Duncan's Neighborhood
An isotropic neighborhood class on the square lattice, and the harmonic functional that certifies it
A cellular-automaton neighborhood on ℤ2 generically imposes the lattice's own directions on the dynamics: a rule built from the eight-cell Moore set spreads influence as a square, not a disk. Duncan's Neighborhood is the class of neighborhoods whose weight depends on Euclidean distance alone, together with the requirement that a measured directional anisotropy fall below tolerance. Isotropy becomes a decidable admissibility condition rather than an aspiration.
1. The lattice and the neighborhood class
Work on the square integer lattice ℤ2. A neighborhood of the origin is a finite weighted set of nonzero offsets. Duncan's Neighborhood is the two-parameter class Duncan(R, w) fixed by a radius R ≥ 1 and a radial kernel w.
The support is every lattice cell within Euclidean distance R of the centre; the centre itself is excluded (its own state enters an update rule separately, not as a neighbour). Membership is by the true metric |v|, not by Chebyshev or Manhattan distance — that choice is what makes the two nearest shells (orthogonal at distance 1, diagonal at √2) distinguishable in the first place.
The weight of a neighbour depends only on its distance, never on its bearing — this is the entire content of "no privileged direction." Normalizing the total weight to 1 decouples the neighborhood's geometry from its overall gain, so changing R or w does not silently rescale a rule built on top of it.
A construction caveat. The reach R′ of a smooth kernel must exceed the support radius R. A raised cosine w(d) = ½(1 + cos(πd / R′)) vanishes at d = R′; if R′ = R it zeroes the outermost occupied shell — catastrophically at R = 1, where all four cells sit exactly on it and every weight is zero. The reference implementation uses R′ = R + 1. (Unweighted kernels, w ≡ 1, have no such issue.)
2. The isotropy functional
Isotropy is measured, not assumed. The key structural fact is that on the square lattice the low-order isotropy is free, and the first genuine obstruction is four-fold.
Second order is automatic
Whenever the support and weights are invariant under the dihedral group D₄ of the square (rotations by 90° and reflections) — which a radial kernel on the symmetric digital disk always is — the second-moment tensor is forced to be isotropic:
The only D₄-invariant symmetric 2-tensors are scalar multiples of the identity, so Q12 = 0 and Q11 = Q22 identically. Moore already passes at this order; second-order isotropy is necessary but far from sufficient.
Fourth order is the obstruction
The first non-trivial anisotropy lives in the fourth moment, whose D₄-invariants split into an isotropic part and a single four-fold-anisotropic part. The anisotropic part is captured compactly in the complex plane, writing z = v₁ + i v₂ = |v| eiθ:
For a D₄-symmetric neighborhood C₄ is real, and its sign records the bias: positive is axis-aligned, negative is diagonal-aligned. A₄ is the normalized magnitude — the fraction of fourth-moment "mass" carried by the four-fold harmonic. (In the project spec this index is written 𝒞; A₄ is used here to make the harmonic order explicit.)
Admissibility. An instance is admissible iff A₄ ≤ ε for a stated tolerance; the reference tolerance is ε = 0.02. Higher harmonics C₄k (eight-fold, twelve-fold, …) are the residual obstructions once C₄ is controlled; full continuous isotropy is the simultaneous vanishing of all C4k.
3. The classical neighborhoods, scored
The von Neumann and Moore neighborhoods are the degenerate unweighted members Duncan(1, 1) and Duncan(√2, 1), and the functional assigns them exact values.
von Neumann-4 → A₄ = 1
Maximal anisotropy: the four offsets are purely axial, so the four-fold harmonic saturates. This is the diamond that von Neumann growth is famous for.
Moore-8 → A₄ = 0.6
The sign is negative: the diagonal cells, at |v|4 = 4, outweigh the axial cells fourfold in the fourth moment, so Moore's bias is toward the diagonals. Equal weighting of two shells at different radii is exactly the defect — and it is why merely down-weighting the diagonal shell cannot reach isotropy: two four-fold-symmetric shells sum to a four-fold-symmetric neighborhood.
