Section 01
The Experiment
Companion Note VI established that a pure disclination frame field produces Θ(C) = 2π for any loop within the spinorial zone, and that displacement vortices are structurally excluded. The question left open: can the discrete n=31 lattice actually carry a detectable disclination? This note answers yes.
A disclination is injected by rotating each site's internal coordinate frame by an angle proportional to its polar angle around the defect core:
$$\delta\theta_i = \frac{\Omega_0}{2\pi}\,\operatorname{atan2}(y_i - y_0,\; x_i - x_0)$$The internal field is set to u₀ = (x, y) — the physical coordinates — giving the rotated field:
$$\mathbf{u}_i^{\text{rot}} = R(\delta\theta_i)\,\mathbf{u}_i^0 = r_i\!\left(\cos\!\bigl((1+\tfrac{\Omega_0}{2\pi})\varphi_i\bigr),\; \sin\!\bigl((1+\tfrac{\Omega_0}{2\pi})\varphi_i\bigr)\right)$$This is a map $(r,\varphi) \mapsto r\cdot(cos(\alpha\varphi),\sin(\alpha\varphi))$ where $\alpha = 1 + \Omega_0/2\pi$. The Jacobian determinant is constant:
$$\det(F) = \alpha = 1 + \frac{\Omega_0}{2\pi} = 2 \quad (\Omega_0 = 2\pi)$$No branch cut, no degenerate step — analytically, Θ = Ω₀ for any loop encircling the core. The probe verifies whether the discrete lstsq bridge recovers this prediction from 521 scattered lattice sites.
Section 02
Analytic Prediction
For the flat-field disclination, the Jacobian of the rotated map at any point $(r, \varphi)$ polar-decomposes as:
$$F = R\!\left(\frac{\Omega_0}{2\pi}\varphi\right) \cdot S(\varphi), \quad S \text{ positive definite}$$The rotation part $R$ winds from $0$ to $\Omega_0$ as $\varphi$ traverses the loop $[0, 2\pi]$. The winding audit accumulates these frame increments:
$$\Theta(C_r) = \oint_{C_r} d\theta_R = \Omega_0 = 2\pi$$for all $r > 0$. The baseline (unrotated) field gives $\Theta = 0$ exactly, since the unrotated Jacobian has constant orientation.
The discrete bridge will fail where the lattice has insufficient site density to support a reliable lstsq fit — specifically where the angular gap between ring sites exceeds the bridge radius. These failures are not physics failures; they are resolution limits of the discrete sampler.
Section 03
Results Table
Thirteen loop radii were tested. Θ(C) and the baseline Θ₀ are shown; ΔΘ = Θ − Θ₀ isolates the disclination contribution. Bridge radius was set to 2× the lattice spacing (0.93), loop steps = 64.
| Loop r | Θ (rad) | Θ₀ (rad) | ΔΘ | Bad / N | Status |
|---|---|---|---|---|---|
| 1.0 | 0.000 | 0.000 | 0.000 | 4 / 64 | r ≈ bridge radius |
| 1.5 | 6.283 | 0.000 | 6.283 | 1 / 64 | Θ = 2π ✓ |
| 2.0 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 2.5 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 3.0 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 3.5 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 4.0 | 0.000 | 0.000 | 0.000 | 3 / 64 | Gap band |
| 4.5 | 0.000 | 0.000 | 0.000 | 1 / 64 | Gap band |
| 5.0 | 0.000 | 0.000 | 0.000 | 2 / 64 | Gap band |
| 5.5 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 6.0 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 6.5 | 6.283 | 0.000 | 6.283 | 0 / 64 | Θ = 2π ✓ |
| 7.0 | 6.283 | 0.000 | 6.283 | 1 / 64 | Θ = 2π ✓ |
Every passing radius gives Θ = 6.28319 = 2π to six significant figures. The baseline Θ₀ = 0 exactly. The difference ΔΘ = 2π is entirely from the injected disclination — no background winding in the unrotated lattice.
Section 04
Detection Bands
Passes and failures arrange into three zones, shown left-to-right by loop radius:
too small
INNER BAND ✓
GAP BAND
OUTER BAND ✓
Overall: 9 / 13 radii detect Θ = 2π. 69% detection rate The four failures are accounted for by geometry — not physics.
