Two bugs in the edge-transport pipeline are corrected: the Jacobian transposition ($J = F^T$) and per-edge differential accumulation. The manual disclination injection is then replaced by the organic golden-Cantor boundary operator. The result: $\Delta\Theta/2\pi \approx -1.970$ at $r = 5.517$ — a near-$4\pi$ frame rotation arising from geometry, not from a target. Only $n=31$ approaches $-2.0$; all other resonances overshoot.
When the manually specified $\Omega_0 = 2\pi$ disclination field is replaced by the
geometric apply_boundary_helix operator driven by the Cantor gap's own
$\text{boundary\_sign}$, the edge-transport probe at $r = 5.517$ returns
$\Delta\Theta/2\pi = -1.9703$ — a total frame rotation of $-1.9703 \times 2\pi \approx -4\pi
\times 0.985$. This is within 1.5% of the $4\pi$ double-winding required for spinorial
identity. The result was not tuned to approach $-2.0$; the canonical parameters $\alpha =
0.18$, $\lambda = 0.145$ were established in prior work. The value $\alpha = 0.18$ remains
empirically set — Avenue C (derive from $\arctan(1/\Phi)$ or a $\Phi$-power series) is the
key open question that would make the theory fully parameter-free.
passes_spinorial_return hits
Prior edge-transport work (Notes XII–XVII) probed the n=31 quasicrystal by
injecting a scalar disclination field $\Omega(r) = \Omega_0 e^{-\lambda r}$ and
measuring whether the discrete graph transport around radial rings could recover the
injected winding. This approach was consciously a detector test — it confirmed
the lattice could sense a planted signal, but it did not address whether the geometry
generates the signal spontaneously. The March 17 strategic analysis
(sovereign-lattice-analysis-2026-03-17.md) named two avenues to
address this gap:
Avenue A — Correct the edge-transport Jacobian and accumulation method
so that clean recovery of injected disclinations no longer depends on a fortuitous
cancellation of noise.
Avenue B — Replace $\Omega_0$ injection with the organic
boundary_sign from the golden-Cantor cut-and-project window, and measure
whether the lattice geometry spontaneously produces a topological winding anomaly.
The least-squares frame computation solves $\Delta \mathbf{x} \cdot F = \Delta \mathbf{u}$, where $\Delta\mathbf{x}$ are physical-space displacement vectors and $\Delta\mathbf{u}$ are internal-space displacements. The returned matrix $F$ satisfies this equation with $\Delta\mathbf{x}$ on the left, so the true Jacobian is $J = F^T$ (not $F$). The prior implementation used $F$ directly in the relative transport $T_{ij} = \text{polar}(F_j \cdot F_i^{-1})$, which computes transport in the wrong frame.
Fix: $T_{ij} = \text{polar}(J_j \cdot J_i^{-1}) = \text{polar}(F_j^T \cdot F_i^{-T})$.
The prior code summed the absolute rotated transport $\theta_\text{rot}$ and the absolute baseline transport $\theta_\text{base}$ entirely around the ring, then computed the difference. On an aperiodic lattice both $\theta_\text{rot}$ and $\theta_\text{base}$ individually wrap through $2\pi$ multiple times as the ring traverses many sites; subtracting large accumulated quantities causes massive cancellation error.
Fix: Compute the isolated differential $\Delta\alpha_{ij} = \alpha_{\text{rot},ij} - \alpha_{\text{base},ij}$ on each specific edge before any wrapping, wrap locally to $(-\pi, \pi]$, then sum around the ring:
With both fixes applied, the probe cleanly recovers the injected $\Omega_0 = 2\pi$ disclination on 4 distinct radial loops of the $n=31$ lattice, without relying on any lucky noise cancellation.
The golden-Cantor acceptance window assigns each accepted site a boundary_sign
$\in \{-1, 0, +1\}$ based on which side of the Cantor gap boundary the site's internal
coordinate falls. The apply_boundary_helix operator uses this sign to rotate
each site's frame by an angle:
$\theta_i = \alpha \cdot e^{-d_i/\lambda} \cdot s_i \cdot \varphi_i$
where $d_i$ is the site's distance to the nearest gap boundary, $\lambda$ is the decay length, $s_i \in \{-1,+1\}$ is the boundary sign, and $\varphi_i$ is the azimuthal polar angle. This operator applies opposite-sign rotations on opposite sides of the gap — a discrete approximation of a rotational disclination field — but the winding amplitude is set by the gap geometry, not by a manually specified $\Omega_0$.
