## Worked example 3: a helical trajectory on a torus with explicit $(m,n)$, explicit length $L_{m,n}$, and discrete frequency scales This example makes the $(m,n)$ data operational. We do not invoke “string postulates.” We use only: - a torus as a closed surface supporting tangent flow, - closure of a flow line as a periodicity condition, - and the already-derived consequence that periodic domains enforce discrete mode spectra. The central deliverables are: - a concrete parametrization of a torus, - a concrete curve class with winding numbers $(m,n)$, - an explicit approximate formula for the curve length $L_{m,n}$, - and the resulting mode frequencies $\omega_k(m,n)$ for perturbations on the closed path. The point is to show exactly how integer winding forces discrete geometry, hence discrete spectral structure. --- ## Torus geometry Fix two radii: - major radius $R$ (distance from center of hole to tube center), - minor radius $r$ (radius of the tube). Standard embedding of a torus in $\mathbb{R}^3$ uses angles $(\phi,\theta)$: - $\phi$ is the toroidal angle (around the hole), - $\theta$ is the poloidal angle (around the tube). Define $$ \mathbf{X}(\phi,\theta)= \begin{pmatrix} (R+r\cos\theta)\cos\phi\\ (R+r\cos\theta)\sin\phi\\ r\sin\theta \end{pmatrix}, \qquad (\phi,\theta)\in[0,2\pi)\times[0,2\pi). $$ Compute tangent basis vectors: $$ \partial_\phi \mathbf{X}= \begin{pmatrix} -(R+r\cos\theta)\sin\phi\\ \ \ (R+r\cos\theta)\cos\phi\\ 0 \end{pmatrix}, \qquad |\partial_\phi \mathbf{X}|=R+r\cos\theta, $$ $$ \partial_\theta \mathbf{X}= \begin{pmatrix} - r\sin\theta\cos\phi\\ - r\sin\theta\sin\phi\\ \ \ r\cos\theta \end{pmatrix}, \qquad |\partial_\theta \mathbf{X}|=r. $$ Also note orthogonality: $$ \partial_\phi \mathbf{X}\cdot \partial_\theta \mathbf{X}=0. $$ Thus the induced metric on the torus is $$ ds^2 = (R+r\cos\theta)^2\,d\phi^2 + r^2\,d\theta^2. $$ --- ## A curve class with winding numbers $(m,n)$ A simple representative curve with toroidal winding $n$ and poloidal winding $m$ is $$ \phi(t)=nt,\qquad \theta(t)=mt,\qquad t\in[0,2\pi]. $$ Closure is automatic because at $t=2\pi$: $$ \phi(2\pi)=2\pi n,\qquad \theta(2\pi)=2\pi m, $$ so both angles return mod $2\pi$. Thus $(m,n)\in\mathbb{Z}^2$ label the homotopy class of the curve. This curve is not the most general torus-knot trajectory, but it is a clean canonical representative for derivations. --- ## Exact arclength integral for this $(m,n)$ curve Compute $ds$ along the curve using the metric. We have $$ d\phi = n\,dt,\qquad d\theta=m\,dt. $$ So $$ ds^2 = (R+r\cos\theta)^2(n\,dt)^2 + r^2(m\,dt)^2. $$ But $\theta(t)=mt$, so $\cos\theta=\cos(mt)$: $$ ds = \sqrt{n^2(R+r\cos(mt))^2 + m^2 r^2}\,dt. $$ Thus the total length is $$ L_{m,n}= \int_0^{2\pi} \sqrt{n^2(R+r\cos(mt))^2 + m^2 r^2}\,dt. $$ This is an explicit formula. It is generally not elementary because of the $\cos(mt)$ inside the square root. But it is already enough to show: - $L_{m,n}$ depends on integers $(m,n)$, - and thus any quantity depending on $L$ is discretized by topology. Still, we can proceed further with useful approximations. --- ## Thin-torus approximation and explicit length estimate Assume a thin torus: $$ r \ll R. $$ Then $R+r\cos(mt)\approx R$ to leading order, and we get $$ ds \approx \sqrt{n^2R^2 + m^2 r^2}\,dt. $$ This is constant in $t$, so $$ L_{m,n}\approx \int_0^{2\pi}\sqrt{n^2R^2 + m^2 r^2}\,dt = 2\pi\sqrt{n^2R^2 + m^2 r^2}. $$ Equivalently, $$ L_{m,n}\approx 2\pi\sqrt{(nR)^2 + (mr)^2}. $$ This is the same “unrolled chart” intuition: - around the torus: distance per turn $\sim 2\pi R$ (toroidal), - around the tube: distance per turn $\sim 2\pi r$ (poloidal), - total path behaves like hypotenuse of a rectangle