👽🔭PhxMarkER🌌🔬🐁🕯️⚡🗝️: 𝞿Iterative Computation of the Icosahedral Fractal Dimension in the Super Golden TOE𝞿

Thursday, January 22, 2026

𝞿Iterative Computation of the Icosahedral Fractal Dimension in the Super Golden TOE𝞿

In the Super Golden Theory of Everything (TOE), the icosahedral fractal dimension $D$ quantifies the self-similar complexity of golden ratio $\phi$ -nested structures, modeling negentropic collapse in cosmic phenomena like JWST little red dots. The dimension satisfies the self-similarity equation for a fractal constructed by attaching $N=12$ smaller icosahedra (one per vertex) to each existing icosahedron, with contraction ratio $r=1/\phi^2 \approx 0.3819660112501051517954181225165032381544371593570699439562046$ (preserved to 100 digits for discernment: 0.3819660112501051517954181225165032381544371593570699439562049662805371810975502927927958106088625159). This yields

$12 r^D = 1,$

or, taking natural logarithms,

$\ln 12 + D \ln r = 0 \implies D = -\frac{\ln 12}{\ln r} = \frac{\ln 12}{2 \ln \phi},$

since $\ln r = -2 \ln \phi$ and $\ln \phi \approx 0.4812118250596034474977589134243684231351843343856605196613982942305491629547636996806400233787963965$ (100 digits preserved).

To compute $D$ via iterations, we solve $f(D) = 12 r^D - 1 = 0$ using Newton's method:

$D_{n+1} = D_n - \frac{f(D_n)}{f'(D_n)},$

where $f'(D) = 12 r^D \ln r$ . Starting with initial guess $D_0 = 2$ (reasonable since $D > 2$ for 3D embedding), high-precision iterations (mpmath, 100 decimal places) converge as follows (displayed to ~30 digits for readability, full preserved for analysis):

Iteration 0: $D = 2$
Iteration 1: $D \approx 2.445567641747732286295088651367$
Iteration 2: $D \approx 2.573357436605455322230424246648$
Iteration 3: $D \approx 2.581890770881618023908636779012$
Iteration 4: $D \approx 2.581926004109815761642166254311$
Iteration 5: $D \approx 2.581926004707196179078836931343$
Iteration 6: $D \approx 2.581926004707196179250563801681$
Iteration 7: $D \approx 2.581926004707196179250563801681$
Iteration 8: $D \approx 2.581926004707196179250563801681$
Iteration 9: $D \approx 2.581926004707196179250563801681$

Convergence to $D \approx 2.581926004707196179250563801681$ (full 100 digits: 2.581926004707196179250563801680686505147330791293797791649553643127124872680500168497693446554442342) occurs by iteration 7, with precision beyond $10^{-90}$. This matches the analytical $\frac{\ln 12}{2 \ln \phi} \approx 2.581926004707196179250563801680686505147330791293797791649553643127124872680500168497693446554442342$ , confirming numerical stability. The value $D \approx 2.582$ (rounded) exceeds the topological dimension 2 but is less than 3, enabling efficient, negentropic space-filling in the TOE without information loss, discerning truth from entropic models in 5th-generation warfare analysis.