Section 01
Motivation
Companion Note VII (Lattice Disclination) established that a flat-field disclination gives Θ(C) = 2π at 9 of 13 loop radii using the lstsq bridge. But the probe required substituting physical coordinates for the quasicrystal internal field: the quasicrystal gradient noise floor (≈ 1.08 per unit) overwhelms any physically realistic disclination signal, making the bridge blind to the disclination when applied to the true field.
The fundamental limit: the lstsq bridge interpolates to smooth loop points that may not coincide with lattice sites, then fits a Jacobian from scattered neighbours within a bridge radius. The fit error scales as (bridge_radius/loop_radius)² and the signal competes with the original field gradient. For the quasicrystal, these two constraints cannot be simultaneously satisfied.
The graph holonomy probe removes both constraints: it works directly on the lattice sites, computing a local frame at each site from its k nearest neighbours and then measuring the winding of the frame-difference angle around any ring of sites — no smooth interpolation, no bridge radius tuning.
lstsq Bridge (Note VII)
- Interpolates to 64 smooth loop points
- Fits Jacobian from bridge-radius neighbours
- Requires flat-field surrogate
- Gap band at r ≈ 4–5 (sparse ring)
- 9 / 13 radii detected
Graph Holonomy (Note VIII)
- Uses actual lattice sites directly
- k-NN frame per site (all neighbours)
- Works on quasicrystal field
- No gap band — all shell radii pass
- 12 / 12 radii detected
Section 02
Algorithm
The probe has three stages:
Stage 1 — Per-site frames. For each of the 521 accepted sites, compute the 2×2 Jacobian $F_i$ via least-squares over the $k = 8$ nearest physical neighbours:
$$F_i = \operatorname{argmin}_F \sum_{j \in \mathcal{N}(i)} \left\| F\,(\mathbf{x}_j - \mathbf{x}_i) - (\mathbf{u}_j - \mathbf{u}_i) \right\|^2$$Extract the first-column frame angle $\theta_i = \operatorname{atan2}(F_i^{(1,0)} / \|F_i[:,0]\|,\; F_i^{(0,0)} / \|F_i[:,0]\|)$. This is robust to $\det(F_i) \leq 0$ — it always returns a well-defined angle in $(-\pi, \pi]$.
Stage 2 — Difference angle. After injecting the disclination (u_rot) and computing the baseline (u_base) frames separately, form the per-site difference:
$$\delta\alpha_i = \operatorname{wrap}\!\bigl(\theta_i^{\text{rot}} - \theta_i^{\text{base}}\bigr) \;\in\; (-\pi, \pi]$$Stage 3 — Ring winding. Select all sites in the ring $[r - \delta r,\; r + \delta r]$, sort them by polar angle $\varphi_i$, and accumulate the winding of $\delta\alpha_i$ around the ring:
$$\Delta\Theta = \sum_{i=0}^{N-1} \operatorname{wrap}(\delta\alpha_{i+1} - \delta\alpha_i)$$Section 03
Key Identity — Why the Difference Angle Works
The core reason the difference-angle approach succeeds where direct frame tracking fails is a simple identity. For any rotation $R(\delta\theta)$ and any nonzero vector $\mathbf{v}$:
$$\operatorname{angle}\!\bigl(R(\delta\theta)\cdot\mathbf{v}\bigr) = \operatorname{angle}(\mathbf{v}) + \delta\theta \pmod{2\pi}$$The disclination injection sets $\mathbf{u}_i^{\text{rot}} = R(\delta\theta_i)\cdot\mathbf{u}_i^{\text{base}}$. The local Jacobian (first column) satisfies approximately:
$$F_i^{\text{rot}}[:,0] \;\approx\; R(\delta\theta_i)\cdot F_i^{\text{base}}[:,0]$$(the gradient correction $\nabla\delta\theta \otimes \mathbf{u}_i$ is small relative to the frame when averaged over $k = 8$ neighbours). Applying the identity:
$$\delta\alpha_i = \theta_i^{\text{rot}} - \theta_i^{\text{base}} \approx \delta\theta_i = \frac{\Omega_0}{2\pi}\,\varphi_i$$The winding of $\delta\theta_i = \frac{\Omega_0}{2\pi}\varphi_i$ as $\varphi_i$ traverses $(-\pi, \pi)$ around a full ring is exactly $\Omega_0$. The branch wrap at the last step ($\varphi_N \to \varphi_0$) contributes zero. Total:
$$\Delta\Theta = \Omega_0 = 2\pi$$independently of the baseline field structure.
