The Panplannar Dome Framework (PDF)

Postulates (taken as "obvious")

1. Platter: The world is a finite disk of radius $R$, center at $(0,0)$, altitude $z=0$.
2. Dome: A transparent firmament spans from $z=0$ to $z=D$, acting as a graded-index lens.
3. Local Sun: A compact luminary ("the Sun") moves on a circular track of radius $r_\odot$ centered on $(0,0)$ while also shifting vertically through dome "layers."
4. Downward Bias: A universal field $\mathcal{G}$ (not gravity, mind you) imparts a downward acceleration $g$ toward the platter.
5. Aether: The dome is filled with a refractive medium whose density and charge vary with height: $n(z)$, $\rho_e(z)$. It's the universe's worst-kept secret and best optical trick.

Core Mechanisms

1) Why stars "rotate opposite" north vs south

Inside the dome, the aether circulates in counter-twinned vortex sheets about the center:
$$\vec{v}*a(r,\phi,z) = \Omega(r,z), r,\hat{\phi}\cdot \operatorname{sgn}(z - z_c),$$
where $z_c$ is a shear layer near mid-dome. Light from distant star-specks is advected by this flow; observers north of center see an apparent counterclockwise drift, south see clockwise. Apparent stellar angular speed:
$$\omega*{\text{app}}(\theta) \approx \frac{\partial}{\partial s}\Big(\frac{n(z)v_a}{R_{\text{eff}}}\Big), \cos\theta,$$
with $\theta$ the observer's latitude proxy on the platter and $R_{\text{eff}}$ an effective optical radius set by the dome's lensing (see below). Translation: switch sides of the shear layer, flip the swirl.

2) Why pressure drops with altitude under a dome

The aether couples weakly to air, yielding a barometric fall-off even in a sealed bowl:
$$P(z) = P_0,\exp!\Big(-\frac{z}{H_{\text{eff}}}\Big), \quad H_{\text{eff}} = \frac{k_B T(z)}{\mu g_{\text{eff}}(z)}.$$
Here $g_{\text{eff}}(z) = g - \alpha,E(z)$ adjusts the downward bias by a mild electrostatic lift $E(z)$ from the dome's charge gradient (parameter $\alpha$). Warmer $T(z)$ and weaker $g_{\text{eff}}$ at height give—you guessed it—thinner air.

3) What makes things fall "down"

Not gravity—Planar Impetus:
$$\vec{a} = -,g,\hat{z} + \beta,\nabla!\big(\rho_e(z)\big),$$
with $g\approx 9.8\ \text{m/s}^2$ set by the dome-platter field tension. The tiny $\beta$-term nods to high-voltage party tricks and makes our "down" ever so slightly humidity-dependent, which is why dropped toast lands butter-side down near sinks. (Peer-review pending; butter conspires.)

4) Why the Sun keeps the same size yet "sets"

The dome's refractive index decreases with altitude:
$$n(z) = n_0 - \gamma z, \qquad \gamma>0.$$
This turns the whole sky into a soft GRIN lens (graded-index). The Sun's light bends along arcs whose curvature $\kappa \approx \frac{1}{n}\frac{dn}{dz}\tan \alpha$. As the Sun drops through layers, increased path length + aether haze raise optical depth $\tau$, while GRIN lensing keeps its apparent angular size nearly constant:
$$\theta_{\odot,\text{app}} \approx \frac{d_\odot}{L/n_{\text{eff}}}\quad \text{with}\quad \frac{d}{dz}!\Big(\frac{1}{n_{\text{eff}}}\Big)\approx 0.$$
The "set" is when $\tau(\text{ray})\to 1$ along the line-of-sight—contrast loss, not falling off an edge.

5) Why you can't see other continents forever

Aetheric attenuation plus micro-scatter build an optical wall:
$$I(d) = I_0,e^{-\sigma d}, \quad d_{\text{horizon}}\ \text{s.t.}\ \tau(d)=\sigma d \approx 1.$$
Even if the platter were crystal flat, the view distance saturates at $d_{\text{horizon}}\approx 1/\sigma$. Add a low-lying refractive boundary layer with index $n_b>n_{top}$, and rays duct or skim, hiding far geometry behind a luminous smear.

