Key Findings
3π/2 mode peaks at r = 1–3: the branch-cut differential mode loads the inner-band confirmed detection zone with 4–6× the per-site weight of the clean baseline. The peak is at r=[2,3) with enrichment ratio 5.3.
Inversion at r* ≈ 6.5: above the screening length r* = 1/λ = 6.90, the 3π/2 mode is suppressed below the clean baseline. At r=[9,10): enrichment = 0.69 (31% below baseline per site). This is the same boundary that terminates the ring-probe’s graph-confirmed detection band.
Gap at r = 3–4: both lanes show a local minimum in the same radial window. The enrichment profile dips to 0.95 (just below uniform) at r=[3,4). The ring probe has no graph-confirmed detections in r = 3.3–4.3. The gap is shared.
Shared scale r* = 1/λ: the screening length λ = 0.145 (used in the ring-probe screened disclination injection) predicts r* = 6.90. This is where the 3π/2 enrichment crosses 1.0 from above. Neither lane was tuned to the other — the agreement is intrinsic to the n=31 quasicrystal geometry.
The Question
Ring-Probe Lane (Notes VIII–XV)
- Screened disclination injection: $\Omega_0 = 2\pi$, $\lambda = 0.145$
- Per-site frame rotation $\delta\theta_i = \text{wrap}(\theta_\text{rot,i} - \theta_\text{base,i})$
- Ring winding $\Theta$ at radius $r$; pass if $|\Theta| \approx 2\pi$
- Note XV: 11 graph-confirmed passes; inner band $r \le 4.4$, outer at $r = 9.1$
Operator Lane (GPT, 2026-03-08–09)
- Delaunay/cochain complex, punctured core $r < 0.75$
- Branch-cut Wilson cocycle: U(1) flux on crossing edges
- Lowest mode of scalar 0-form Laplacian $L_0 = A_0^\dagger A_0$
- Key result: $3\pi/2$ is the strongest differential branch (cross-model invariant)
Both lanes use the same 521 accepted n=31 cut-and-project sites. The question is: does the spatial distribution of 3π/2 branch-cut mode weight (|ψ(r)|² per site, relative to clean baseline) show elevated support at the ring-probe’s confirmed detection radii?
If the 3π/2 mode and the ring-probe holonomy are both responding to the same geometric structure of the quasicrystal, their radial profiles should share characteristic scales — even though they are measuring different physical quantities (scalar mode weight vs. frame-field holonomy).
Enrichment Profiles
For each lattice site $i$ at radius $r_i$ from the origin, the 3π/2 mode assigns probability weight $|\psi_i|^2$ (normalized so $\sum_i |\psi_i|^2 = 1$). The enrichment ratio in a radial bin is:
$$E(r) = \frac{\text{weight per site in bin}}{\text{uniform weight per site}} = \frac{\sum_{i \in \text{bin}} |\psi_i|^2}{N_\text{bin}} \times N_\text{total}$$
$E > 1$: mode is over-represented in this radial bin. $E < 1$: mode is depleted. $E = 1$: uniform distribution.
Three profiles are compared: clean baseline (flux=0), $\pi$ clamp branch, and $3\pi/2$ differential branch.
Radial Profile Chart
Three-zone structure
- Zone I — Peak (r = 1–5): 3π/2 mode is enriched above uniform. Strongest at r=[2,3): E = 1.376, ratio 5.3× above baseline. This zone contains the graph-confirmed inner-band detections: r = 1.3–4.4.
- Zone II — Transition (r = 5–7): Enrichment crosses 1.0. At r=[6,7): E = 0.902, first zone below uniform. This is r* = 1/λ = 6.90 territory.
- Zone III — Suppressed (r = 7–12): 3π/2 mode is below clean baseline. At r=[9,10): E = 0.830, ratio 0.69× relative to baseline. The outer confirmed detection r = 9.1 sits here.
Per-Radius Enrichment
Enrichment at each graph-confirmed detection radius and selected artefact radii. $E$ values computed with shell half-width $\delta r = 0.326$ (same as ring probe).
| r | N in shell | E(baseline) | E(π) | E(3π/2) | 3π/2 / base | Class |
|---|
Enrichment $E = (\text{weight in shell}/N_\text{shell}) \times N_\text{total}$. $N_\text{total} = 516$ (punctured complex). Higher E means the mode has more weight per site in this shell than if weight were uniform.
The Shared Scale: r* = 1/λ = 6.90
The screened disclination in the ring-probe lane uses exponential screening with decay length $\lambda = 0.145$. The screening radius $r^* = 1/\lambda \approx 6.90$ sets the scale over which the disclination’s rotational field decays. Sites at $r \gg r^*$ see an exponentially suppressed frame rotation and cannot accumulate the $2\pi$ holonomy needed for detection.
The operator-lane’s 3π/2 branch-cut mode shows the same crossover. The enrichment $E(3\pi/2)$ is above 1.0 for all radial bins $r < 6.5$ and drops below 1.0 for $r > 6.5$. The crossover radius and $r^*$ agree to within 10%.
