From Unity to Number Duality
Year: 2026
Type: preprint
Abstract: This master consolidates the "Unity to Number Duality" and "Spectral Duality" architectures into a single, rigorous predictive framework. The foundational premise is that the generation of the discrete integer lattice is governed by the absolute conservation of discrete parity—a number must be exactly 100% Odd or Even. When mapped through a Hopf intertwiner into continuous $\mathfrak{so}(3)$ phase space, this discrete probability norm ($\mathcal{N}$) establishes the topological Casimir invariant. The framework yields the master equation bridging discrete arithmetic to the continuous Riemann spectrum: $$\mathcal{N}(u) = \sqrt{R(u)^2} \propto \cosh(\Delta(LOS) \cdot u)$$ where $\Delta(LOS)$ is the deviation of any spectral zero from the geometric 0.5 critical axis. Because $\cosh(x)$ grows exponentially for any $x > 0$, an off-line Riemann zero forces the macroscopic wave amplitude to diverge toward infinity. Because the fundamental discrete integer lattice cannot warp to absorb infinite probability ($\mathcal{N} = 1$ is an unbreakable invariant), the continuous geometry physically suppresses the wave by forcing $\Delta(LOS) \to 0$. The Riemann Hypothesis is thereby formally resolved not as a statistical coincidence, but as the strict mechanical limit-state required to preserve the unitarity of discrete arithmetic.
Author: Desmond, Timothy
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222279292
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