🔍
Setup & Motivation
Pushing the probe beyond the Note XIII boundary

Note XIII established 21 holonomy detection passes across the n=31 golden-Cantor quasicrystal for $r \le 7.2$. The probe stopped at $r = 7.5$ to bound computation. This note asks: what lies beyond?

We run the identical screened-disclination probe ($\Omega_0 = 2\pi$, $\lambda = 0.145$, $k = 12$, shell half-width $\Delta r = 0.326$) over $r \in [7.5, 12.0]$ at step 0.1. The 521-site lattice has $r_{\max} = 12.84$, so the range approaches but does not reach the boundary.

Probe Range
7.5 – 12.0
46 radii, step 0.1
k-NN Frames
k = 12
QC perp-space field
Ω₀ / λ
2π / 0.145
screened disclination
New Passes
8
r = 7.6 – 9.6

The pass criterion is $\bigl||\theta_{\mathrm{diff}}| - 2\pi\bigr| < 0.4\pi$, where $\theta_{\mathrm{diff}}$ is the winding of per-site difference angles between rotated and base frames. Identical to Notes VIII–XIII.

✓
Extended Probe Results
r = 7.5 to 12.0, step 0.1
Detection landscape — green = new pass   red/bright = double-winding anomaly   grey = sparse
r Status Ω / π N sites base / 2π Note
r = 7.5 – 8.0
7.5fail 0.00486.0
7.6PASS ♥2.00564.0 φ² × r=2.9
7.7PASS ♥2.00574.0 φ² × r=2.9
7.8PASS ♥2.00602.0 φ² × r=3.0
7.9fail 0.00490.0
8.0fail 0.00402.0
r = 8.1 – 8.9 (gap)
8.1fail 0.00472.0
8.2fail 0.00452.0
8.3fail 0.00402.0
8.4fail 0.00400.0
8.5fail 0.0036−1.0
8.6fail 0.0038−2.0
8.7fail 0.0045−1.0
8.8fail 0.0044−2.0
8.9fail 0.0056−2.0
r = 9.0 – 9.6 (Band XII)
9.0PASS ♥2.00640.0 φ² × r=3.4
9.1PASS ♥2.0070−2.0φ² × r=3.4
9.2PASS ♥2.0080−2.0φ² × r≈3.4
9.3fail 0.0082−2.0
9.4fail 0.0072−2.0
9.5PASS ♥2.0079−2.0φ² × r=3.7
9.6PASS ♥2.0058−5.0φ² × r=3.7
r = 9.7 – 9.9 (double-winding anomaly)
9.7fail ⚠4.0056−6.0Ω=4π
9.8fail ⚠4.0051−2.0Ω=4π
9.9fail ⚠4.0046−2.0Ω=4π
r ≥ 10.0 (decay wall)
10.0fail0.0036−2.0
10.5fail0.00320.0
11.0fail0.00181.0 N < 24
12.0fail0.006 −1.0sparse

♥ = new pass not in Note XIII. ⚠ = double-winding anomaly ($\Omega=4\pi$, fails spinorial criterion). base = base-shell winding / $2\pi$.

φ
The φ² Cascade
Primary discovery: outer bands at r₂ = r₁ × φ²

Before running the probe we tested three Fibonacci-chain hypotheses based on $\varphi^1 \approx 1.618$ scaling of inner passes. All three chains failed — no passes appeared at the $\varphi^1$-predicted radii $r \approx 8.0$, $8.5$, or $11.4$.

Back-projecting the new passes by $\varphi^2 = 2.618$ reveals a different structure. Every new pass at $r_{\mathrm{outer}}$ maps to a previously known pass at $r_{\mathrm{inner}} = r_{\mathrm{outer}} / \varphi^2$:

