> TO BE READ: rough AI draft pending detailed human review. # Appendix A1. Isotropy of the Exterior Shell as a Topologically Protected Energy Minimum Chapter 10 asserts that the exterior curl-flux of a net-chiral toroidal closure is distributed isotropically over large enclosing shells, so that its surface density falls as $1/r^2$. This appendix proves that claim in three steps. 1. A non-uniform flux distribution on an enclosing shell creates a restoring force, derived directly from the self-refraction principle of Chapter 7b. 2. The uniform distribution is the unique minimum of the exterior shell energy subject to fixed total flux, by a direct variational argument. 3. The winding numbers $(m,n)$ are topological invariants of the source-free flow. Their rigidity prevents any source-free perturbation from moving the closure out of its $(m,n)$ sector, so the energy minimum is globally stable. ## A1.1 Setup and Notation Let $\mathcal{T}$ denote a toroidal closure with net chiral winding $(m,n)$. From the divergence-of-curl argument of Chapter 10, the total organized curl flux $$ \Phi_{(m,n)} := \int_{S_r} (\nabla\times\mathbf{F})\cdot\hat{\mathbf{n}}\,dA $$ is the same for every enclosing shell $S_r$ of radius $r$ centered on the closure, and is nonzero when the net chiral winding is nonzero. Write the flux density per unit solid angle on $S_r$ as $\phi(\hat{\mathbf{r}},r)$, defined by $$ (\nabla\times\mathbf{F})\cdot\hat{\mathbf{n}}\big|_{S_r} = \frac{\phi(\hat{\mathbf{r}},r)}{r^2}, $$ so that the conservation law reads $$ \int_{S^2} \phi(\hat{\mathbf{r}},r)\,d\Omega = \Phi_{(m,n)}, \qquad \forall r > r_0, $$ where $r_0$ is a radius enclosing $\mathcal{T}$ and $d\Omega$ is the standard solid-angle measure on the unit sphere. From the amplitude-squared relation of Chapter 7a, the local exterior energy density on $S_r$ is proportional to the square of the local field amplitude. The organized $(m,n)$ sector carries amplitude proportional to the curl flux per unit area, so $$ u_{(m,n)}(\hat{\mathbf{r}},r) = \kappa\,\frac{\phi(\hat{\mathbf{r}},r)^2}{r^4}, $$ for a positive constant $\kappa$ fixed by the normalization of the two-aspect split. The total organized exterior energy on the shell is therefore $$ U_r = \int_{S^2} u_{(m,n)}\,r^2\,d\Omega = \frac{\kappa}{r^2} \int_{S^2} \phi(\hat{\mathbf{r}},r)^2\,d\Omega. $$ ## A1.2 Non-uniform Flux Produces a Restoring Force Suppose $\phi$ is not constant on $S_r$. Define its tangential gradient on $S_r$ by $\nabla_\perp\phi$. By the self-refraction principle of Chapter 7b, a region of higher local energy density loads the flow more strongly, producing a lower effective advance rate: $$ c_\mathrm{eff}(\hat{\mathbf{r}},r) = \frac{c}{n_\mathrm{eff}(\hat{\mathbf{r}},r)}, \qquad n_\mathrm{eff} \propto \phi(\hat{\mathbf{r}},r). $$ A tangential gradient of $c_\mathrm{eff}$ bends transport toward the more strongly loaded side. The transverse refractive force per unit volume exerted on the exterior flow is (Chapter 12, ray equation) $$ \mathbf{f}_\perp = \nabla_\perp n_\mathrm{eff} \propto \nabla_\perp\phi. $$ This force acts on the vortex tubes constituting the exterior continuation of $\mathcal{T}$. A nonzero $\nabla_\perp\phi$ therefore drives tangential redistribution of flux on $S_r$. The force vanishes if and only if $$ \nabla_\perp\phi = 0 \qquad\text{on }S_r, $$ that is, when $\phi$ is constant over $S_r$. The direction of the force is toward regions of higher $\phi$, so it moves flux away from depleted sectors and toward enriched ones. This is a restoring force toward uniformity, not away from it. ## A1.3 Uniform Distribution is the Unique Energy Minimum **Proposition.** Among all non-negative functions $\phi: S^2 \to \mathbb{R}_{\geq 0}$ satisfying $$ \int_{S^2} \phi\,d\Omega = \Phi_{(m,n)} =: C > 0, $$ the functional $$ I[\phi] := \int_{S^2} \phi^2\,d\Omega $$ attains its minimum uniquely at $$ \phi_* = \frac{C}{4\pi} = \mathrm{const}. $$ **Proof.** Write $\phi = \phi_* + \delta\phi$, where $\phi_* = C/4\pi$ and $\int_{S^2}\delta\phi\,d\Omega = 0$. Then $$ I[\phi] = \int_{S^2}(\phi_*+\delta\phi)^2\,d\Omega = 4\pi\phi_*^2 + 2\phi_*\underbrace{\int_{S^2}\delta\phi\,d\Omega}_{=\,0} + \int_{S^2}(\delta\phi)^2\,d\Omega. $$ Since $(\delta\phi)^2 \geq 0$ pointwise, with equality if and only if $\delta\phi = 0$ identically, $$ I[\phi] \geq 4\pi\phi_*^2 = \frac{C^2}{4\pi} = I[\phi_*], $$ with equality if and only if $\phi = \phi_*$ almost everywhere on $S^2$. $\square$ **Remark.** This is equivalent to the Cauchy-Schwarz inequality $$ C^2 = \left(\int_{S^2}\phi\,d\Omega\right)^2 \leq \left(\int_{S^2} 1^2\,d\Omega\right)\!\left(\int_{S^2}\phi^2\,d\Omega\right) = 4\pi\,I[\phi], $$ with equality iff $\phi$ is proportional to $1$, i.e., constant. Both proofs are elementary and reach the same strict minimum. **Consequence.