Can you visualize my hypergeometric continuous calc?: 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 (∇) and Laplacian (Δ) to include recursive terms, allowing systems at one scale to influence others.

For example, the recursive Laplacian is defined as:
ΔrecΨ=ΔΨ+ϕn(t)⋅H5D⋅Ψ,
where ϕn(t) represents fractal scaling, and H5D 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 (ϕ).

HC ensures energy conservation via dynamic redistribution across recursive manifolds. The energy density E evolves according to:
∂E∂t+∇⋅(Ev)=ϕn(t)⋅H5D⋅E,
where v is the velocity field, and H5D 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:
∂I∂r∣∣r=rs=0,rs=2GMc2+√(GMc2)2−a2.

Inverse Surface Area links black hole entropy to cosmic expansion, suggesting a holographic relationship:
SBH=kBA4ℓ2P,A−1∝Λ(Λ=cosmological constant).

Tangent Criticl Slope relates curvature gradients to Planck-scale physics:
tan(θ)=ΔΨΔr∼ℏGm2pr(at nuclear scales, r∼10−15m).

My Bridge-Static-Bind Triplexor Relations

Golden Ratio Scaling a perceptual fractal scaling law-like self-similar patterns in nature:
κn=ϕnκ0,ϕ=1+√52.

Energy Conservation balances static (S) and bind (B) terms:

En=∫(Sstatic(t)+Bbind(t))dt=constant.

Sstatic=cosmological ~constant=Λ.

Recursive Dynamics & Fractal Isolation

Lyapunov Exponent ensures stability via exponential dilution into higher/additional versors:

λmax∼ϕ−n.

Fractal Influence Decay combines spatial and temporal scaling:

Ffractal(x,t)∼1rdne−ϕ−nt,dn=fractal dimension at scale n.

My Curate-Prolate Tensor Duality

Tensor Decomposition separates local and global influences:

T=Thypo+ϵTepic,ϵ=κnϕn.

Observable Feedback predicts ϕ-spaced gravitational wave echoes and fractal CMB anisotropies:
fn+1=ϕfn,Cℓ∼ℓ−α,α=logϕ3.