# The Physics of Energy Flow — Figure Captions These are caption-first figure concepts for TPOEF. Each caption is written to be usable both as a book caption and as the basis for image generation. ## 1. Something Exists 1. Present configuration of energy density over a region of space. A smooth field $u(\mathbf{r})$ shown as continuous contours or a low relief surface, with no particles or point objects, only distributed presence. ## 2. Energy Flows 2. Two registered configurations of the same field. An initial density $u_1(\mathbf{r})$ and a later registered density $u_2(\mathbf{r})$, with arrows showing continuous reconfiguration rather than disappearance and reappearance. ## 3. Continuity 3. Energy bookkeeping in a finite volume. A bounded region $V$ with inflow and outflow of energy flow $\mathbf{S}$ across its boundary, and the interior density $u$ changing only through that boundary transport. ## 4. No Sources or Sinks 4. Closed accounting without primitive creation. A control volume whose energy changes only by transport across its boundary, with no internal source or sink terms drawn inside the region. ## 5. Divergence-Free Flow 5. Continuous source-free flow lines in empty space. A smoke-ring-like bundle of closed field lines with no endpoints, contrasting with a forbidden pattern that begins or ends at isolated points. ## 6. Curl Preserves Flow 6. Local rotation preserves source-free structure. A divergence-free vector field turning around itself under curl evolution, with circulation maintained and no new endpoints introduced. ## 7. Double Curl Transport Closure 7. Complementary electric and magnetic aspects of one advancing flow. A local energy flow $\mathbf{S}$ with transverse $\mathbf{E}$ and $\mathbf{B}$, orthogonal to one another, and $\mathbf{E}\times\mathbf{B}$ aligned with the direction of transport. 8. Double rotation as self-advancing transport. One field rotating the other, and being rotated in return, shown as a helical advance rather than a single recirculating swirl. ## 8. Topology and Discreteness 9. Sphere versus torus for continuous circulation. A sphere with an unavoidable defect in any smooth tangential flow, beside a torus supporting continuous closed circulation without enforced stagnation points. 10. Two independent non-contractible winding cycles on a torus. A toroidal flow labeled by integer winding numbers $(m,n)$, with one cycle around the tube and one around the hole. 11. Spin as angular momentum of toroidal circulation. A self-closing toroidal flow with angular momentum vector $\mathbf{L}$ about the mode's center, where winding class and handedness fix the sign and class of the circulation. ## 9. Mass as Trapped Energy 12. Mass as resistance of confined flow to redirection. A stable toroidal energy configuration being redirected as a whole, emphasizing that the entire closed circulation must reorganize when its motion changes. ## 10. Charge as Circulation 13. Near field and far field of a circulating toroidal configuration. Detailed closed local flow near the torus, with a signed through-hole flux across the torus aperture and a definite angular momentum about the mode's center, transitioning at large distance to a spherically spreading inverse-square projection. ## 11. Schrödinger as Narrow-Band Maxwell 14. Carrier wave and envelope. A narrow-band electromagnetic wave packet with a rapid carrier oscillation inside a slow envelope, showing how the envelope tracks the effective Schrödinger dynamics. ## 12. Newton as Flux Accounting 15. Momentum flux through a surface. A closed surface surrounding a localized configuration, with stress or momentum flux crossing the boundary and net force emerging from the imbalance. ## 13. Gravity as Refraction 16. Gravity as symmetric electromagnetic slowing. A concentrated background energy configuration surrounded by a halo of raised effective $\varepsilon_{\text{eff}}$ and $\mu_{\text{eff}}$, with a passing ray bending toward the slower region. 17. Electric-only half-medium versus full electromagnetic medium. Two side-by-side schematics: one with only $\varepsilon$ perturbed giving the Newtonian half-value, and one with both $\varepsilon$ and $\mu$ perturbed giving the full deflection.