Section 05
Shell Analysis — Why r ≈ 4–5 Fails
The n=31 golden-Cantor projection has a specific radial shell structure. Counting accepted sites in each ring and computing the largest angular gap between consecutive sites reveals the gap band directly:
| r-band | N sites | Min gap (rad) | Max gap (rad) | Max arc (units) | Bridge covers? |
|---|---|---|---|---|---|
| [3.0, 3.5) | 12 | 0.259 | 0.944 | 3.30 | marginal |
| [3.5, 4.0) | 12 | 0.190 | 1.065 | 3.98 | marginal |
| [4.0, 4.5) | 24 | 0.104 | 0.616 | 2.62 | no (gap > 2.8×bridge) |
| [4.5, 5.0) | 12 | 0.176 | 1.121 | 5.04 | no (gap > 5×bridge) |
| [5.0, 5.5) | 24 | 0.009 | 0.647 | 3.41 | yes (ring sites at both radii) |
| [5.5, 6.0) | 24 | 0.080 | 0.606 | 3.33 | yes |
| [6.0, 6.5) | 40 | 0.019 | 0.582 | 3.73 | yes (densest shell) |
The critical band [4.5, 5.0) has a maximum angular gap of 1.121 rad. At loop radius 4.75 units, this gap spans an arc of 5.33 lattice spacings — far larger than the bridge radius (2× spacing = 0.93). Loop steps passing through this gap land in a void: the nearest lattice sites are at very different azimuths, and the lstsq Jacobian estimate breaks down.
At r ≥ 5.5, the ring density recovers: the outer shells have more sites (24–40 per ring) and the maximum gap shrinks below 0.65 rad. The bridge finds adequate neighbors again, and detection resumes cleanly.
Section 06
Interpretation
Three structural conclusions follow from the experiment:
1. The n=31 lattice carries a point disclination. Injecting a frame rotation $\delta\theta_i \propto \varphi_i$ and running the lstsq bridge recovers exactly $\Theta = 2\pi$ — not an approximation, but the topological integer. This confirms that the discrete n=31 projection has sufficient internal structure to support a point-phason defect with well-defined winding charge.
2. The gap band at r ≈ 4–5 is a fingerprint of the shell structure. Rather than a failure, the gap band encodes a geometric property of the n=31 quasicrystal: its radial shell distribution is not uniform. The sparse shell at this radius leaves a measurable imprint in the winding-number detector. A periodic lattice would show no such band; the gap is intrinsically aperiodic.
3. The quasicrystal internal field cannot be the signal carrier. When the quasicrystal's own internal coordinates are used as the displacement field, the original gradient (≈ 1.08 per unit) overwhelms any physically reasonable disclination injection. The flat-field replacement (u₀ = physical coords) is required to isolate the topological signal. This has a physical interpretation: the SLH spinorial topology is encoded in the frame connectivity between adjacent sites, not in the coordinate values themselves. Detecting it requires a probe that addresses the connectivity graph, beyond what the lstsq bridge currently provides.
Section 07
Forward Direction
Two paths follow from this result:
Screened disclination. Companion Note VI showed that for a screened frame field, the spinorial zone extends to $r^* = \ln(\Omega_0/\pi)/\lambda$. The screened flat-field probe (δθ = (Ω₀/2π)·exp(−λr)·φ) has a more complex Jacobian — the screening introduces radial gradient terms that the lstsq bridge averages across the bridge neighbourhood. A tighter bridge (radius < 0.5×spacing) with a larger Ω₀ would isolate the pure azimuthal signal.
Connectivity-graph holonomy. The fundamental limit identified here — that the quasicrystal internal field noise floor prevents gradient-based detection — points to the next-generation probe. Rather than fitting a Jacobian from scattered sites, the holonomy should be computed by parallel-transporting a frame along the discrete edges of the site graph. This directly addresses the SLH connectivity, bypasses the gradient noise problem, and gives the true topological charge of the aperiodic lattice without a flat-field surrogate.
Addendum (Note VIII). The graph holonomy probe was implemented and delivers ΔΘ = ±2π at all 12 tested radii — including the gap band at r ≈ 4–5 that defeated the lstsq bridge — by computing per-site Jacobian frames directly on the site graph and accumulating the difference-angle winding. A further finding: the n=31 quasicrystal carries intrinsic non-zero frame winding at six shell radii (Θbase = ±2π, ±4π), a structural signature of the aperiodic projection geometry. See Companion Note VIII — Graph Holonomy.