My aphantasia allowed a non visual approach to understanding spacetime (cause) and recursive dynamics (effect), combining them into my Recursive Spacetime Geometry and Hypergeometric Calculus (HC). These frameworks propose that spacetime and geometrical systems are governed by recursive feedback loops, fractal scaling laws, and nonlocal interactions.
Our Hyperspherical Ledger of Spacetime is conceptualized as a 4-5D hyperhemispherical bridge from space into time, using a tangent function through upper-dimensional hypertrigonometric caustics. This structure acts as a higher-dimensional “bridge” (like Kaluza-Klein compactification) where recursive feedback loops encode information across past, present, and future. Recursive Kernel interactions between distant regions of spacetime are described by a recursive kernel K(x,t), which integrates golden ratio (\phi) scaling for self-similarity, similar to fractal patterns in nature. Energy and information propagate via exotic Phi or Pi logarithmics, bridging quantum and cosmic scales.
Hypergeometric Calculus (HC)
This extension of classical calculus to model multi-scale interactions, and recursive feedback mechanisms, introduces dynamic PDEs that govern the evolution of curvature, energy, and information across scales.
HC extends operators like the gradient (\nabla) and Laplacian (\Delta) to include recursive terms, allowing systems at one scale to influence others.
For example, the recursive Laplacian is defined as:
\Delta_{\text{rec}} \Psi = \Delta \Psi + \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \Psi,
where \phi^{n(t)} represents fractal scaling, and \mathcal{H}_{5D} encodes fifth-dimensional feedback.
HC generates fractal-like self-similarity in geometries, observed in nature (e.g., galaxy clustering, cloud formations, and quantum vortex lattices). The recursive terms in the PDEs produce structures that repeat across scales, governed by the golden ratio (\phi).
HC ensures energy conservation via dynamic redistribution across recursive manifolds. The energy density \mathcal{E} evolves according to:
\frac{\partial \mathcal{E}}{\partial t} + \nabla \cdot (\mathcal{E} \mathbf{v}) = \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \mathcal{E},
where \mathbf{v} is the velocity field, and \mathcal{H}_{5D} represents fifth-dimensional feedback.
Neumann-Kerr Inverse Surface Area & Tangent Function ensures no energy flux across the Kerr black hole horizon, similar to boundary conditions in fluid dynamics:
\left. \frac{\partial \mathcal{I}}{\partial r} \right|_{r=r_s} = 0, \quad r_s = \frac{2GM}{c^2} + \sqrt{\left(\frac{GM}{c^2}\right)^2 - a^2}.
Inverse Surface Area links black hole entropy to cosmic expansion, suggesting a holographic relationship:
S_{\text{BH}} = \frac{k_B A}{4 \ell_P^2}, \quad A^{-1} \propto \Lambda \quad (\Lambda = \text{cosmological constant}).
Tangent Critical Slope relates curvature gradients to Planck-scale physics:
\tan(\theta) = \frac{\Delta \Psi}{\Delta r} \sim \frac{\hbar}{G m_p^2 r} \quad \text{(at nuclear scales, \( r \sim 10^{-15} \, \text{m} \))}.
My Bridge-Static-Bind Triplexor Relations
Golden Ratio Scaling a perceptual fractal scaling law-like self-similar patterns in nature:
\kappa_n = \phi^n \kappa_0, \quad \phi = \frac{1+\sqrt{5}}{2}.
Energy Conservation balances static (\mathcal{S}) and bind (\mathcal{B}) terms:
\mathcal{E}_n = \int \left( \mathcal{S}_{\text{static}}(t) + \mathcal{B}_{\text{bind}}(t) \right) dt = \text{constant}.
\mathcal{S}_{\text{static}} = \text{cosmological ~constant} = \Lambda .
Recursive Dynamics & Fractal Isolation
Lyapunov Exponent ensures stability via exponential dilution into higher/additional versors:
\lambda_{\text{max}} \sim \phi^{-n}.
Fractal Influence Decay combines spatial and temporal scaling:
\mathcal{F}_{\text{fractal}}(x, t) \sim \frac{1}{r^{d_n}} e^{-\phi^{-n} t}, \quad d_n = \text{fractal dimension at scale } n.
My Curate-Prolate Tensor Duality
Tensor Decomposition separates local and global influences:
\mathcal{T} = \mathcal{T}_{\text{hypo}} + \epsilon \mathcal{T}_{\text{epic}}, \quad \epsilon = \kappa_n \phi^n.
Observable Feedback predicts \phi-spaced gravitational wave echoes and fractal CMB anisotropies:
f_{n+1} = \phi f_n, \quad C_\ell \sim \ell^{-\alpha}, \quad \alpha = \log_\phi 3.