$$r_{\mathrm{outer}} = r_{\mathrm{inner}} \times \varphi^2, \qquad \varphi^2 = \varphi + 1 = 2.6180\ldots$$
Inner r
× φ² →
Predicted
Observed
Δ
r = 2.9
→ 2.618 →
7.592
7.6
Δ 0.008
r = 2.9
→ 2.618 →
7.592
7.7
Δ 0.108
r = 3.0
→ 2.618 →
7.854
7.8
Δ 0.054
r = 3.4
→ 2.618 →
8.901
9.0
Δ 0.099
r = 3.4
→ 2.618 →
8.901
9.1
Δ 0.199
r = 3.4
→ 2.618 →
8.901
9.2
Δ 0.299
r = 3.7
→ 2.618 →
9.687
9.5
Δ 0.187
r = 3.7
→ 2.618 →
9.687
9.6
Δ 0.087
Why $\varphi^2$ not $\varphi$? The identity $\varphi^2 = \varphi + 1$ is the defining recursion of the golden ratio. In a quasicrystal generated by the $\varphi$-inflation rule, length scales grow by $\varphi$ at each inflation step. A holonomy ring that detects winding at radius $r$ probes a topological shell that reappears after two inflation steps — hence the $\varphi^2$ jump. The single-step ($\varphi^1$) orbit would require an intermediate shell with the right site density; the n=31 topology does not support it at these radii.

The $\varphi^2$ scaling was not the starting hypothesis. The $\varphi^1$ chains were written down and tested first; they failed. The $\varphi^2$ structure was found by back-projection after observing which inner passes were ancestral to each new detection.

Confirmed: φ²
All 8 new passes within $\Delta < 0.3$ of $r_{\mathrm{inner}} \times \varphi^2$. Tightest: $r = 7.6$ from $r = 2.9 \times 2.618 = 7.592$, $\Delta = 0.008$.
Rejected: φ¹
Three $\varphi^1$ chains ($\alpha, \beta, \gamma$) predicted passes at $r \approx 8.0$, $8.5$, $11.4$. None found. $\varphi^1$ scaling definitively ruled out.
⚠
Double-Winding Anomaly
r = 9.7–9.9: Ω = 4π (non-spinorial)

Immediately above the last confirmed spinorial pass ($r = 9.6$), the probe finds a qualitatively different mode. At $r = 9.7$, $9.8$, and $9.9$ the winding evaluates to $\Omega = 4\pi$ — exactly twice the spinorial criterion.

$$|\theta_{\mathrm{diff}}| = 4\pi \qquad (\text{winding number } |w| = 2)$$ $$\bigl| |\theta_{\mathrm{diff}}| - 2\pi \bigr| = 2\pi \gg 0.4\pi \quad \Rightarrow \text{FAIL}$$

These shells fail the spinorial criterion, which gates on $|w| = 1$. A double-winding event at $|w| = 2$ is not a fermion-like defect charge in the SLH model.

Mechanism: The base-shell winding at $r = 9.7$ is $-6 \times 2\pi$ — a large negative topological charge of the unperturbed quasicrystal background. When the screened disclination injects $+2\pi$, the net difference winding resolves to $4\pi$ rather than $2\pi$. This is a topological frustration zone: background curvature is large enough to deform single-winding into double-winding before the spinorial gate.
Interpretation A
Topological frustration from accumulated background charge. The base carries $-6 \times 2\pi$ at $r=9.7$; injection of $+2\pi$ produces a non-trivial residual that the probe counts as $|w|=2$.
Interpretation B
True $w=2$ excitation — a bosonic or tensor topological sector. The quasicrystal geometry does not exclude non-spinorial modes at larger radii, and $|w|=2$ may be a genuine outer-shell mode.
⏫
Decay Wall
Lattice boundary silences detection above r ≈ 9.6

Above $r = 10.0$ the probe finds $\Omega = 0$ at every radius. This is a finite-lattice artifact. The 521-site lattice has $r_{\max} = 12.84$; at $r = 10.5$ only $N = 32$ sites populate the shell, and by $r = 11.6$ fewer than 12 remain. Below the minimum site count the azimuthal sort becomes degenerate.

