Spinorial Holonomy on Aperiodic Substrates:
A No-Go Result for the Smooth Bridge Coframe
We ask whether the Fibonacci 4D→2D cut-and-project substrate can support spinorial holonomy — the topological condition required for a genuine fermion sector. A computational and analytic argument shows that the current smooth least-squares bridge coframe cannot achieve this: the coframe field inherits the regularity of the fit and cannot concentrate the $2\pi$ frame rotation required for a genuine spinor sign flip. Two structural corrections are identified that would restore the possibility.
What does it take for a substrate to carry spinors?
The Sovereign Lattice Hypothesis proposes a covariant digital-physical framework in which curvature, metric, and matter emerge from the parity structure of an aperiodic lattice. The Dirac Basin companion note (v0.2.5) documents the program to realise a fermion sector within this framework, and reports a regression: the best previous parity-breaking near-lock did not generalise beyond the specific 21-step loop family in which it was found.
This note steps back from numerical search and asks the prior mathematical question: can the substrate class, in principle, support the topological structure that spinors require? This is a question about the coframe field, not the connection or the equations of motion. It must be answered before further parameter sweeps are meaningful.
Level 1 — Orientability ($w_1 = 0$)
The frame field must have consistent global orientation: no net sign flip around any closed loop. Previous parity audits suggest this is satisfied for the current substrate — the parity = +1 results are consistent with an orientable effective geometry.
Level 2 — Spin structure ($w_2 = 0$)
The frame bundle must admit a lift to $\mathrm{Spin}(2)$. For any orientable 2-manifold, $w_2 = 0$ automatically. This level is not the obstruction — if the substrate is orientable, a spin structure exists.
Level 3 — Spin connection holonomy
Given a spin structure, the induced connection must have non-trivial holonomy $H_\mathrm{spin}(C) = -1$ around some loop. This is the operative criterion. It can fail even when Levels 1 and 2 are satisfied.
Constructing the frame field from the residual projection
Let $\Lambda \subset \mathbb{R}^2$ be the accepted physical-space sites from the golden-Cantor 4D→2D cut-and-project construction, with internal-space residuals $\delta : \Lambda \to \mathbb{R}^2$. The discrete coframe at site $p_i \in \Lambda$ is the local deformation gradient:
The polar decomposition $F_i = R_i S_i$ isolates the rotation part:
holonomy_audit.py) computed a
weighted signed turn fraction: $\sum_i \sigma_i \omega_i / 2\pi$
where $\sigma_i = \mathrm{sgn}(\det F_i)$ is the parity sign.
This is NOT the spinorial winding number. It conflates parity memory (an
aperiodic handedness signature) with frame-angle accumulation (the holonomy
relevant to spinors). The correct quantity is $\Theta(C)$ defined in §III.
When does a spinor return as $-\psi$?
For a closed loop $C = (p_{i_0}, \ldots, p_{i_{N-1}}, p_{i_0})$, the connection increment along each bond is the principal-value frame angle difference:
The lifted holonomy in $\mathrm{Spin}(2) \cong U(1)$ is:
$\Theta(C) \approx 0$ on every probed loop
Two systematic computations were run using the corrected winding number diagnostic
(winding_number_audit.py):
Standard centroid loop (21 steps)
$\Theta(C) \approx 0$ rad.
$\varphi_\mathrm{rms} = 0.057$ rad.
Spin holonomy phase: $0°$ (trivial).
Cantor boundary probe
156 boundary sites identified (within 0.12 of a Cantor gap).
150 loops: sites × $N \in \{4,5,6,7,8\}$.
Non-trivial $\Theta$ (>0.05 rad): 0 of 150.
N-scaling test
Loop steps: $N \in \{21, 42, 63, 84, 105\}$.
Parity factor: $+1$ at all $N$.
$\Theta \approx 0$ throughout — no convergence signal.
| Loop family | $N$ | Sites tested | $\Theta_\mathrm{max}$ (rad) | $\Theta / 2\pi$ | Spinorial? |
|---|---|---|---|---|---|
| Centroid ring | 21 | 1 | ≈ 0.000 | ≈ 0.000 | ❌ No |
| Cantor depth-0 boundary | 4–8 | 30 | < 0.05 | < 0.008 | ❌ No |
| Cantor depth-1 boundary | 4–8 | 30 | < 0.05 | < 0.008 | ❌ No |
| N-scaling multiples of 21 | 21–105 | 5 | ≈ 0.000 | ≈ 0.000 | ❌ No |
Caveat 1 — Loop coverage: The computation tests 150 specific loops on one realisation of the substrate. It does not prove that every possible loop on the full tiling hull is trivial. The correct statement is: the audited coframe field is effectively flat on all tested loop families. The possibility of non-trivial winding on untested loop configurations — in particular, very large loops encircling extended phason defect clusters — is not excluded by these results.
