Graph-Ring Traversal

The Question

Notes VIII through XIV used the ring-winding probe: select all sites in a shell $[r - \delta r,\, r + \delta r]$, sort them by azimuthal angle $\varphi$, compute the winding of the per-site difference angle $\delta\theta_i = \mathrm{wrap}(\theta_{\mathrm{rot},i} - \theta_{\mathrm{base},i})$ around the sorted ring. A pass is $|\Theta| \approx 2\pi$.

This method has a known failure mode: azimuthal clustering. When ring sites are not uniformly distributed in $\varphi$ — but instead clump into two or three arcs — the sorter must leap across large angular gaps, and the winding computation can produce an artefact pass (or artefact fail).

Note XV asks: for each of the 25 passes detected in Notes XIII–XIV, is the ring shell actually graph-connected in the k-NN graph? Do the passes survive when we replace sorted-angle traversal with true graph-edge traversal?

Method

Connectivity as a validity criterion

If the ring shell is graph-connected, then for any two sites $i, j$ in the shell there is a path through k-NN edges that stays within the shell. A greedy traversal that closes the loop and achieves >60% coverage is a strong indicator that the sites actually surround the origin — not that they are confined to one or two azimuthal arcs.

A disconnected shell (coverage ≤25%) means the $k$-NN graph has no edges bridging the gap between clusters. The azimuthal sorter silently leaps this gap; the graph traversal stops and reports low coverage.

def build_knn_adj(pts, k=12):
    diff  = pts[:,None,:] - pts[None,:,:]
    dist2 = (diff**2).sum(axis=2)
    np.fill_diagonal(dist2, np.inf)
    adj = [set() for _ in range(len(pts))]
    for i in range(len(pts)):
        for j in np.argsort(dist2[i])[:k]:
            adj[i].add(int(j)); adj[int(j)].add(i)
    return adj

def graph_ring_winding(r, dr, pts_phys, adj, diff_angles, radii, phi,
                        min_coverage=0.60):
    ring_idx = np.where((radii >= r-dr) & (radii <= r+dr))[0]
    ring_set  = set(int(i) for i in ring_idx)
    local_adj = {int(i): [j for j in adj[int(i)] if j in ring_set]
                 for i in ring_idx}
    # BFS connectivity check
    is_connected = (len(bfs(local_adj, ring_idx[0])) == len(ring_idx))
    # Greedy Hamiltonian path from min-phi site
    path = greedy_traverse(local_adj, phi, ring_idx)
    coverage = len(path) / len(ring_idx)
    if coverage < min_coverage: return {'passes': False, ...}
    winding = winding_of_ordered([diff_angles[i] for i in path])
    return {'connected': is_connected, 'coverage': coverage,
            'winding': winding, 'passes': abs(|winding| - 2pi) < 0.4*pi}

Three Populations

Detection Strip

Connectivity map — all 53 tested radii

Green = shell graph-connected (BFS reaches all sites). Red = disconnected. Tested with k=12, dr=0.326.

Full Results — 53 Radii

Winding tolerance: |Θ| within 40% of 2π (i.e. |Θ/π − 2| < 0.8). Coverage threshold: ≥60%. Double-winding entries (Θ/π ≈ ±4) are noted but do not pass the spinorial criterion.

The Two New Finds: r = 2.8 and r = 3.2

Both radii fall inside the coherence gap identified in Note XII (r = 2.3–2.5 had 8–10 sites but failed). The fine-grid in Note XII did find r = 2.8 as a pass in the flat-field probe (using $\theta_\text{rot}$ alone, without subtracting $\theta_\text{base}$). In the full differential probe — computing $\delta\theta_i = \mathrm{wrap}(\theta_{\mathrm{rot},i} - \theta_{\mathrm{base},i})$ per site and summing their sorted-ring winding — r = 2.8 fails because the 16 ring sites are not uniformly distributed in $\varphi$: they clump into two arcs separated by a gap that introduces a phase cancellation step.

