Complete mode expansions of the declared guide and cavity classes: transverse profile completeness, the rectangular reflection class and the smooth PEC cavity eigenbasis

[Clause (a)] The families $\{h_{mn}\}$ and $\{e_{mn}\}$ are orthogonal complete systems in $L^2(R_{ab})$ by [F1], which gives the expansion statement. Each member with $(m,n)\mathrm{\ne }(0,0)$ has zero integral over $R_{ab}$ by the orthogonality of [F1] against $h_{00}=1$, so the closed linear span $S$ of $\{h_{mn}:(m,n)\mathrm{\ne }(0,0)\}$ is contained in the closed zero-mean subspace $Z=\{f\in L^2(R_{ab}):{\int }_{R_{ab}}f=0\}$; conversely, if $f\in Z$, the expansion theorem [F1] applied to the complete orthonormal family obtained by normalizing all the $h_{mn}$ writes $f$ as the norm limit of its finite partial sums, and the coefficient of $h_{00}$ is $⟨f,h_{00}⟩\mathrm{/}\mathrm{\parallel }{h}_{00}{\mathrm{\parallel }}^2={\int }_{R_{ab}}f\mathrm{/}ab=0$, so those partial sums use only indices $(m,n)\mathrm{\ne }(0,0)$ and $f\in S$. Hence $S=Z$: the remaining Neumann profiles are complete in the zero-mean subspace and not in all of $L^2(R_{ab})$ because the constant function is orthogonal to them and nonzero. The constant $h_{00}=1$ stays in the full mathematical basis, while the physical TE candidate family excludes it by the Maxwell index restriction of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff (c), so no nonzero positive-frequency TE reconstruction is asserted from the constant scalar profile; the Dirichlet profiles keep their full scalar completeness.

[Clause (b)] By [F2] the finite separated Fourier modes are dense in the rectangular reflection class at every time, every initial datum in the class has a unique $C(\mathbb{R};L^2)$ weak solution of the lossless first-order system in the class, and the energy $\frac{1}{2}{\int }_Q(\epsilon \mathrm{\mid }E{\mathrm{\mid }}^2+\mathrm{\mid }B{\mathrm{\mid }}^2\mathrm{/}\mu )$ is constant along the solution; this distributional system is the explicitly adopted weak material evolution of the reflection model. The energy equals ${\int }_{Q}u$ for the weak stored-energy density $u=\frac{1}{2}(\epsilon \mathrm{\mid }E{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }B{\mathrm{\mid }}^2)$ of [F5], which is defined almost everywhere and nonnegative; its integrability and constancy are furnished by the supplier, and no nonlinear weak Poynting identity is asserted. Finally, the four macroscopic equations with $D=\epsilon E$ and $H=B\mathrm{/}\mu$ follow distributionally from the two evolution equations and the divergence constraints of the class: $\mathrm{div}D=\epsilon \mathrm{div}E=0$ and $\mathrm{div}B=0$ are the class constraints, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ is the second evolution equation, and $\mathrm{curl}H={\mu }^{-1}\mathrm{curl}B={\mu }^{-1}(\epsilon \mu ){\mathrm{\partial }}_{t}E={\mathrm{\partial }}_{t}D$ uses the first evolution equation ${\mathrm{\partial }}_{t}E=(\epsilon \mu {)}^{-1}\mathrm{curl}B$.

[Clause (c), spectral content] Assume the full Axiom of Choice as in [F3], let $\mathrm{\Omega }$ be a bounded $C^{\mathrm{\infty }}$ cavity and let $X,V,H$, $\{u_j\}$, ${\lambda }_j\ge 0$ be the classes and eigenbasis of [F3]. Then the curl form on $V$ has the complete orthonormal $L^2$ eigenbasis with nonnegative finite-multiplicity eigenvalues tending to infinity when $H$ is infinite dimensional and a finite-dimensional zero eigenspace retained; the eigen-equation holds distributionally with the PEC trace, each $u_j$ has a $C^{\mathrm{\infty }}$ representative on the closure, and the full-$X$ variational identity holds for every $v\in X$. With $v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}=(\epsilon \mu {)}^{-1\mathrm{/}2}>0$, the associated second-order electric wave has displacement in $C(\mathbb{R};V)$ and velocity in $C(\mathbb{R};H)$ for arbitrary initial data in $V\times H$, finite modal sums converge in these norms uniformly on compact times, and its zero eigenmodes may be affine in time. These are the spectral and wave assertions of [F3], taken with the exact hypotheses of the suppliers.

