Duncan's Neighborhood

A radial neighborhood class on the square lattice, and the harmonic functional that gates its fourth-order anisotropy

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. What becomes decidable is the fourth-order term, which is the first one the square lattice makes non-trivial. That is a gate rather than a certificate of isotropy, and § 2 states exactly what it does and does not settle.

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. That is less than "no privileged direction," and the difference is the reason the rest of this page exists. A radial kernel treats every lattice site at equal Euclidean radius alike, but it has no say in which radii are occupied or at what angles. At R = 1 the only available offsets are the four axial ones, and the neighborhood is maximally four-fold anisotropic while being perfectly radial. The lattice supplies the direction that w declines to. 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.

What admissibility is not. A₄ ≤ ε bounds the fourth-order term and says nothing about the rest, so an admissible instance is not thereby isotropic. The Moore support makes this concrete rather than hypothetical. All eight of its offsets lie at multiples of π/4, so ei8θ = 1 at every one of them and A₈ = 1 for any radial kernel on that support, whatever the weights. A neighborhood can therefore sit at A₄ = 0 and at maximal eight-fold anisotropy simultaneously. Read A₄ as a diagnostic for the lattice's first obstruction, not as a certificate that the obstruction is gone.

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 down-weighting the diagonal shell does remove it at fourth order. With per-site weights a on the axial shell and b on the diagonal, C₄ = 4a − 16b, which vanishes at a = 4b; normalized, w(1) = 0.2 and w(√2) = 0.05 give A₄ = 0 exactly, on eight cells, with w positive and non-increasing as § 1 requires.

That instance is admissible and it is not isotropic. Its eight-fold index is A₈ = 1, the maximum, for the reason given in § 2: every Moore offset lies at a multiple of π/4. Re-weighting the two shells does not remove the lattice's directional structure, it relocates the structure to a harmonic the gate does not inspect. This is the clearest available demonstration of what A₄ measures and what it leaves alone.

4. The verified spectrum, and non-monotonicity

Computed values from the reference implementation (offsets exact; raised cosine with reach R′ = R+1). Rows admissible at fourth order (A₄ ≤ 0.02) in green; the classical controls in red. The A₈ columns are reported alongside because admissibility is a fourth-order statement, and a reader entitled to the gate is entitled to see what it leaves standing.

No row in this table is isotropic. The two orders move independently, and the table is the clearest evidence of it. R = 3 under the raised cosine reaches A₄ = 0.009 while sitting at A₈ = 0.611. R = 4 reaches A₄ = 0.003, near-perfect at fourth order, at A₈ = 0.169. Meanwhile R = √13 is the best eight-fold performer in the table and fails the fourth-order gate outright at A₄ = 0.220. Suppressing one harmonic is not evidence about the next one.

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.

What improves with radius is the limit rather than any finite step toward it. As the disk fills, the sum tends to an integral that annihilates the four-fold harmonic:

because the angular integral vanishes. The displayed identity is the motivation and not the proof: it shows the continuum harmonic is annihilated, while convergence of the discrete normalized sums A₄(R) is a separate question about how well the lattice sums approximate the integral. Computation supports the claim without settling it analytically. Across every distinct shell with R2 > 1000, unweighted, the largest A₄ is 0.016 and the largest A₈ is 0.029. Finite R retains a fluctuating discrete remainder; the tuned kernel and a well-chosen radius suppress it. The cheapest admissible member in the families tabulated above is Duncan(√5, 1), twenty cells, unweighted, A₄ = 0.014. It is not the cheapest admissible member of the class, which is the eight-cell re-weighted Moore set of § 3, at A₄ = 0 and A₈ = 1. What Duncan(√5, 1) has that the eight-cell instance does not is angular coverage, and its eight-fold index is correspondingly lower at A₈ = 0.51 rather than 1. Its success at fourth order 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 u 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, and the admissibility gate keeps that one out.

It does not keep all of them out, and the scope of the guarantee should be stated rather than assumed. Directional structure at eighth order and above is supplied by the lattice on the same terms and passes the gate untouched. The substrate's position is therefore that four-fold bias, the lattice's first and largest obstruction, is excluded by measurement, and that higher-order bias is bounded only by whatever radius and kernel happen to have been chosen. Raising the gate to A₈ as well is available and it is not free: the smallest unweighted disk holding both A₄ and A₈ under 0.02 is R2 = 106, at 340 cells against the twenty of Duncan(√5, 1). Whether that cost is worth paying is a substrate design question this page does not decide.

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.