Caveat 2 — Bridge dependence: The no-go argument depends entirely on the regularity of the least-squares bridge. It is a no-go for this pipeline, not for the substrate class. Replace the smooth bridge with a sharp construction that preserves the acceptance-boundary discontinuity at the coframe level (Path B, §VI), and the theorem's scope does not apply. The claim is therefore: smooth bridge + smooth acceptance window → flat effective coframe → no spinorial holonomy.
The acceptance boundary is not a disclination
The golden-Cantor acceptance window creates sharp boundaries between accepted and rejected sites in internal space. These boundaries are the natural candidates for disclinations: sites near a Cantor gap have very different internal-space projections from their nearest neighbours that happen to lie on the other side of the gap.
However, the extraction bridge constructs the coframe by a smooth least-squares fit over spatial neighbours in physical space. That fit regularises the discontinuity before frame angles are computed. The jump in internal-space projection across a Cantor gap boundary translates not into a spike in $\varphi_i$ at the boundary site, but into a smooth gradient spread over the neighbourhood radius.
Let $\delta : \Lambda \to \mathbb{R}^2$ be the residual field and let $F_i$ be estimated by any smoothing operation (e.g., least-squares fit, kernel smoothing) with neighbourhood radius $r > 0$. If $\delta$ is bounded by the acceptance window radius $r_\perp$ and the residual gradient is bounded by $\|\nabla \delta\| \leq r_\perp / r$, then: $$|\varphi_i| \leq \arctan\!\Bigl(\frac{r_\perp}{r}\Bigr), \quad |\Theta(C)| \leq N \cdot \arctan\!\Bigl(\frac{r_\perp}{r}\Bigr)$$ For $r_\perp = 0.98$, $r \approx 0.35$ (median nearest-neighbour spacing), $N = 21$: $|\Theta| \lesssim 21 \times 1.23 \approx 25.8\,\mathrm{rad}$ in the worst case with all increments aligned, but $\Theta \approx 0$ when the residual field is smooth and bounded, because increments cancel around closed loops in smooth fields. A $2\pi$ disclination requires the residual field to be singular (topologically non-trivial) — which a smooth bridge cannot produce.
What the substrate needs to carry spinors
The no-go result constrains, not destroys, the program. Two structural corrections would each independently restore the possibility of spinorial holonomy.
Introduce a wedge disclination explicitly into the lattice: remove a $\theta_0$-wedge of material and re-glue the faces. In the continuum theory of defects, this produces a disclination of Frank vector $\Omega = \theta_0 / 2\pi$. For $\theta_0 = 2\pi$ (full wedge removal), the Frank vector is $1$ and the frame field has a genuine $2\pi$ singularity. The frame angle $\varphi_i$ then winds by $2\pi$ around a small loop encircling the defect core, satisfying the spinorial criterion exactly.
Replace the smooth least-squares bridge with a construction that preserves the acceptance boundary discontinuity at the coframe level. One approach: assign $F_i$ not from neighbourhood averaging but from the local discrete difference of the residual between $p_i$ and its nearest accepted neighbours. At a Cantor gap boundary, this difference is large (one neighbour is just inside, one just outside the window), and $F_i$ becomes nearly singular at the boundary site. The frame angle $\varphi_i$ then spikes at the boundary, and a loop encircling the boundary site accumulates $\Theta \to 2\pi$ as the window sharpens.
Connection to the Atiyah-Singer index
In the long run, the SLH fermion program requires not just a coframe with $2\pi$ disclinations but a discrete Dirac operator $D$ with non-zero index. The Atiyah-Singer index theorem relates this to the topology of the frame bundle:
How to refer to this note
Croydon-McRae, S. T. (2026). Spinorial Holonomy on Aperiodic Substrates: A No-Go Result for the Smooth Bridge Coframe (Research note v0.1.0). Te Kete Ako Research.