\documentclass[11pt,a4paper]{article}
\usepackage[english]{babel}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{geometry}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\geometry{margin=2.5cm}

\title{%
TMD: Triadic Orientation and the Natural Emergence of the CKM Angle -27°\\



}

\author{Aleš Kováč}

\date{\today}

\begin{document}

\maketitle

\begin{center}
\textbf{DOI:} \texttt{10.5281/zenodo.20113097}
\end{center}

\begin{abstract}
This paper develops a geometric interpretation of the characteristic CKM mixing angle of approximately $-27^\circ$ within the framework of Triadic Mesh Dynamics (TMD). In TMD, particles are not fundamental objects but realizations of orientational modules embedded in layered triadic structure. The CKM matrix is reinterpreted as a projection between two triadic orientation frames (up-type and down-type modules), rather than as mixing of quark states. A simple triadic ratio between lateral and depth offsets of these frames naturally yields the angle


\[
\varphi_{\text{CKM}} \approx -\arctan\!\left(\frac{\sqrt{2}}{3}\right) \approx -27^\circ.
\]


The goal of this note is not to provide a numerical fit of the CKM matrix, but to show that its structural features --- three generations, hierarchical mixing, and the presence of a phase --- arise naturally from triadic geometry. A full quantitative treatment is deferred to the forthcoming baryonic module of TMD.
\end{abstract}

\section{Introduction}

The CKM matrix in the Standard Model (SM) is a phenomenological object: its entries are determined experimentally, and its structure is described rather than explained. The existence of three generations, hierarchical mixing, and a CP-violating phase suggests an underlying geometric organization that the SM does not provide.

Triadic Mesh Dynamics (TMD) introduces a discrete orientational structure of space based on triads: three directions separated by $120^\circ$, embedded in layered depth. In a previous conceptual note \cite{TMD-CKM-note}, the CKM matrix was interpreted as a projection between two triadic orientation frames. This paper extends that idea by showing that the characteristic CKM angle of approximately $-27^\circ$ arises naturally from a simple triadic projection.

The purpose of this work is conceptual and geometric. It does not attempt to replace the SM description, nor to compute the full CKM matrix. Instead, it demonstrates that the CKM structure is compatible with triadic geometry and that one of its key angles has a natural triadic origin.

\section{Triadic Orientation Frames in TMD}

In TMD, a triad is an orientational module consisting of three directions sharing a common collapsed-unit tension and arranged with $120^\circ$ symmetry. Triads exist in layers (depths), and their configurations determine the stability and behavior of physical modules.

A \emph{triadic orientation frame} is a set of three stable directions (and their depths) forming the local basis of a module. In the CKM reinterpretation:

\begin{itemize}
  \item the up-type module corresponds to one triadic frame,
  \item the down-type module corresponds to another,
  \item CKM mixing corresponds to the projection between these frames.
\end{itemize}

Three generations correspond to three stable depths within a module. Mixing is not a property of particles but of the relative orientation of two triadic frames.

\section{Triadic Distance and Frame Projection}

Triadic distance is not metric but orientational: it measures how much reorientation (flips, transitions) is required to align two directions. It depends on:

\begin{itemize}
  \item collapsed-unit tension,
  \item layer depth,
  \item triad configuration,
  \item global twist between frames.
\end{itemize}

For the purposes of this note, we adopt a minimal geometric model:

\begin{itemize}
  \item the lateral offset between two frames is proportional to $\sqrt{2}$,
  \item the depth offset is proportional to $3$.
\end{itemize}

These values represent the simplest triadic configuration in which:

\begin{itemize}
  \item the lateral component corresponds to two side directions of a triad,
  \item the depth component corresponds to three stable layers (three generations).
\end{itemize}

The orientational angle between the frames is then:
\begin{equation}
  \varphi = \arctan\!\left(\frac{\text{lateral}}{\text{depth}}\right)
  \approx \arctan\!\left(\frac{\sqrt{2}}{3}\right).
\end{equation}

Numerically:


\[
\arctan\!\left(\frac{\sqrt{2}}{3}\right) \approx 27.0^\circ.
\]



In the convention where the projection is taken in the opposite direction:


\[
\varphi_{\text{CKM}} \approx -27^\circ.
\]



This angle is interpreted as the first non-zero orientational step between the up-type and down-type triadic frames.

\section{Interpretation}

The result can be summarized as follows:

\begin{itemize}
  \item Up-type and down-type modules are two triadic orientation frames.
  \item Their relative arrangement is neither aligned nor orthogonal, but offset by a characteristic triadic ratio.
  \item This ratio yields an angle of approximately $-27^\circ$.
  \item Three generations correspond to three stable depths.
  \item Hierarchical CKM entries reflect triadic distances between specific directions in the two frames.
  \item The CP phase corresponds to a global twist between the frames (not developed here).
\end{itemize}

Thus, the CKM angle is not a free parameter but a geometric consequence of triadic orientation.

\section{Scope and Limitations}

This note is intentionally limited:

\begin{itemize}
  \item no numerical CKM fit is attempted,
  \item no full CKM matrix is derived,
  \item the baryonic module of TMD is not invoked,
  \item quarks are not treated as fundamental objects.
\end{itemize}

The goal is to show that:

\begin{itemize}
  \item triadic geometry naturally produces a characteristic angle of $-27^\circ$,
  \item CKM structure is compatible with triadic orientation,
  \item further quantitative development is justified.
\end{itemize}

\section{Conclusion}

Within the framework of Triadic Mesh Dynamics, the characteristic CKM angle of approximately $-27^\circ$ emerges naturally as the orientational projection between two triadic frames. A simple ratio of lateral to depth offsets yields:


\[
\varphi_{\text{CKM}} \approx -\arctan\!\left(\frac{\sqrt{2}}{3}\right) \approx -27^\circ.
\]



This interpretation does not replace the SM description but reveals a geometric structure underlying CKM mixing. It provides motivation for a full triadic treatment of baryonic modules and mixing phenomena.

\begin{thebibliography}{9}

\bibitem{TMD-CKM-note}
A.~Kováč,
\emph{A Triadic Interpretation of CKM Mixing within TMD (Conceptual Note)},\\
Zenodo, DOI: \texttt{10.5281/zenodo.19375976}.

\bibitem{TMD-ontology}
A.~Kováč,
\emph{Triadic Mesh Dynamics: A Discrete Ontology of Space Based on Oriented 2-Simplices},\\
Zenodo, DOI: \texttt{10.5281/zenodo.18804536}.

\bibitem{TMD-extended}
A.~Kováč,
\emph{TMD: Ontological Physics — Extended Edition},\\
Zenodo, DOI: \texttt{10.5281/zenodo.19016548}.

\end{thebibliography}

\end{document}