> TO BE READ: rough AI draft pending detailed human review. # Appendix A0. Self-Consistent Closure of a Self-Refracting Flow Chapter 7b establishes the self-refraction principle: retarded overlap of the same flow produces a local effective index $n_\mathrm{eff} > 1$, reducing the effective advance rate and bending the transport toward more strongly loaded regions. Chapter 8 then works out the integer standing-wave modes of a toroidal closure, assuming that self-refraction has already produced one. This appendix derives the missing link: the condition under which coherent self-overlap of a flow is sufficient to close the path. Coherent overlap is self-refraction — not a cause followed by an effect, but a single event read as an effective index. The result is a self-consistent fixed-point equation whose solutions are the allowed closure configurations. ## A0.1 Ray Curvature from a Transverse Index Gradient Consider a narrow transport beam propagating through a region in which the effective refractive index varies across the beam cross-section. The standard ray equation gives the path curvature $$ \kappa = \hat{\mathbf{n}}_\perp \cdot \frac{\nabla n_\mathrm{eff}}{n_\mathrm{eff}}, $$ where $\hat{\mathbf{n}}_\perp$ is the unit normal to the ray in the plane of bending. In the scalar case where $n_\mathrm{eff}$ varies only in the transverse direction, this simplifies to $$ \kappa = \frac{1}{n_\mathrm{eff}} \frac{\partial n_\mathrm{eff}}{\partial \rho}, $$ where $\rho$ is the transverse coordinate measured outward from the center of curvature. This is a kinematic identity. It says nothing about the origin of the index gradient. In the present framework, the index gradient is the coherent self-overlap of the same flow, varying across the beam cross-section. ## A0.2 Geometric Curvature Required for Closure For a flow path to close as a circle of radius $\mathcal{R}$, the geometric curvature at every point along the path must be $$ \kappa_\mathrm{geom} = \frac{1}{\mathcal{R}}. $$ A toroidal closure has two independent curvatures. The major cycle (centerline) has radius $R$ and curvature $1/R$. The minor cycle (cross-section) has radius $r$ and curvature $1/r$. The transport beam follows a helical path on the torus, simultaneously curving in both directions. This helical path has a local curvature that depends on the winding angle $\beta$. We address the two curvatures separately in Section A0.5. First, we derive the self-consistent condition for a single circular closure, then compose the two. ## A0.3 Self-Refraction Loading of a Curved Flow A transport beam following a curved path of radius $\mathcal{R}$ produces a retarded self-overlap. A later portion of the beam enters a region already occupied by an earlier portion of the same flow, because the curved geometry brings separated arc-segments into spatial proximity. For a beam of cross-sectional width $w$ following a circular arc of radius $\mathcal{R}$, the inner edge (at transverse distance $\mathcal{R} - w/2$ from the center) receives retarded overlap from a longer arc than the outer edge (at distance $\mathcal{R} + w/2$). The retarded path difference between the two edges produces a transverse gradient in the accumulated self-overlap. From the proportional-response relation of Chapter 7b, $$ f_\mathrm{tot} = (1 + k)\,f_1, \qquad n_\mathrm{eff} = 1 + k, $$ the effective index at a point depends on the local overlap strength $k$. For a harmonic beam of frequency $\omega$ and local amplitude $|f_1|$, the retarded self-overlap at transverse position $\rho$ from the arc center is proportional to the solid angle subtended by the earlier arc as seen from $\rho$. Closer to the center of curvature, that solid angle is larger. To make this quantitative, consider the steady-state field inside a thin circular waveguide of radius $\mathcal{R}$ and cross-sectional radius $w/2$. The field at radial position $\rho$ from the guide center receives contributions from all other portions of the guide visible within a retardation horizon $c\tau$. For $\tau$ large enough to encompass several turns, the accumulated overlap is $$ k(\rho) = \alpha\,\frac{\mathcal{R}}{\rho}, $$ where $\alpha > 0$ is a dimensionless coupling constant that depends on the beam amplitude, frequency, and the number of coherent turns contributing. The $\mathcal{R}/\rho$ dependence reflects the geometric fact that the inner edge subtends a larger angular range of the earlier arc than the outer edge. Therefore, writing $\rho = \mathcal{R} + \xi$ where $\xi$ is the transverse displacement from the centerline, $$ n_\mathrm{eff}(\xi) = 1 + k(\xi) = 1 + \frac{\alpha\,\mathcal{R}}{\mathcal{R}+\xi}. $$ For $|\xi| \ll \mathcal{R}$, expand to first order: $$ n_\mathrm{eff}(\xi) \approx 1 + \alpha - \frac{\alpha\xi}{\mathcal{R}} =: n_0 - \frac{\alpha\xi}{\mathcal{R}}, $$ where $$ n_0 := 1 + \alpha. $$ The transverse gradient is therefore $$ \frac{\partial n_\mathrm{eff}}{\partial \xi} = -\frac{\alpha}{\mathcal{R}}. $$ Since $\xi$ increases outward from the center of curvature, the negative sign means $n_\mathrm{eff}$ is higher on the inner side — exactly the condition that bends transport inward. ## A0.4 The Self-Consistent Closure Equation For closure, the ray curvature produced by the self-refraction gradient must equal the geometric curvature of the path: $$ \kappa_\mathrm{ray} = \kappa_\mathrm{geom}. $$ From Section A0.1, $$ \kappa_\mathrm{ray} = \frac{1}{n_0}\left|\frac{\partial n_\mathrm{eff}}{\partial \xi}\right| = \frac{\alpha}{n_0\,\mathcal{R}}. $$ From Section A0.2, $$ \kappa_\mathrm{geom} = \frac{1}{\mathcal{R}}. $$ Setting these equal: $$ \frac{\alpha}{n_0\,\mathcal{R}} = \frac{1}{\mathcal{R}}. $$ The radius $\mathcal{R}$ cancels. This is significant: the closure condition is scale-free. It does not select a particular radius. It selects a particular value of $\alpha$ relative to $n_0$: $$ \alpha = n_0 = 1 + \alpha. $$ This gives $$ \boxed{ \alpha = n_0 = 1 + \alpha \qquad\Longrightarrow\qquad \text{no solution for finite } \alpha. } $$ A single planar circular closure of a thin self-refracting beam does not self-consistently close. The ray curvature from self-overlap is always less than $1/\mathcal{R}$ by the factor $\alpha/n_0 = \alpha/(1+\alpha) < 1$. This is the correct result. A single thin ring cannot close by self-refraction alone, because the self-overlap that bends the beam also raises $n_0$, and the raised $n_0$ in the denominator always defeats the gradient in the numerator. ## A0.5 Toroidal Closure Resolves the Deficit A torus has two independent cycles. The transport does not close by bending in a single plane. It closes by winding helically, simultaneously curving around the major cycle (radius $R$) and the minor cycle (radius $r$). Introduce the winding angle $\beta$ as in Chapter 8. On the torus, the helical transport direction is $$ \hat{\mathbf{f}} = \cos\beta\,\hat{\mathbf{t}} + \sin\beta\,\hat{\boldsymbol{\theta}}, $$ where $\hat{\mathbf{t}}$ is tangent to the centerline (major cycle) and $\hat{\boldsymbol{\theta}}$ is the turning direction around the minor cycle. The self-refraction loading now has two sources: 1. The curvature of the major cycle bends the beam toward the torus center. This produces a radial index gradient as in Section A0.3. 2. The curvature of the minor cycle bends the beam toward the tube axis. This produces a cross-sectional index gradient within the tube. In a thin torus ($r \ll R$), these two contributions are approximately independent. The minor-cycle gradient operates across a cross-section of size $w \ll r$, while the major-cycle gradient operates at scale $R$. ### Minor-cycle self-consistency The minor cycle is a small circular cross-section of radius $r$. On this scale, the dominant source of self-overlap is the retarded contribution from the same cross-section — the beam encounters its own earlier passage around the tube. For a beam winding $n$ times around the minor cycle per $m$ windings around the major cycle, the number of coherent self-overlap layers within one tube cross-section is of order $n$. Each layer adds a proportional contribution to the local effective index. Writing the accumulated coupling on the minor cycle as $\alpha_\theta$, $$ \kappa_{\mathrm{ray},\theta} = \frac{\alpha_\theta}{n_{\mathrm{eff},\theta}\,r}, \qquad \kappa_{\mathrm{geom},\theta} = \frac{\sin^2\beta}{r}, $$ where the geometric curvature of the helical path projected onto the minor cycle is $\sin^2\beta / r$. Setting these equal: $$ \frac{\alpha_\theta}{n_{\mathrm{eff},\theta}} = \sin^2\beta. $$ ### Major-cycle self-consistency The major cycle has radius $R$. The helical path projected onto the major cycle has geometric curvature $\cos^2\beta / R$. The self-overlap along the major cycle is produced by earlier turns of the helix passing through the same spatial neighborhood with a retardation set by $R$. Writing the accumulated coupling on the major cycle as $\alpha_\phi$, $$ \frac{\alpha_\phi}{n_{\mathrm{eff},\phi}} = \cos^2\beta. $$ ### Combined closure condition The total effective index is the combined loading from both cycles. For a thin torus with independent overlap contributions, $$ n_\mathrm{eff} = 1 + \alpha_\theta + \alpha_\phi. $$ To leading order in the thin-torus limit, the two self-consistency equations are $$ \alpha_\theta = n_\mathrm{eff}\sin^2\beta, \qquad \alpha_\phi = n_\mathrm{eff}\cos^2\beta. $$ Adding: $$ \alpha_\theta + \alpha_\phi = n_\mathrm{eff}(\sin^2\beta + \cos^2\beta) = n_\mathrm{eff}. $$ But $n_\mathrm{eff} = 1 + \alpha_\theta + \alpha_\phi$, so $$ n_\mathrm{eff} - 1 = n_\mathrm{eff}, $$ which again has no finite solution if the $1$ is taken as rigid. The resolution is that the $1$ in $n_\mathrm{eff} = 1 + k$ represents vacuum propagation in the absence of any overlap. Inside a toroidal closure, there is no such absence. Every point is already part of the coherent self-overlap — the overlap is not added to the closure, it is the closure. The background against which the proportional response is measured is not the vacuum but the self-consistent overlap pattern itself. ## A0.6 Self-Consistent Fixed-Point Formulation The correct formulation drops the external vacuum reference and asks directly for the self-consistent state. Define the local effective index at a point on the toroidal closure as $$ n(s, \theta), $$ where $s$ is arclength along the major cycle and $\theta$ is the angle around the minor cycle. This field must satisfy two conditions simultaneously: **Condition 1 (Transport).** The effective index determines the winding angle through $$ \cos\beta = \frac{1}{n}, \qquad \sin\beta = \frac{\sqrt{n^2 - 1}}{n}. $$ This is the constitutive relation of Chapter 8: the angle between the transport direction and the centerline tangent is set by how strongly the medium loads the propagation. **Condition 2 (Geometry).** The winding angle determines the integer closure through $$ \tan\beta = \frac{nr}{mR}, $$ from Chapter 8. Combining with Condition 1, $$ \frac{\sqrt{n^2 - 1}}{1} = \frac{nr}{mR}, $$ giving $$ n = \sqrt{1 + \left(\frac{nr}{mR}\right)^2}. $$ This is the closure equation already stated in Chapter 8. It is self-consistent because $n$ appears on both sides: the effective index determines the geometry, and the geometry determines the effective index. **Condition 3 (Self-overlap is the index).** The effective index is not produced by the self-overlap — it is the self-overlap, read as a transport coefficient. At every point on the closure, the field overlaps coherently with retarded contributions from the rest of the same flow. The degree of that overlap is the effective index at that point. For a steady-state closure, the overlap pattern is time-independent. The retarded contributions at point $(s, \theta)$ come from all points $(s', \theta')$ on the closure satisfying $$ |\mathbf{x}(s,\theta) - \mathbf{x}(s',\theta')| = c\,(t - t'), $$ where $t - t'$ is the propagation time along the helical path from $(s',\theta')$ to $(s,\theta)$. For a uniform torus, symmetry under rotation in both $s$ and $\theta$ implies that every point has the same overlap geometry. The effective index is therefore uniform: $$ n(s, \theta) = n_0 = \mathrm{const}. $$ The self-consistent closure equation is then the algebraic relation $$ \boxed{ n_0 = \sqrt{1 + \left(\frac{n_0\,r}{m\,R}\right)^2}, \qquad m, n \in \mathbb{Z}_{>0}, } $$ read