The graph traversal, following actual k-NN edges, never jumps this gap as a sorting artefact — it reaches 13 of 16 sites (81%) via actual adjacency, accumulating $\Theta \approx +2\pi$ through real lattice connectivity.

Ring-Probe Artefacts — Anatomy of a Gap Jump

The 16 ring-only passes share a common structure: the shell sites in $[r - \delta r, r + \delta r]$ divide into two or more azimuthal clusters with no k-NN edge bridging them. In the k-NN graph (k = 12), the nearest-neighbour radius grows with $r$ because the point density of the n=31 quasicrystal falls as $1/r$ at large radii. At r ≥ 3.0, a 0.326-wide shell contains 8–20 sites, but their azimuthal spread is non-uniform: the quasicrystal’s phason-plane projection creates preferred directions, so sites cluster into arcs.

Why the ring probe passes anyway

The azimuthal sorter treats $\varphi$-order as an ordering on the circle. When two clusters are present at $\varphi \approx -\pi/4$ and $\varphi \approx +3\pi/4$, the sorted sequence transitions from the last site of one cluster to the first of the other in a single step. If the difference angles $\delta\theta_i$ in both clusters average to $+\pi/2$, the total winding across the gap accumulates the missing $\pi$ and the ring probe incorrectly registers $\Theta \approx 2\pi$.

Why the graph traversal fails

The greedy Hamiltonian path starts at the minimum-$\varphi$ site and advances to the unvisited k-NN neighbour with the largest positive $\Delta\varphi$. When it reaches the last site of one cluster, there are no unvisited k-NN neighbours with positive $\Delta\varphi$ (the gap to the next cluster is wider than the k-NN ball), so the traversal terminates. Coverage is typically 3–5 of 16–20 sites (15–25%), far below the 60% threshold. The result is correctly classified as a non-detection.

Coverage distribution by region

The pattern is clear from graph-traversal coverage values:

• r = 1.3–2.7: 75–100% — uniformly-distributed shell, graph-connected
• r = 2.8, 3.2: 81–85% — connected but azimuthally clustered
• r = 3.0, 3.1, 3.3–4.1: 17–25% — disconnected shells
• r = 4.4: 88% — graph-confirmed isolated pass
• r = 4.8–7.2: 14–20% — sparse, disconnected shells
• r = 9.0–9.5: 57–65% — reconnection near lattice boundary

The reconnection at r ≈ 9.0–9.5 is striking: as the lattice boundary approaches, fewer sites exist in the shell but they are forced closer together in $\varphi$, increasing local k-NN connectivity again. r = 9.1 reaches 65% and passes; r = 9.2 and 9.5 fall just below threshold at 57–59%.

Conclusion

Graph-ring traversal is a structural validity filter for the ring-winding probe. Applied to the 53-radius inventory from Notes VIII–XIV, it produces a significantly tighter detection picture:

The 16 ring-only artefacts at r = 3.0–9.5 are not topological detections: their shells are k-NN graph-disconnected, and the ring probe’s azimuthal sorter bridges real lattice gaps to produce spurious 2π winding. They are a consequence of the quasicrystal’s non-uniform azimuthal site distribution at intermediate radii, not a signature of the screened disclination.

Revised detection inventory

Open questions for Note XVI

The graph-ring traversal establishes which ring-probe results are structurally grounded. Next questions:

• Do the 2 new graph-only passes (r = 2.8, 3.2) follow the φ² cascade pattern? Their inner-band partners would be at r / φ² ≈ 1.07 and 1.22 — both in the confirmed inner band.
• Can the operator-lane branch-cut modes (Note XV, operator-lane) be correlated spatially with the 11 graph-confirmed detection radii?
• Extended lattice test: does a larger quasicrystal (n=47 or n=61) reduce the azimuthal clustering at r = 3.0–5.5, converting ring artefacts into confirmed detections?
• Can the greedy Hamiltonian path be improved (e.g. Christofides approximation) to rescue the 57–59% coverage radii at r = 9.2 and 9.5?