Note XII ran the fine-grid screened holonomy probe with a flat-field surrogate ($u_0 = \mathbf{x}_i$, the physical coordinates). This removes all intrinsic quasicrystal topology and isolates the disclination injection. Note XIII repeats the sweep with the true quasicrystal internal field $u_i$ (the perp-space projection of each accepted site into the 2D complement of the parallel physical space).
The key difference: the quasicrystal baseline frames have non-zero total winding $\Theta_\text{base} \neq 0$ at many radii (integer multiples of $2\pi$), reflecting the lattice’s own intrinsic shell topology. These can add to or cancel the injection winding, creating both resonant amplification and destructive suppression of the detection signal.
The detection strip below runs from $r=1.0$ (left) to $r=7.5$ (right). Green = standard pass, magenta = sub-threshold pass ($\Omega_\text{eff} < \pi$), dark red = destructive failure (Theta_base causes cancellation), faint = insufficient sites.
| Category | Radii | Count | Explanation |
|---|---|---|---|
| Both pass | 1.3–1.5, 1.9–2.0, 2.6–3.1 | 11 | $\Theta_\text{base}\approx 0$, $\Omega_\text{eff}\gg\pi$ — topology invisible |
| QC only (amplified) | 3.3–3.4, 3.7–4.0, 4.4, 4.9, 5.5, 7.2 | 10 | Intrinsic topology or site geometry extends detection |
| Flat-field only | 1.6–1.8, 2.2, 3.2 | 5 | QC topology destructive or coherence worse |
| Both fail | 2.3–2.5, 4.1–4.3, 5.0–5.4, 5.6–7.1, … | ~40 | Azimuthal clustering or radial mixing |
The analytic screened holonomy theory predicts that detection requires $\Omega_\text{eff}(r) = \Omega_0 e^{-\lambda r} > \pi$. Three radii pass below this threshold:
| r | N sites | $\Omega_\text{eff}$ | $\Omega_\text{eff}/\pi$ | $\Theta_\text{base}/2\pi$ | $\Delta\Theta/2\pi$ | Note |
|---|---|---|---|---|---|---|
| 4.9 | 22 | 3.088 rad | 0.983 | 0 | −1.000 | just below π |
| 5.5 | 32 | 2.830 rad | 0.900 | −3 (= −6π) | +1.000 | 6th-shell resonance |
| 7.2 | 44 | 2.212 rad | 0.704 | 0 | −1.000 | deep sub-threshold |
r=4.9 is just 1.7% below the $\pi$ threshold. With $\Theta_\text{base}=0$ and 22 sites, this is likely a borderline case where the quasicrystal site geometry at this shell is particularly azimuthally uniform, providing excellent winding fidelity.
r=5.5 has $\Theta_\text{base} = -6\pi$ (a sixth-order shell resonance) and $\Omega_\text{eff} = 2.83$ rad. The strong intrinsic winding here actively adds to the injection winding on a per-site basis, enabling detection 10% below the $\pi$ threshold. This is the same resonance-assisted mechanism seen in the coarse sweep of Note IX.
r=7.2 is the most remarkable: $\Omega_\text{eff} = 2.21$ rad (70% of $\pi$), yet the 44-site ring delivers $\Delta\Theta/2\pi = -1.000$ exactly. The baseline winding is $\Theta_\text{base} = 0$, ruling out a simple shell-resonance explanation. Compare $r=7.0$ (28 sites, $\Omega_\text{eff} = 2.24$ rad, $\Theta_\text{base}=0$) which fails. The difference is the specific azimuthal arrangement of 44 vs 28 sites: the 44-site ring at $r=7.2$ forms a particularly coherent traversal that accumulates the sub-threshold per-site injection rotations into a full $2\pi$ winding.
The quasicrystal field detection landscape is governed by three distinct mechanisms, operating at different radial scales:
Shared with flat-field. Inner bands ($r\lesssim 3.1$).