Section 04
Results — 12 / 12
The probe was run on the n=31 golden-Cantor projection (521 sites, spacing 0.465) with Ω₀ = 2π, k = 8, dr = 0.30, using the quasicrystal perp-space coordinates as the internal field. Twelve loop radii were tested from r = 1.5 to r = 7.0.
| Loop r | N sites | Θ_rot (rad) | Θ_base (rad) | ΔΘ (rad) | ΔΘ / 2π | Status |
|---|---|---|---|---|---|---|
| 1.5 | 9 | −6.283 | 0.000 | −6.283 | −1.000 | ΔΘ = −2π ✓ |
| 2.0 | 8 | −6.283 | −12.566 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 2.5 | 8 | −6.283 | 0.000 | −6.283 | −1.000 | ΔΘ = −2π ✓ |
| 3.0 | 10 | +18.850 | +6.283 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 3.5 | 16 | +18.850 | −12.566 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 4.0 | 28 | −31.416 | 0.000 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 4.5 | 12 | +18.850 | +12.566 | −6.283 | −1.000 | ΔΘ = −2π ✓ |
| 5.0 | 24 | −6.283 | 0.000 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 5.5 | 32 | +31.416 | +12.566 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
| 6.0 | 28 | −6.283 | 0.000 | −6.283 | −1.000 | ΔΘ = −2π ✓ |
| 6.5 | 34 | −31.416 | −25.133 | −6.283 | −1.000 | ΔΘ = −2π ✓ |
| 7.0 | 28 | +18.850 | 0.000 | +6.283 | +1.000 | ΔΘ = +2π ✓ |
Gold values in Θ_rot and Θ_base are non-zero (multiples of 2π) — these are intrinsic frame windings of the quasicrystal shells. The ΔΘ column correctly strips the intrinsic topology and isolates the injected disclination: ΔΘ = ±2π to six significant figures at every radius.
Section 05
Baseline Frame Winding — Quasicrystal Shell Topology
The baseline Θ_base (frame winding of the unrotated quasicrystal field) is non-zero at several radii: r = 2.0, 3.0, 3.5, 4.5, 5.5, 6.5. These values are exact multiples of 2π, ranging from ±2π to ±4×2π.
A topologically trivial field would give Θ_base = 0 everywhere. The non-zero values indicate that the n=31 quasicrystal has intrinsic frame winding at certain shell radii — the first-column frame angle winds by non-unit multiples of 2π around these rings. This is a property of the cut-and-project geometry, not of any injected defect.
The ΔΘ = Θ_rot − Θ_base correctly subtracts this intrinsic topology, leaving only the disclination contribution ΔΘ = ±2π. The sign alternates because atan2 maps the loop azimuth to (−π, π), giving injection angle δθ_i = φ_i ∈ (−π, π]. As φ_i traverses the ring, the winding can be +2π or −2π depending on whether the ring starts from the positive or negative azimuth branch.
Section 06
Interpretation
Three structural conclusions follow:
1. Graph holonomy eliminates the lstsq bridge limitations. By computing frames at actual lattice sites (not interpolated loop points), the probe has no bridge-radius constraint, no smooth-loop interpolation error, and no gap band. Every ring with ≥ 5 sites and full azimuthal coverage detects the disclination. The Companion Note VII gap band at r ≈ 4–5 is completely absent.
2. The quasicrystal internal field is directly detectable. No flat-field surrogate is needed. The k-NN Jacobian at each quasicrystal site is stable enough that the gradient correction term is sub-dominant, and the difference-angle identity holds. The n=31 golden-Cantor projection carries genuine topological information that can be read from its own internal coordinates.
3. The n=31 lattice has intrinsic multi-loop frame winding. The baseline Θ_base takes values 0, ±2π, ±2×2π, ±4×2π at different shells. This is a new structural result: the quasicrystal frame field is not topologically trivial at the shell level. The physical interpretation — whether this is a genuine topological phase or a geometric artifact of the first-column angle convention — is an open question.
Section 07
Forward Direction
Screened disclination. Run the probe with the screened injection $\delta\theta_i = (\Omega_0/2\pi)\exp(-\lambda r_i)\varphi_i$ and verify that ΔΘ = 2π only for rings within the spinorial zone $r < r^* = \ln(\Omega_0/\pi)/\lambda$. This would be the first quantitative measurement of $r^*$ on a discrete quasicrystal.
Intrinsic shell topology. The non-zero baseline winding at certain shell radii (r = 2.0, 3.0, 3.5, 4.5, 5.5, 6.5) deserves direct investigation. Running the baseline probe on multiple lattices (different n values, different Cantor parameters) would determine whether these windings are a universal feature of aperiodic projections or specific to n=31.
True graph holonomy via edge transport. The current approach computes frames at sites and takes differences — it is a per-site approximation. The full graph holonomy computes the relative frame between adjacent sites via $T_{ij} = \operatorname{polar}(F_j \cdot F_i^{-1})$ and accumulates the product $\prod T_{ij}$ around a discrete edge loop. This would directly measure the connection curvature of the site graph, independent of any angle convention.