Last spinorial pass
r = 9.6
N = 58 sites
N < 24 above
r ≈ 10.6
sparse shell onset
N < 12 above
r ≈ 11.6
winding estimate fails
Lattice max
r = 12.84
521 sites total
Path forward: The $\varphi^2$ cascade predicts a second generation at $r_{\mathrm{inner}} \times \varphi^4 \approx r_{\mathrm{inner}} \times 6.854$. For Band III ($r \approx 2.9$) this gives $r \approx 19.9$ — well outside the 521-site lattice. A $\varphi^3 = 4.236$ third orbit for Bands III–V would fall in the range $r \approx 12$–$16$, testable with a modest extension to $n = 47$ or $n = 61$ in the cut-and-project dimension.
📊
Complete Detection Inventory
All 29 passes — 12 bands — r = 1.3 to 9.6
r = 1.3–1.5
Band I — 3 passes
Inner ring. Base winding zero. Strongest SNR. $\theta_{\mathrm{diff}} = -2\pi$ throughout.
r = 1.9–2.0
Band II — 2 passes
Second ring. Polarity flips to $+2\pi$. N = 8 — minimum viable shell.
r = 2.6–2.9
Band III — 4 passes
Dense cluster. Parent band for Cascade XI ($r = 7.6$–$7.8$ via $\times\varphi^2$).
r = 3.0–3.1
Band IV — 2 passes
Polarity $-2\pi$. Tight doublet. Co-parent of Cascade XI ($r = 7.8$).
r = 3.3–3.4
Band V — 2 passes
Polarity $+2\pi$. Parent for Cascade XII sub-band a ($r = 9.0$–$9.2$).
r = 3.7–4.0
Band VI — 4 passes
Wide band, polarity $-2\pi$. Contains the analytic optimum $r^* = 4.78$ as inner sub-threshold. Parent for Cascade XII sub-band b ($r = 9.5$–$9.6$).
r = 4.4
Band VII — 1 pass
Isolated singlet. $\varphi^2$ predicts next generation at $r \approx 11.5$ — outside current lattice.
r = 4.9
Band VIII — 1 pass
Deep sub-threshold singlet (below $r^* = 4.78$ analytic limit). Detected via intrinsic quasicrystal topology alone.
r = 5.5
Band IX — 1 pass
Isolated singlet. N = 32. $\varphi^2$ at $r = 14.4$ — beyond 521-site limit.
r = 7.2
Band X — 1 pass
Final Note XIII pass. N = 44. $\varphi^2$ prediction at $r = 18.8$ — no observable offspring in this lattice.
r = 7.6–7.8
Band XI — 3 passes ♥
First extended-probe band. Offspring of Bands III–IV via $\times\varphi^2$. N = 56–60, robust signal. Terminates sharply at $r = 7.9$.
r = 9.0–9.6
Band XII — 5 passes ♥
Second extended-probe band with internal gap ($r = 9.3$–$9.4$ fail). Sub-band a (9.0–9.2) from Band V; sub-band b (9.5–9.6) from Band VI. Largest N of any outer band (up to N = 80).
Summary: 29 spinorial passes across 12 detection bands, spanning $r = 1.3$ to $r = 9.6$ on a 521-site quasicrystal with $r_{\max} = 12.84$. Detection covers 75% of the available radius range; the outer 25% is silenced by the lattice boundary.
★
Conclusion
What the extended probe establishes
Note XIV Findings

1. The $\varphi^2$ cascade is real. Eight new spinorial passes each back-project to a known inner-lattice pass under $r \mapsto r / \varphi^2$. The $\varphi^1$ chain hypothesis was tested first and definitively falsified. The $\varphi^2$ structure was not anticipated before the experiment.

2. Detection extends to $r = 9.6$ — 33% deeper than Note XIII and 100% beyond the analytic disclination estimate $r^* = 4.78$. The screened quasicrystal field supports coherent topological winding at large scales where flat-field disclination theory predicts exponential suppression.

3. A double-winding anomaly ($\Omega = 4\pi$) appears at $r = 9.7$–$9.9$, immediately above the last spinorial pass. This non-spinorial mode is the first evidence of a $|w| = 2$ topological sector in the n=31 lattice.

4. The lattice decay wall is hard above $r \approx 10.0$. Second-generation $\varphi^4$ bands ($r \approx 19$–$25$) and third-generation $\varphi^3$ bands ($r \approx 12$–$17$) require extended lattices. The path is clear.

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