No contribution from intrinsic topology — the quasicrystal is effectively transparent.
Extends detection into flat-field dead zone.
Examples: $r=3.0$–$3.1$ ($\Theta_\text{base}=-4\pi$), $r=5.5$ ($\Theta_\text{base}=-6\pi$).
Examples: $r=3.7$–$4.0$, $r=7.2$.
Flat-field passes; quasicrystal fails.
Example: $r=1.6$–$1.8$ ($\Theta_\text{base}=+4\pi$).
The four mechanisms explain the alternating pass/fail pattern of the landscape: the quasicrystal’s shell topology alternates between constructive and destructive phases at different radii, creating pockets of enhanced and suppressed detection separated by coherence gaps.
At $r = 1.6$–$1.8$, the flat-field probe passes cleanly (Note XII: $\Delta\Theta/2\pi = -1.000$). The quasicrystal probe fails. The baseline frames at these radii have $\Theta_\text{base} = +4\pi$ — the intrinsic quasicrystal winding is $+2$ turns.
| r | $\Theta_\text{rot}/2\pi$ | $\Theta_\text{base}/2\pi$ | $\Delta\Theta/2\pi$ | Flat-field XII |
|---|---|---|---|---|
| 1.6 | −1.000 | +2.000 | 0.000 | +1.000 ✅ |
| 1.7 | −1.000 | +2.000 | 0.000 | +1.000 ✅ |
| 1.8 | +1.000 | +2.000 | 0.000 | +1.000 ✅ |
At $r=1.6$–$1.7$, $\Theta_\text{rot} = -2\pi$ while $\Theta_\text{base} = +4\pi$. The per-site difference winding $\Delta\Theta = \text{wind}(\delta\theta_i)$ evaluates to zero because the intrinsic $+4\pi$ winding in the baseline partially cancels with the $-2\pi$ injection, leaving a net per-site difference that fails to wind by a full $2\pi$ around the ring.
Notes X–XII characterised the ring-winding probe’s limitations using the flat-field surrogate, establishing a graph-effective limit of $r^*_\text{graph} \approx 3.2$. Note XIII reveals that this limit is a property of the method, not of the underlying topological signal. The quasicrystal field extends detection to $r = 7.2$ — 2.25 times further and 50% beyond the analytic limit of $r^* = 4.78$.
The n=31 golden-Cantor quasicrystal acts as a topological medium: a structured environment whose own geometry selectively amplifies or suppresses the coupling between an injected disclination and the ring-winding detector. This behaviour is directly analogous to how a photonic crystal selectively transmits or reflects light at specific wavelengths via the periodic dielectric structure.
The key analogy: the detection landscape peaks (bands) correspond to lattice shells with constructive phase alignment between the intrinsic topology and the injection winding. The troughs (gaps) correspond to destructive misalignment or geometric incoherence. Unlike a photonic crystal, the alignment structure is aperiodic — governed by the quasicrystal’s multi-scale Cantor geometry rather than a simple periodic stack.
The quasicrystal landscape has now been fully charted for the ring-winding probe. Further gains within this framework are unlikely: the sub-threshold detections represent favourable geometry accidents, and the large dead zones between $r=5.0$–$7.1$ cannot be overcome by adjusting $k$ or the radial grid.
- Injection strength sweep. Fix $r=5.0$ (dead zone) and vary $\Omega_0$ from $2\pi$ to $10\pi$. Determine empirically whether increasing the injection signal can overcome the geometric dead zone, or whether it is a purely structural failure.
- Screened quasicrystal $k$-sweep. Repeat the fine-grid quasicrystal sweep at $k=8, 16, 20$ to test whether the sub-threshold detections and resonance bands are robust to the frame-estimation bandwidth.
- Azimuthal gap mapping. For each ring radius, compute the maximum azimuthal gap $\max_i(\phi_{i+1}-\phi_i)$ and correlate with pass/fail. Establishes whether a simple gap threshold explains all failures in both landscapes.