Unified Dynamics

**Unified Dynamics (UD)** is a comprehensive theoretical framework that derives the fundamental structure of the universe from a single geometric primitive: the **tetrahedron**, represented as the complete graph **K4**. Developed by independent researcher **Benjamin Walker Mayes**, the theory posits that physical laws are not arbitrary but are the inevitable combinatorial consequences of this minimal 3D closure. ### **Core Axioms and the Zero-Parameter Goal** The central ambition of UD is to eliminate the Standard Model's 19+ "free parameters" by deriving them through pure geometry with **zero adjustable constants**. * **Packet Substrate:** Reality is modeled as a discrete network of "packet units" on a fractal lattice. * **The K4 Seed:** Every persistent structure must pass through the "seed threshold" of the K4 tetrahedron, which is the unique self-dual minimal 3D closure. * **The Power of Pi:** The theory identifies the **Regge deficit angle** of the tetrahedron as exactly **$\pi$**, which serves as the universal normalization for gauge couplings. ### **Derivation of Physical Constants** By counting the boundary elements of the tetrahedron, UD derives the strengths of the fundamental forces: * **Gauge Groups:** $SU(3)$ (Strong) emerges from the 4 faces, $SU(2)$ (Weak) from the 6 edges, and $U(1)$ (Electromagnetism) from the 4 vertices. * **Coupling Strengths:** These are calculated as the ratio of boundary elements to $\pi$. For example, the **Strong Coupling Constant $\alpha_s(m_Z)$** is derived as $129/(112\pi^2) \approx 0.1167$. * **Weak Mixing Angle:** Predicted to be exactly **$3/13 \approx 0.2308$**. * **Fine Structure Constant:** The inverse value $\alpha^{-1}$ is derived as **$13\pi^2 \approx 128.3$**. ### **Matter and Generations** The theory provides a geometric explanation for why there are exactly **three generations** of matter. * **Symmetry Breaking:** The quotient group $A_4/V_4$ yields the cyclic group **$Z_3$**, forcing exactly three generations into existence. * **The Koide Ratio:** The relationship between lepton masses is derived as a "geometric saturation bound" of **2/3**, matching experimental data to within 0.001%. * **Fermions as Defects:** Matter is defined as a **framing defect** (a $Z_2$ orientation flip) within the tetrahedral vacuum. ### **Cosmology and Dimensionality** UD utilizes an **Iterated Function System (IFS)** to scale from the Planck scale to the Hubble scale, generating a **Sierpiński tetrahedron** fractal lattice. * **Spectral Index:** The theory predicts a cosmological spectral index $n_s = 185/192 \approx 0.9635$, which aligns with Planck satellite observations. * **Why 3D?:** UD argues that **three spatial dimensions** are uniquely stable; in higher dimensions ($d \ge 4$), the geometric "binding energy" is insufficient, causing structures to rupture back into 3D remnants. * **Dark Matter:** The framework identifies **neutral composite K4 closures** as dark matter candidates, predicting a relic fraction of approximately 19%. ### **Status and Falsifiability** The framework has produced **16 predictions within 1.5%** of measured values using only one input scale—the electron mass (or $\nu_0 \approx 83$ keV) to set units. The theory is highly falsifiable; for instance, the discovery of a **fourth fermion generation** or a significant deviation in the derived value of $\sin^2 \theta_W$ would invalidate the core architecture.