Birefringence in the A=1 lattice

A finding from the Dirac derivation

How directional dependence in the lattice’s propagation showed up as a side effect of deriving the Dirac equation, and what it implies for what we can predict and measure.

The result in one paragraph

The bipartite octahedral lattice does not propagate signals at the same speed in every direction. When you push the discrete update toward its continuum limit and ask it to look like the Dirac operator, the propagator’s energy eigenvalues turn out to depend on the direction of the wavevector, not just its magnitude — the lattice is birefringent, like a uniaxial crystal. There is a single special direction, the optical axis V_1+V_2+V_3 = (1,1,-1), along which the dispersion is flat to all orders; in the plane perpendicular to it, the leading quadratic term carries an anisotropic coefficient of 4/3. A wave packet launched oblique to that axis projects onto two propagator eigenmodes of equal magnitude but different phase, so they travel at different group velocities and the packet splits in two over distance. That splitting — one pulse becoming two — is the canonical signature of birefringence, and here the birefringent medium is spacetime itself. The effect is fixed by the lattice geometry alone, and it is sharp enough to be falsifiable.

How it came up

It was not something we went looking for. It fell out of the rotational -invariance step of the Dirac derivation in Paper I (§, the continuum-limit subsection §).

The original draft leaned on a clean-looking “frame condition,” \sum_{v\in\text{RGB}} v\,v^{\mathsf T} = 3\,\mathbb{I}, to argue that the discrete propagator has no preferred direction and so reduces to the isotropic Dirac operator. That identity is the load-bearing line for the \mathrm{SO}(3) rotational-invariance claim. It is also false for the actual basis vectors. A five-line SymPy check settles it immediately:

import sympy as sp
V1 = sp.Matrix([ 1,  1,  1])
V2 = sp.Matrix([ 1, -1, -1])
V3 = sp.Matrix([-1,  1, -1])
M  = sum(v * v.T for v in [V1, V2, V3])
print(M.tolist())              # [[3, -1, 1], [-1, 3, 1], [1, 1, 3]]
print(M.equals(3 * sp.eye(3))) # False

The off-diagonal entries are not zero. Preparing a referee-proof appendix for exactly this step is what surfaced the problem, and chasing it through the eigenvalues of the spinor propagator is what turned a bug into a prediction. The anisotropy is not an artifact we introduced to make the Dirac limit work — it is a side-condition the geometry imposes, and the honest continuum limit keeps it.

What it looks like

The math

The single-tick spinor propagator in Fourier space is

T(\mathbf{k}) = \begin{pmatrix} i\sin(\omega/2) & \cos(\omega/2)\,H_\text{RGB}(\mathbf{k}) \\ \cos(\omega/2)\,H_\text{CMY}(\mathbf{k}) & i\sin(\omega/2) \end{pmatrix}, \qquad H_\text{RGB}(\mathbf{k}) = \tfrac13\!\sum_{v\in\text{RGB}} e^{i\mathbf{k}\cdot v}, \quad H_\text{CMY} = \overline{H_\text{RGB}}.

Its trace is k-independent, \operatorname{tr}T = 2i\sin(\omega/2), and the eigenvalues are

\lambda_\pm(\mathbf{k}) = i\sin(\omega/2)\,\pm\,\cos(\omega/2)\,|H_\text{RGB}(\mathbf{k})|.

So the entire directional content lives in |H_\text{RGB}|^2. Expanding near \mathbf{k}=0:

|H_\text{RGB}(\mathbf{k})|^2 = 1 - \mathbf{k}^{\mathsf T} M_\text{eff}\,\mathbf{k} + O(k^4), \qquad M_\text{eff} = \tfrac19\begin{pmatrix} 8 & -4 & 4 \\ -4 & 8 & 4 \\ 4 & 4 & 8 \end{pmatrix}.

The eigenvalues of M_\text{eff} are \{\,4/3,\ 4/3,\ 0\,\}. The zero eigenvalue points along (1,1,-1). That is the optical axis, and the flat direction is not just a leading-order accident: set \mathbf{k}=s\,(1,1,-1)/\sqrt3 and all three dot products \mathbf{k}\cdot V_i become equal, so |H_\text{RGB}|^2 = 1 exactly, at all orders in k. Along the axis the two eigenmodes never separate; off the axis they do.

There is a second, dual copy of the same structure. The gauge sector — the photon kinetic term induced by the bipartite plaquette sum — produces its own quadratic form \mathbf{Q} on \mathbf{B}-space with eigenvalues \{4,4,16\}. Same optical axis (1,1,-1), opposite shape: where the kinematic ellipsoid is prolate (stretched along the axis, dispersion flat there), the gauge ellipsoid is oblate (squashed along the axis, kinetic term steepest there). One axis, two anisotropies, two independent falsifiable signatures.

The picture

Dual anisotropy on a single optical axis. Both surfaces are level sets x^{\mathsf T}Ax = 3. Left (kinematic sector): the frame matrix M with eigenvalues \{4,4,1\} — a prolate ellipsoid whose long axis is the optical axis (1,1,-1); kinematic dispersion is flat along that axis and steep perpendicular to it. Right (gauge sector): the plaquette tensor \mathbf{Q} with eigenvalues \{4,4,16\} — an oblate ellipsoid whose short axis is the same (1,1,-1); the induced photon kinetic term is steepest along the axis. Both shapes inherit the special direction from the same RGB/CMY bipartite geometry.

