Defect Seeding — Companion Note VI

I

The Prediction

Companion Note V (Transport Law) derived the boundary-local helix as the unique solution to a screened Laplace equation and extended it to 2D via the Green's function $G(r) = K_0(\lambda r) / (2\pi)$. Near a point phason source at $\mathbf{x}_0$, the rotation angle becomes logarithmic:

$$\theta(\mathbf{x}) \;\approx\; -\frac{\alpha_0}{2\pi}\,\ln|\mathbf{x} - \mathbf{x}_0| \quad (r \ll 1/\lambda)$$

As $\mathbf{x}$ traverses a loop $\mathcal{C}$ encircling the source, the total accumulated frame rotation is predicted to be

$$\Theta(\mathcal{C}) \;=\; \alpha_0 \cdot e^{-\lambda r}$$

where $r$ is the loop radius and $\lambda$ is the inverse phason screening length. For $\alpha_0 = 2\pi$ (a single spinorial dislocation), this satisfies the spinorial criterion $\Theta \equiv 2\pi \pmod{4\pi}$ near the core.

Companion Note VI tests this numerically. Three modes are run: a pure disclination (analytic frame rotation), a displacement vortex ($F = I + \nabla u$, analytic), and a discrete lattice injection via the lstsq bridge. Parameters: $\alpha_0 = 2\pi$, $\lambda = 0.145$, $1/\lambda = 6.897$.

II

Mode 1 — Pure Disclination

A disclination is a defect in the frame field itself: the local frame rotates by $\Omega = \alpha_0 e^{-\lambda r}$ as we traverse the loop. At each of the $N = 360$ loop steps, the tetrad is a pure rotation matrix:

$$F_k \;=\; R\!\left(\Omega \cdot \frac{k}{N}\right) \;=\; \begin{pmatrix} \cos(\Omega k/N) & -\sin(\Omega k/N) \\ \sin(\Omega k/N) & \cos(\Omega k/N) \end{pmatrix}$$

This is orientation-preserving ($\det F_k = 1$) everywhere on the loop. The winding audit accumulates the unwrapped frame angle increments.

r	Ω = α₀ e^−λr	Θ (rad)	Θ / 2π	Status
0.06897	6.2207	6.28319	1.0000	✅ spinorial
0.20690	6.0975	6.28319	1.0000	✅ spinorial
0.68966	5.6853	6.28319	1.0000	✅ spinorial
1.37931	5.1442	6.28319	1.0000	✅ spinorial
3.44828	3.8109	6.28319	1.0000	✅ spinorial
4.82759	3.1201	0.00000	0.0000	▪ Ω < π
6.89655	2.3115	0.00000	0.0000	▪ Ω < π
13.79310	0.8503	0.00000	0.0000	▪ Ω < π
34.48276	0.0423	0.00000	0.0000	▪ Ω < π

Result: 5/9 loops spinorial. The transition is at r* ≈ 4.78, where Ω(r*) = π. The reported Θ is exactly 2π for all passing radii, not the analytic Ω. This is explained in Section V.

III

Mode 2 — Displacement Vortex

A displacement vortex is the other natural candidate: apply a vortex displacement field in internal space with a scalar potential $u_x = (b/2\pi)\,\mathrm{arg}(\mathbf{x})$. The gradient gives the tetrad

$$F \;=\; I + \nabla u \;=\; \begin{pmatrix} 1 - B\sin\phi & B\cos\phi \\ 0 & 1 \end{pmatrix}, \quad B = \frac{b}{2\pi r}$$

The winding criterion requires $B > 2$ (so the complex vector $\mathbf{z} = (2 - B\sin\phi) + i(-B\cos\phi)$ winds around the origin). But $\det F = 1 - B\sin\phi$, which is negative near $\phi = \pi/2$ whenever $B > 1$.

Structural exclusion: The spinorial regime ($B > 2$, i.e. $r < b/4\pi$) requires $B > 2 > 1$, which forces $\det F < 0$ on part of the loop. The winding audit flags these as bad steps and assigns $\phi = 0$, corrupting $\Theta$. The two regimes — spinorial and orientation-preserving — are mutually exclusive for a displacement vortex.

Numerically: at $r = 0.069$ (where $B = 4.6$), 155 of 360 steps have $\det F \leq 0$. The winding audit returns $\Theta = 0$ despite $B > 2$. At all larger radii $B < 1$ and $\Theta = 0$ by the flat criterion. No spinorial detection is possible.

This rules out the displacement vortex as a model for the SLH spinorial defect. The correct model must be a disclination — a rotation in the frame field, not a translation of the displacement field.

IV

Mode 3 — Lattice Injection

The discrete n=31 golden-Cantor projection (521 accepted sites, limit 8) receives the displacement vortex $u_x \mathrel{+}= (b/2\pi)\,\mathrm{atan2}(y - y_0, x - x_0)$ at all accepted sites, then the lstsq bridge estimates the tetrad at each loop step.

Scale constraint: For the bridge to detect the vortex, the Burgers magnitude $b$ must satisfy $b > 4\pi \times \mathrm{spacing}$. With median spacing $= 0.465$, this requires $b > 5.85$. The probe uses $b = 2.0$, falling short by a factor of $\times 2.9$. All 9 radii return $\Theta = 0$, as predicted by the scale analysis.

The deficit is not a failure of the prediction — it is a measurement of the minimum Burgers magnitude needed for the bridge to resolve the vortex. The discrete bridge averages over a neighborhood of radius $r_\text{bridge} = 0.35$; this averaging smooths out the vortex gradient when the vortex core ($r_c = b/4\pi = 0.159$) is smaller than the averaging scale.

