Constant-Ohmic PEC modal decay and exact energy-class completeness

[Rectangular damped modal system] Assume Countable Choice. Fix the box $Q$ and the reflection class of [F2], and let $(E,B)$ be a $C(\mathbb{R};L^2)$ distributional solution of the damped first-order system of [F1] in that class. The finite separated Fourier modes are dense in the class by [F2], and for a wavevector $k\in (\pi \mathrm{/}{L}_1)\mathbb{Z}\times (\pi \mathrm{/}{L}_2)\mathbb{Z}\times (\pi \mathrm{/}{L}_3)\mathbb{Z}$ the coefficient system of the damped system reads ${\widehat{E}}_t=i{v}^{2}k\times \widehat{B}-\gamma \widehat{E}$, ${\widehat{B}}_t=-ik\times \widehat{E}$: this is [F2]'s recorded coefficient system with the additional scalar term $-\gamma \widehat{E}$ of [F1], and the algebraic constraint $k\cdot (k\times w)=0$ preserves $k\cdot \widehat{E}=k\cdot \widehat{B}=0$, while the finite Fourier modes carry the parities of the class because the reflection rules are diagonal in the separated basis.

[Smooth PEC cavity spaces and the magnetic decomposition] Assume the full Axiom of Choice as in [F3], let $\mathrm{\Omega }$ be a bounded $C^{\mathrm{\infty }}$ domain and let $X,V,H$, $\{u_j\}$, ${\lambda }_j\ge 0$ be the classes and eigenbasis of [F3]. Put $M:=\{B\in L^2(\mathrm{\Omega };{\mathbb{R}}^3):{\int }_{\mathrm{\Omega }}B\cdot \mathrm{\nabla }\psi =0\text{ for every smooth scalar }\psi \text{ on }\bar{\mathrm{\Omega }}\}$, the weak normal-zero divergence-free space, and $K_M:=\{B\in M:\mathrm{curl}B=0\text{ distributionally}\}$. For each positive eigenvalue put $w_j:=\mathrm{curl}{u}_j\mathrm{/}\sqrt{{\lambda }_j}$. Then the $w_j$ are orthonormal, $w_j\in M$, and $M={\overline{\mathrm{span}\{w_j:{\lambda }_j>0\}}}^{L^2}\oplus K_M$. Orthonormality: the full-$X$ identity of [F3] with the test $v=u_l\in V\subseteq X$ gives $(\mathrm{curl}{u}_j,\mathrm{curl}{u}_l)={\lambda }_j(u_j,u_l)={\lambda }_j{\delta }_{jl}$, so $(w_j,w_l)={\delta }_{jl}$. Membership and the forward inclusion: for a smooth scalar $\psi$, [F8] gives $\mathrm{div}(u_j\times \mathrm{\nabla }\psi )=\mathrm{\nabla }\psi \cdot \mathrm{curl}{u}_j-u_j\cdot \mathrm{curl}\mathrm{\nabla }\psi =\mathrm{\nabla }\psi \cdot \mathrm{curl}{u}_j$, and its integral over $\mathrm{\Omega }$ is the boundary integral ${\int }_{\mathrm{\partial }\mathrm{\Omega }}(n\times u_j)\cdot \mathrm{\nabla }\psi =0$ because $u_j$ is smooth on the closure with $n\times u_j=0$ [F3]; hence $w_j\in M$, and the closed span of the $w_j$ lies in $M$. For an element $u\in V$, the $V$-norm identity of [F3] with $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_V^2=1+{\lambda }_j$ gives the expansion $u={\sum }_j(u,u_j)u_j$ in $L^2$, and applying the bounded operator $\mathrm{curl}$ on $V$ gives $\mathrm{curl}u={\sum }_j(u,u_j)\sqrt{{\lambda }_j}{w}_j$ in $L^2$; hence $\overline{\mathrm{curl}V}=\overline{\mathrm{span}\{w_j\}}$, and both lie in the closed subspace $M$. Decomposition: the orthogonal decomposition theorem [F5] applied to the closed subspace $\overline{\mathrm{span}\{w_j\}}$ of $M$ writes $B\in M$ as the sum of its projections onto that span and onto its orthogonal complement inside $M$; it remains to identify the complement with $K_M$. If $B\in M$ is orthogonal to every $\mathrm{curl}v$ with $v\in V$ and $\phi$ is smooth and compactly supported in $\mathrm{\Omega }$, then the smooth Dirichlet solver [F4] with coefficient identity, zero boundary trace and source $-\mathrm{div}\phi$ supplies a smooth $f$ on the closure with $\mathrm{\Delta }f=\mathrm{div}\phi$; then $\phi -\mathrm{\nabla }f$ is smooth on the closure with $\mathrm{div}(\phi -\mathrm{\nabla }f)=0$ and with vanishing tangential trace on $\mathrm{\partial }\mathrm{\Omega }$ (the first term is compactly supported, and $f{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$ forces the tangential derivatives of $f$, hence $n\times \mathrm{\nabla }f$, to vanish there), so $\phi -\mathrm{\nabla }f$ lies in $V$; since $\mathrm{curl}(\phi -\mathrm{\nabla }f)=\mathrm{curl}\phi$ by [F8], we obtain $(B,\mathrm{curl}\phi )=0$; as $\phi$ is an arbitrary compactly supported smooth test, this is exactly the distributional statement $\mathrm{curl}B=0$, i.e. $B\in K_M$. Conversely, if $B\in K_M$ then for each eigenfield $u_j$ the collar cutoff of [F9] yields compactly supported smooth approximants ${\eta }_{\delta }{u}_j$ with ${\eta }_{\delta }{u}_j\to u_j$ and $\mathrm{curl}({\eta }_{\delta }{u}_j)\to \mathrm{curl}{u}_j$ in $L^2$ (the boundary layer has width $2\delta$ and $n\times u_j=0$ makes the boundary term vanish in the limit), so the distributional pairing $\int B\cdot \mathrm{curl}({\eta }_{\delta }{u}_j)=0$ passes to the limit and gives $(B,\mathrm{curl}{u}_j)=0$, hence $(B,w_j)=0$; by continuity of the pairing and closedness of the span, $B$ is orthogonal to $\overline{\mathrm{span}\{w_j\}}$, which is the claimed complement and finishes the decomposition.

