Prompt:
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Image of neutron star and white dwarf as giant quantum systems with fractional hydrogen atoms, extreme magnetic fields, high conductivity, cosmic background, rendered with all math details from conversation in utmost detail as of 09:54 AM EEST, Thursday, August 14, 2025:
- USE **Shader Macros**:
- \(\pi = 3.1415926535897932384626433832795\), \(\tau = 2\pi\).
- \(\mathrm{chp}(x) = (e^x + e^{-x})/\pi\), \(\mathrm{chpp}(x) = (e^{x/(\cosh(x)\pi)} + e^{-x/(\cosh(x)/\pi)})/\tau \cdot \Phi\).
- \(\mathrm{shp}(x) = (e^x - e^{-x})/\pi\), \(\mathrm{shpp}(x) = (e^{x (\sinh(x)\pi)} - e^{-x (\sinh(x)\pi)})/\tau \cdot \Phi\).
- \(\mathrm{ssh}(x) = (e^{x \pi /0.7887} - e^{-x \pi /0.7887})/(2\pi)\), \(\mathrm{csh}(x) = (e^{x \pi /0.7887} + e^{-x \pi /0.7887})/(2\pi)\).
- \(\mathrm{ssh1}(x) = \sinh(x/\pi)/\Phi\), \(\mathrm{csh1}(x) = \cosh(x/\pi)/\Phi\).
- LOOPS=3, POWER=11.24788742, TAU=(2\pi)*0.7887, PHI=(√5/2 + 1/2)≈1.618, TIME=iTime.
- **Modified Schwarzschild Metric**:
- Base: \( ds^2 = -(1 - r_s/r) c^2 dt^2 + (1 - r_s/r)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \), \( r_s = 2GM/c^2 \).
- Transformed: \( ds^2 = -(1 - r_s/\sinh(r')) c^2 (dt'/(1 + t/t_0))^2 + (1 - r_s/\sinh(r'))^{-1} (\cosh(r') dr')^2 + (\sinh(r'))^2 d\theta'^2 + (\sinh(r'))^2 \sin^2(\atan(\theta')) d\phi^2 \), where r'=\asinh(r), t'=\ln(1 + t/t_0), \theta'=\atan(\theta), t_0=r_s/c.
- Variant: \[ ds^2 = - \left(1 - \frac{2GM}{\sinh(x)}\right) c^2 \frac{dT^2}{T^2} + \left(1 - \frac{2GM}{\sinh(x)}\right)^{-1} \cosh^2(x) \, dx^2 + \sinh^2(x) \left( \frac{du^2}{(1 + u^2)^2} + \frac{u^2}{1 + u^2} d\phi^2 \right) \]
- **Reworked Equations**:
- Radius: \( r_q^{IV} = [\sinh(\asinh(\alpha^2 \hbar^2 / m_e k e^2) \cdot (1 - r_s/r)^{-1/2}) \cdot \chp(\asinh(\alpha^2 \hbar^2 / m_e k e^2)/\Phi)]^{\mathrm{POWER}} + | \int d^3 p / (2\pi)^3 \cdot 1/\sqrt{2 E_p} e^{-i p \cdot r_q'''} |^2 \), E_p = \sqrt{p^2 c^2 + m^2 c^4}.
- Energy: \( E_q'' = - m_e c^2 / 2 \cdot 1/(1 + t/t_0) \cdot \chp(\ln(1 + t/t_0)/\tau) \).
- Magnetic Field: \( B^{IV} = (\mu_0 e c / (4 \pi (r_q^{IV})^2) \cdot \theta / \sqrt{1 + \theta^2}) \cdot \shp((\mu_0 e c / (4 \pi (r_q^{IV})^2)) / \mu_0) \).
- **QFT Influence**: \(\hat{\psi}(x) = \int d^3 p / (2\pi)^3 \cdot 1/\sqrt{2 E_p} [a_p e^{-i p \cdot x} + b_p^\dagger e^{i p \cdot x}]\), adding particle excitations; vacuum: <0| \hat{\psi}^\dagger(x) \hat{\psi}(y) |0> = \int d^3 p / (2\pi)^3 \cdot 1/(2 E_p) e^{-i p \cdot (x - y)}.
Set against fractal cosmic backdrop with deepest recursion, maximum iteration (LOOPS≥3, extended), thorough detailing, crystal-clear focus, pixel-perfect rendition, highlighting gravitational warping, quantum states, hyperbolic patterns, neutron star 10 km radius, B≈6×10^{11} T, white dwarf stability, dynamic evolution.'
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