scaled by $(n,m)$. This shows explicitly how integers turn into geometric length. --- ## Effective forward speed on the torus We now define “forward” as the toroidal direction (increasing $\phi$). This is the natural macroscopic direction for an energy flow circulating around the hole. Along the curve, the $\phi$-motion per $dt$ is $d\phi=n\,dt$. The physical toroidal arc element is $(R+r\cos\theta)\,d\phi$. In the thin approximation, this is approximately $R\,d\phi$. So the physical toroidal displacement over $dt$ is approximately: $$ d\ell_{\text{tor}} \approx R\,d\phi = Rn\,dt. $$ Meanwhile the arclength travelled is approximately: $$ ds \approx \sqrt{n^2R^2 + m^2 r^2}\,dt. $$ So the projection factor is $$ \frac{d\ell_{\text{tor}}}{ds} \approx \frac{Rn}{\sqrt{n^2R^2 + m^2 r^2}}. $$ If local transport along the path is at speed $c$: $$ \frac{ds}{dt}=c, $$ then the effective toroidal transport speed is $$ v_{\text{tor}}(m,n) = \frac{d\ell_{\text{tor}}}{dt} = \frac{d\ell_{\text{tor}}}{ds}\frac{ds}{dt} \approx c\frac{Rn}{\sqrt{n^2R^2 + m^2 r^2}}. $$ Thus $$ v_{\text{tor}}(m,n) \approx \frac{c}{\sqrt{1+\left(\frac{mr}{nR}\right)^2}}. $$ This is the torus analogue of the helix result $v=c\cos\theta$, with $$ \cos\theta_{\mathrm{eff}}(m,n) = \frac{Rn}{\sqrt{n^2R^2 + m^2r^2}}. $$ So winding in the poloidal direction forces a reduction in effective toroidal transport. --- ## Discrete mode spectrum on the closed trajectory Now treat the flow line as a 1D closed domain of length $L_{m,n}$. Small transverse excitations $\xi(s,t)$ satisfy, at leading order, $$ \partial_t^2\xi - c^2\partial_s^2\xi = 0, $$ with periodicity $$ \xi(s+L_{m,n},t)=\xi(s,t). $$ Therefore Fourier modes are $$ \xi(s,t)=\sum_{k\in\mathbb{Z}}\xi_k(t)\,e^{i2\pi ks/L_{m,n}}, $$ and the normal frequencies are $$ \omega_k(m,n)=\frac{2\pi c}{L_{m,n}}|k|. $$ Using the thin-torus length approximation gives a fully explicit scale: $$ \omega_k(m,n) \approx \frac{2\pi c}{2\pi\sqrt{n^2R^2+m^2r^2}}|k| = \frac{c}{\sqrt{n^2R^2+m^2r^2}}|k|. $$ Thus the frequency scale is discretized by: - $k\in\mathbb{Z}$ from periodicity on a loop, - $(m,n)\in\mathbb{Z}^2$ from topological closure class on the torus. This is the precise mechanism behind “discreteness from topology” in the program. Nothing quantum has been assumed. --- ## Where “tension” and “inertia” enter here From the thin-tube extraction: $$ T=\int_{\Sigma_s} u\,dA, \qquad \mu=\frac{T}{c^2}. $$ On any closed trajectory the same holds. So for a torus-wound tube: - $T$ depends on the cross-sectional energy profile, - $\mu$ follows, - and the closed length $L_{m,n}$ fixes the discrete mode spacings. Thus the object is characterized by: - local quantities: $T$, $\mu$, - global quantities: $(m,n)$, $L_{m,n}$, - and internal spectrum: $\omega_k(m,n)$. All are computed from localization plus topology. --- ## Summary of the torus winding worked example 1. Torus embedding yields metric: $$ ds^2=(R+r\cos\theta)^2d\phi^2 + r^2d\theta^2. $$ 2. Canonical winding curve: $$ \phi(t)=nt,\qquad \theta(t)=mt,\qquad t\in[0,2\pi]. $$ 3. Exact length: $$ L_{m,n}= \int_0^{2\pi} \sqrt{n^2(R+r\cos(mt))^2 + m^2 r^2}\,dt. $$ 4. Thin-torus explicit approximation: $$ L_{m,n}\approx 2\pi\sqrt{n^2R^2+m^2r^2}. $$ 5. Effective toroidal transport speed (projection): $$ v_{\text{tor}}(m,n) \approx c\frac{Rn}{\sqrt{n^2R^2+m^2r^2}} = \frac{c}{\sqrt{1+\left(\frac{mr}{nR}\right)^2}}. $$ 6. Discrete mode spectrum on the closed trajectory: $$ \omega_k(m,n)=\frac{2\pi c}{L_{m,n}}|k| \approx \frac{c}{\sqrt{n^2R^2+m^2r^2}}|k|. $$ This is the explicit bridge from: - divergence-free closed flow on a torus to - integer winding to - discrete geometrical length to - discrete mode spectrum.