Flat-Field Screened Holonomy
The Graph Detection Limit

Section 02

Analytic Prediction

With the flat internal field the continuum spinorial criterion applies without modification. The screened injection has effective strength $\Omega_{\mathrm{eff}}(r) = \Omega_0\,e^{-\lambda r}$ at radius $r$. Spinorial detection requires $\Omega_{\mathrm{eff}} > \pi$, giving

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

Seven of the 12 probe radii lie inside $r^*$; the outer five do not. Because the baseline field is trivially flat ($\Theta_{\mathrm{base}} = 0$ everywhere), no shell-resonance amplification is possible. This probe should deliver the purest available measurement of whether the discrete graph holonomy method can actually resolve $r^*$.

Section 03

Results

The flat-field probe detects $\Delta\Theta = 2\pi$ at only 3 of 12 radii. All three passes are in the innermost cluster ($r = 1.5, 2.0, 2.5$); every radius from $r=3.0$ outward produces $\Delta\Theta = 0$ exactly, including four radii well inside $r^* = 4.78$. The Note IX quasicrystal comparison is included for direct contrast.

⚠ = unexpected result.   ★ = quasicrystal-only pass (resonance amplification from Note IX).   Θ values shown as multiples of 2π.   Flat-field baseline: $\Theta_{\mathrm{base}} = 0$ at all radii.

Section 04

The Topological Snap

A striking feature of the three passing radii is that $\Delta\Theta/2\pi = -1.000$ exactly, even though the screened injection amplitude at $r=1.5$ is only $C = e^{-\lambda \cdot 1.5} \approx 0.80$ — not 1.0. Naively one might expect a graded response where $\Delta\Theta/2\pi \approx -C$.

The explanation is topological. The ring winding is computed as the cumulative sum of wrapped consecutive differences in frame angle as the probe sites traverse the ring:

$$\Theta_{\mathrm{rot}} = \sum_{\text{sites}} \mathrm{wrap}\!\left(\theta^{\mathrm{rot}}_{i+1} - \theta^{\mathrm{rot}}_i\right).$$

When the screened injection is coherent across the ring — meaning each per-site frame angle tracks the azimuthal pattern $\theta^{\mathrm{rot}}_i \approx -C\,\varphi_i$ without jumps — successive differences are small and the sum accumulates smoothly. The wrap operation then collapses the total to exactly $-2\pi$ regardless of whether $C$ is 0.80 or 1.00, because the angular winding of a monotonically varying function over a full circle is always an integer multiple of $2\pi$.

When the injection is incoherent — per-site noise overwhelms the azimuthal signal — individual differences scatter randomly and the wrapped sum cancels to zero. There is no intermediate state: detection is binary.

Section 05

The Discrete Graph Detection Limit

The flat-field result establishes a concrete bound: the graph holonomy method with $k=8$ nearest neighbours can only detect the screened disclination at $r \leq 2.5$, failing at $r \geq 3.0$ despite $\Omega_{\mathrm{eff}} > \pi$. This defines the discrete graph detection limit $r^*_{\mathrm{disc}}$, estimated between 2.5 and 3.0:

$$r^*_{\mathrm{disc}} \approx 2.7 \quad \ll \quad r^* = 4.78.$$

Mechanism: per-site noise vs. injection signal

Each site’s frame angle is computed by fitting a 2×2 Jacobian to its $k=8$ nearest-neighbour displacements via least squares. On the aperiodic quasicrystal lattice the neighbours are never perfectly distributed, so the lstsq frame carries inherent noise $\sigma_{\theta}$ that scales with local irregularity. In the flat-field regime the baseline noise is constant (the identity Jacobian is exact), but the rotated frame noise grows because the screened injection mixes azimuthal and radial gradients across the $k$-NN neighbourhood.

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

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

The azimuthal term $e^{-\lambda r}\nabla\varphi$ provides the coherent winding signal. The radial term $-\lambda e^{-\lambda r}\varphi\,\hat{r}$ is an incoherent radial perturbation that enters the lstsq Jacobian fit and rotates each site’s frame away from the azimuthal direction. The signal-to-noise ratio is approximately

$$\mathrm{SNR}(r) \;\sim\; \frac{e^{-\lambda r}/r}{\lambda\,e^{-\lambda r}} = \frac{1}{\lambda r}.$$

At $r=2.5$: $\mathrm{SNR} \approx 1/(0.145 \times 2.5) = 2.76$.   At $r=3.0$: $\mathrm{SNR} \approx 1/(0.145 \times 3.0) = 2.30$. The coherence threshold sits between these values. Beyond $r \approx 2.7$ the radial gradient term wins and the per-site frames scatter incoherently, driving the winding to zero.

