Divisor (algebraic geometry)
View article: A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications
A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications Open
Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ and De~Moivre's formula describe rotations generated by an imaginary unit $i$ satisfying $i^2=-1$. In associative and alternative hypercomplex algebras, however, arbitrary elements $A$ …
View article: A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications
A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications Open
Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ and De~Moivre's formula describe rotations generated by an imaginary unit $i$ satisfying $i^2=-1$. In associative and alternative hypercomplex algebras, however, arbitrary elements $A$ …
View article: Structure of Perfect Numbers
Structure of Perfect Numbers Open
This paper proves the nonexistence of odd perfect numbers by revealing the deep structural nature of perfection. We show that perfect numbers are not arithmetic coincidences but arise solely from binary symmetry: two coherent geometric seq…
View article: Structure of Perfect Numbers
Structure of Perfect Numbers Open
This paper proves the nonexistence of odd perfect numbers by revealing the deep structural nature of perfection. We show that perfect numbers are not arithmetic coincidences but arise solely from binary symmetry: two coherent geometric seq…
View article: An Expository Note on MacMahon's Partition Statistics and the Elementary Detection of Powers of Two
An Expository Note on MacMahon's Partition Statistics and the Elementary Detection of Powers of Two Open
This expository note presents a novel application of MacMahon's partition statistics to elementary number theory. Building on recent work by Craig, van Ittersum, and Ono (2024), which utilized partition functions to detect primality, this …
View article: A Perfectoid Framework for Moduli of Log Fano Varieties
A Perfectoid Framework for Moduli of Log Fano Varieties Open
This paper develops a novel perfectoid framework for the study of moduli spaces of log Fano varieties. Log Fano varieties, characterized by their anti-canonical divisor being ample relative to a boundary, play a crucial role in birational …
View article: A Perfectoid Framework for Moduli of Log Fano Varieties
A Perfectoid Framework for Moduli of Log Fano Varieties Open
This paper develops a novel perfectoid framework for the study of moduli spaces of log Fano varieties. Log Fano varieties, characterized by their anti-canonical divisor being ample relative to a boundary, play a crucial role in birational …
View article: An Expository Note on MacMahon's Partition Statistics and the Elementary Detection of Powers of Two
An Expository Note on MacMahon's Partition Statistics and the Elementary Detection of Powers of Two Open
This expository note presents a novel application of MacMahon's partition statistics to elementary number theory. Building on recent work by Craig, van Ittersum, and Ono (2024), which utilized partition functions to detect primality, this …
View article: Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling
Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling Open
DOI: 10.5281/zenodo.17824151Title: Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling** Author: Miles Enoch Tracy contact: [email protected] ORCID: https://orcid.…
View article: Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling
Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling Open
DOI: 10.5281/zenodo.17824151Title: Unity Is One: The Riemann Hypothesis as a Geometric Theorem of Perfect Error Correction via 24‑Fold Toroidal Cardioid Tiling** Author: Miles Enoch Tracy contact: [email protected] ORCID: https://orcid.…
View article: Bézout-Genus Duality: The Structural Core of Riemann-Roch
Bézout-Genus Duality: The Structural Core of Riemann-Roch Open
This paper explores the profound relationship between Bézout's theorem and the concept of genus within the foundational framework of the Riemann-Roch theorem for algebraic curves. We propose and elucidate a "Bézout-Genus Duality" as the in…
View article: Bézout-Genus Duality: The Structural Core of Riemann-Roch
Bézout-Genus Duality: The Structural Core of Riemann-Roch Open
This paper explores the profound relationship between Bézout's theorem and the concept of genus within the foundational framework of the Riemann-Roch theorem for algebraic curves. We propose and elucidate a "Bézout-Genus Duality" as the in…
View article: A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications
A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications Open
Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ and De~Moivre's formula describe rotations generated by an imaginary unit $i$ satisfying $i^2=-1$. In associative and alternative hypercomplex algebras, however, arbitrary elements $A$ …
View article: The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1}
The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1} Open
We extend the Period-Basin Bijection to n symbolic sequences. The underlying space isnon-metrizable but uniformizable; periodic structure is extracted via entourage filtration onthe uniform structure, not distance. Joint periods in n seque…
View article: Logarithmic de Rham Stacks and Non-Abelian Hodge Theory
Logarithmic de Rham Stacks and Non-Abelian Hodge Theory Open
In this article, we introduce the logarithmic de Rham stack of a pair (X, D), for X a smooth variety X over a field k of positive characteristic p, and D a strict normal crossings divisor on X. Using this stack, we prove a new version of l…
