A Lorentz shield for discrete space
Today the preprint A Lorentz Shield for Discrete Space went on Zenodo: doi:10.5281/zenodo.23754164.
The problem. Any lattice picks out preferred directions and a preferred frame. Collins et al. (2004) showed that quantum corrections drag such Planck-scale violations of Lorentz symmetry down to everyday energies without the suppression one would expect. Observations of gamma-ray bursts see no such effect. So a discrete space needs a shield.
The picture. Think of a frozen lake with holes in the ice. The water moves freely underneath; from above you see it only in the holes. In the model the "water" is a field that propagates in a continuous bulk with one extra dimension (P1); the "ice" is our three-dimensional lattice — the tetrahedral–octahedral honeycomb (TOT) — and the "holes" are the centres of its triangular faces, which we call apertures (P2). Every field couples to the lattice through the same apertures (P3).
What is established — geometry only. Face centres are fixed points of the lattice's subdivision: no vertex or edge of any finer level ever lands on them. A TOT cell holds eight apertures, at least a third of an edge apart, and the aperture sets of successive levels are nested.
What is conditional. If the lattice only samples a smooth field (and the operator that does this exactly is still to be built), lattice dispersion vanishes below the band limit. If universality survives quantum corrections (not yet proven), the leading Lorentz violation reduces to a common metric that a change of coordinates removes. If the dynamics is CPT-symmetric, the operators responsible for a linear energy-dependent photon delay and vacuum birefringence are forbidden to all orders.
Predictions. No linear energy-dependent delay of photons, no first-order vacuum birefringence — and a small quadratic delay whose strength depends on direction with octahedral (ℓ = 4) symmetry. The last one is the signature: a continuous space would not have it.
Status: a mechanism with conditional results and a four-step calculation programme, not a finished derivation. Scripts for the geometry are in the Zenodo record.