Appendix AM - Structural Derivation of Coulomb's Law in PBG

1. Objective

Derive Coulomb’s inverse-square force law from coherence anchoring dynamics in the Phase-Biased Geometry (PBG) framework — no field theory, no tuning.

2. Modal Shell Structure

Electron Envelope

The electron is modeled as a coherence shell with an extended, smooth radial profile:

Proton Envelope

The proton is a tightly bound triplet modal shell:

Regularisation

Prevents divergence at the origin.

3. Phase Winding Structure

Each mode carries a radial phase:

• Proton: $$ \phi_p(r) = +k_p r, \quad k_p = \frac{1}{r_p} $$
• Electron: $$ \phi_e(r - R) = -k_e (r - R), \quad k_e = \frac{1}{r_e} $$

Phase difference:

This results in a dominant destructive interference, producing attraction.

4. Anchoring Cost Functional

The total anchoring cost between two offset shells is:

This represents the coherence distortion cost required to maintain mutual phase alignment at separation $$ R $$.

5. Anchoring Force

Force is the cost gradient, scaled by the mode’s anchoring stiffness:

6. Anchoring Gradient Constant

Derived structurally from the electron’s coherence energy:

With:

• $$m_e=9.11\times {10}^{-31}\text{ kg}$$
• $$c=3\times {10}^8\text{ m/s}$$
• $$a_0=5.29\times {10}^{-11}\text{ m}$$

So:

7. Numerical Evaluation

At $$ R = 1 , \text{\AA} $$:

• PBG Anchoring Force:

  $$F_{\text{PBG}}\approx -5.48\times {10}^{-7}\text{N}$$

• Classical Coulomb Force:

  $$F_{\text{Coulomb}}\approx 2.31\times {10}^{-10}\text{N}$$

• Ratio:

  $$\frac{F_{\text{PBG}}}{F_{\text{Coulomb}}}\approx -2377$$

8. Interpretation

• The force is attractive (correct sign),
• The slope is $$ \sim 1/R^2 $$ (correct structure),
• The magnitude is within 3 orders of the observed Coulomb force,
• No tuning required — only shell structure and phase winding.

9. Conclusion

Coulomb’s law arises in PBG as the cost gradient of modal coherence overlap,
with structural constants derived from electron modal energy and phase anchoring.

No charges, no fields — only coherence.

Appendix AL | [Index](./Appendix Master) | Appendix AK