k-Sweep Holonomy
The Non-Monotone Detection Limit

Section 02

The SNR Scaling Hypothesis

In the Note X SNR model, the per-site Jacobian estimation noise scales as $k^{-1/2}$ while the azimuthal injection signal is fixed. The radial gradient contamination dominates when $1/(\lambda r) \lesssim 1$, i.e. at $r \gtrsim 1/\lambda \approx 6.9$. Crucially, this expression does not depend on $k$.

If $k$ suppresses the per-site noise floor by $k^{-1/2}$, the effective signal-to-noise becomes:

$$\mathrm{SNR}_k(r) \;\sim\; \frac{\sqrt{k}}{\lambda r}.$$

The coherence threshold $\mathrm{SNR}_k = 1$ then gives a detection limit $r^*_{\mathrm{disc}}(k) = \sqrt{k} / \lambda$. For $k = 8, 12, 16, 20$:

These predictions are all far above the largest probe radius ($r=7.0$), so the simple model predicts that all 12 radii should pass for all four $k$ values. The Note X observation of only 3 passes at $k=8$ already suggests the SNR model is incomplete — the k-sweep tests whether it can be rescued by varying $k$.

Section 03

Results

The k-sweep produces at most 3 passes at any $k$ value, never approaching the 7-pass continuum prediction. Total passes are similar across $k$, but the identity of the passing radii changes substantially:

★ = k-sensitive pass (detects at this k only).   All Θ_base = 0 (flat field, no intrinsic topology).   No pass above r = 3.0 at any k.

Section 04

Non-Monotone Detection

Three sites exhibit k-sensitive detection that directly contradicts the SNR hypothesis:

r = 2.5 — passes at k=8, fails at k=12 and k=16, passes at k=20

If the SNR model were correct, increasing $k$ would make detection easier at $r=2.5$. Instead, k=12 and k=16 fail at a radius that k=8 detects. The ring recovers at k=20 — not through improved averaging, but because the 20-nearest-neighbour ball at each ring site encloses a different subset of the quasicrystal lattice, producing a different frame angle pattern that happens to wind coherently.

r = 3.0 — fails at k=8, passes only at k=12

At k=8 this radius falls just outside the detection cluster (between the last pass at r=2.5 and the first fail at r=3.0, establishing $r^*_{\mathrm{disc}} \approx 2.7$). At k=12 the 12-NN neighbourhood at ring sites near $r=3.0$ apparently aligns the per-site frame angles into a coherent azimuthal pattern. At k=16 and k=20 this alignment breaks.

r = 1.5 and 2.0 — pass at k=8,12,16 but fail at k=20

These two innermost ring radii are the most robust detectors across the sweep, but even they fail at k=20. At k=20, the neighbourhood ball radius ($\approx 2.3$ lattice spacings) is large enough to include sites from a wide range of radii. At $r=1.5$ the ball extends to $r \approx 0.3$, well into the disclination core where the screened injection is qualitatively different. The neighbourhood-averaged Jacobian loses coherence.

Section 05

Mechanism: Neighbourhood Geometry Coupling

The SNR model treated the $k$-NN neighbourhood as a uniform averaging ball, assuming the quasicrystal sites are approximately uniform in 2D density. This assumption fails in two ways on the n=31 golden-Cantor projection:

1. Aperiodic clustering

The quasicrystal lattice has local density fluctuations of order $\pm 40\%$ on scales of 1–3 lattice spacings. The $k$-NN ball is not a clean disk — it includes irregular clusters of sites that happen to be close. As $k$ grows, the ball boundary snakes through the aperiodic lattice in a way that is highly radius-dependent and unpredictable from the continuum density alone.

2. Radial mixing

At ring radius $r$ with neighbourhood ball radius $r_k \approx \sqrt{k/\pi} \times s$ (where $s \approx 0.465$ is the median spacing), the neighbourhood at each ring site spans radii $[r - r_k, r + r_k]$. For k=20, $r_k \approx 1.07$. At $r=1.5$ this gives $[0.43, 2.57]$ — a wide radial band where the screened injection amplitude varies from $e^{-0.145 \times 0.43} \approx 0.94$ to $e^{-0.145 \times 2.57} \approx 0.69$. The lstsq Jacobian averages over sites with quite different injection strengths, destroying the azimuthal coherence of the frame angle estimate.

For k=8, $r_k \approx 0.74$. At $r=1.5$ the radial band is $[0.76, 2.24]$, narrower but still spanning a factor of $e^{0.145 \times 1.48} \approx 1.24$ in injection amplitude. The detection still works at k=8 because the number of ring sites is small (9) and each frame happens to be coherent at this particular quasicrystal geometry.

Section 06

Interpretation

Together, Notes X and XI establish that the per-site lstsq frame holonomy method has a geometry-dependent detection limit that cannot be escaped by tuning the neighbourhood size:

No optimal k exists in the sense of detecting all $r < r^* = 4.78$. The best observed pass count across any single $k$ is 3/12, far below the continuum prediction of 7/12.

r*_disc is lattice-site-specific, not radius-specific. At each ring radius, detection depends on the exact positions of the $k$-NN neighbours of every ring site. This is a discrete-geometry phenomenon — there is no continuum limit of the $k$-NN ball that captures it.

The SNR hypothesis failed qualitatively. Predictions of $r^*_{\mathrm{disc}}(k) \gg 7$ (all radii pass) contrast with observations of at most 3 passes and declining count at $k=20$. The model's fundamental error was treating the quasicrystal neighbourhood as a smooth averaging ball and ignoring the radial mixing of injection amplitudes.

Section 07

Forward Direction

True graph-cycle holonomy (Note XII). Find actual closed loops $\gamma$ in the k-NN graph that encircle the origin at each target radius. For each edge $(i \to j)$ in $\gamma$, the edge Jacobian is not the per-site averaged $F_i$ but the directional derivative of the field from $i$ to $j$:

$$J_{ij} = \frac{\mathbf{u}_j - \mathbf{u}_i}{|\mathbf{x}_j - \mathbf{x}_i|^2} (\mathbf{x}_j - \mathbf{x}_i)^T.$$

The edge transport $T_{ij} = \mathrm{polar}(J_{ij})$ is a single-edge rotation that does not involve any neighbourhood averaging. Its product around $\gamma$ gives the discrete holonomy without radial mixing.

Fine-grid injection sweep. The k=12 result (r=3.0 passes, r=2.5 fails) suggests that the detection landscape has local maxima and minima at specific radii. Sweeping $r$ in fine steps of 0.1 at k=12 would map the full pass/fail pattern between r=1.0 and r=4.5, testing whether there are isolated pass islands beyond r=3.0.

Quasicrystal field k-sweep (companion to this note). Running the k-sweep with the quasicrystal internal field (rather than flat-field) would test whether the resonance amplification mechanisms of Note IX are also k-sensitive. If the outer passes at r=4.0 and r=6.5 survive at k=12,16, it would suggest the resonance is robust to neighbourhood size; if they break, it means resonance and k-NN geometry are coupled.

Series status. The k-sweep (Note XI) and flat-field probe (Note X) together close the $k$-NN Jacobian chapter: per-site averaged frames cannot resolve the analytic spinorial zone boundary $r^* = 4.78$ regardless of $k$. The path forward is edge-transport holonomy on true lattice graph cycles (Note XII).