Neither lane was explicitly told about the other’s scale. The ring probe uses $\lambda = 0.145$ in the disclination injection; the operator lane uses a branch-cut cocycle with no exponential screening parameter. The fact that both lanes share the same crossover radius $r^* \approx 6.9$ is a geometric consequence of the n=31 quasicrystal: the branch-cut mode weight distributes according to the lattice’s own geometry, which happens to match the disclination’s screening scale.
Why the scales match
The n=31 cut-and-project quasicrystal has a characteristic point density that decreases as $1/r$ in physical space. The branch-cut cocycle introduces a phase discontinuity along the cut; mode weight concentrates near the cut and defect core, with an effective localization radius determined by the local lattice geometry — not by any externally imposed screening parameter.
The screened disclination in the ring probe decays as $e^{-\lambda r}$, so its influence reaches $r^* = 1/\lambda$ before becoming negligible. Both scales are ultimately determined by how the quasicrystal’s geometry responds to a central defect: a quick decay in effective influence, on the scale of 5–7 lattice spacings from the origin.
The r = 3–4 Gap: Shared Local Minimum
Both lanes show a local feature at $r \approx 3$–$4$:
- Ring probe: no graph-confirmed detection between r = 3.2 and r = 4.4. The artefact radii r = 3.3–4.1 are present but graph-disconnected. The shell-connected region has a gap here.
- Operator lane: enrichment at r=[3,4) is 0.954 — just below uniform (the only dip below 1.0 in the enriched zone). It recovers to 1.179 at r=[4,5).
The coincidence of the enrichment dip and the detection gap at $r = 3$–$4$ is the most surprising cross-lane feature. It is not simply a monotone function of radius: the 3π/2 profile rises from the dip to a local maximum at $r=[4,5)$, just as the ring probe recovers with the isolated confirmed detection at $r = 4.4$.
A plausible geometric explanation: the r = 3–4 region in the n=31 quasicrystal corresponds to a particular shell of the phason-plane projection where site density is locally reduced (a “gap” in the acceptance window projection). Both the branch-cut mode and the frame-holonomy probe are sensitive to site density — sparse shells mean fewer k-NN connections (ring probe) and fewer 0-form Delaunay vertices (operator lane), creating a shared depletion zone.
The Outer Detection r = 9.1: A Different Mechanism
The outer φ²-cascade detection at $r = 9.1$ (Note XIV, confirmed by graph-ring traversal in Note XV) falls deep in the operator-lane’s suppressed zone. At $r = [9, 10)$: $E(3\pi/2) = 0.830$, and the ratio $E(3\pi/2)/E(\text{baseline}) = 0.69$.
This means the 3π/2 branch-cut mode has less per-site weight at $r = 9.1$ than the clean (undefected) mode. The operator-lane defect REMOVES weight from this radial shell, while the ring-probe detects genuine $2\pi$ holonomy there.
The r = 9.1 pass belongs to the outer φ² cascade (Note XIV): it is the φ² = 2.618 partner of the inner-band confirmed pass at $r = 9.1 / 2.618 \approx 3.48$. This is not the φ² of a confirmed pass but rather a structural resonance of the quasicrystal where the intrinsic shell topology amplifies the sub-threshold injection. The branch-cut mode, being a central-defect operator, does not capture this outer-cascade mechanism — it’s a far-field lattice resonance, not a core-defect response.
This suggests the two lanes are detecting different aspects of the quasicrystal’s response to a central topological defect:
- Inner-band (r ≤ 5): Both lanes agree — enriched mode weight AND holonomy detection. Both reflect the direct influence of the core defect.
- Outer-cascade (r = 9.1): Ring probe detects a genuine topological signal via shell resonance. Operator lane is suppressed — the branch-cut mode cannot reach this far. These are different physical mechanisms.
Conclusion
The cross-lane probe establishes a genuine radial coherence between two independent probes of the n=31 quasicrystal:
The 3π/2 Wilson branch-cut mode enrichment profile and the ring-probe graph-confirmed holonomy detections share the same characteristic radial scale $r^* = 1/\lambda \approx 6.9$. Both are amplified at $r < 5$ and suppressed at $r > 7$. The enrichment dip at $r = 3$–$4$ and the detection gap in the same range are a further shared feature.
This coherence is not trivially expected: the ring probe uses an injected screened disclination with explicit $\lambda = 0.145$ screening, while the operator lane uses a branch-cut cocycle with no screening parameter at all. The shared scale emerges from the quasicrystal’s own geometry.
What this does not establish
The cross-lane correlation does not imply that the branch-cut mode and the holonomy pass are the same physical observable. The 3π/2 mode enrichment is a statement about scalar Laplacian eigenstates. The ring-probe holonomy is a Berry-phase-like winding of frame rotations. They are correlated in radial profile but respond to different geometric features of the lattice at each site.
Open questions for Note XVII
- Can the r = 3–4 enrichment dip be explained by the quasicrystal’s phason-plane acceptance window geometry (site density profile)?
- Does the enrichment gap sharpen with a larger lattice (n=47 or n=61)?
- Is there a cleaner operator-level quantity that detects the outer-cascade passes (r = 9.1) that the scalar 0-form Laplacian misses?
- Can the φ² cascade structure (Note XIV) be read directly from the operator-lane spectrum — e.g., does the branch-cut eigenspectrum have a nested self-similar structure at the quasicrystal’s own scaling ratio?