yields explicit infinite families of solutions for all integers satisfying n otequiv 1, 7 pmod{12} . Our principal contribution is the discovery of a unified parametric framework that establishes the conjecture's equivalence to the existence of positive integer solutions to the equation m(4t - 1 - m) = 2n satisfying an elementary parity condition. This framework not only provides a constructive proof for the most obstinate residue classes but also reduces the computational verification from a three-dimensional search to a two-dimensional one, thereby furnishing both theoretical completeness and computational efficacy.
Category: Number Theory