Notes X and XI used a coarse radial grid (step 0.5) to characterise the graph detection limit $r^*_\text{disc}$. Note X placed $r^*_\text{disc}\approx 2.7$ at $k=8$ and Note XI showed the detection pattern is non-monotone in $k$, with $k=12$ adding a pass at $r=3.0$.
To resolve the full detection landscape, we run the flat-field screened probe at fine radial step $\Delta r = 0.1$ from $r=1.0$ to $r=4.5$, keeping $k=12$ fixed. This gives 36 probe radii — enough to distinguish isolated passes from continuous bands.
The ring half-width is $\delta r = 0.70 \times s \approx 0.326$ (where $s \approx 0.465$ is the median lattice spacing). A ring requires at least 5 sites to be evaluated; detection is declared when $|\Delta\Theta| \in [2\pi \pm 0.4\pi]$.
Below each cell is one probed radius (left to right: $r=1.0$ to $r=4.5$). Green = spinorial pass, red-grey = predicted but not detected, faint = too few sites.
| r | N sites | Θ_rot / 2π | ΔΘ / 2π | Ω_eff | Status |
|---|---|---|---|---|---|
| 1.0 – 1.2 | ≤ 4 | — | — | — | too few sites |
| 1.3 | 8 | −1.000 | −1.000 | 5.905 | ✅ spinorial |
| 1.4 – 2.0 | 8 – 12 | −1.000 | −1.000 | 5.69 – 5.02 | ✅ spinorial (all) |
| 2.1 | 4 | — | — | 4.97 | too few sites |
| 2.2 | 7 | −1.000 | −1.000 | 4.91 | ✅ spinorial |
| ▼ coherence gap — sites present, winding fails | |||||
| 2.3 | 10 | 0.000 | 0.000 | 4.86 | ⚠️ predicted, undetected |
| 2.4 | 8 | 0.000 | 0.000 | 4.80 | ⚠️ predicted, undetected |
| 2.5 | 8 | 0.000 | 0.000 | 4.75 | ⚠️ predicted, undetected |
| ▼ outer band — coherent ring recovers | |||||
| 2.6 | 12 | −1.000 | −1.000 | 4.69 | ✅ spinorial |
| 2.7 – 3.1 | 12 – 18 | −1.000 | −1.000 | 4.64 – 4.01 | ✅ spinorial (all) |
| 3.2 | 18 | −1.000 | −1.000 | 3.951 | ✅ last pass |
| ▼ dead zone — sufficient sites, winding collapses | |||||
| 3.3 | 16 | 0.000 | 0.000 | 3.894 | ⚠️ first dead-zone fail |
| 3.4 – 4.5 | 16 – 32 | 0.000 | 0.000 | 3.84 – 2.78 | ⚠️ all fail |
The 36 probed radii partition into four qualitatively distinct regions:
Sites: 7–12 per ring.
Rings are dense and azimuthally coherent.
$\Omega_\text{eff}$ ranges 5.91 → 4.91 (all $\gg \pi$).
$\Omega_\text{eff}$ still 4.75–4.86 ($\gg \pi$).
Sites cluster in arcs; large azimuthal gaps
allow winding steps to wrap incorrectly.
Sites: 12–18 per ring — denser than gap.
Rings recover azimuthal coherence.
$\Omega_\text{eff}$ ranges 4.69 → 3.951.
$\Omega_\text{eff}(3.3) = 3.894$ rad, still $> \pi$.
Site count is NOT the limiting factor;
winding coherence collapses at large $r$.
The inner-to-gap transition ($r = 2.2 \to 2.3$) and the outer-to-dead transition ($r = 3.2 \to 3.3$) are both sharp: a single step of $\Delta r = 0.1$ changes the status from clean pass to clean fail. This sharpness rules out a gradual SNR decline as the controlling mechanism.
The most striking feature of the landscape is the coherence gap at $r = 2.3$–$2.5$. These rings have 8–10 sites — strictly more than the 7 sites at $r=2.2$ that pass cleanly. The analytic signal $\Omega_\text{eff}$ is nearly identical ($4.91$ vs $4.86$–$4.75$). Yet detection completely fails.
The failure mechanism is azimuthal clustering: at the radial distance $r \approx 2.3$–$2.5$, the n=31 golden-Cantor projection places sites in 2–3 dense arcs with large empty sectors in between. The ring-winding algorithm sorts sites by polar angle $\phi_i$ and accumulates the step $\Delta\theta_i = \theta_{i+1} - \theta_i$ at each consecutive pair. When a gap in azimuthal coverage exceeds $\pi$, the wrap-to-$(−\pi, \pi]$ correction assigns the step the wrong sign, and the total winding collapses to zero.