[Scalar characteristic classification] For parameters $\gamma \ge 0$, $v>0$, $\lambda \ge 0$, the solutions of $a^{\mathrm{\prime }\mathrm{\prime }}+\gamma a^{\mathrm{\prime }}+v^2\lambda a=0$ are classified by $\mathrm{\Delta }:={\gamma }^2\mathrm{/}4-v^2\lambda$: if $\mathrm{\Delta }>0$ the characteristic roots are $-\gamma \mathrm{/}2\pm \sqrt{\mathrm{\Delta }}$ and the general solution is $A{e}^{(-\gamma \mathrm{/}2+\sqrt{\mathrm{\Delta }})t}+B{e}^{(-\gamma \mathrm{/}2-\sqrt{\mathrm{\Delta }})t}$; if $\mathrm{\Delta }<0$ the roots are $-\gamma \mathrm{/}2\pm i\sqrt{-\mathrm{\Delta }}$ and the general solution is $e^{-\gamma t\mathrm{/}2}(A\cos (\sqrt{-\mathrm{\Delta }}t)+B\sin (\sqrt{-\mathrm{\Delta }}t))$; if $\mathrm{\Delta }=0$ it is $(A+Bt)e^{-\gamma t\mathrm{/}2}$; and if $\lambda =0$ the solutions are $A+B{e}^{-\gamma t}$ for $\gamma >0$ and $A+Bt$ for $\gamma =0$. These are finite real computations with the square roots of [F7] and the real exponential and trigonometric solutions certified by [F7]; no choice principle is used.

