Appendix S — Derivation 19: Chiral Anchoring and Parity Violation

Overview

In the Standard Model, weak interactions violate parity: they act only on left-handed fermions. This fundamental asymmetry is imposed by construction, through chiral couplings in the Lagrangian.

In modal dynamics, chirality emerges from structural anchoring.
There is no imposed left–right asymmetry; rather, parity violation arises from the coherence field’s asymmetric response to internal phase handedness.

This appendix derives:

The origin of left–right structural asymmetry
Why certain modes anchor preferentially with one handedness
How parity violation and helicity selection emerge from first principles

Each mode is defined by a coherence function:

ψ (x, t) = ρ (x, t) e^{i ϕ (x, t)}

The chirality of a mode refers to the handedness of its internal phase progression. In three dimensions, this is a geometric property of the phase surface:

Left-chiral modes: $\nabla \times \nabla ϕ > 0$ (positive twist)
Right-chiral modes: $\nabla \times \nabla ϕ < 0$ (negative twist)

These phase windings determine how a mode interacts with the coherence field $B (x)$ .

2. Anchoring Cost Asymmetry

The total anchoring cost of a mode is:

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

For modes with chiral structure, the spatial term becomes sensitive to handedness:

| \nabla ψ |^{2} = | \nabla ρ + i ρ \nabla ϕ |^{2}

Cross terms of the form:

Im (\nabla ρ \cdot \nabla ϕ^{*}) \sim ρ \cdot (\nabla \times \nabla ϕ)

may be nonzero depending on the coherence field geometry.

In a structured field $B (x)$ with directional bias (e.g. early universe coherence gradient), this term produces a cost asymmetry between left- and right-chiral modes.

3. Spontaneous Parity Violation

If:

C_{left} [ψ_{L}] < C_{right} [ψ_{R}]

then only left-chiral modes stably anchor in that environment.

This leads to:

Structural parity violation
Chiral selection in interactions
Suppression or exclusion of one handedness

This is not imposed by symmetry breaking—it is the natural coherence response to anchoring cost gradients.

4. Weak Interaction Coupling

In Appendix I, we showed that weak-like modal transitions occur via structural decay or re-anchoring.

Chiral anchoring now adds:

A filter that prevents right-chiral modes from participating
A bias that drives transitions through left-handed phase channels only

This reproduces the observed weak interaction asymmetry:

Only left-handed neutrinos couple
Helicity correlations emerge naturally from structural anchoring alignment

5. Helicity and Propagation

For a moving latent mode (e.g. a neutrino), chirality aligns with helicity (spin direction relative to motion) when anchoring response is dominant.

Modes whose chirality conflicts with coherence gradients experience:

Increased decoherence cost
Higher instability
Exclusion from structural transitions

This explains:

The absence of right-handed neutrinos
Helicity selection in beta decay
Directional asymmetries in mode evolution

6. Cosmic Origin of Chiral Bias

In early modal environments with strong coherence gradients:

A global $B (x)$ field with net orientation naturally arises
Anchoring costs are structurally asymmetric from the start
This locks in a universal chiral preference without symmetry breaking

Thus, the universe's parity asymmetry is fossilised coherence bias.

Conclusion

Chirality is not an imposed quantum label.
It is the handedness of phase alignment in a structured coherence field.
Parity violation arises because anchoring cost gradients respond asymmetrically to internal twist.

Appendix R | [Index](./Appendix Master) | Appendix T