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[7] ai.viXra.org:2512.0086 [pdf] submitted on 2025-12-25 22:21:54
Authors: J. W. McGreevy
Comments: 10 Pages.
Geometric Resolution Quantum Field Theory (GRQFT) is a framework that derives the Standard Model of particle physics, general relativity, and a proof of the Riemann Hypothesis from the sequential exhaustion of the arithmetic orbifold Spec(Z) ∪ {∞}. The exhaustion process approximates the circle at infinity with regular n-gons, producing a failure gap ∆(n) = π 2/(54n2 ) that defines the failure class [fn] = coker(fn) ∈ Het1 ´ (O,Z). This failure class is the orthogonal complement and harmonic conjugate (via Cauchy—Riemann equations) to the graded dimensions Vn of the moonshine module V ♮ . Monster group invariance on V ♮ forces the inner product ⟨[fn], vfixed⟩ = 0 for the Monster-fixed Weyl vector vfixed, implying that all zeros of the analytically continued [fn] lie on the balanced circle |n| = e 1/2, corresponding to Re(s) = 1/2 for the Riemann zeta function — thus proving the Riemann Hypothesis. Gauge symmetries emerge from fiber twists: the order-4 µ4 at τ = i yields SU(2)L × U(1)Y , triality at ρ yields SU(3)c with three generations, and octonion non-associativity yields E8 gravity. The Higgs field is the radial mode on the exceptional divisor E2, electroweak symmetry breaking from its resolution, and the Einstein field equations from the geodesic n(t) in the moduli space of elliptic curves. The Weierstrass ℘-function emerges as the failure potential, sourcing Coulomb-like interactions and Rydbergspectra. Planck duality (momentum/energy as circumferences, length/time as radii) andtorsion from non-associativity complete the theory, with supersingular primes as resonancepoints of maximal symmetry. GRQFT realizes arithmetic holography, with exhaustion asrenormalization group flow and Monster symmetry as bulk invariance.
Category: Number Theory
[6] ai.viXra.org:2512.0068 [pdf] submitted on 2025-12-18 21:39:11
Authors: Stephen R. Campbell
Comments: 38 Pages.
This paper introduces Mersenne Block Dynamics, a structural framework for analyzing the accelerated Collatz or Syracuse map on odd integers. The approach decomposes orbits based on the 2-adic valuation of the successor of an odd integer, effectively measuring the length of the trailing run of ones in its binary expansion, termed the Mersenne tail. This decomposition partitions the dynamics into deterministic blocks where the tail length decreases by exactly one bit at each step, creating a rigid wedge pattern in the binary representation. The framework defines a coarse-grained block map that transitions directly between the starts of successive blocks, isolating all arithmetic complexity into a specific exit exponent. The study derives explicit closed-form transition identities and exact time-scale bookkeeping for these block jumps. Furthermore, it establishes that the block length and exit parameters follow independent geometric distributions in terms of natural density. Under a heuristic assumption of orbit mixing, this intrinsic statistical model predicts a net negative expected logarithmic drift, recovering the classical probabilistic prediction for the Collatz conjecture within a precise structural coordinate system.
Category: Number Theory
[5] ai.viXra.org:2512.0063 [pdf] submitted on 2025-12-17 20:30:41
Authors: Jonas Kaiser
Comments: 7 Pages.
We study the Collatz iteration restricted to odd integers and exhibit a concrete emph{core set}(X) inside the forward-invariant set(Y={6n+1,,6n+5:ninmathbb{N}_0}) (odd integers not divisible by $3$).For the odd Collatz map(f_c(u)=(3u+1)/2^{v_2(3u+1)}) we prove that the restriction (f_cvert_X) is a bijection onto $Y$.This yields a redundant-free ``Collatz sieve'': every value in $Y$ has a unique emph{core predecessor} in $X$.For this particular core, the induced dynamics emph{inside} $X$ is strictly decreasing.As a consequence, $X$ decomposes (without duplicates) into disjoint infinite one-sided chains indexed by a set of emph{heads}(Hsubset X): every element of $Xsetminus{1}$ lies on exactly one head chain, and moving one step up a chain increases the time spent inside $X$ by one.