4. The verified spectrum, and non-monotonicity
Computed values from the reference implementation (offsets exact; raised cosine with reach R′ = R+1). Admissible rows (A₄ ≤ 0.02) in green; the classical controls in red.
| R | |v|2 | cells | A₄ (w ≡ 1) | A₄ (raised cosine) |
|---|---|---|---|---|
| 1 | 1 | 4 | 1.000 | 1.000 |
| √2 | 2 | 8 | 0.600 | 0.397 |
| 2 | 4 | 12 | 0.619 | 0.371 |
| √5 | 5 | 20 | 0.014 | 0.023 |
| 2√2 | 8 | 24 | 0.482 | 0.249 |
| 3 | 9 | 28 | 0.074 | 0.009 |
| √10 | 10 | 36 | 0.173 | 0.083 |
| √13 | 13 | 44 | 0.220 | 0.078 |
| 4 | 16 | 48 | 0.089 | 0.003 |
| 5 | 25 | 80 | 0.123 | 0.050 |
A₄ is not monotone in R. Enlarging the disk does not steadily improve isotropy: Duncan(2) at A₄ = 0.619 is worse than Moore, because the added cells (±2,0),(0,±2) are purely axial and reinforce the four-fold harmonic; R = √10 and R = 5 rebound upward for the same kind of reason. Admissibility therefore cannot be inferred from radius — it must be checked. This is precisely what makes the functional a gate rather than a formality.
The behaviour that is monotone is the limit. As the disk fills, the sum tends to an integral that annihilates the four-fold harmonic:
because the angular integral vanishes. Finite R retains a fluctuating discrete remainder; the tuned kernel and a well-chosen radius suppress it. The cheapest admissible member found is Duncan(√5, 1) — twenty cells, unweighted, A₄ = 0.014 — whose success is structural: the knight's-move offsets (±1,±2),(±2,±1) sit at angles (≈26.57°, 63.43°, …) off both axis and diagonal, so they populate the angular gaps the four-fold harmonic exploits. Notably, at this radius the unweighted set beats the raised cosine (0.014 vs 0.023): kernel weighting is a tool for suppressing A₄, not a guarantee of lowering it.
5. Role in the substrate; scope
Within the Bio-Kernel substrate the neighborhood is a purely spatial relation: it fixes what a cell reads, carrying no timing, and composes with the asynchronous update scheme orthogonally (the α dial for time, the isotropy index for space). The reason isotropy is enforced at all is the substrate's governing question — whether a persisting structure produced its own order or inherited the lattice's. A four-fold-biased neighborhood is a supplied structure; the admissibility gate keeps it out.
The mathematics here is not new, and it should not be mistaken for it. Choosing lattice weights so that moment tensors are isotropic to a given order is the foundation of the lattice Boltzmann method (moment isotropy to fourth, sixth, eighth order); weighted "masks" over large neighborhoods for isotropic excitable-media automata date to Weimar–Tyson–Watson (1992) and Markus–Hess (1990); isotropic finite-difference stencils are the same construction. Duncan's Neighborhood contributes the packaging: the functional A₄ as an explicit admissibility gate, the classical neighborhoods recovered as scored degenerate members, and the reading of anisotropy as supplied structure. It invents no mathematics.
On the name. Duncan's Neighborhood is named in honor of Duncan. The name is a dedication, not an attribution: the technique above is prior art, so the eponym honors a person, not a claim of mathematical priority. What it names is what this project does contribute — the admissibility gate A₄, the classical neighborhoods scored as failing members, and the reading of anisotropy as supplied structure. In the field's own terms it is an isotropic, radial-kernel neighborhood (built elsewhere under other names, as above); "Duncan's Neighborhood" is this project's handle for it, not a distinct construct.