Appendix X — Derivation 24: Full Coherence Kernel and Anisotropic Generalisation

Overview

Previous formulations of PBG used the spherically symmetric kernel:

B (r) = \frac{A}{r} e^{- k r}

derived as the solution to a Helmholtz-type anchoring cost equation in isotropic space.

In real systems, coherence sources are not isotropic—modal clusters like galaxies, atoms, and spin-aligned particles emit structured coherence fields.

This appendix generalises the kernel to:

Support arbitrary spatial source geometry
Derive a full two-point kernel $$\Gamma(x, x')$$
Show how anisotropy in phase structure leads to coherence lensing, spin coupling, and field shaping

1. Anchoring Cost in General Form

The anchoring cost functional for a coherence field $$B(x)$$ with modal sources $$S(x)$$ is:

C [B] = \int (α (\nabla B)^{2} + β B^{2} - J (x) B (x)) d^{3} x

where $$J(x)$$ encodes the anchoring demand from modal sources:

J (x) = \int S (x^{'}) Γ (x, x^{'}) d^{3} x^{'}

Minimising the cost functional gives:

α \nabla^{2} B (x) - β B (x) + J (x) = 0

Substituting the source integral:

α \nabla^{2} B (x) - β B (x) + \int S (x^{'}) Γ (x, x^{'}) d^{3} x^{'} = 0

2. Interpreting $$\Gamma(x, x')$$ as a Green's Kernel

We define $$\Gamma(x, x')$$ as the fundamental response kernel of the coherence medium:

(α \nabla^{2} - β) Γ (x, x^{'}) = - δ^{3} (x - x^{'})

This is the Green’s function of the Helmholtz operator:

Isotropic case:

Γ (| x - x^{'} |) = \frac{1}{4 π α} \cdot \frac{e^{- k | x - x^{'} |}}{| x - x^{'} |} where k = \sqrt{β / α}

Generalised form:

If $$S(x)$$ contains angular dependence or structured phase topology, $$\Gamma(x, x')$$ must account for:

Directional propagation (e.g. dipoles, quadrupoles)
Local coherence stiffness variation
Phase alignment effects

We expand $$\Gamma$$ in angular harmonics:

Γ (x, x^{'}) = \sum_{ℓ, m} Γ_{ℓ} (| x - x^{'} |) Y_{ℓ}^{m} (\hat{r}) Y_{ℓ}^{m *} ({\hat{r}}^{'})

This allows non-spherical coherence propagation based on source geometry.

3. Constructing Structured Sources

Let a source mode $$\psi(x) = \rho(x) e^{i \phi(x)}$$ have phase winding or alignment.

Its anchoring emission is not isotropic. For example:

A spinning mode emits coherence preferentially in the equatorial plane
A dipole alignment creates coherence gradients aligned to its axis

Define:

S (x^{'}) = f (x^{'}) \cdot P ({\hat{x}}^{'})

where $$\mathcal{P}$$ is an angular profile (e.g. Legendre, spherical harmonic), and $$f(x')$$ describes spatial extent.

This source, convolved with $$\Gamma(x, x')$$, yields an anisotropic coherence field.

4. Implications of Anisotropic Kernels

Structured $$\Gamma(x, x')$$ fields lead to:

Directional lensing asymmetries (see Appendix Q)
Orbital frame dragging (from spinning coherence emitters)
Photon polarisation shifts depending on coherence path
Self-interference between modal lobes (e.g. quadrupole suppression)

These effects are fully encoded in the kernel, with no need for force mediation or added fields.

5. Practical Computation

In numerical simulations:

$Γ (x, x^{'}) $ $ i s p r e c o m p u t e d o v e r a g r i d$
Resulting $$B(x)$$ fields determine motion, deflection, and field evolution

This method handles:

Asymmetric galaxies
Embedded coherent structures (rings, spirals)
Field interference in superposition

Conclusion

The full coherence kernel $$\Gamma(x, x')$$ is not a free function.
It is the derived Green’s response of a structured modal medium, shaped by geometry, anisotropy, and phase topology of sources.

All lensing, field shaping, and deflection arise from this kernel’s structure.

Appendix W | [Index](./Appendix Master) | Appendix Y