[Rectangular class: global solutions, energy and convergence] For every initial pair in the rectangular class of [F2] the damped system of [F1] has a unique $C(\mathbb{R};L^2)$ distributional solution in the class, finite Fourier mode sums converge in energy norm uniformly on compact time intervals and are dense at every time, and the coefficient equations of step 1.1 hold for every wavevector. Indeed, for each $k$ the right-hand side of the finite-dimensional system of step 1.1 is globally state-Lipschitz (it is affine), so [F7] gives a unique global solution through the initial coefficients; differentiate the per-mode energy $\epsilon \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }\widehat{B}{\mathrm{\mid }}^2$. With $C:=\bar{\widehat{E}}\cdot (k\times \widehat{B})$ and $\bar{\widehat{B}}\cdot (k\times \widehat{E})=-\bar{C}$, the curl contribution is $(2\mathrm{/}\mu )\mathrm{Re}(iC+i\bar{C})=0$ by the scalar triple-product identity of [F8] and complex conjugation [F9]; this gives $\frac{d}{dt}(\epsilon \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }\widehat{B}{\mathrm{\mid }}^2)=-2\sigma \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2\le 0$, so each modal energy is nonincreasing forward and bounded by $e^{2\gamma T}$ times its initial value on $[-T,T]$ (multiply by $e^{2\gamma t}$ and differentiate); Parseval for the complete class [F2] therefore controls the energy-norm of every omitted tail uniformly on compact times. Finite solutions converge in $C_t{L}^2$ to a limit in the class, and the linear distributional equations pass to the limit against compactly supported smooth periodic tests by Cauchy–Schwarz [F5] on a compact time support, exactly as in the lossless argument of [F2]; the constraints $k\cdot \widehat{E}=k\cdot \widehat{B}=0$ and the parities are preserved by step 1.1, so the limit lies in the same class. Uniqueness: a $C_t{L}^2$ weak solution has continuous coefficients whose distributional derivatives equal the displayed continuous right-hand sides; for each coefficient equation, subtract the oriented primitive of the continuous right-hand side, which is $C^1$ with that derivative by [F6], use the $C^1$ regular-distribution compatibility of [F6] to identify its distributional derivative, conclude that the difference has zero distributional derivative, hence is constant by [F6], and match the initial value; global uniqueness of the finite system identifies the coefficients, and completeness of the Fourier class identifies the fields.

