# Scope and stance This document does not introduce new primitives. We assume only: - a source-free region, - continuous electromagnetic fields, - and Maxwell’s vacuum dynamics as the experimentally abstracted transport law. Everything else below is derived. No particles are postulated. No “forces” are postulated. No independent conservation axioms are postulated. # Foundations ## Maxwell dynamics in a source-free region In vacuum, the fields satisfy $$ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0, $$ $$ \nabla \times \mathbf{E} = -\partial_t \mathbf{B}, \qquad \nabla \times \mathbf{B} = \mu_0 \epsilon_0\,\partial_t \mathbf{E}. $$ Define the constant $$ c^2 = \frac{1}{\mu_0 \epsilon_0}. $$ These four equations are the dynamical content we start from. # Step 1: Energy continuity (Poynting theorem) ## Energy density and flux Define electromagnetic energy density $$ u = \frac{\epsilon_0}{2}\,|\mathbf{E}|^2 + \frac{1}{2\mu_0}\,|\mathbf{B}|^2, $$ and energy flux (Poynting vector) $$ \mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E}\times \mathbf{B}. $$ ## Derivation of local continuity Take the dot product of Ampère–Maxwell with $\mathbf{E}$ and Faraday with $\mathbf{B}/\mu_0$: $$ \mathbf{E}\cdot(\nabla\times \mathbf{B}) = \mu_0\epsilon_0\,\mathbf{E}\cdot \partial_t \mathbf{E}, $$ $$ \frac{1}{\mu_0}\mathbf{B}\cdot(\nabla\times \mathbf{E}) = -\frac{1}{\mu_0}\mathbf{B}\cdot \partial_t \mathbf{B}. $$ Subtract the second from the first and use the vector identity $$ \nabla\cdot(\mathbf{E}\times \mathbf{B}) = \mathbf{B}\cdot(\nabla\times \mathbf{E}) - \mathbf{E}\cdot(\nabla\times \mathbf{B}), $$ to obtain $$ \partial_t \left( \frac{\epsilon_0}{2}|\mathbf{E}|^2 + \frac{1}{2\mu_0}|\mathbf{B}|^2 \right) + \nabla\cdot \left( \frac{1}{\mu_0}\mathbf{E}\times \mathbf{B} \right) =0. $$ That is, $$ \partial_t u + \nabla\cdot \mathbf{S} = 0. $$ Interpretation: this is a strict local bookkeeping identity implied by Maxwell evolution. It says changes in field energy density are accounted for by flux divergence. # Step 2: Momentum continuity and force as flux ## Momentum density Define electromagnetic momentum density $$ \mathbf{g} = \epsilon_0\,\mathbf{E}\times \mathbf{B}. $$ Using $\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times \mathbf{B}$ and $c^2=1/(\mu_0\epsilon_0)$ gives the exact relation $$ \mathbf{g}=\frac{\mathbf{S}}{c^2}. $$ ## Maxwell stress tensor Define the Maxwell stress tensor $\mathbf{T}$ with components $$ T_{ij} = \epsilon_0 \left( E_i E_j -\frac{1}{2}\delta_{ij}|\mathbf{E}|^2 \right) + \frac{1}{\mu_0} \left( B_i B_j -\frac{1}{2}\delta_{ij}|\mathbf{B}|^2 \right). $$ ## Local momentum balance (source-free) A standard computation using Maxwell’s equations and vector identities yields $$ \partial_t \mathbf{g} + \nabla\cdot \mathbf{T} = \mathbf{0}. $$ This is the momentum continuity equation for the field in a source-free region. Interpretation: momentum changes in a region are accounted for by stress flux across the boundary. ## Integrated form (force as momentum flux) Let $V$ be a fixed volume with boundary $\partial V$ and outward normal $\mathbf{n}$. Integrate and apply the divergence theorem: $$ \frac{d}{dt}\int_V \mathbf{g}\,dV = -\int_{\partial V} \mathbf{T}\,\mathbf{n}\,dA. $$ The right-hand side is the net momentum flux through the boundary. This is the precise origin of “force” in this program: - “force on a subsystem” is not postulated, - it is defined as the boundary flux of momentum. # Step 3: Newton-like laws for localized electromagnetic configurations We now extract mechanics from the continuity identities. ## Localized configuration (“knot” or “object”) Assume a field configuration whose energy and momentum are concentrated in a bounded region that moves without dispersing appreciably over the time interval of interest. Define total energy and momentum: $$ E(t)=\int_{\mathbb{R}^3} u(\mathbf{x},t)\,dV, \qquad \mathbf{P}(t)=\int_{\mathbb{R}^3} \mathbf{g}(\mathbf{x},t)\,dV. $$ From the continuity equations, if the flux vanishes sufficiently fast at infinity (localized configuration), then $$ \frac{dE}{dt}=0, \qquad \frac{d\mathbf{P}}{dt}=\mathbf{0}. $$ This is conservation of energy and momentum as a derived property of source-free Maxwell evolution plus localization. ## Center-of-energy trajectory Define the center-of-energy position $$ \mathbf{R}(t) = \frac{1}{E}\int_{\mathbb{R}^3}\mathbf{x}\,u(\mathbf{x},t)\,dV. $$ Differentiate and use energy continuity $\partial_t u = -\nabla\cdot \mathbf{S}$: $$ \frac{d\mathbf{R}}{dt} = \frac{1}{E}\int_{\mathbb{R}^3}\mathbf{S}(\mathbf{x},t)\,\frac{dV}{c^2} = \frac{1}{E}\int_{\mathbb{R}^3}\mathbf{g}(\mathbf{x},t)\,dV = \frac{\mathbf{P}}{E}. $$ Thus, for a localized source-free configuration, $$ \frac{d\mathbf{R}}{dt}=\frac{\mathbf{P}}{E}, \qquad \frac{d^2\mathbf{R}}{dt^2}= \mathbf{0}. $$ This is inertial motion: constant center-of-energy velocity follows from derived momentum conservation. ## “Force” and interaction If the configuration is not isolated because other field structure crosses the boundary of a chosen region $V$, then $$ \frac{d}{dt}\int_V \mathbf{g}\,dV = -\int_{\partial V} \mathbf{T}\,\mathbf{n}\,dA, $$ so changes in subsystem momentum are caused by stress flux across its boundary. No additional force law is required. Interaction is momentum exchange through the field. # Step 4: Effective tension and inertia of a thin flux tube Assume a configuration whose energy is concentrated in a thin tube around a smooth closed curve $X(s)$, with arclength parameter $s\in[0,L]$. Let $\Sigma_s$ be a small cross-section transverse to the tube at position $s$. ## Tension (energy per unit length) Define line energy density $$ T = \int_{\Sigma_s} u\,dA. $$ Then total energy in the tube is $$ E = \int_0^L T\,ds. $$ If $T$ is approximately constant along the tube, $$ E = T L. $$ This defines the effective tension: it is not assumed; it is energy localization per unit length. ## Inertial line density Using $\mathbf{g}=\mathbf{S}/c^2$, the line momentum density along the local tangent direction $\hat{\mathbf{t}}(s)$ is $$ p(s)=\int_{\Sigma_s} \mathbf{g}\cdot \hat{\mathbf{t}}\,dA. $$ In null-like transport inside the tube (energy flow locally at the maximal rate), one has the identity $$ |\mathbf{S}| = c\,u \quad\Rightarrow\quad |\mathbf{g}| = \frac{u}{c}. $$ Under the same thin-tube localization, this yields the effective inertial line density $$ \mu = \frac{T}{c^2}. $$ This is the clean field-theoretic origin of “mass density” for a one-dimensional effective object. # Step 5: Toroidal organization forces integer winding Assume the energy-flow lines lie on an invariant toroidal surface $T^2$ and form a smooth tangent flow. A torus has two independent non-contractible cycles. Closed flow lines must wind around both cycles by integers. The resulting winding data are coprime integers $$ (m,n)\in\mathbb{Z}^2. $$ Interpretation: - the pair $(m,n)$ is not inserted, - it is forced by global single-valuedness and closure on $T^2$. This is the topological origin of discrete classes of configurations. # Step 6: Discrete modes from periodicity A closed tube has a circular arclength domain $s\sim s+L$. Any small perturbation variable $\xi(s,t)$ on the tube that satisfies a linear wave equation on the circle has Fourier modes: Assume $$ \partial_t^2 \xi - v^2\,\partial_s^2 \xi = 0, \qquad \xi(s+L,t)=\xi(s,t). $$ Then expand $$ \xi(s,t)=\sum_{k\in\mathbb{Z}} a_k e^{i(2\pi k s/L)} f_k(t), $$ and obtain the discrete frequency set $$ \omega_k = \frac{2\pi v}{L}|k|. $$ Discreteness is forced by topology (periodicity), not by quantization axioms. In the program, the remaining question is the origin of confinement and stability of the tube; in the trilogy this is tied to emergent refraction and self-trapping. # Step 7: What can be reconstructed from energy density and flux There are two distinct questions: - Representation: given transport observables, can we represent them with fields? - Dynamics: how do the fields evolve? ## Reconstruction statement (representation) Given a scalar energy density $u>0$ and a flux $\mathbf{S}$ satisfying $$ |\mathbf{S}|\le c\,u, $$ one can construct at least one pair of fields $(\mathbf{E},\mathbf{B})$ such that $$ u = \frac{\epsilon_0}{2}|\mathbf{E}|^2 + \frac{1}{2\mu_0}|\mathbf{B}|^2, $$ $$ \mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times \mathbf{B}. $$ The construction is not unique. The two missing local degrees of freedom correspond to polarization (duality freedom). ## What requires Maxwell dynamics Reconstruction does not determine how the configuration evolves. The evolution law that preserves divergence-free structure and yields the above continuity identities is Maxwell curl dynamics. # What has been derived (no added postulates) From source-free Maxwell evolution plus localization/topology assumptions stated explicitly, we derived: - energy continuity, - momentum continuity and stress flux, - inertial motion of center-of-energy for isolated configurations, - interaction as momentum exchange through boundary stress, - effective tension and inertial line density for thin tubes, - integer winding on toroidal organization, - discrete mode spectra on closed loops, - and the precise boundary between representation and dynamics. # References - Light speed as an emergent property of electromagnetic superposition: Polarization without matter. DOI: https://writing.preferredframe.com/doi/10.5281/zenodo.18396637 - String Theory Derivation in a Maxwell Universe. 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