** The total exterior shell energy $U_r \propto I[\phi]/r^2$ is minimized uniquely when $\phi$ is uniform over $S_r$. Any non-uniform distribution carries strictly higher exterior energy and, by Section A1.2, is subject to a restoring force. Uniform flux is therefore a critical point, and the proposition shows it is a strict minimum, not a saddle. ## A1.4 Topological Protection: Local Minimum is Global Minimum The argument so far establishes that uniform flux is a strict local minimum of $U_r$ within the space of flux distributions satisfying fixed total $C$. It remains to rule out escape to a lower-energy configuration in a different winding class. **Definition.** The winding numbers of the $(m,n)$ flow on $\mathcal{T}$ are $$ m = \frac{1}{2\pi}\oint_{\gamma_1} d\varphi, \qquad n = \frac{1}{2\pi}\oint_{\gamma_2} d\theta, $$ where $\gamma_1$ and $\gamma_2$ are the two independent non-contractible cycles of the torus, and $\varphi$, $\theta$ are the respective phase angles of the flow around each cycle. Because the flow is continuous and single-valued on each cycle, $m$ and $n$ are integers. **Lemma.** Under any continuous source-free deformation of $\mathcal{T}$, the winding numbers $m$ and $n$ are preserved. **Proof.** The winding number $m$ is the degree of the map $\gamma_1 \to S^1$ defined by the phase of the flow amplitude $f$ around $\gamma_1$. This degree is a homotopy invariant: it cannot change under a continuous deformation of the map. A deformation of the map corresponds here to a continuous source-free evolution of the flow. The only way $m$ can change is if the map $\gamma_1 \to S^1$ becomes discontinuous at some instant — that is, if the flow amplitude $|f|$ vanishes somewhere on $\gamma_1$, destroying the phase. But $|f|^2 = u > 0$ everywhere on the closure, because a zero of $|f|$ on $\mathcal{T}$ is a local sink, forbidden by source-free continuity. Therefore $|f| > 0$ at all times, the phase map is everywhere defined and continuous, and $m$ cannot change. The identical argument applies to $n$. $\square$ **Theorem.** The uniform-flux equilibrium of the $(m,n)$ closure is globally stable: no source-free perturbation of any amplitude can move the system to a configuration with different winding numbers. **Proof.** By the Lemma, any source-free evolution preserves $(m,n)$. Any configuration reachable from the $(m,n)$ equilibrium under source-free dynamics therefore belongs to the same winding class. Within that class, Section A1.3 shows the uniform-flux distribution is the unique minimum of $U_r$. The system cannot exit the $(m,n)$ class, and within that class the equilibrium is the unique minimum. $\square$ **Remark.** The distinction between local and global stability is resolved entirely by topology. Without topological protection, one would need to check whether the closure can tunnel through a zero-amplitude configuration to reach a lower-energy winding class. The source-free constraint forbids that configuration from arising. The winding is frozen, and the energy minimum is therefore global. ## A1.5 Exterior Shell Density From the uniform flux $\phi_* = C/4\pi = \Phi_{(m,n)}/4\pi$ and the energy-density relation of Section A1.1, the exterior organized energy density on $S_r$ is $$ u_{(m,n)}(r) = \kappa\,\frac{\phi_*^2}{r^4} = \frac{\kappa\,\Phi_{(m,n)}^2}{16\pi^2 r^4}. $$ The total organized energy on $S_r$ is $$ U_r = 4\pi r^2\cdot u_{(m,n)}(r) = \frac{\kappa\,\Phi_{(m,n)}^2}{4\pi r^2}. $$ Writing $Q^2 := \kappa\,\Phi_{(m,n)}^2/4\pi$, the shell energy density is $$ \boxed{ u_{(m,n)}(r) = \frac{Q^2}{4\pi r^4}, \qquad \sigma(r) := \frac{U_r}{4\pi r^2} = \frac{Q^2}{(4\pi)^2 r^4}. } $$ The shell density falls as $1/r^4$ in energy and as $1/r^2$ in the amplitude (field-strength) reading. It is isotropic — independent of $\hat{\mathbf{r}}$ — by the result of Section A1.3, and that isotropy is globally stable by Section A1.4. The quantity $Q$, proportional to $\Phi_{(m,n)}$, is signed by the handedness of the net chiral winding and is an integer multiple of its elementary value. This is the exterior scalar that Chapter 11 identifies with charge. ## A1.6 Summary The three steps of this appendix establish: 1. **Restoring force** (Section A1.2). A non-uniform flux distribution on an enclosing shell creates a tangential self-refraction force proportional to $\nabla_\perp\phi$, driving flux toward uniformity. 2. **Strict energy minimum** (Section A1.3). Among all distributions with fixed total flux $C$, the uniform distribution $\phi = C/4\pi$ uniquely minimizes the exterior shell energy. Any deviation raises the energy strictly. 3. **Global stability by topology** (Section A1.4). The winding numbers $(m,n)$ are homotopy invariants of the source-free flow. They cannot change under any continuous source-free evolution because a change would require a zero of $|f|$ on the closure, which is a sink and is forbidden. The energy minimum is therefore global, not merely local. Together these three steps ground the isotropy assertion of Chapter 10 as a derived result rather than an assumption, and identify the $1/r^2$ shell density as the necessary exterior form of a topologically stable net-chiral toroidal closure.