A detectable discrete experiment requires either a much larger Burgers vector ($b \gtrsim 6$) or a much finer lattice (limit $\geq 20$, giving spacing $\lesssim 0.1$). Both are computationally accessible but outside the scope of the current probe.

V

The Topological Threshold

In Mode 1, the reported $\Theta$ is exactly $2\pi$ for all passing radii, not the analytic $\Omega = \alpha_0 e^{-\lambda r}$. This is not a numerical error — it is the correct topological behavior. Here is why.

The winding audit accumulates increments $\Delta\phi_k = \mathrm{principal\_angle}(\phi_{k+1} - \phi_k)$. For the disclination with $N$ steps, all forward bonds contribute $\Omega/N$. The closing bond (step $N-1$ back to step $0$) contributes $\mathrm{principal\_angle}(-\Omega + \Omega/N)$:

$$\Theta = (N-1)\cdot\frac{\Omega}{N} + \mathrm{principal\_angle}\!\left(-\Omega + \frac{\Omega}{N}\right)$$

When $\Omega > \pi$: the closing bond wraps, contributing $(-\Omega + \Omega/N + 2\pi)$. The total is $\Omega - \Omega + 2\pi = 2\pi$. When $\Omega < \pi$: no wrap, closing bond contributes $-\Omega + \Omega/N$. Total $\approx 0$.

The winding audit is a topological counter: it returns exactly $0$ or $2\pi$ (or integer multiples), never a fractional value. The spinorial criterion is satisfied if and only if $\Omega > \pi$, i.e.

$$\alpha_0\, e^{-\lambda r} > \pi \;\;\Longleftrightarrow\;\; r < r^* = \frac{\ln(\alpha_0/\pi)}{\lambda} \;=\; \frac{\ln 2}{\lambda} \;\approx\; \frac{0.693}{\lambda}$$

For $\alpha_0 = 2\pi$, $\lambda = 0.145$: $r^* = \ln 2 / 0.145 = 4.78$. The five passing radii ($r = 0.069, 0.207, 0.690, 1.379, 3.448$) all satisfy $r < 4.78$; the four failing radii ($r = 4.828, 6.897, 13.793, 34.483$) do not. The probe data is in exact agreement with this analytic threshold.

VI

Physical Diagnosis

The three modes together deliver a coherent picture:

1. The correct defect type is a disclination. A disclination is a rotation defect in the frame field: the local frame rotates by $\Omega$ as the loop traverses the defect. This is distinct from a displacement dislocation, where the material field $u$ acquires a vortex but the frame orientation stays smooth (except for regions of negative Jacobian). Only the disclination gives $\Theta = 2\pi$ in an orientation-preserving frame.

2. The spinorial zone is r < ln(2)/λ ≈ 0.693/λ. With $\lambda = 0.145$ this is $r^* \approx 4.78$ — roughly 70% of the screening length $1/\lambda = 6.9$. Inside this zone, the disclination charge $\Omega > \pi$ and the winding audit returns $\Theta = 2\pi$ (topologically robust). Outside it, $\Omega < \pi$ and $\Theta = 0$.

3. A displacement vortex cannot produce a spinorial signature. The orientation-preserving condition ($\det F > 0$) and the spinorial condition ($B > 2$) require $B \in (1, 2)$ — an empty intersection. The Transport Law paper (Companion Note V) described a "logarithmic vortex" near $r \to 0$, which was correct in the sense that $G(r) = K_0(\lambda r)$ has logarithmic divergence; but the frame rotation (not the displacement) is what creates the spinorial signature.

4. The discrete bridge requires scale separation. For the lattice mode to detect the disclination, the Burgers vector must exceed $4\pi \times$ lattice spacing. At n=31 with limit 8 (spacing 0.465), this requires $b > 5.85$. This is a well-defined experimental condition, not an open-ended caveat.

VII

Forward Direction

Three experiments follow directly from this result:

Lattice disclination (immediate, 2–4h). Instead of injecting a displacement vortex, directly inject a frame rotation vortex: rotate the internal coordinates of accepted site $i$ by $\delta\theta_i = (\Omega_0/2\pi)\,\mathrm{atan2}(y_i - y_0, x_i - x_0)$. This is a genuine disclination in the discrete lattice. The lstsq bridge should then recover $\Theta \approx 2\pi$ for loops with $r < r^*$.

Spinorial zone mapping (medium, 1–2 days). Sweep $\lambda$ values (i.e., Cantor gap widths) and measure $r^*(\lambda) = \ln 2 / \lambda$. This maps the spinorial zone as a function of the phason screening length — a testable prediction for any specific quasicrystal geometry.

Dislocation vs. disclination census (open). The SLH boundary helix (Companion Note V) rotates internal coordinates in opposite directions on each side of the Cantor gap. This is structurally a disclination field (rotation defect), not a dislocation field. Confirming that the Cantor-gap structure is a disclination — not just a displacement jump — would complete the physical identification and directly connect the sharp bridge no-go result to the correct defect taxonomy.

Addendum (Companion Note VII). The lattice disclination probe confirms that the discrete n=31 bridge recovers Θ = 2π at 9 of 13 loop radii. A sparse-ring gap at r ≈ 4–5 interrupts detection and encodes the n=31 shell structure as a topological fingerprint. See Companion Note VII — Lattice Disclination for the full result.

Defect SeedingThe Spinorial Zone