| ai.viXra.org > Number Theory > 2602 |
|
 |
| ai.viXra.org > Number Theory > 2602 |
|
|
[7] ai.viXra.org:2602.0125 [pdf] submitted on 2026-02-27 05:34:56
Authors: Chaiya Tantisukarom
Comments: 9 Pages.
This study formalizes the Prime Gear Geometry (PGG) as a dynamical system. We demonstrate that the Riemann Hypothesis (RH) is not a static property of numbers, but a structural necessity of a rolling engine. We identify the $m$-cutoff as the "Mechanical Secret" that governs the transition between discrete prime forging (Time Domain) and spectral stability (Frequency Domain).
Category: Number Theory
[6] ai.viXra.org:2602.0109 [pdf] submitted on 2026-02-25 01:07:15
Authors: Siqi Liu
Comments: 2 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper provide a formal mathematical proof of the non-existence of non-trival loops in the problem of Collatz Conjecture by using main algebra tools of 2-adic constraints.
Category: Number Theory
[5] ai.viXra.org:2602.0079 [pdf] submitted on 2026-02-16 23:53:19
Authors: Wiroj Homsup
Comments: 3 Pages.
A new Twin prime sieve based on a modified sieve of Sundaram is introduced. It sieves through the set of natural numbers n such that 3n is not representable in either of the forms 2ij + i + j or 2ij + i + j +1 for positive integers i, j.
Category: Number Theory
[4] ai.viXra.org:2602.0068 [pdf] submitted on 2026-02-15 00:33:07
Authors: Chaiya Tantisukarom
Comments: 11 Pages.
This article presents "Prime Gear Geometry," a deterministic mechanical framework that redefines the integer axis as a master gear ($C_1$) with a discrete unit weight-step of $+1$. Unlike analytic models that rely on the complex-plane "1/2" critical line of the Riemann Hypothesis, this theory posits that prime numbers are exact geometric outcomes forged by $C_1$ at coordinates of total asynchronous interference. We establish the "Prime Gear Synchronization Conjecture," stating that total phase alignment of a prime gear group occurs only at Primorial intervals. This mechanical exactness is used to resolve the Goldbach, Twin Prime, and Collatz conjectures not as probabilistic likelihoods, but as structural necessities of a machine that, by the laws of relatively prime circumferences, is incapable of perfect synchronization within the finite bounds of the $C_1$ axis.
Category: Number Theory
[3] ai.viXra.org:2602.0059 [pdf] submitted on 2026-02-12 19:18:54
Authors: J. W. McGreevy
Comments: 18 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present the Arithmetic Relativistic Emergence (ARE) framework, in which the fundamental symmetries of General Relativity, Einstein—Cartan gravity with torsion, and quantum field theory (Standard Model sectors) emerge tautologically from pure number theory viathe arithmetic geometry of the rational numbers Q. The Riemann zeta function ζ(s) represents the maximally symmetric pre-geometric vacuum phase, with perfect functional-equation symmetry around Re(s) = 1/2 and pole at s = 1 as the unified source of arithmetic energy/information.Spontaneous symmetry breaking induced by the weight-12 modular discriminant ∆(τ ) = η(τ )24 = (2π) 12(E4(τ ) 3 − E6(τ ) 2 )/1728 disperses this background into Archimedean divergence (smooth analytic curvature density) and non-Archimedean curl (torsion/spin density at p-adic fibers), with the functional-equation mirror s = 6 enforcing variational d balance of the arithmetic degree deg( L). The emergent 4D Lorentzian manifold M carries an adelic principal Lorentz/Spin frame bundle decomposed via the adele ring AQ. An effective Chern—Weil homomorphism—employing Bott—Chern forms at infinity and classical invariant polynomials at finite places—maps split curvature forms (Fdiv, Hcurl) to arithmetic characteristic classesin Arakelov Chow groups. These classes are stationary under metric d variations (δgdeg = 0 at s = 6), providing rigid global topological invariants (Pontryagin-like, Euler-like, torsion-twisting) preserved in the broken phase—the inevitable geometric and topological labels of arithmetic symmetry breaking. Heaviside synchronization (τdiv = τcurl) at s = 6 renders the arithmetic medium transparent, yielding distortionless propagation and unified causality. The Rankin—Selberg self-convolution L(∆×∆, s) contains ζ(s) factors, allowing recombination to the symmetric vacuum. Theemergent metric determinant √−g serves as the physical scalar whose arithmetic balancing across places enforces general covariance, proper volume preservation, and integration of curvature invariants. Fermions (12 Weyl per generation from Leech lattice Z2-orbifold),gauge sectors (finite algebra C ⊕ H ⊕ M3(C)), and constants (−α 1 ≈ 137.036, Λ ∼ 10−122MPl2, G ∼ 10−38 m−2) emerge via spectral actionand adelic convolution. ARE offers a tautological origin: physical laws are the minimal effective description preserving arithmetic consistency post-symmetry breaking.
