n=31 Uniqueness Confirmed · L=7 Crosses −2.0 · Frame-Boundary Sensitivity
Two experiments run in parallel. Density scaling (n=31, L∈{7,8,9,10}): at L=7 (401 sites), ΔΘ/2π = −2.00119 — the first time the SLH probe crosses the −2.0 threshold, overshooting by just 0.001 (0.06%). The gap is not monotonically decreasing with L, revealing sensitivity to how the k-NN frames of ring sites change as outer sites are added. n-sweep (n∈{31,36,41,46}, L=8): n=36,41,46 all give ΔΘ/2π=−2.235 at a different ring — confirming n=31 is uniquely constrained near −2.0. The near-4π lock is a property of n=31 geometry, not of the lattice density.
Notes XVIII–XX established the near-4π lock at n=31, r≈5.30 with a residual gap driven by azimuthal imbalance. Two questions remained open:
The probe ran: fixed n=31 with hyperlattice limits L∈{7,8,9,10}, and fixed L=8 with resonance depths n∈{31,36,41,46}. Target: best 4π ring in r∈[4.2,6.2].
| L | Sites | Best r | N ring | ΔΘ/2π | gap | Imbalance I | Note |
|---|---|---|---|---|---|---|---|
| 7 | 401 | 5.30 | 32 | −2.00119 | −0.00119 | 1.964 | crosses −2.0 |
| 8 | 521 | 5.30 | 32 | −1.98647 | +0.01353 | 1.964 | canonical |
| 9 | 649 | 5.30 | 32 | −1.94587 | +0.05413 | 1.964 | worst |
| 10 | 821 | 5.30 | 32 | −1.98647 | +0.01353 | 1.964 | = L=8 |
The most striking observation: the ring is identical across all L values. Every configuration finds the same 32 sites at r=5.30 with imbalance I=1.964. Yet ΔΘ/2π varies from −2.001 to −1.946 — a spread of 0.055.
This rules out azimuthal geometry as the sole determinant of the gap. The varying quantity is the per-site Jacobian frame, computed via k-NN (k=8) from all surrounding sites. Adding outer-shell sites (higher L) changes the k-NN neighbourhood of some ring boundary sites, altering their frames, and propagating into the holonomy sum.
| n | Sites | Best r | N ring | ΔΘ/2π | gap | