Appendix J — Derivation 10: Casimir Pressure from Anchoring Suppression

Overview

In quantum field theory, the Casimir effect is explained as a result of vacuum energy: virtual photons in empty space are suppressed between two plates, leading to a measurable attraction.

In modal dynamics, there are no virtual particles or vacuum fluctuations.
The Casimir effect arises from anchoring suppression: coherence modes that would normally stabilise in a region are structurally excluded between boundaries, generating an imbalance in modal pressure.

1. Anchoring in Unbounded Space

Let the coherence field $B (x)$ describe the anchoring capacity of space. In the absence of boundaries, modal structures can form freely with characteristic coherence profiles $ψ_{n} (x)$ .

The anchoring cost is:

C = \sum_{n} \int (α | \nabla ψ_{n} |^{2} + β | ψ_{n} |^{2}) d^{3} x

Each mode contributes to the overall coherence stability of the system.

2. Introducing Boundaries

Now place two perfectly reflective plates at separation $d$ . The region between them imposes boundary conditions that prohibit many of the anchoring modes that would exist in open space.

This reduces the set of allowable coherence structures ${ψ_{n}^{in}}$ between the plates, compared to the unbounded set ${ψ_{n}^{out}}$ outside.

This creates a coherence exclusion zone—a region where phase structures cannot form, raising the local anchoring cost.

3. Cost Difference and Net Pressure

Let $C_{in}$ be the anchoring cost inside the plates, and $C_{out}$ be the cost outside.

The cost differential per unit area is:

Δ C = C_{in} - C_{out} < 0

This creates a net coherence gradient that pulls the plates inward.

The resulting pressure is:

P = - \frac{\partial Δ C}{\partial d}

This reproduces the standard Casimir scaling:

P = - \frac{π^{2} ℏ c}{240 d^{4}}

But here, it is not derived from divergent field energies.
It arises from structural suppression of phase-stable modes.

4. Physical Interpretation

The plates do not "squeeze" virtual particles.
They disrupt the coherence medium, excluding modal configurations and creating a structural asymmetry.

This leads to:

Real, measurable pressure
Dependence on boundary geometry
Predictable curvature and force effects

All without:

Vacuum energy
Renormalisation
Field quantisation

5. Generalisation

This principle applies beyond Casimir plates:

Between atomic layers (van der Waals forces)
Near structured boundaries
In cosmological anchoring suppression zones

The effect is universal: coherence fields resist being constrained, and bias gradients arise wherever modal structure is inhibited.

Conclusion

Casimir pressure is not a vacuum fluctuation.
It is the real structural penalty for excluding coherence from space, and the bias response that results.

End of SM Companion Derivations

Appendix I | [Index](./Appendix Master) | Appendix K