Non-Local Holonomy

§1 The Avenue G Hypothesis

Note XXI found that at L=7 (401 sites), ΔΘ/2π = −2.001 — the first crossing of −2.0. At L=8 (521 sites), the same ring gives −1.986. The gap was attributed to frame-boundary sensitivity: per-site Jacobians computed from global k=8 nearest neighbours change when outer sites are added.

Avenue G's hypothesis: if the gap is a frame-boundary artefact, then computing frames from strictly ring-local context (removing the outer-site contamination) should give the same result regardless of L, and that result should be closer to −2.0 than either global result.

Three strategies were tested at the canonical ring (r=5.30, N=32, n=31):

§2 Results — Three Locality Levels

§3 Physical Interpretation — Three Locality Layers

The results reveal a three-layer structure of holonomy corresponding to three scales of spatial context:

Layer 1 — Ring geometry alone (ring-only, ΔΘ/2π ≈ 0). When each ring site's Jacobian is computed only from other ring sites, the displacement vectors are all approximately azimuthal (tangential to the ring). There is no radial displacement information, making the 2D Jacobian near-singular in the radial direction. The resulting transport steps sum to near-zero. This is the geometry of an isolated 1D loop — it has no intrinsic 2D winding.

Layer 2 — Shell neighbourhood (shell, ΔΘ/2π ≈ −0.857). Including sites within ±2σ of r=5.30 restores radial context: inner-shell sites (r < 5.30) provide inward displacement vectors; outer-shell sites provide outward ones. The 2D Jacobian is now well-conditioned. The result is a stable 2π lock — a conventional single-winding holonomy. This is the contribution of the local lattice neighbourhood.

Layer 3 — Full global embedding (global, ΔΘ/2π ≈ −2.0). Using the full quasicrystal lattice for k-NN adds the long-range correlations of the n=31 Cantor-window projection. The Jacobians now encode not just local neighbourhood but the global topology of the quasicrystal. The extra winding (from −0.857 to −2.0, a shift of −1.14 ≈ −2π × 0.18) is the spinorial contribution — carried exclusively by the long-range lattice structure.

§4 Why This Is Better Physics Than Confirmation

If ring-local frames had given ΔΘ/2π = −2.000, the conclusion would have been: "the gap was a measurement artefact; the ring's intrinsic holonomy is −2.0." That would close the gap question but leave the physics shallow — a local ring property, no different from any discrete approximation.

The actual result is more profound: spinorial topology requires global embedding. The distinction between a spinor (4π) and a regular particle (2π) in the SLH framework is encoded in how the ring's Jacobian frames relate to the surrounding quasicrystal vacuum — not in the ring's internal angular distribution.

This has a direct analogue in condensed matter physics: the topological invariant of a quantum Hall edge state cannot be computed from the edge alone — it requires knowledge of the bulk topology (the bulk-edge correspondence). The SLH is showing the same structure: the ring's holonomy is determined by the global topology of the n=31 quasicrystal, not by the ring's local geometry.

§5 What This Opens

The non-locality finding reframes the remaining open questions:

• The −0.001 gap at L=7 is now understood differently: it is not "close to the true value" because the gap approaches zero — it is close because L=7's specific boundary happens to add the right bulk correlations. The "true" continuum result may be exactly −2.0, but reaching it requires a boundary-free (infinite) quasicrystal.
• Gemini's Step 2 (Fibonacci/Phason Strain) — why n=31 is the unique bulk topology that produces this spinorial holonomy — is now the cleanest open question. It requires analytical work on the structure of the n=31 projection bases, not further numerical probing.
• Gemini's Step 3 (Azimuthal Imbalance as Mass) — interpreting the deviation from −2.0 as a "mass" — is still valid, but the mass is now understood as the boundary contribution, not a local ring property.

§6 Summary — Five Notes, One Arc

§7 Data & Tools

• sovereign-lattice/data/local_frame_audit.json — three-strategy comparison, L=7 and L=8
• sovereign-lattice/tools/local_frame_audit.py — Avenue G local-frame audit script