Six of eight: is the causal lattice a sparse slice?

A reader’s question about the unit-cell model — could lattices of different dimension share one topology, with ours keeping fewer of its links?

The bipartite octahedral lattice uses six of the eight cube body-diagonals as causal links — and the two it omits are exactly the optical axis. A look at whether the 3D causal lattice is best read as a sparse subset of something denser, and what would settle it.

Looking at the new unit-cell model, a reader asked a sharp question: suppose lattices of different dimension could exist in parallel on the same adjacency topology — is the 3D causal lattice just using a less dense subset of the available nodes?

It is worth saying plainly what that object is. The lattice in the model is not a picture of physics — it is the mathematically defined geometric object whose intrinsic constraints the program uses to induce physics (its Statement of Intent). So the reader’s question is not idle: it asks how much structure that object actually spends — whether the physics we recover leans on all of the lattice, or only a thrifty part of it.

The geometry has something concrete to say, and it points at one of the framework’s open questions. Let me separate what the lattice actually shows from where the reading turns speculative.

Six directions, and the one that is missing

A node’s six causal neighbours sit along the basis vectors the engine uses — V_1=(1,1,1), V_2=(1,-1,-1), V_3=(-1,1,-1) (the RGB / even sublattice) and their negatives -V_1, -V_2, -V_3 (the CMY / odd sublattice). Every one of those is a body-diagonal of the unit cube: a vector to a corner (\pm1,\pm1,\pm1).

But a cube has eight corners, and the lattice uses only six of them. Which two does it leave out? Subtract the six from the eight and exactly two remain:

(1,1,-1) \quad\text{and}\quad (-1,-1,1) = \pm(1,1,-1).

That is the optical axis — the direction V_1+V_2+V_3 = (1,1,-1) along which Paper I predicts the lattice’s residual birefringence and flat dispersion, the axis that ties together the framework’s strongest falsifiable prediction. The lattice has no direct causal link along the very axis where its symmetry restoration is weakest. It is blind in the one direction it tells you to go looking.

I want to be careful about the status of that observation. The six vectors and the optical axis are Paper I; the framing of the optical axis as the two body-diagonals the lattice omits is a consequence I am drawing from the engine’s vectors, not a theorem stated in the paper. In fact Paper I takes the opposite view of the same number: it calls coordination six “forced by the bipartite octahedral geometry” — six is presented as intrinsic, not as a sparse selection from eight. Which is exactly where the reader’s question bites.

“Less dense” in what, exactly?

Before pushing on the subset idea, it helps to pin down which density.

• Nodes. Over a full pair of ticks the dynamics touch every node — the bipartite split only alternates which sublattice is active. So the lattice is not really sparser in nodes.
• Directions. At each tick a node hops along just three active vectors (one sublattice). That is where the thinness lives — and it is load-bearing. Three active directions correspond to three spatial dimensions and the three spatial gamma matrices. In this framework, the dimension of space is the count of active directions.

So if there is a sense in which our lattice is “a less dense subset,” it is about coordination, not node count: three live directions per tick, drawn from a cube that offers more. The reader’s instinct survives the refinement — it just relocates from nodes to links.

Parallel dimensions, two ways

There are two distinct ways “lattices of different dimension on one topology” can be made precise, and the framework already gestures at both.

Spatially — the diamond family. The octahedral lattice is one member of a family of higher-dimensional diamond lattices,¹ the theme of an earlier essay, Plato’s Cave and the DCL projector. A D-dimensional member activates more directions per tick; raise D and the induced walk spreads thinner across more neighbours. Different members could in principle share an ambient node set and differ only in which links are live — which is the reader’s picture exactly: ours as a low-coordination slice of a richer graph. The quantity that tells the members apart is the spectral dimension d_s, the return-probability and spreading law of the walk. That is the discriminator you could measure from inside the cave.

Internally — and this is the closer match to the question. Paper II keeps the same spatial nodes but extends the per-site amplitude to \mathbb{C}^{12} = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3, and finds the Standard-Model gauge algebra as a factor-product projection of a much larger, 71-dimensional symmetry of that extended amplitude. That is, almost literally, multiple structures coexisting on one topology, with the physics we observe being a projection — a subset — of a denser object. In the gauge sector, the reader’s intuition is not far from the thesis of Paper II itself.

What would settle it

Here is the honest boundary. Whether the 3D causal lattice is best read as a sparse slice of a denser or higher-dimensional structure or as intrinsically six-coordinate is, for the spatial picture, an open interpretive question — a “which projector?” question, not a PASS row. The unused-diagonal observation is solid as geometry, but it does not by itself decide between “the lattice omits two links” and “the lattice simply has six.” The internal version (Paper II’s projection) stands on firmer ground; the spatial diamond-family version is exploratory, and the claim map flags the relevant optical-axis predictions as STUB.

Two things could move it from interpretation toward measurement:

1. Spectral dimension. A high-energy propagation measurement that pins d_s would say which member of the diamond family — which density of live directions — is compatible with the shadows.
2. The optical axis. If the omitted-diagonal reading is more than a coincidence, then the predicted (1,1,-1) birefringence is the fingerprint of the link the lattice left out — anisotropy concentrated in precisely the direction with no causal edge. A concordant multi-channel signal along that axis would be hard to read any other way.

The unit-cell model is what made the question askable — you can now turn the thing over and notice that one diagonal is conspicuously absent. That a single missing edge lands exactly on the framework’s one falsifiable axis is, at the least, a coincidence worth chasing. Consistent with the program’s Statement of Intent, I am not claiming the lattice is a slice of a hidden denser world — only that the geometry makes the question well-posed, and hands us two ways to test it.

References

• Menendez, Geometry First (Paper I) — the basis vectors V_1, V_2, V_3, fixed coordination six, and the optical-axis (1,1,-1) birefringence prediction.
• Menendez, Geometry Forces Physics (Paper II) — the \mathbb{C}^{12} extension and the factor-product projection of the 71-dimensional automorphism algebra.
• Plato’s Cave and the DCL projector — the higher-dimensional diamond family and the spectral-dimension discriminator.
• Unit-cell model · Claim map.

Footnotes

1. A caveat on rigour: the construction of this family is not yet formally stated. Paper I establishes coordination six as the unique solution for \mathbb{Z}^3, so the 3D lattice is special rather than a sampled member; and at least two higher-dimensional generalisations compete — a cube-diagonal one (the D-cube’s 2^{D-1} diagonal pairs, of which the lattice would keep D) versus a simplex / tetrahedral one (the (D{+}1)-vertex D-simplex, the sense in which a crystallographic “diamond” generalises). They already diverge at four dimensions: the tesseract’s 16 corners versus the 5-cell’s 5 vertices. Pinning down which family is the right one — and its spectral-dimension signature — is the subject of a planned paper in the series.