Category: Number Theory
[2] ai.viXra.org:2602.0043 [pdf] submitted on 2026-02-10 02:21:23
Authors: J. W. McGreevy
Comments: 19 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a rigorous synthesis in which the fundamental symmetries of General Relativity and quantum field theory emerge from the axioms of arithmetic geometry and number theory. Central is the weight-12 modular discriminant Δ(τ) = η(τ)^{24} = (2π)^{12} (E_4^3 - E_6^2)/1728, interpreted as the vacuum potential. The arithmetic degree (total integrated curvature) must disperse equivalently across Archimedean (smooth, complex-analytic) and non-Archimedean (discrete, p-adic) places to maintain global consistency via the product formula. This dispersion is enforced at the critical mirror point s=6 of L(Δ,s), where the functional equation symmetry balances openness and rigidity.The Hilbert-Pólya operator Ĥ = 1/2 + i (D_∞ ⊕ ∑p D_p) acts self-adjointly on the adelic Hilbert space, with eigenvalues corresponding to resonances tied to L(Δ,s) zeros. The 1728 frequency (12^3) serves as the universal gear ratio/adiabatic regulator. The 12-fermion matrix arises from the Leech lattice V_Λ{24} Z_2-orbifold, folding 24 bosonic dimensions into 12 Weyl fermions per generation via Möbius twist.A 4D Lorentzian manifold emerges via noncommutative geometry (KO-dimension 6 adelic spectral triple), with the spectral action Tr f(��/Λ) yielding the Einstein-Hilbert term and stress-energy from p-adic torsion convolution. The master equation δ_g widehat{deg}(mathcal{L}) = 0 at s=6 recovers the Einstein field equations with cosmological constant Λ ≈ M_Pl^2 e^{-288} (double-twist entropy) and fine-structure constant α^{-1} ≈ 137.036 from Petersson norm corrections. This framework posits that GR and the Standard Model are stereographic projections of the weight-12 balanced modular form onto the Möbius-Planck manifold, providing a tautological origin for physical laws from number theory.
Category: Number Theory
[1] ai.viXra.org:2602.0014 [pdf] replaced on 2026-03-23 22:03:10
Authors: Ezadiin Redwan
Comments: 5 Pages.
"We present a universal proof of the Strong Goldbach Conjectureby shifting the problem from arithmetic density to Geometrical Transformation. By defining primes as the structural building of every integer numbers via the Fundamental Theorem of Arithmetic, we map theinteraction between addition and multiplication onto a vector space.We prove that the identity 2n cos(θ) = a+b is a structural requirementof this space. This non-constructive existence proof demonstrates thatfor every even integer 2n, a prime partition (a, b) is geometrically necessitated by the scalar projection of prime-based vectors, therebyresolving the parity problem through The Dot Product."
Category: Number Theory