Section 06

Contrast with Note IX: Resonance Enhancement

Comparing the two screened probes side by side reveals what the quasicrystal’s own topology contributes to holonomy detection:

The flat field misses $r=1.5$ in Note IX and gains $r=1.5$ and $r=2.5$ that Note IX misses — even where both probes use exactly the same injection. The difference is the baseline field. The quasicrystal’s per-site frame angles are not isotropic; they carry directional structure inherited from the 4D cut-and-project geometry. This structure sometimes destructively interferes with the azimuthal injection signal (inner failures in Note IX) and sometimes constructively amplifies it (outer passes).

Shell resonance as range extension

The two outer passes in Note IX ($r=4.0$ and $r=6.5$) have no flat-field analogue: without $\Theta_{\mathrm{base}} \neq 0$, there is no topological lever to amplify the sub-threshold screened injection. These passes are genuinely new physics — detection events enabled entirely by the quasicrystal’s intrinsic multi-loop shell winding, which acts as a pre-amplifier for weak signals.

The flat-field experiment is therefore the null control for the resonance lattice hypothesis: it confirms that the outer passes in Note IX cannot arise from the injection alone, and that quasicrystal topology is at least a necessary additional ingredient in this probe family.

Section 07

Interpretation

Together, Notes IX and X establish a two-layer picture of discrete graph holonomy detection for screened disclinations on the n=31 golden-Cantor lattice:

Layer 1: geometric probe limit. The $k$-NN least-squares frame method has an intrinsic coherence cutoff $r^*_{\mathrm{disc}} \approx 2.7$ for the current parameters. Below this radius the screened injection remains coherent across the ring and the winding snaps to $\pm 1$. Above it, the injection’s radial gradient contaminates the per-site frames and the ring winding collapses to zero. This cutoff is a measurement artefact, not physics: it sets the floor below which any probe using the same method will detect the spinorial zone, regardless of field.

Layer 2: lattice topology amplification. On the quasicrystal field, radii with intrinsic $\Theta_{\mathrm{base}} \neq 0$ can detect sub-threshold injections via shell resonance. This extends effective detection range beyond $r^*_{\mathrm{disc}}$ at specific radii, creating the resonance lattice. The extension is not uniform or monotone; it depends on the sign and magnitude of $\Theta_{\mathrm{base}}$ at each shell, which in turn encodes the projection geometry of the 4D Cantor cut.

The analytic $r^*$ as an asymptotic limit. Neither probe detects a clean transition at $r^* = 4.78$. The flat-field probe cuts off far earlier ($r^*_{\mathrm{disc}} \approx 2.7$); the quasicrystal probe scatters detections across the full radial range. The analytic $r^*$ is recovered only in the limit of an ideal continuum probe with infinite radial resolution — an asymptote that the discrete graph method approaches as $k \to \infty$, $\delta r \to 0$.

Section 08

Forward Direction

Raising $r^*_{\mathrm{disc}}$ by increasing $k$. The detection limit scales as $r^*_{\mathrm{disc}} \sim 1/(\lambda k^{1/2})$ if the noise averages down as $k^{-1/2}$. Running the flat-field probe with $k = 12$ and $k = 16$ would test this scaling and establish whether $r^*_{\mathrm{disc}}$ can be lifted to the analytic $r^* = 4.78$.

Edge-transport frames. Replacing per-site lstsq frames with the exact edge-transport matrix $T_{ij} = \mathrm{polar}(F_j \cdot F_i^{-1})$ accumulated around a discrete edge loop eliminates the neighbourhood-averaging artefact entirely. In this regime $r^*_{\mathrm{disc}}$ should equal $r^*$ for any coherent injection, providing the first truly exact measurement of the spinorial zone boundary on a discrete lattice.

Screened colour mode in Lattice Explorer. The per-site flat-field diff angles (Note X) and quasicrystal diff angles (Note IX) can both be added to the Lattice Explorer as additional colour modes, making the contrast between the two probes immediately visual: the flat-field mode will show a smooth azimuthal gradient in the inner core fading to noise at $r \approx 2.7$, while the quasicrystal mode will show structured resonance patterns at the outer shells.

Lowering $\lambda$ to widen the spinorial zone. Reducing the screening rate (e.g. $\lambda = 0.07$, doubling $r^*$ to $\approx 9.9$) would move the analytic spinorial zone well beyond the current probe radii, making multiple outer shells fall inside $r^*$ and giving the resonance amplification mechanism more shells to work with. This would test whether the resonance lattice becomes denser as $r^*$ grows, or whether shell geometry limits it.

Series status. The flat-field probe completes the two-experiment screened holonomy comparison (Notes IX and X). The next natural step is edge-transport holonomy (Note XI), which would replace the lstsq frame approximation with exact per-edge parallel transport and deliver the definitive test of whether the discrete quasicrystal graph encodes a clean spinorial zone boundary.