[Clause (c), first-order modal equation] Let $J$ be an open time interval and let a strong first-order solution obey the hypotheses of the Statement: $E\in C^1(J;H)\cap C(J;V)$, $B\in C^1(J;L^2)\cap C(J;H(\mathrm{curl}))$, with the first-order equations holding as $L^2$ identities and both distributional divergence constraints, the magnetic normal-zero condition ${\int }_{\mathrm{\Omega }}B(t)\cdot \mathrm{\nabla }\psi =0$ for every smooth scalar $\psi$ on the closure, and $E(t)\in V$ encoding the electric tangential-zero data; no magnetic membership in $X$ is assumed. Put $a_j(t):=⟨E(t),u_j⟩$. To pair $\mathrm{curl}B$ against the smooth eigenfield $u_j$, use the collar cutoff of [F4]: write $s$ for the inward normal coordinate in a tubular neighbourhood of the compact boundary, so that $\mathrm{\mid }n\times u_j\mathrm{\mid }\le Cs$ in the collar because $u_j$ is smooth on the closure with $n\times u_j=0$, and let ${\eta }_{\delta }$ be the standard smooth step with ${\eta }_{\delta }=0$ for $s\le \delta$ and $1$ for $s\ge 2\delta$. Then ${\eta }_{\delta }{u}_j\to u_j$ and $\mathrm{curl}({\eta }_{\delta }{u}_j)\to \mathrm{curl}{u}_j$ in $L^2$ as $\delta \to 0$, since the derivative-cutoff term is supported on a layer of volume $O(\delta )$ and is bounded by $C\mathrm{/}\delta$ times $O(s)$; the distributional identity of [F4] on the compactly supported smooth test ${\eta }_{\delta }{u}_j$ gives $⟨\mathrm{curl}B,{\eta }_{\delta }{u}_j⟩=⟨B,\mathrm{curl}({\eta }_{\delta }{u}_j)⟩$, and Cauchy–Schwarz with the two limits gives $⟨\mathrm{curl}B,u_j⟩=⟨B,\mathrm{curl}{u}_j⟩$ for the actual graph-regular field $B$. Hence $a_j^{\mathrm{\prime }}=⟨E^{\mathrm{\prime }},u_j⟩=v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2⟨\mathrm{curl}B,u_j⟩=v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2⟨B,\mathrm{curl}{u}_j⟩$ on $J$. The assumed $B\in C^1(J;L^2)$ permits one further derivative: $a_j^{\mathrm{\prime }\mathrm{\prime }}=v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2⟨B^{\mathrm{\prime }},\mathrm{curl}{u}_j⟩=-v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2⟨\mathrm{curl}E,\mathrm{curl}{u}_j⟩=-v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2{\lambda }_j⟨E,u_j⟩$, where the last equality is the full-$X$ eigenvariational identity of [F3] tested at $v=E(t)\in V\subset X$; thus $a_j^{\mathrm{\prime }\mathrm{\prime }}+v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2{\lambda }_j{a}_j=0$, and $a_j$ is $C^2$ because $E$ and $B$ are $C^1$ in their norms. If ${\lambda }_j=0$, the same identity at $v=u_j$ gives $\mathrm{\parallel }\mathrm{curl}{u}_j{\mathrm{\parallel }}_2^2=0$, so $\mathrm{curl}{u}_j=0$ and $a_j^{\mathrm{\prime }}=v_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}^2⟨B,\mathrm{curl}{u}_j⟩=0$: the electric zero modes are constant under first-order Maxwell evolution, while the unrestricted second-order wave of step 1.3 retains its affine zero modes. This distinguishes first-order Maxwell data from arbitrary second-order $V\times H$ data; the pairing imposed no magnetic electric-type trace, and the divergence constraints and the declared weak magnetic normal condition remain part of the model.

[Clause (d)] Clause (a) is step 1.1, clause (b) is step 1.2, and clause (c) is steps 1.3 and 2.1. Completeness is asserted only for the transverse $L^2$ profiles and for the two declared cavity classes with their exact hypotheses; this lossless theorem does not itself supply the longitudinal expansion or the dissipative evolution, which are supplied separately by Complete longitudinal Fourier energy evolution in a rectangular PEC guide and Constant-Ohmic PEC modal decay and exact energy-class completeness. None of the statements supplies arbitrary lossy, dispersive, open-end scattering, non-smooth or exterior problems, or arbitrary non-rectangular guide problems beyond the declared smooth-cavity spectral class; no trace regularity for arbitrary $H(\mathrm{curl})$ fields and no identification of the weak classes with every maximal tangential-trace realization is asserted, and the zero eigenvalues and zero modes are retained throughout. The inherited Axiom of Choice and Countable Choice cover the spectral, Fourier and collar suppliers, and no lossy or open-domain step was made.