[Smooth PEC cavity: modal system, variational equations and uniqueness] Assume the full Axiom of Choice as in [F3]. Let $H\times M$ be the energy class of steps 1.2, with $M=\overline{\mathrm{span}\{w_j\}}\oplus K_M$. For coefficients $a_j=(\cdot ,u_j)$, $b_j=(\cdot ,w_j)$ declare the first-order modal system $a_j^{\mathrm{\prime }}=v^2\sqrt{{\lambda }_j}{b}_j-\gamma a_j$, $b_j^{\mathrm{\prime }}=-\sqrt{{\lambda }_j}{a}_j$ for ${\lambda }_j>0$, and $a_j^{\mathrm{\prime }}=-\gamma a_j$ for the electric zero modes, and declare the magnetic kernel data constant, $(B,K_M{)}^{\mathrm{\prime }}=0$ for every kernel test. Endow the paired-magnetic space with the completion $D_M$ of finite $w_j$-sums plus $K_M$ for the norm $\mathrm{\parallel }w{\mathrm{\parallel }}_{D_M}^2:=\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{curl}w{\mathrm{\parallel }}_2^2$, and consider the weak variational equations $⟨E_t,\varphi ⟩=v^2⟨B,\mathrm{curl}\varphi ⟩-\gamma ⟨E,\varphi ⟩$ for $\varphi \in V$ and $⟨B_t,\psi ⟩=-⟨E,\mathrm{curl}\psi ⟩$ for $\psi \in D_M$. Then every initial pair in $H\times M$ has a unique $C(\mathbb{R};H\times M)$ solution of this declared realization, it satisfies the modal equations distributionally, and finite electric and paired-magnetic modal sums together with the static kernel component converge uniformly on compact time intervals in energy norm. Existence and uniqueness are finite-dimensional: each paired system is a globally Lipschitz constant-coefficient system solved by [F7], and the kernel data are constant, so a candidate solution is the norm-limit of its finite modal sums; the sums converge because the energy ${\sum }_j(\epsilon \mathrm{\mid }{a}_j{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }{b}_j{\mathrm{\mid }}^2)$ is controlled as in step 2.1 and the kernel component is retained exactly. Each finite sum with the static kernel satisfies the first-order equations distributionally in the interior: $\mathrm{curl}{w}_j=\sqrt{{\lambda }_j}{u}_j$ follows from [F3], while the electric zero eigenfields and magnetic kernel have zero curl. Uniform compact-time $L^2$ convergence passes these linear equations to the limit against compactly supported smooth spacetime tests by Cauchy–Schwarz [F5]. For the variational equations, the finite sums satisfy them against finite spectral tests by the full-$X$ identity of [F3], and the density of the finite spectral tests in $V$ and in $D_M$ passes the identities to the limit by Cauchy–Schwarz [F5], since uniform energy convergence on compact times controls both pairings. Conversely, testing the variational equations against $u_j$ (for $\varphi$) and $w_j$ or kernel vectors (for $\psi$) recovers the modal equations, so the weak realization and the modal system are equivalent in the stated class; uniqueness follows from the scalar ODE uniqueness and the complete decomposition of step 1.2. For strong first-order solutions with $E\in C_t^{1}H\cap C_{t}V$ and $B\in C_t^1{L}^2\cap C_{t}H(\mathrm{curl})$, the same pairing gives $a_j^{\mathrm{\prime }}=v^2(B,\mathrm{curl}{u}_j)-\gamma a_j$ and, differentiating with $B^{\mathrm{\prime }}=-\mathrm{curl}E$ and the full-$X$ identity, $a_j^{\mathrm{\prime }\mathrm{\prime }}+\gamma a_j^{\mathrm{\prime }}+v^2{\lambda }_j{a}_j=0$ with $a_j^{\mathrm{\prime }}=-\gamma a_j$ at ${\lambda }_j=0$. No maximal rough tangential-trace realisation is claimed.

[Exact energy balance] In both declared classes, for every $t_0<t_1$, $\mathcal{U}(t_1)+\sigma {\int }_{t_0}^{t_1}\mathrm{\parallel }E(t){\mathrm{\parallel }}_2^{2}dt=\mathcal{U}(t_0)$ and ${\mathcal{U}}^{\mathrm{\prime }}(t)=-\sigma \mathrm{\parallel }E(t){\mathrm{\parallel }}_2^2\le 0$. In the rectangular class, sum the finite per-mode identities of step 2.1 over the wavevectors of a finite modal projection, integrate over $[t_0,t_1]$ with [F6], and pass to the limit using the uniform energy-norm convergence of step 2.1 to pass both the energy and the continuous $\mathrm{\parallel }E(t){\mathrm{\parallel }}_2^2$-integral terms; in the smooth class, do the same with the paired modal energies of step 3.1 and the retained kernel component, whose curl vanishes. The derivative form then follows from the integral form by [F6]; the boundary term of a smooth local Poynting identity is not used, so no unprovided nonlinear weak local identity is asserted.

