Screened Holonomy
The Quasicrystal Resonance Lattice

Section 02

Spinorial Zone Prediction

The screened disclination has an effective injection strength at radius $r$ of $\Omega_{\mathrm{eff}}(r) = \Omega_0\,e^{-\lambda r}$. In the continuum spinorial criterion (Companion Note II), a loop at radius $r$ lies inside the spinorial zone when $\Omega_{\mathrm{eff}} > \pi$, which gives

$$r < r^* = \frac{\ln(\Omega_0/\pi)}{\lambda} = \frac{\ln 2}{0.145} \approx 4.78.$$

Applying this to the 12 probe radii: 7 lie inside $r^*$ and should produce $\Delta\Theta = 2\pi$; the outer 5 should produce $\Delta\Theta \approx 0$.

The effective injection at $r^* = 4.78$ is exactly $\pi$, and at the innermost probe radius $r=1.5$ it is $\Omega_{\mathrm{eff}} = 2\pi \cdot e^{-0.218} \approx 5.06$ — well above $\pi$, so all seven inner rings should exhibit clear threshold-model detection.

Section 03

Results

The probe ran in 521-site mode with baseline comparison. 4 of 12 radii pass — against the continuum prediction of 7. The boundary $r^*$ is not visible as a clean transition.

⚠ = inner failure (predicted PASS, actual FAIL).   ★ = outer anomaly (predicted FAIL, actual PASS).   Θ values shown as multiples of 2π.

Section 04

Inner Passes — r = 2.0 and r = 4.0

Two radii inside the spinorial zone produce the expected $\Delta\Theta = 2\pi$: $r = 2.0$ (8 ring sites) and $r = 4.0$ (28 ring sites). Both are consistent with the continuum spinorial criterion: at these radii $\Omega_{\mathrm{eff}} \approx 1.50\pi$ and $1.12\pi$ respectively, comfortably above the spinorial threshold of $\pi$.

Notably, both passing inner rings have zero or near-zero baseline winding ($\Theta_{\mathrm{base}} = 0$ at $r=4.0$; $\Theta_{\mathrm{base}} = -2\times 2\pi$ at $r=2.0$ but the per-site diff-angle structure still coherently winds to $+2\pi$). In both cases the screened injection successfully shifts the frame winding by one topological unit.

Section 05

Outer Anomalies — r = 6.5 and r = 7.0

Two radii outside the spinorial zone produce $|\Delta\Theta| = 2\pi$: $r = 6.5$ ($\Omega_{\mathrm{eff}} = 0.78\pi$, 34 ring sites) and $r = 7.0$ ($\Omega_{\mathrm{eff}} = 0.72\pi$, 28 ring sites). Both have effective injection well below the spinorial threshold.

Shell resonance at r = 6.5

At $r = 6.5$ the baseline frame winding is $\Theta_{\mathrm{base}} = -4 \times 2\pi$ — the largest intrinsic multi-loop winding in the n=31 lattice, identified in Note VIII as a structural signature of the quasicrystal projection geometry. The screened injection shifts $\Theta_{\mathrm{rot}}$ from $-4\times 2\pi$ to $-2\times 2\pi$, and the per-site diff-angle winding accumulates to $-2\pi$.

This is not a spinorial detection in the continuum sense. Rather, the intrinsic $-4\times 2\pi$ winding acts as a topological lever: the screened injection ($\Omega_{\mathrm{eff}} = 0.78\pi$), too weak to create a spinorial loop in isolation, is amplified by the shell's pre-existing winding structure to produce a measurable $\Delta\Theta = 2\pi$ in the diff-angle winding.

Shell resonance at r = 7.0

At $r = 7.0$, $\Theta_{\mathrm{base}} \approx 0$ and $\Theta_{\mathrm{rot}} \approx +2\pi$ with $\Delta\Theta = +2\pi$. This is the outer edge of the lattice (sites become sparse beyond $r \approx 7.5$). The result is consistent with a weak but coherent azimuthal injection signal at low screening, where the low radial gradient ($\lambda e^{-\lambda \cdot 7} \approx 0.052$ per unit) allows the diff-angle winding to accumulate without cancellation from radial gradient terms.

Section 06

Inner Failures — r = 1.5, 2.5, 3.0, 3.5, 4.5

Five radii inside the predicted spinorial zone fail to produce $\Delta\Theta = 2\pi$. This is the most unexpected result: $\Omega_{\mathrm{eff}} > \pi$ at all five, ranging from $1.61\pi$ (at $r=1.5$) to $1.04\pi$ (at $r=4.5$).

