Appendix AD — Derivation 29: Chiral Anchoring and Structural Parity Asymmetry

Appendix AD — Derivation 29

Chiral Anchoring & Structural Parity Asymmetry

Phase-Biased Geometry (PBG) has no imposed chiral couplings, yet weak-like parity violation emerges naturally.
The key is a handed phase winding interacting with a directional coherence gradient.

1 Phase–winding modes

A helical mode is written

ψ (x) = ρ (x) \exp [i (k \cdot x + θ_{\pm} (x))],

where

$θ_{+}$ = left-handed twist
$θ_{-}$ = right-handed twist

The anchoring field obeys the static Helmholtz result
$α \nabla^{2} B - β B = ρ_{anchor}$
so $\nabla B$ generally points away from dense regions.

2 Directional anchoring tension

Define the directional anchoring energy

T_{\pm} = \int ρ^{2} (\nabla θ_{\pm} \cdot \nabla B) d^{3} x .

If $\nabla\theta_\pm $aligns with $\nabla B$
→ $T_{\pm} < 0$ → favoured anchoring.
If it opposes $\nabla B$
→ $T_{\pm} > 0$ → penalised anchoring.

Because $θ_{+}$ and $θ_{-}$ have opposite twists,
they generally yield different $T$ .

3 Anchoring cost asymmetry

Total energies

C_{\pm} = C_{0} + T_{\pm}, Δ C = C_{+} - C_{-} .

If $Δ C < 0$ → left-helical modes prevail.
If $Δ C > 0$ → right-helical modes prevail.

In the early-universe coherence gradient (Appendix R) one finds
$| \nabla B | \neq 0$ and typically
$T_{+} < T_{-}$ → left-handed bias.

4 Physical consequences

PBG statement	Observed SM analogue
Only helical modes with minimal anchoring cost enter coherence-driven interactions	Left-handed fermions couple to the weak force
Opposite-helicity modes decohere rapidly	Absence of right-handed neutrinos
Bias strength tracks $\nabla B$ (denser eras ⇒ stronger parity violation)	Electroweak scale sets size of SM asymmetry

5 Observables & tests

CMB polarisation: small TE-correlation chirality predicted from residual large-scale $\nabla B$ .
Laboratory chiral media: tailored coherence gradients should favour one helical photon mode over the other → test with anisotropic cavities.
Neutrino experiments: slight environment-dependent helicity drift expected if $| \nabla B |$ can be modulated (dense matter vs. vacuum).

6 Summary

Parity violation in PBG is geometric:
a phase-winding’s handedness couples unequally to directional coherence gradients, producing a built-in chiral bias without axial gauge terms.

Appendix AC - Coherence Class Chi | [Index](./Appendix Master) | Appendix AE - Continuum Mechanics