View article: Logarithmic de Rham Stacks and Non-Abelian Hodge Theory
Logarithmic de Rham Stacks and Non-Abelian Hodge Theory Open
In this article, we introduce the logarithmic de Rham stack of a pair (X, D), for X a smooth variety X over a field k of positive characteristic p, and D a strict normal crossings divisor on X. Using this stack, we prove a new version of l…
View article: The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1}
The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1} Open
We extend the Period-Basin Bijection to n symbolic sequences. The underlying space isnon-metrizable but uniformizable; periodic structure is extracted via entourage filtration onthe uniform structure, not distance. Joint periods in n seque…
View article: The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1}
The n-Dimensional Period-Basin Bijection: Geometry Encodes Arithmetic on N^{n-1} Open
We extend the Period-Basin Bijection to n symbolic sequences. The underlying space isnon-metrizable but uniformizable; periodic structure is extracted via entourage filtration onthe uniform structure, not distance. Joint periods in n seque…
View article: Iterative Binary Divisibility Criterion for Odd Divisors
Iterative Binary Divisibility Criterion for Odd Divisors Open
A deterministic algorithm for testing the divisibility of an integer N by a given odd divisor d is proposed. The algorithm uses exclusively the operations of addition of d and division by 2 (right bitwise shift), thereby avoiding the costl…
View article: The Structural Core of Riemann-Roch: A Bézout-Genus Duality
The Structural Core of Riemann-Roch: A Bézout-Genus Duality Open
The Riemann-Roch theorem stands as a cornerstone in algebraic geometry, providing a fundamental relationship between the dimension of spaces of meromorphic functions or sections of line bundles and the topological and geometric properties …
View article: The Structural Core of Riemann-Roch: A Bézout-Genus Duality
The Structural Core of Riemann-Roch: A Bézout-Genus Duality Open
The Riemann-Roch theorem stands as a cornerstone in algebraic geometry, providing a fundamental relationship between the dimension of spaces of meromorphic functions or sections of line bundles and the topological and geometric properties …
View article: Iterative Binary Divisibility Criterion for Odd Divisors
Iterative Binary Divisibility Criterion for Odd Divisors Open
A deterministic algorithm for testing the divisibility of an integer N by a given odd divisor d is proposed. The algorithm uses exclusively the operations of addition of d and division by 2 (right bitwise shift), thereby avoiding the costl…
View article: Universal Divisor Superposition (UDS) – Edition 2
Universal Divisor Superposition (UDS) – Edition 2 Open
We present an extremely simple orthonormal basis {|n⟩}n≥1 of a subspace of ℓ²(ℕ) defined by the single formula |n⟩ = 1/√d(n) ∑_{d|n} |ϕ_d⟩, where d(n) is the number of positive divisors of n and {|ϕ_k⟩} is the canonical basis. Immediate co…
View article: Universal Divisor Superposition (UDS) – Edition 2
Universal Divisor Superposition (UDS) – Edition 2 Open
We present an extremely simple orthonormal basis {|n⟩}n≥1 of a subspace of ℓ²(ℕ) defined by the single formula |n⟩ = 1/√d(n) ∑_{d|n} |ϕ_d⟩, where d(n) is the number of positive divisors of n and {|ϕ_k⟩} is the canonical basis. Immediate co…
View article: Evaluating the tame Brauer group of open varieties over local fields
Evaluating the tame Brauer group of open varieties over local fields Open
In this document we let $U$ be a smooth variety of pure dimension $d$ over a local field $k_v$ with unit ball $\mathcal{O}_v$ and residue field $\mathbb{F}$ of characteristic $p>0$ and we set $n$ to be a positive integer such that $p\nmid …
View article: Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.0)
Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.0) Open
📄 Overview This paper presents the first numerical evidence that deterministic arithmetic systems can exhibit fast scrambling dynamics, a hallmark of black hole quantum chaos. We investigate the Out-of-Time-Order Correlator (OTOC) for the …
View article: A Stronger Riemann-Roch and Sharp Genus Bounds for Moduli Spaces of Curves
A Stronger Riemann-Roch and Sharp Genus Bounds for Moduli Spaces of Curves Open
This paper introduces a strengthened version of the classical Riemann-Roch theorem, explicitly tailored for application within the complex geometry of moduli spaces of curves. While the original theorem provides a fundamental link between …
View article: Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.1)
Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.1) Open
📄 Overview This record releases version v1.1 of the paper: “Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian”. The paper investigates whether a fully deterministic arithmetic sys…
View article: A Stronger Riemann-Roch and Sharp Genus Bounds for Moduli Spaces of Curves
A Stronger Riemann-Roch and Sharp Genus Bounds for Moduli Spaces of Curves Open
This paper introduces a strengthened version of the classical Riemann-Roch theorem, explicitly tailored for application within the complex geometry of moduli spaces of curves. While the original theorem provides a fundamental link between …
View article: Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.1)
Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian (v1.1) Open
📄 Overview This record releases version v1.1 of the paper: “Fast Scrambling in Arithmetic Holography: Measuring the Lyapunov Exponent of the Additive Divisor Hamiltonian”. The paper investigates whether a fully deterministic arithmetic sys…