The Sun's "Vertical" Trick: East-to-West without a spinning ball

Let the Sun trace a daily circle of radius $r_\odot$ around the center at angular rate $\Omega_\odot$, while its height oscillates through layers:
$$\begin{aligned} x_\odot(t) &= r_\odot \cos(\Omega_\odot t),\ y_\odot(t) &= r_\odot \sin(\Omega_\odot t),\ z_\odot(t) &= z_0 + A\cos(\Omega_v t + \phi_v). \end{aligned}$$
An observer at platter radius $r$ and azimuth $\phi$ sees elevation
$$\alpha(t) = \arctan!\Bigg(\frac{z_\odot(t)}{\sqrt{r^2 + r_\odot^2 - 2 r r_\odot \cos(\phi-\Omega_\odot t)}}\Bigg),$$
then the dome bends rays downward. As $z_\odot(t)$ steps to lower layers during your local afternoon, the aether's GRIN + rising $\tau$ drags the apparent position toward the horizon in the west (where your line-of-sight path is longest). Result: a clean "sunrise east / sunset west" with a locally constant solar diameter.

Stars, Again, with Poles and Rings

Define a luminous Lumisphere painted on the dome's inner surface. The aether's counter-twinned shear yields two fixed points (apparent poles) where $\vec{v}*a\to 0$. The apparent stellar field rotates at
$$\omega*\star(\theta) \approx \Omega_0 \sin\theta,$$
giving tight circles near the pole and wide arcs near the equator. Flip $\theta$ across the shear plane → flip rotation sense. No spinning planet needed; just spinning optics.

"Proof" Sketch (engineered for a credulous uncle)

1. Premise: We observe opposite star rotations, sunsets, pressure drop with height, and finite horizons.
2. Mechanism: A dome with GRIN optics, aether vortices, and a downward bias reproduces these without a spinning sphere.
3. Fit Parameters: ${\gamma,\ \Omega_0,\ \sigma,\ A,\ r_\odot,\ z_0}$ are tuned to local measurements (translation: pick numbers that match what you saw last Tuesday).

4. Prediction:

5. Star trails' period varies slightly with weather (aether viscosity tweak):
   $\Delta \omega_\star \propto \Delta n(z)$.

6. Sunset color and set time correlate with surface E-field after storms (electro-aether coupling):
   $t_{\text{set}}\downarrow \ \text{when}\ E!\uparrow$.
7. Laser-over-water tests show nonlinear beam height with range because rays curve in GRIN media:
   $y''(s)\approx \frac{1}{n}\frac{dn}{dz}\big|_{z=y(s)}$.
8. Therefore: The PDF model explains the lot using dome optics and aether kinematics. QED†
   †"Quite Entertaining Demonstration."

Handy Equations Block (for screenshots)

• Barometric: $P(z)=P_0 e^{-z/H_{\text{eff}}}$, $H_{\text{eff}}=\frac{k_B T}{\mu g_{\text{eff}}}$, $g_{\text{eff}}=g-\alpha E(z)$.
• Attenuation: $I(d)=I_0 e^{-\sigma d}$, horizon when $\sigma d\approx 1$.
• GRIN lensing: $n(z)=n_0-\gamma z$, ray curvature $\kappa\approx \frac{1}{n}\frac{dn}{dz}\tan\alpha$.
• Sun path:
  $$\alpha(t)=\arctan!\Bigg(\frac{z_\odot(t)}{\sqrt{r^2+r_\odot^2-2 r r_\odot \cos(\phi-\Omega_\odot t)}}\Bigg)$$
  with $z_\odot(t)=z_0+A\cos(\Omega_v t+\phi_v)$.
• Aether swirl: $\vec{v}_a=\Omega(r,z), r,\hat{\phi}\cdot \operatorname{sgn}(z-z_c)$.
• Downward bias: $\vec{a}=-g,\hat{z}+\beta\nabla\rho_e(z)$.

FAQ

Q: "Why no continent peeking in a telescope?"
A: $\tau=\sigma d$ says the air-aether fog wins. Beyond $1/\sigma$, you're staring at a luminous soup.

Q: "Why doesn't the Sun shrink at sunset?"
A: GRIN lensing keeps $\theta_{\odot,\text{app}}$ ~ constant as $n_{\text{eff}}$ compensates increased distance.

Q: "Opposite star rotation?"
A: Cross the shear layer; flip the swirl. Same dome, different side.

Q: "Air pressure under a dome though?"
A: Weight + temperature gradient + weak electro-lift → the same exponential law you'd expect in any stratified fluid.

Boilerplate Summary

The Panplannar Dome Framework reproduces sunsets, star trails, horizons, and pressure gradients with a graded-index dome, a twin-vortex aether, and a mild downward bias. Its equations are compact, its parameters tunable, and its predictions cheekily testable. If reality fails to comply, adjust $\gamma$, blame humidity, and declare victory.