Category: Number Theory
[4] ai.viXra.org:2512.0047 [pdf] submitted on 2025-12-12 21:35:03
Authors: Khazri Bouzidi Fethi
Comments: 8 Pages.
We present an unconditional framework linking the distribution of prime numbers to the zeros of the Riemann zeta function, with fully explicit and computable error bounds. The core of our method is the stratified constant C_{N,P}(s), which converges unconditionally to 2pi. By isolating the contribution of the Riemann-von Mangoldt error term R(T), we derive the first explicit unconditional constraint on its weighted sum, yielding a quantitative coherence test for zero distribution. The framework is extended to Dirichlet L-functions, providing a new measure for Chebyshev bias. We also develop high-precision computational methods for pi(x) beyond x > 10^{12} and validate our results numerically, achieving relative errors as low as 3.2 times 10^{-10}.
Category: Number Theory
[3] ai.viXra.org:2512.0045 [pdf] submitted on 2025-12-11 00:54:58
Authors: Stephen R. Campbell
Comments: 113 Pages. https://doi.org/10.5281/zenodo.17887464
We develop a unified Mersenne block dynamics framework for the accelerated Collatz (Syracuse) map on odd integers, and push it from structural analysis to a concrete finite-certificate criterion. Each odd x is decomposed into a Mersenne tail and an even prefix, giving rise to Mersenne blocks and a residue graph that control the evolution of trajectories. Using a ledger of visits to residue classes together with a height-aware prefix-carry factor, we derive a carry-controlled drift inequality over windows of W Mersenne blocks. Thisyields a finite-certificate criterion: if one can exhibit a finite residuegraph and associated data satisfying a single explicit inequality, together with a finite verification for all odd x < N0, then every trajectory reaches 1. We instantiate this framework with an explicit mod 64 Mersenne-block residue graph, dynamic programming computations of the relevant drift invariant, and a small-n verification. These data are packaged into machine-readable certificate artifacts; we provide explicit certificates whose correctness can be both mechanically and manually verified, and whose validity would give a complete proof of the Collatz conjecture within this framework.
Category: Number Theory
[2] ai.viXra.org:2512.0025 [pdf] submitted on 2025-12-07 01:23:11
Authors: Kevin Fidelis
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We observe an explicit algebraic relationship between certain initial values under the Collatz map. For a starting integer (X) and a chosen step count (m), numbers of the form [Y = X + k cdot 3^u cdot 2^e ] share the same parity sequence as (X) for the first (m) steps, where (e) is the number of even steps in (X)'s first (m) iterations. The difference (Delta_s = C^{(s)}(Y) - C^{(s)}(X)) evolves as [Delta_s = k cdot 3^{u+o_s} cdot 2^{e-e_s}, ] where (o_s, e_s) count odd/even steps up to (s). This relationship persists until the exponent of 2 in (Delta_s) becomes negative. The result is an elementary algebraic identity with no implication for the Collatz conjecture.
Category: Number Theory
[1] ai.viXra.org:2512.0013 [pdf] submitted on 2025-12-03 21:26:44
Authors: Bahbouhi Bouchaib
Comments: 19 Pages.
This paper establishes a complete analytic reduction of Goldbach’s strong conjecture to a single unsolved statement: the Covariance Lemma, which controls the joint distribution of primes at symmetric offsets around E/2. All other components of Goldbach’s problem—including the existence of primes in short symmetric intervals of width proportional to (log E)²—are already unconditionally resolved by explicit results on primes in short intervals, notably those of Dusart [Dusart 2010, Dusart 2018], as well as classical density theorems grounded in the Prime Number Theorem. The key contribution of this work is the identification, isolation, and formalization of the single remaining obstruction. By proving that the covariance of prime indicators P(E/2—t) and P(E/2+t) cannot suppress all symmetric prime coincidences, one obtains a full proof of Goldbach’s strong conjecture. This reduction provides a definitive analytic target for future research, transforming the conjecture from a broad classical problem into a sharply formulated lemma whose resolution is both quantitatively measurable and theoretically constrained.
Category: Number Theory