# Emergence of Maxwell Fields and Inverse-Square Forces from a Causal-Energy Network ## 1. Introduction We consider a model where reality emerges from a single, indivisible **Node** $N$, internally composed of causally interacting **subnodes** $\{n_i\}$. There is no background space or time; instead: * **Time** emerges from the partial ordering $n_i \succ n_j$. * **Distance** emerges by counting causal steps. * **Observers within the node**, by performing internal experiments, recover Maxwell's equations and associated physical laws. Our goal: show how classical electromagnetic behavior, matter quantization, and gravitational-like forces emerge as experimental abstractions inside this structure. ## 2. Causal Network and Emergent Observables ### 2.1 Cause-Effect Ordering Define $n_i \succ n_j$ if a change in subnode $n_i$ triggers a change in $n_j$. This defines a **partial order**, the basis of time. ### 2.2 Distance via Causal Chain Length Let $L_{ij}$ be the minimal number of steps in a causal chain from $n_i$ to $n_j$. Define: $d(n_i, n_j) \propto L_{ij}.$ ### 2.3 Emergent Space and Embedding Subnodes form a graph with causal distances $d(n_i, n_j)$. If large subsets embed in $\mathbb{R}^3$ with low distortion, space appears 3D. ## 3. Experimental Abstraction: Emergence of Maxwell Fields Any observer within this causal subnode space can construct **clocks** and **rods** from cyclic cause-effect loops. From repeated experiments with signals and interactions, they reconstruct: $\nabla \cdot \vec{E} = 0, \quad \nabla \cdot \vec{B} = 0, \quad \nabla \times \vec{E} = -\partial_t \vec{B}, \quad \nabla \times \vec{B} = \mu_0 \varepsilon_0 \partial_t \vec{E}.$ These are abstracted **Maxwell vacuum laws**—universally observed because they describe the large-scale behavior of causal-energy flow. ## 4. Quantized Matter from Toroidal Standing Waves ### 4.1 Topological Energy Modes Energy confined in a toroidal topology with radii $R, r$ supports standing waves with winding numbers $n_1, n_2$. Coherence requires: $\lambda_1 = 2\pi R / n_1, \quad \lambda_2 = 2\pi r / n_2.$ ### 4.2 Energy Quantization Lowest mode: $n_1 = n_2 = 1$. Higher modes: $E_{n} = E_0 / n^2,$ matches Rydberg formula without invoking charge or mass. ### 4.3 Mass as Field Inertia Stable circulation of electromagnetic energy resists acceleration. Mass emerges from internal **momentum flux**, not as fundamental substance. ## 5. Gravity from Energy Density: Light Speed as Local Function From Maxwell’s wave equation: $c(x) = 1/\sqrt{\mu(x)\varepsilon(x)}.$ Increased energy density (from compressed matter) modifies $\mu, \varepsilon$, reducing $c$. This **mimics curvature**: * Light bends due to spatial variation in $c(x)$. * Redshift arises from local field energy differences. No spacetime curvature is invoked. Gravitational effects result from the dielectric nature of clustered electromagnetic energy. ## 6. Inverse-Square Force from Standing-Wave Confinement Two energy lumps at distance $r$ support standing-wave modes. Energy of mode $m$: $W_m(r) \propto \frac{1}{r}.$ Total energy is conserved. Differentiating: $F(r) = -\frac{dW}{dr} \propto -\frac{1}{r^2}.$ Thus, inverse-square attraction arises from **wave geometry and energy conservation**, not mass. ## 7. Interpretation and Coherence * Observers abstract Maxwell fields from internal cause-effect patterns. * Matter quantization and inertia emerge from stable energy loops. * Gravity is a **light-speed modulation** from local energy density. * $1/r^2$ attraction arises from standing-wave energy slope. All physical law becomes **emergent structure** on a stable, energy-conserving causal network. ## 8. Conclusion We have shown that within a causal-energy network with no external spacetime: * Maxwell's equations emerge from experimental abstraction. * Quantized matter and mass arise from coherent field configurations. * Gravity results from electromagnetic dielectric gradients. * Inverse-square forces follow from wave energy constraints. This provides a minimal, complete, emergent foundation for classical field theory and the structure of matter. ## Acknowledgments We thank the internal consistency of nature for permitting such derivations. ## References 1. Rodriguez, A. & Rodriguez, A. (2025). *Hydrogen Atom Quantization from Purely Classical Maxwell Electromagnetic Fields*. WIP. 2. Rodriguez, A. & Rodriguez, A. (2025). *Gravity as a Dielectric in Electromagnetism*. WIP. 3. Palma, A. & Rodriguez, A. (2025). *Emergent -1/r^2 Force in a Pure Maxwell Universe*. PDF. 4. Palma, A. & Rodriguez, A. (2025). *A Cause-Effect Model for Emergent Time and Distance*. RG DOI: 10.13140/RG.2.2.35753.30569 5. Palma, A. & Rodriguez, A. (2025). *Dimension and Space as Emergent Properties of Distance*. RG DOI: \[pending] ---