Appendix S — Chiral Anchoring and Parity Violation

(concept sketch)

1 Phase chirality of a mode

For
$\psi =\rho e^{i\varphi }$

• Left-chiral: $(\mathrm{\nabla }\times \mathrm{\nabla }\varphi )>0$
• Right-chiral: $(\mathrm{\nabla }\times \mathrm{\nabla }\varphi )<0$

The sign of this internal twist is the mode’s handedness.

2 Anchoring cost asymmetry

Spatial term of the locked energy density (Foundations §1):

The cross-term
$2\rho \mathrm{\nabla }\rho \cdot \mathrm{\nabla }\varphi \propto {\rho }^2(\mathrm{\nabla }\times \mathrm{\nabla }\varphi )$
changes sign with chirality in a background coherence gradient
$\mathrm{\nabla }\rho \ne 0$.
Thus
${\mathcal{C}}_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}\ne {\mathcal{C}}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$.

3 Structural parity violation

If ${\mathcal{C}}_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}<{\mathcal{C}}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$, only left-chiral modes
anchor stably:

• right-handed envelopes decohere via the calibrated sink
  ${\mathrm{\Gamma }}_{\mathrm{d}\mathrm{e}\mathrm{c}}$ (Appendix F).
• interactions filter to left-handed channels ― reproducing weak V–A
  selection without an explicit chiral gauge field.

4 Implications

Quantifying the bias requires a single overlap integral; notebook
chiral-asymmetry.ipynb (in prep) will supply δ and error bars.

5 Numeric anchor for the chiral bias

We fix the single dimension-less bias parameter δ by matching the
measured left–right asymmetry in neutron β–decay.

• Experimental electron asymmetry (PDG 2024)

  $$A_e^{\mathrm{o}\mathrm{b}\mathrm{s}}=-0.1184\pm 0.0004.$$

• PBG model: the relative anchoring probability difference for
  left- vs right-chiral modes emitted in a weak decay is

  $$A_e^{\mathrm{P}\mathrm{B}\mathrm{G}}=\frac{P_L-P_R}{P_L+P_R}=-2\delta .$$

Setting $A_e^{\mathrm{P}\mathrm{B}\mathrm{G}}=A_e^{\mathrm{o}\mathrm{b}\mathrm{s}}$ gives

This calibrated value is now used wherever the interference term

appears (see Gauge‐Symmetry appendix and weak-decay notes). No other
free parameters are introduced.

*Update log (2025-06-10): inserted δ anchor; Appendix S is now fully
numerical.

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