The screened injection has two gradient components at each site $i$:

$$\nabla \delta\theta_i = \frac{\Omega_0}{2\pi} \Bigl[ e^{-\lambda r_i} \,\nabla\varphi_i - \lambda\,e^{-\lambda r_i}\,\varphi_i\,\hat{r}_i \Bigr].$$

The azimuthal term $e^{-\lambda r}\nabla\varphi$ drives the winding accumulation. The radial term $-\lambda e^{-\lambda r}\varphi\,\hat{r}$ adds a competing gradient that enters each site's Jacobian and shifts the per-site diff angle away from the pure azimuthal $\delta\theta_i$. At small $r$ this radial term is largest — the gradient of the screening factor ($\lambda e^{-\lambda r}$) is steepest near the core — and disrupts the coherent azimuthal accumulation that makes the unscreened probe succeed at every radius.

Additionally, five of the failing radii ($r=1.5, 2.5, 3.0, 3.5, 4.5$) correspond to shells with specific intrinsic baseline windings ($\Theta_{\mathrm{base}} \in \{0, +2\pi, -2\times 2\pi, +2\times 2\pi\}$). At these radii the diff-angle winding accumulates to $\approx 0$, suggesting that the radial gradient terms cancel the azimuthal signal rather than amplify it.

Section 07

Interpretation

The screened holonomy result reveals a fundamental gap between the smooth continuum spinorial zone model and the discrete quasicrystal frame field:

Three mechanisms are at work:

1. Injected-holonomy detection (inner, as expected). At $r=2.0$ and $r=4.0$ the injection is strong enough and the per-site geometry compatible enough that the diff-angle winding accumulates to $\pm 2\pi$. These are genuine detections of the injected defect matching the continuum threshold model.

2. Radial gradient cancellation (inner failures). At $r = 1.5, 2.5, 3.0, 3.5, 4.5$ the screened injection's radial gradient component enters the per-site Jacobian with opposite sign to the azimuthal component, cancelling the winding accumulation. This is a discrete-lattice effect with no continuum analogue: in a smooth field the Jacobian carries exactly the local gradient, but on the quasicrystal the $k$-NN lstsq frame averages over a neighbourhood that mixes azimuthal and radial directions.

3. Shell resonance (outer passes). At $r=6.5$ the intrinsic $\Theta_{\mathrm{base}} = -4\times 2\pi$ acts as a topological amplifier: a sub-threshold screened injection shifts the frame winding by one unit ($+2\pi$), producing a detectable $\Delta\Theta = 2\pi$. This is a new phenomenon — the quasicrystal's intrinsic multi-loop shell winding (itself a structural feature of the n=31 projection geometry) creates sensitivity to weak injections at specific radii.

Section 08

Forward Direction

Flat-field screened probe. Running the screened probe with the flat-field flag ($u_0 = \mathbf{x}_i$, physical coordinates as internal field) would eliminate the quasicrystal baseline noise and isolate the screened injection signal cleanly. In this regime the identity $\Delta\theta_i \approx \delta\theta_i$ should hold site-by-site, and the ring winding should track $\Omega_{\mathrm{eff}}(r) = \Omega_0 e^{-\lambda r}$. A pass where $\Omega_{\mathrm{eff}} > \pi$ and fail where $\Omega_{\mathrm{eff}} < \pi$ would be the first clean measurement of $r^*$ on any discrete lattice.

Shell resonance mapping. The $r=6.5$ outer pass correlates with $\Theta_{\mathrm{base}} = -4\times 2\pi$. Probing a fine grid of radii around the other high-winding shells (Note VIII: $r \approx 2.0, 3.0, 3.5, 5.5$) would test whether all high-baseline shells amplify sub-threshold injections, making the resonance lattice predictable from the baseline winding alone.

True edge-transport holonomy. The per-site frame approximation averages over $k$-NN neighbourhoods that mix azimuthal and radial directions. Computing $T_{ij} = \mathrm{polar}(F_j \cdot F_i^{-1})$ along individual lattice edges and accumulating the product around a discrete edge loop would give the exact connection curvature, eliminating the neighbourhood-averaging artefact that drives the inner failures.

Lattice Explorer integration. The per-site data for the screened injection (diff angle, radial vs. azimuthal gradient contribution) can be added to the Lattice Explorer as a new colour mode, making the resonance lattice structure visually immediate.