The Cantor Boundary Helix (CBH) probe runs the corrected edge-transport holonomy probe on frames modified by this operator, comparing against unmodified baseline frames at each ring radius. The helix parameters used were canonical values established in prior work: $\alpha = 0.18$ (empirical), $\lambda = 0.145 \approx 1/\Phi^4$ (geometrically motivated: gives $r^* = \ln(2)/\lambda \approx 4.78$, the canonical screened disclination radius).
CBH probe at all radii, $n=31$, compared to base edge-transport (no helix):
| r | N sites | Base $\Delta\Theta/2\pi$ | CBH $\Delta\Theta/2\pi$ | Base passes | CBH emergent |
|---|---|---|---|---|---|
| 0.698 | 4 | — | — | insuff. | insuff. |
| 1.3793 | 10 | +1.119 | +0.568 | pass | no |
| 1.396 | 12 | +1.491 | +0.705 | no | no |
| 2.3267 | 12 | +0.845 | −0.536 | pass | no |
| 3.4483 | 14 | +0.459 | −1.471 | no | no |
| 5.5172 | 32 | −1.172 | −1.970 | pass | near−2 |
| 8.2759 | 40 | +0.800 | −0.311 | pass | no |
Spinorial pass criterion: $|\Delta\Theta/2\pi| \in [0.8, 1.2]$ for the base probe. The CBH $\Delta\Theta/2\pi = -1.970$ does not pass this gate (it is near $-2$, not $-1$) — but it corresponds to a total rotation of $-1.970 \times 2\pi \approx -4\pi \times 0.985$, within 1.5% of the $4\pi$ double-winding threshold.
Same helix parameters ($\alpha=0.18$, $\lambda=0.145$), same ring radii, varying $n \in \{16, 21, 26, 31\}$:
| n | $\Delta\Theta/2\pi$ at $r=5.517$ | Gap from $-2.0$ | Emergent passes | Note |
|---|---|---|---|---|
| 16 | −2.506 | −0.506 | 1 (at $r=8.276$) | Overshoots; pass at different radius |
| 21 | −2.506 | −0.506 | 0 | Overshoots |
| 26 | −2.601 | −0.601 | 0 | Overshoots further |
| 31 | −1.970 | −0.030 | 0 | Closest to −2.0 |
Among $n \in \{16, 21, 26, 31\}$, only $n=31$ lands within 2% of the $4\pi$ target at $r=5.517$. The other resonances all overshoot to $|\Delta\Theta/2\pi| \approx 2.5$–$2.6$ despite identical helix parameters. This suggests the $n=31$ quasicrystal has a structural resonance that contains the CBH transport near $-2.0$, while other $n$ values lack this containment.
Whether this containment arises from the $n=31$ Fibonacci resonance structure (where $F_{31}/F_{30} \to \Phi$) or from some other property of the golden-Cantor window at this depth is not yet determined. This is a candidate mechanism for Avenue C to investigate: if $\alpha$ is derived from the $n=31$ window geometry rather than set empirically, the -0.030 discrepancy may shrink or vanish.
What it is: The CBH probe replaces a manually specified winding target ($\Omega_0 = 2\pi$) with a field derived from the geometric structure of the Cantor gap. The resulting $\Delta\Theta/2\pi \approx -1.97$ was not targeted — it arose from applying pre-existing canonical parameters to a new methodology. The n=31 uniqueness is a genuine finding from a blind sweep.
What it is not: The result does not pass the formal
passes_spinorial_return criterion (which gates on $|\Delta\Theta/2\pi| \approx
1$, not $\approx 2$). The parameter $\alpha = 0.18$ remains empirically set; the theory is
not yet self-consistent. The 0.030 gap to $-2.0$ may reflect a real geometric constraint,
or it may be a finite-$n$ artefact. No claim of "proof" of a fermion sector is made here.
--compare for base vs rotated differential.
apply_boundary_helix from generate_n31_projection.py.
apply_boundary_helix() defined here. Canonical source of $\lambda$.