Appendix A Modal Evolution from Anchoring Cost

Overview

This derivation shows how the time evolution of a mode $ψ (x, t)$ arises from a coherence cost minimisation principle. The result generalises the Schrödinger equation and replaces it with a structurally grounded modal evolution law.

We begin with the anchoring cost functional and derive the governing dynamics via variational methods.

1. The Anchoring Cost Functional

A mode $ψ (x, t)$ is a complex-valued coherence function with amplitude and phase:

ψ (x, t) = ρ (x, t) e^{i ϕ (x, t)}

Its evolution is governed by an anchoring cost $C [ψ]$ , reflecting the tension imposed by spatial and temporal coherence strain:

C [ψ] = \int (γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{3} x

$γ$ penalises temporal phase distortion (temporal stiffness)
$α$ penalises spatial phase gradients (spatial stiffness)
$β$ penalises unanchored coherence density

This functional represents the instantaneous anchoring stress. To find stable or persistent evolution, we integrate across time to form an action:

S [ψ] = \int C [ψ] d t

2. Variational Principle

To determine the natural evolution of $ψ (x, t)$ , we require that the action is stationary under small variations:

δ S [ψ] = 0

That is, the real modal evolution follows the path of least coherence disruption across spacetime.

3. Functional Derivative

We compute the Euler–Lagrange equation for a complex field. Let $L$ be the Lagrangian density:

L = γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}

We take functional derivatives with respect to $ψ^{*}$ :

[
\frac{\delta \mathcal{S}}{\delta \psi^*} =
-\gamma \frac{\partial^2 \psi}{\partial t^2}

\alpha \nabla^2 \psi

\beta \psi
= 0
]

This yields the modal evolution equation:

γ \frac{\partial^{2} ψ}{\partial t^{2}} = α \nabla^{2} ψ - β ψ

4. Interpretation

This is a second-order in time, linear differential equation—a wave-like modal evolution law.

It replaces the Schrödinger equation and reflects the principle that:

Modal structure evolves to preserve phase alignment under internal and external anchoring pressure.

Key features:

Time evolution is structural, not probabilistic.
Spatial propagation reflects coherence diffusion under phase tension.
Modal inertia emerges from the $γ$ term: large $γ$ means slow response to temporal change—i.e., mass.

5. Limiting Case: Schrödinger Form

In the slowly varying regime (i.e., $ψ$ oscillates rapidly but changes slowly in time), assume:

ψ (x, t) = ϕ (x, t) e^{- i ω t}

Plug into the second-order equation, discard $\partial^{2} ϕ / \partial t^{2}$ terms:

- 2 i ω γ \frac{\partial ϕ}{\partial t} = α \nabla^{2} ϕ - β ϕ

Divide by $2 ω γ$ :

i \frac{\partial ϕ}{\partial t} = - \frac{α}{2 γ ω} \nabla^{2} ϕ + \frac{β}{2 γ ω} ϕ

This has the form of a Schrödinger equation with:

Effective mass $\propto γ ω / α$
Potential energy term $\propto β$

Hence, Schrödinger dynamics arise as an approximate regime of the full modal evolution equation.

Conclusion

We have derived modal time evolution from first principles using only the anchoring cost structure. This framework:

Eliminates probabilistic wavefunctions
Derives mass and motion from structural resistance
Allows for full field-free dynamics

The next derivation builds on this by analysing the structure and origin of the coherence field $B (x)$ itself.

[Index](./Appendix Master) | Appendix B

Appendix A — Derivation 1: Modal Evolution from Anchoring Cost