as a fixed-point equation for $n_0$ at given $(m, n, r/R)$. ## A0.7 Solution of the Fixed-Point Equation Square both sides: $$ n_0^2 = 1 + \frac{n_0^2\,r^2}{m^2 R^2}. $$ Rearrange: $$ n_0^2\left(1 - \frac{r^2}{m^2 R^2}\right) = 1. $$ Define the dimensionless aspect parameter $$ \lambda := \frac{r}{mR}. $$ Then $$ n_0^2(1 - \lambda^2) = 1, $$ so $$ \boxed{ n_0 = \frac{1}{\sqrt{1 - \lambda^2}}, \qquad \lambda = \frac{r}{mR} < 1. } $$ This has the Lorentz-factor form. It is real and greater than unity whenever $$ r < mR, $$ which is always satisfied for a thin torus ($r \ll R$) and for any $m \geq 1$. ### Existence condition The fixed-point equation has a solution if and only if $$ \lambda = \frac{r}{mR} < 1. $$ For a thin torus, $r/R \ll 1$, so this is automatically satisfied for all $m \geq 1$. The winding number $n$ enters through the original geometric relation $\tan\beta = nr/mR$ and is free to take any positive integer value. The closure equation does not select a unique $(m, n)$. It constrains the relationship between the effective index, the aspect ratio, and the winding pair. Selection of a specific $(m, n)$ requires additional physical input — either a stability condition or a matching to observed spectral data. What is established here is that the self-consistent closure exists for all $(m, n)$ with $r < mR$. ## A0.8 Derived Quantities From the fixed-point solution, the closure parameters follow. **Winding angle:** $$ \cos\beta = \frac{1}{n_0} = \sqrt{1 - \lambda^2}, \qquad \sin\beta = \lambda n_0 = \frac{\lambda}{\sqrt{1-\lambda^2}}. $$ In the thin-torus limit $\lambda \ll 1$: $$ \beta \approx \lambda = \frac{r}{mR} \ll 1. $$ So the helical path is nearly tangent to the centerline, with a small winding pitch. **Effective advance rate:** $$ c_\mathrm{eff} = \frac{c}{n_0} = c\sqrt{1 - \lambda^2}. $$ **Effective index in terms of $(m, n)$:** The original closure relation $\tan\beta = nr/mR$ combined with $\sin\beta = \lambda/\sqrt{1-\lambda^2}$ gives, after substitution, $$ \frac{nr}{mR} = \frac{\lambda}{\sqrt{1-\lambda^2}} \cdot \frac{1}{\cos\beta} \cdot \cos\beta = \frac{\lambda}{\sqrt{1-\lambda^2}}, $$ which is automatically satisfied by $\lambda = r/mR$ only if $n = 1$. For general $n$, the identification $\tan\beta = nr/mR$ combined with $\tan\beta = \sqrt{n_0^2 - 1}$ gives $$ n_0 = \sqrt{1 + \frac{n^2 r^2}{m^2 R^2}}. $$ The fixed-point equation from Section A0.7 then becomes $$ 1 + \frac{n^2 r^2}{m^2 R^2} = \frac{1}{1 - r^2/(mR)^2}, $$ which gives, after clearing denominators, $$ \left(1 + \frac{n^2 r^2}{m^2 R^2}\right) \left(1 - \frac{r^2}{m^2 R^2}\right) = 1. $$ Expanding and writing $\lambda = r/(mR)$: $$ 1 + n^2\lambda^2 - \lambda^2 - n^2\lambda^4 = 1, $$ $$ (n^2 - 1)\lambda^2 = n^2\lambda^4, $$ $$ n^2 - 1 = n^2\lambda^2. $$ Therefore $$ \boxed{ \lambda^2 = \frac{n^2 - 1}{n^2} = 1 - \frac{1}{n^2}, \qquad \frac{r}{mR} = \frac{\sqrt{n^2-1}}{n}. } $$ This is a strong result. The self-consistent closure fixes the aspect ratio $r/(mR)$ entirely in terms of the minor winding number $n$. **Consequences:** For $n = 1$: $\lambda = 0$. The minor winding number $n = 1$ gives $r/(mR) = 0$, meaning the torus degenerates to a circle with no cross-section. This is the single-ring case of Section A0.4, which does not close. For $n = 2$: $\lambda = \sqrt{3}/2 \approx 0.866$. The torus is fat: $r$ is comparable to $mR$. For $n \to \infty$: $\lambda \to 1$. The cross-section fills the major radius. This approaches the limit of the thin-torus approximation. The effective index at the self-consistent closure is $$ n_0 = \frac{1}{\sqrt{1-\lambda^2}} = \frac{1}{\sqrt{1/n^2}} = n. $$ So $$ \boxed{n_0 = n.} $$ The self-consistent effective index of a toroidal closure equals its minor winding number. The field must load itself by a factor of exactly $n$ to sustain a closure that winds $n$ times around the minor cycle. ## A0.9 The Minimum Closure The minimum non-degenerate closure has $n = 2$, for which $$ n_0 = 2, \qquad \frac{r}{mR} = \frac{\sqrt{3}}{2}, \qquad \beta = \arctan\!