[Associated second-order branch] Independently of first-order compatibility, the same spectral basis gives the damped associated electric wave for every $q_0\in V$, $q_1\in H$, with $q\in C(\mathbb{R};V)$, $q^{\mathrm{\prime }}\in C(\mathbb{R};H)$ and modal convergence uniform on compact times: its coefficients satisfy $a_j^{\mathrm{\prime }\mathrm{\prime }}+\gamma a_j^{\mathrm{\prime }}+v^2{\lambda }_j{a}_j=0$ with prescribed initial coefficients $a_j(0)=(q_0,u_j)$, $a_j^{\mathrm{\prime }}(0)=(q_1,u_j)$, and at ${\lambda }_j=0$ they are ${\alpha }_j+{\beta }_j(1-e^{-\gamma t})\mathrm{/}\gamma$ for $\gamma >0$ and ${\alpha }_j+t{\beta }_j$ for $\gamma =0$. Indeed, for ${\lambda }_j>0$ the function $Q_j=\mathrm{\mid }{a}_j^{\mathrm{\prime }}{\mathrm{\mid }}^2+v^2{\lambda }_j\mathrm{\mid }{a}_j{\mathrm{\mid }}^2$ satisfies $Q_j^{\mathrm{\prime }}=-2\gamma \mathrm{\mid }{a}_j^{\mathrm{\prime }}{\mathrm{\mid }}^2\le 0$ by [F7], and the positive eigenvalues, being finite in multiplicity and tending to infinity, have a positive lower bound, so compact-time exponential bounds on $Q_j$ control the $V$ displacement and $H$ velocity tails by the weighted $V$-norm identity of [F3]; the finitely many zero modes have the displayed solutions by step 1.3, and the finite-sum FTC estimate of [F6] passes the $H$ derivative to the limit and identifies the variational second-order equation against compact-time tests. Such arbitrary second-order data need not arise from the first-order Maxwell system: its zero modes additionally satisfy $a_j^{\mathrm{\prime }}=-\gamma a_j$, and the broader wave is retained alongside the physical first-order branch. All square roots and real derivatives are those of [F7].

[Decay, zero sectors and long-time limits] For $\sigma >0$ (so $\gamma >0$) every positive mode decays: if ${\mathrm{\Delta }}_j={\gamma }^2\mathrm{/}4-v^2{\lambda }_j>0$ the two real roots $-\gamma \mathrm{/}2\pm \sqrt{{\mathrm{\Delta }}_j}$ are negative; if ${\mathrm{\Delta }}_j<0$ the complex roots have real part $-\gamma \mathrm{/}2<0$, and the oscillatory factor is bounded by the cosine and sine identities of [F7]; if ${\mathrm{\Delta }}_j=0$ the factor $t{e}^{-\gamma t\mathrm{/}2}$ tends to zero by [F7]. The positive-mode derivatives $a_j^{\mathrm{\prime }}$ also tend to zero from the same explicit formulas, so $b_j=(a_j^{\mathrm{\prime }}+\gamma a_j)\mathrm{/}(v^2\sqrt{{\lambda }_j})\to 0$ for each positive mode. Electric zero modes obey $a_j(t)=a_j(0)e^{-\gamma t}$ by step 1.3, and magnetic kernel data stay constant by step 1.2, so in the smooth class the solution tends in energy norm as $t\to +\mathrm{\infty }$ to $(0,P_{K_M}B(0))$, where $P_{K_M}$ is the orthogonal projection onto $K_M$ of step 1.2; the finite-head/finite-tail argument of step 3.1 passes the modewise convergence to the whole energy norm. For $\sigma =0$ the energy is constant by step 4.1, the electric zero modes are static by step 1.3 and the original lossless branch is recovered. The zero sectors are retained, not divided by their eigenvalues, and no universal decay to zero in the presence of magnetic kernel data is claimed.

[Scope] Clauses (a) and (b) are steps 2.1 and 3.1: the rectangular class has a unique finite-energy distributional solution with dense finite modes at every time, and the smooth PEC class has the declared weak realization with a unique solution and energy-norm convergence of finite modal sums; the decomposition of the magnetic space, including arbitrary kernel data, is step 1.2. Clause (c) is steps 4.1 and 5.1. Clause (d) is step 4.2. Clause (e): the argument used only the declared constant Ohmic branch, the ideal PEC boundary model, the finite-energy classes and the recorded spectral and elliptic suppliers; Countable Choice suffices for the rectangular branch, the smooth branch retains its full Choice premise, no arbitrary loss tensor, memory law, changing interface, open-end, rough-wall or empirical claim is made, and weak energy balance was proved by modal limits without asserting an unprovided nonlinear weak local Poynting theorem.