# 222. Boundary Unloading by Superposition Appendices 219 and 221 showed that a lower-loading region can act as a faster transport corridor. The missing step is the unloading mechanism itself. For passive source-free Maxwellian transport, that mechanism is already available. It is the combination of: - boundary determination of the passive interior, - linear superposition of source-free fields, - and the quadratic form of electromagnetic energy density. So the fundamental point is not yet engineering. It is simply this: given boundary control of a passive region, one can subtract as well as reinforce the interior field. ## 222.1 Linear Dependence on Boundary Data Let $\Omega$ be a passive bounded region with smooth closed boundary $\partial\Omega$. Let $B$ denote the complete boundary transport data fed through $\partial\Omega$ over the relevant causal interval. For source-free linear Maxwellian transport, the interior resolved field depends linearly on that boundary data. Writing $$ \mathcal F[B] = (\mathbf E[B],\mathbf H[B]), $$ linearity gives $$ \mathcal F[B_1+B_2]=\mathcal F[B_1]+\mathcal F[B_2]. $$ Therefore boundary control can add or subtract admissible interior field components. This is the first fundamental fact. ## 222.2 Exact Energy Reduction of a Selected Component Take any boundary-driven interior field component $$ (\mathbf E_0,\mathbf H_0) $$ in the target region. Now choose a second boundary control that excites the same component with opposite phase and relative amplitude $\lambda$, where $$ 0\le \lambda \le 1. $$ That is, $$ (\mathbf E_1,\mathbf H_1) = -\lambda(\mathbf E_0,\mathbf H_0). $$ By superposition, the total field becomes $$ (\mathbf E_\lambda,\mathbf H_\lambda) = (1-\lambda)(\mathbf E_0,\mathbf H_0). $$ The electromagnetic energy density is $$ u = \frac12\bigl(\varepsilon E^2+\mu H^2\bigr), $$ and the Poynting flux is $$ \mathbf S = \mathbf E\times\mathbf H. $$ So for this same-mode subtraction, $$ u_\lambda=(1-\lambda)^2u_0, \qquad \mathbf S_\lambda=(1-\lambda)^2\mathbf S_0. $$ This reduction is exact. It is not heuristic. It follows from the quadratic form of energy density and the bilinear form of flux. At $\lambda=1$, the selected component is canceled completely. ## 222.3 Guided-Mode Version In a tube or corridor geometry, let the dominant passive mode have the form $$ (\mathbf E_0,\mathbf H_0) = A\,(\mathbf e,\mathbf h)(x_\perp)\,e^{i(\beta z-\omega t)}. $$ If the boundary actuation injects the same mode with amplitude $$ -\lambda A, $$ then the resulting mode amplitude inside the target region is $$ (1-\lambda)A. $$ So the interior loading of that mode is reduced by the exact factor $$ (1-\lambda)^2. $$ This is the fundamental boundary-unloading mechanism for a transport corridor: mode-wise subtraction by phase-opposed superposition. The exact boundary pattern needed to realize the desired subtraction is the engineering problem. The unloading mechanism itself is already forced by the linearity of the passive interior. ## 222.4 Relation to Local Transport Speed Appendices 214 and 219 work in the symmetric constitutive class $$ \varepsilon=\varepsilon_0\alpha, \qquad \mu=\mu_0\alpha, \qquad k=\frac{c}{\alpha}. $$ Within that class, lower local loading means larger local transport speed. Therefore any boundary program that lowers the resolved background load in a region raises the local transport speed there relative to the more heavily loaded case. That is the fundamental basis of a high-speed transport corridor. The important limit is also clear. Exact cancellation of the selected carrier component gives zero load in that component; it does not by itself provide a working guide or an infinite-speed theorem. Corridor design therefore uses controlled unloading, or unloading of a background component while preserving a separate signal-bearing structure. ## 222.5 Final Statement Given boundary control of a passive region, one can subtract admissible Maxwellian modes as well as reinforce them. Because the field equations are linear and the electromagnetic energy density is quadratic, phase-opposed boundary control lowers interior energy exactly. So the corridor idea does have a clean first-principles derivation: - passive interior fields are boundary-determined, - boundary-determined fields superpose linearly, - opposite-phase excitation subtracts a selected interior mode, - and that subtraction lowers the local loading that sets transport speed. Appendix 221 then gives the corresponding lensing and guidance consequences once such a loading profile has been engineered.