New essay: lattice birefringence
A new essay, Birefringence in the A=1 lattice, is now on the essays page. It is a discovery write-up: the lattice’s directional dependence was not something we set out to find — it fell out of the rotational-invariance step while deriving the Dirac equation in Paper I.
The short version: the bipartite octahedral lattice does not propagate signals at the same speed in every direction. Preparing a referee-proof appendix for the Dirac limit surfaced that a load-bearing “frame condition” — used to argue the propagator has no preferred direction — is simply false for the actual basis vectors. Chasing that through the propagator eigenvalues turned a bug into a prediction: a single optical axis (1,1,-1), shared by the kinematic and gauge sectors, along which the dispersion is flat while oblique wave packets split in two. The honest replacement for the false claim is crystallographic — the lattice carries O_h symmetry at the operator level, with full rotational invariance recovered only on O_h-averaged observables.
This is an analytic result, not an experimental confirmation, and the essay keeps it there. Both birefringence rows in Paper I’s audit table — kinematic and gauge-sector — are marked STUB: derived symbolically, not yet tested numerically. The decisive open question is whether the per-tick \mathcal{A}=1 renormalisation washes the anisotropy out of the observable amplitude in the continuum limit. A planned companion paper, Balanced Equations and Birefringent Channels on the A=1 Lattice, is where that test would move the rows off STUB.
Pointers:
- Read the essay: Birefringence in the A=1 lattice
- Substrate paper: Paper I — Geometry First
- All essays: essays page