The two ellipsoids are duals of each other on a shared axis. That shared axis is the whole content of the prediction: if the lattice is real, the kinematic and gauge sectors must point the same way.

Why it matters

It supports two audit rows — but only as far as the audit honestly goes. In Paper I’s audit table, Lattice birefringence (kinematic) (M_\text{eff} eigenvalues \{4/3,4/3,0\}) and Gauge-sector birefringence (\mathbf{Q} eigenvalues \{4,4,16\}) are both marked STUB: derived analytically, not yet tested numerically. This essay is a write-up of an analytic result, not an experimental confirmation, and the language is kept at that altitude deliberately. Both rows are slated for a planned companion paper — Balanced Equations and Birefringent Channels on the A=1 Lattice — which recasts the lattice dynamics as site-level balance equations and develops the P9 prediction as a bipartite split of the probability gradient \nabla\rho into ordinary and extraordinary modes along the optical axis (1,1,-1). That is the paper that will move these rows off STUB; this essay is the geometric preview of its central object.

The decisive open question is whether the anisotropy survives. The lattice’s \mathcal{A}=1 conservation law is enforced by a per-tick renormalisation (enforce_unity_spinor) that sits between consecutive applications of the propagator. The bare propagator is birefringent — that is settled symbolically. What is not settled is whether that renormalisation step washes the anisotropy out of the observable amplitude in the continuum limit. If it does, birefringence is a property of the bare operator but not of anything you can measure, and the prediction has to be qualified. If it does not, the prediction stands as written. That is the difference between “discreteness artifact” and “real, falsifiable signature,” and it is exactly the test that has not been run yet.

It replaces a weaker, false claim with a stronger, true one. The paper used to assert \mathrm{SO}(3) Lorentz invariance was forced by the frame condition. That was wrong as written. The honest replacement is crystallographic: the lattice has O_h symmetry, not \mathrm{SO}(3), at the operator level, and full rotational invariance is recovered only on O_h-averaged observables. Trading a false isotropy claim for a true anisotropy prediction is a net gain — most discrete-spacetime proposals would happily take a sharp falsifiable handle in exchange for an over-strong symmetry claim.

Downstream subprojects inherit the axis. The same (1,1,-1) direction shows up in the P7 photon-dispersion refinement (direction-resolved GRB time-of-flight), the P9 multi-channel concordance prediction (kinematic, gauge, gravitational, and atomic channels all sharing one axis), the gauge-derivation companion paper, and the gravity sector, whose 1/r^2 recovery picks up direction-dependent discrete corrections along the same axis. If the lattice is real, all of these have to agree on which way (1,1,-1) points in the sky.

Open / next

• Is the magnitude fixed by the substrate alone, or does the conservation law correct it? The bare-propagator coefficient is 4/3 in the perpendicular plane, fixed by geometry. Whether the \mathcal{A}=1 renormalisation rescales, cancels, or preserves it in the observable dispersion is unknown. This is the same enforce_unity_spinor question as above, stated as a magnitude rather than a yes/no.

• What would distinguish lattice birefringence from ordinary material birefringence? Two handles. First, universality of the axis: a real crystal’s optical axis is set by the crystal; the lattice predicts a single cosmic axis shared across photons, gravity, and atomic-clock anisotropies (P9). Second, scaling: the splitting goes as (a/\lambda)^2, so it grows with photon energy — gamma-ray and TeV photons over cosmological baselines (Fermi GBM, IceCube) are the sensitive channel, not a laboratory laser. The cheapest existing test is to re-bin a public GRB time-of-flight catalog by sky direction relative to a candidate axis: bursts near the axis should disperse less than perpendicular ones. The current bound a \lesssim 10^{-19}\,\text{m} assumes isotropy and so has averaged any such signal away.

• Is there an exp_NN a reader could run? Not yet — and that is the honest gap. The decisive numerical test is small and well-specified: propagate a free wave packet (\omega \approx 0.1019) along (1,1,-1) and along a perpendicular direction on a \sim 33^3 grid for ~1000 ticks, and compare group velocities. Equal velocities ⇒ renormalisation restored isotropy; different velocities ⇒ birefringence is observable. That run is what would move both audit rows off STUB, and it is the first piece of the planned Balanced Equations and Birefringent Channels companion paper, where the ordinary/extraordinary split along (1,1,-1) is the central diagram.

References

• Paper I — Geometry First: Quantum Mechanics, Gravity, and the Origin of the Standard Model from a Single Conservation Law. The birefringence result lives in the kinematics section (continuum / Dirac-limit subsection, M_\text{eff} and its eigenvalues), its P7 direction -resolved refinement and P9 optical-axis concordance in the predictions section, and the gauge-sector dual (\mathbf{Q}, eigenvalues \{4,4,16\}) in the induced-gauge-action appendix. See the Paper I landing page for the abstract, current version, and the canonical Zenodo deposit.
• Working notes (derivation and falsification handles): notes/lattice_birefringence_prediction.md and notes/frame_condition_isotropy_memo.md — the eigenvalue computation that surfaced the optical axis and the memo that traced the false frame condition through the paper.
• Planned companion paper — Balanced Equations and Birefringent Channels on the A=1 Lattice (follow-on seed #14). Recasts the lattice as site-level balance equations and develops the kinematic and gauge-sector birefringence rows (currently STUB) into the P9 ordinary/extraordinary split along (1,1,-1). In preparation; not yet released.