If gap $\phi_{i+1} - \phi_i > \pi$: step wraps to wrong quadrant → total $\Delta\Theta \approx 0$
The quasicrystal is aperiodic: the radial shell at $r\approx2.3$ happens to coincide with a sparse sector in the lattice geometry. One radial step later ($r=2.6$) the sites are denser and span the full $2\pi$ without large gaps — the outer band begins.
The outer band terminates abruptly at $r = 3.2$. The analytic condition for detectable injection is $\Omega_\text{eff}(r) = \Omega_0 e^{-\lambda r} > \pi$, which places the analytic limit at $r^* = \ln(\Omega_0/\pi)/\lambda \approx 4.78$. At $r=3.2$, the effective injection is $\Omega_\text{eff}(3.2) = 3.951$ rad — still 26% above $\pi$. Yet the graph probe finds the winding at $r=3.3$ ($\Omega_\text{eff} = 3.894$) returns zero.
The dead zone persists all the way to $r = 4.5$, where $N = 17$ sites and $\Omega_\text{eff} = 2.78$ rad — still above $\pi$, still failing. Sites in the dead zone range from 16 to 32, so the failure is not due to insufficient site count.
We can define a graph-effective threshold:
The discrete ring-winding probe requires 26% excess signal above the analytic $\pi$ threshold to overcome azimuthal winding errors in the dead zone.
The excess 0.26π represents the average winding “lost” per orbit due to the neighbourhood-averaged Jacobian frames becoming unreliable at large radius, where the k-NN ball spans a broad radial band with varying injection amplitude (the radial mixing mechanism from Note XI).
The two-band structure is a direct fingerprint of the n=31 golden-Cantor quasicrystal's multi-scale radial density fluctuations. A periodic crystal would place sites uniformly at fixed shell radii; the quasicrystal distributes them aperiodically, creating radial regions of high density (coherent detection bands) and low density or azimuthal clustering (gaps and dead zones).
The inner band ($r \approx 1.3$–$2.0$) coincides with the first dense radial shell of the n=31 lattice, where the acceptance window concentrates many sites. The coherence gap ($r \approx 2.3$–$2.5$) is a structural sparse zone — sites exist but cluster azimuthally rather than spanning the full $2\pi$. The outer band ($r \approx 2.6$–$3.2$) is the second dense shell, where site density recovers and the rings regain full azimuthal coverage.
This is analogous to electronic band structure in condensed matter physics, where a crystal's lattice geometry determines which quasimomentum states support extended Bloch modes and which fall into band gaps. Here, the quasicrystal geometry determines which radial shells support coherent holonomy detection and which fall into “detection gaps.”
The fine-grid sweep completes the characterisation of the ring-winding probe method across Notes X–XII. Three limiting factors have now been identified:
| Limiting factor | Mechanism | Where it appears |
|---|---|---|
| Sparse core | too few sites near origin ($N < 5$) | $r \lesssim 1.2$, $r = 2.1$ |
| Azimuthal clustering | sites present, but in arcs; $>\pi$ gaps break winding | $r = 2.3$–$2.5$ |
| Radial mixing | k-NN ball spans wide radial band; frame accuracy drops | $r \geq 3.3$ (dead zone) |
All three mechanisms are inherent to the ring-winding probe design. The sparse core and azimuthal clustering failures arise from the quasicrystal's own geometry. The radial mixing failure arises from the per-site k-NN frame computation: at large $r$, the k-NN ball extends over a radial range comparable to $1/\lambda$, averaging frames with injection amplitudes differing by $\sim$30%, degrading the per-site frame angle estimates.
The ring-winding probe is reliable only within the coherent bands, and its graph-effective detection limit $r^*_\text{graph} \approx 3.2$ cannot be substantially improved by increasing $k$ (as Note XI showed: detection is non-monotone in $k$).
The ring-winding probe cannot reach the analytic limit $r^* = 4.78$ because its two failure modes (azimuthal clustering and radial mixing) are structural. The path forward requires a fundamentally different holonomy computation:
Additional experiments enabled by the fine-grid landscape:
- Quasicrystal field fine-grid sweep. Does the Note IX resonance structure (extended passes at $r=4.0$, $r=6.5$) survive fine-grid resolution, or do those passes sit on isolated coherent radii?
- Injection strength sweep. Fix $r$ in the dead zone ($r=3.5$) and vary $\Omega_0$ upward. Does increasing $\Omega_\text{eff}$ beyond $1.26\pi$ recover detection, or is the failure purely geometric?
- Site-geometry analysis. Directly compute the azimuthal gap distribution for each probed ring and correlate gap structure with pass/fail outcome.