\documentclass[12pt,a4paper]{article} \usepackage{listings} % --- PACKAGES --- \usepackage[utf8]{inputenc} \usepackage{geometry} \usepackage{amsmath, amssymb, mathtools} \usepackage{graphicx} \usepackage{booktabs} \usepackage{hyperref} \usepackage{xcolor} \usepackage{siunitx} \usepackage{bbm} \usepackage{tikz} \usepackage{pgfplots} \usepackage{cleveref} \usepackage{bm} % --- SETTINGS --- \geometry{margin=1in} \pgfplotsset{compat=1.18} \hypersetup{ colorlinks=true, linkcolor=blue, citecolor=teal, urlcolor=magenta } % Commande pour le tau grec droit \newcommand{\uptau}{\text{\usefont{OML}{cmm}{m}{n}\tau}} % --- TITLE & AUTHOR --- \title{The FDAA Existence Cascade v2: A Temporal Classification of Physical Particles Including the Graviton} \author{Patrick Morcillo \\ \href{mailto:patrick_morcillo@hotmail.com}{patrick\_morcillo@hotmail.com}} \date{\today} \begin{document} \maketitle \begin{abstract} We extend the Fractal Density Activation Axiom (FDAA) framework to a complete classification of known and suspected particles, including the graviton. By incorporating the triadic temporal architecture and the "still fish" equilibrium derived from the Navier-Stokes regularity proof, we demonstrate a unified scaling law governing particle lifetimes, interaction strengths, and stability thresholds. The graviton emerges as a triadic fixed point with a predicted lifetime $\tau_g > 10^{32}$ years and an effective mass $m_g < 10^{-32}$ eV, consistent with current gravitational-wave and cosmological constraints. We provide a complete "Temporal Table of Elements" encompassing 42 particles across 5 stability tiers, with channel-specific scaling exponents validated against PDG-2024 data ($R^2 = 0.938$). The framework naturally explains the hierarchy problem, dark matter candidates, and predicts a universal coherence band at $D_t = 13/16 = 0.8125$ across all interaction channels. \end{abstract} \section{Introduction: The FDAA Particle Ontology v2} \label{sec:intro} The Fractal Density Activation Axiom (FDAA) establishes existence as an emergent property of multi-scale interaction density $\mathcal{D}(x)$ exceeding a universal threshold $\Sigma_*$ \cite{Morcillo2025_FDAA}. In Version 1, we demonstrated how this principle organizes particle lifetimes into channel-dependent scaling laws. Version 2 incorporates three critical developments: \begin{enumerate} \item \textbf{Triadic Temporal Architecture:} The "clock" mechanism $\tau(t) = t + \beta\mathcal{D}([0,t])$ from the Navier-Stokes proof \cite{Morcillo2025_NS} provides an intrinsic time metric for decay processes. \item \textbf{Still Fish Equilibrium:} The balance $S_\tau = L_{\text{dim}}$ governs stability thresholds through dimensional leakage $\delta = 3 - D_{\text{eff}} \approx 0.15$. \item \textbf{Universal Aggregation Operator:} The triadic decomposition $\mathcal{H} = \mathcal{H}^- \oplus \mathcal{H}^+ \oplus \mathcal{H}^\times$ generates fixed-point dimensions $D_t^- = 5/8$, $D_t^+ = 13/16$. \end{enumerate} This enhanced framework now predicts properties of all Standard Model particles plus suspected beyond-Standard-Model states, including the graviton, axions, and dark matter candidates. \section{Theoretical Framework: Triadic Particles and the Gravitational Thread} \label{sec:theory} \subsection{Triadic Decomposition of Particle States} Every particle state $|\psi\rangle$ decomposes into three irreducible threads: \begin{align} |\psi\rangle &= |\psi^-\rangle \oplus |\psi^+\rangle \oplus |\psi^\times\rangle \\ \mathcal{D}(\psi) &= \sum_{i\in\{-,+,\times\}} w_i \langle \mathbf{E}^i(\psi), \mathbf{I}^i(\psi) \rangle \end{align} where: \begin{itemize} \item $|\psi^-\rangle$: \textbf{Carrier thread} (bare mass/charge, $D_t = 5/8$) \item $|\psi^+\rangle$: \textbf{Envelope thread} (field interactions, $D_t = 13/16$) \item $|\psi^\times\rangle$: \textbf{Coupling thread} (force mediation) \end{itemize} \subsection{The Graviton as a Triadic Fixed Point} The graviton $G$ emerges as the coupling thread $\tau^\times$ for the gravitational interaction: \begin{equation} G \equiv \psi^\times_{\text{grav}} = \mathcal{U}(\tau^-_{\text{EM}}, \tau^+_{\text{weak}}, \tau^\times_{\text{strong}}) \end{equation} Under the universal aggregation operator $\mathcal{U}$, the graviton achieves the fixed-point dimension: \begin{equation} D_t(G) = \frac{13}{16} = 0.8125 \end{equation} This corresponds to a spatial dimension via parabolic coupling: \begin{equation} D_x(G) = (3 + z) - \kappa - zD_t = \frac{17}{8} = 2.125 \end{equation} with $z = 2/3$, $\kappa = 1$. \subsection{Still Fish Equilibrium for Particle Stability} The lifetime $\tau_p$ of particle $p$ is determined by the balance equation: \begin{equation} \underbrace{\alpha \|\nabla_\tau \Phi_p\|^2}_{S_\tau} = \underbrace{\beta \delta \frac{\mu_p}{\tau_{\text{relax}}}}_{L_{\text{dim}}} \end{equation} where $\Phi_p$ is the coherence potential, $\delta = 0.15$ is the un

Zenodo (CERN European Organization for Nuclear Research) 2026 Open access