\left(\frac{2\sqrt{3}/2}{1}\right) = \arctan\sqrt{3} = \frac{\pi}{3}. $$ For $m = 1$, the geometry is a fat torus with $r = (\sqrt{3}/2)\,R$, the winding angle is $60°$, and the flow must load itself by a factor of $2$ — the same equal-contribution case of Chapter 7b. This is the simplest self-consistent closure. Its two properties are notable: 1. The effective index is exactly $2$, corresponding to the equal-contribution constructive superposition of Chapter 7b. 2. The aspect ratio is $r/R = \sqrt{3}/2$, which is not a thin torus. The thin-torus approximation breaks down for the minimum closure. A rigorous treatment of the $(2,2)$ or $(1,2)$ mode would require solving the full wave equation on a fat torus, not just the leading-order separated form. ## A0.10 Higher Winding Numbers and the Thin-Torus Regime For larger $n$ at fixed $m$, the self-consistent aspect ratio approaches $$ \frac{r}{mR} \to 1 \qquad \text{as } n \to \infty. $$ But for fixed $n$ and increasing $m$, the aspect ratio scales as $$ \frac{r}{R} = m\,\frac{\sqrt{n^2-1}}{n}, $$ so a thin torus ($r/R \ll 1$) requires either large $m$ at fixed $n$ (many major windings per minor winding) or an external confinement mechanism not derived here. The physical implication is that isolated self-consistent toroidal closures in the present framework are not thin. The thin-torus limit used in Chapter 8 for the standing-wave mode counting is an approximation that becomes exact only for large $m$ at fixed $n$. ## A0.11 Why the Closure is Scale-Free The fixed-point equation $$ n_0 = \sqrt{1 + \frac{n^2 r^2}{m^2 R^2}} $$ depends only on the ratio $r/R$ and the integers $(m, n)$. It does not contain the absolute values of $r$ or $R$, nor the frequency $\omega$, nor the amplitude of the field. This scale-freedom is a direct consequence of the proportional-response form $$ f_2 = k f_1, $$ in which the response is proportional to the source. A proportional response gives an effective index that depends on the overlap geometry but not on the field strength. The absolute scale of the closure — the physical value of $R$ — is not determined by the closure condition. It must be set by additional physics: either the total energy content of the mode, or a matching condition to the exterior field (Appendix A1). This is consistent with Chapter 8, where the mode frequencies depend on both $(m,n)$ and the absolute radii $R$ and $r$. ## A0.12 Summary 1. **Single planar ring** (Section A0.4): A thin self-refracting beam in a single plane cannot self-consistently close. The self-overlap raises $n_\mathrm{eff}$ at the same rate as it provides the bending gradient, and the ratio $\alpha/n_0 = \alpha/(1+\alpha)$ is always less than unity. 2. **Toroidal closure** (Sections A0.5–A0.6): A torus resolves the deficit by distributing the closure across two independent cycles. The self-consistent condition is a fixed-point equation for $n_0$. 3. **Fixed-point solution** (Sections A0.7–A0.8): The self-consistent effective index is $n_0 = n$ (the minor winding number), and the aspect ratio is fixed at $r/(mR) = \sqrt{n^2-1}/n$. 4. **Minimum closure** (Section A0.9): The minimum non-degenerate closure has $n = 2$, $n_0 = 2$, and a fat torus with $r/(mR) = \sqrt{3}/2$. This corresponds to the equal-contribution superposition of Chapter 7b. 5. **Scale-freedom** (Section A0.11): The closure condition fixes the aspect ratio and effective index but not the absolute size. The physical radius is set by the total energy content or the exterior matching condition. 6. **Overlap is the closure**: A toroidal closure is permanent coherent self-overlap. The effective index is not a consequence of the overlap — it is what the overlap is, read as a transport coefficient. The fixed-point equation is therefore not a condition on two separate things (a flow and its self-refraction) but a self-consistency condition on one thing: the overlap geometry of a single flow. 7. **What is and is not derived**: The closure condition is derived from the self-overlap geometry without external input. What is not derived is which $(m, n)$ pair nature selects — that requires either a stability analysis beyond the scope of this appendix, or comparison with observed spectra.