New essay: is the causal lattice a sparse slice?

Prompted by the new unit-cell model — the lattice uses six of the eight cube body-diagonals as causal links, and the two it omits are exactly the optical axis. An essay on reading the 3D lattice as a subset of something denser, and what would settle it.

A reader, turning over the new unit-cell model, asked whether the 3D causal lattice might be a less dense subset of some richer topology — lattices of different dimension sharing one set of nodes. The new essay, Six of eight: is the causal lattice a sparse slice?, takes the question seriously.

The geometry offers a concrete hook. A node’s six causal neighbours are body-diagonals of the unit cube — but a cube has eight corners, and the lattice uses only six. The two it leaves out are exactly \pm(1,1,-1): the optical axis along which the framework predicts its residual birefringence. The lattice has no causal link in the very direction it tells you to go looking.

From there the essay separates the verified from the speculative: the six vectors and the optical axis are Paper I, but reading that axis as the omitted diagonal is an observation, not a theorem — and Paper I actually calls coordination six intrinsic, not a subset. It then lays out two ways “parallel dimensions on one topology” can be made precise (the spatial diamond family, discriminated by spectral dimension; and Paper II’s internal \mathbb{C}^{12} projection, the firmer case), and what measurement would settle which reading holds.

Pointers:

• Read the essay: Six of eight
• The model that prompted it: unit-cell viewer
• Companion: Plato’s Cave and the DCL projector