The n=31 quasicrystal has an approximately uniform radial site density (oscillating ±33% around a plateau). Sites-per-shell does not distinguish the 11 graph-confirmed ring-probe detections from the 16 ring-probe artefacts. Azimuthal imbalance — the ratio of the widest angular gap to the mean gap — does.
An imbalance threshold of $\mathcal{I} = 2.0$ correctly classifies 14 of 16 ring-probe artefacts ($\mathcal{I} \ge 2.0$) while retaining all 11 confirmed detections, with one borderline case (r=9.1, $\mathcal{I}=2.07$, confirmed). The two artefacts below threshold (r=3.0, 3.1, $\mathcal{I}=1.20$) fail on graph-coverage rather than azimuthal clustering.
The r=[3,4) enrichment dip seen in Note XVI (3π/2 mode weight 5% below uniform) is confirmed here as a genuine eigenfunction node — site density in that bin is 86% of the r=[4,5) reference, near-normal, so the suppression cannot be attributed to fewer sites.
For each 1-unit radial bin $[r_{\text{lo}}, r_{\text{hi}})$, the site density is $\rho = N_{\text{sites}} / \pi(r_{\text{hi}}^2 - r_{\text{lo}}^2)$. Relative density $\hat\rho = \rho / \rho_{\text{ref}}$ uses the r=[4,5) bin as reference ($\rho_{\text{ref}} = 4/\pi \approx 1.273$ sites/unit area).
The n=31 projection generates alternating dense and sparse shells. This is a structural property of the quasicrystal, not a bias. The oscillation amplitude is modest: all bins in r=[1,10) have $\hat\rho \in [0.80, 1.33]$ except the r=[7,8) overdense ring ($\hat\rho = 1.33$, driven by the outer φ² cascade clusters).
For a shell of $N$ sites, sort the sites by azimuthal angle $\varphi_i = \text{atan2}(y_i, x_i)$. Compute the $N$ angular gaps $\Delta\varphi_k = \varphi_{k+1} - \varphi_k$ (mod $2\pi$), with the last gap wrapping back to the first site. Define:
$$\mathcal{I} = \frac{\max_k \Delta\varphi_k}{2\pi / N}$$
$\mathcal{I} = 1$ means all sites are uniformly spaced. $\mathcal{I} \gg 1$ means at least one large arc is unoccupied, which causes the ring-winding probe to miscount winding steps (a single large $\Delta\varphi$ step can be wrapped to $\pm\pi$ regardless of the true angular traversal).
These bins contain both a graph-confirmed detection and a ring-probe artefact at similar site density. Within each bin, azimuthal imbalance separates the two.
Artefacts have more sites (N=32 vs 24) yet fail. Density not predictive; imbalance is.
r=9.1 confirmed sits between two artefacts in the same density shell. r=9.2 is the hardest case (ℑ=2.13, borderline).
r=3.0/3.1 artefacts have low imbalance (ℑ=1.20) — they fail the graph-traversal on coverage (disconnected subgraph), not clustering. The second failure mode.
Two mechanisms produce ring-probe artefacts:
Site count is neither necessary nor sufficient. The n=31 density varies slowly; the quasicrystal geometry, not the projection density, governs which shells form connected rings vs arc clusters.
Note XVI reported that the 3π/2 Wilson branch-cut mode achieves enrichment $E = 0.954$ at r=[3,4) — the only bin below the uniform baseline in the r=[1,6) range. A naive explanation would be: fewer sites in that bin, so less mode weight.
The density profile rules this out. The r=[3,4) bin contains 24 sites at relative density $\hat\rho = 0.857$ (only 14% below the r=[4,5) reference), while the r=[2,3) bin — which has the highest enrichment ratio of any bin ($E=5.3\times$ per site) — has only 16 sites at $\hat\rho = 0.800$. If density drove enrichment, the r=[2,3) bin would have less enrichment than r=[3,4), not 5× more.
The dip at r=[3,4) is a genuine zero of the eigenfunction between the near-field lobe ($r < 3$, branch-cut flux channelled through the Delaunay core) and the intermediate distribution ($r \approx 4$–5, Yukawa tail). Both lanes (ring probe and operator) see a structural break at this scale.
| r | N sites | Classification | Max gap | Imbalance ℑ | Distribution |
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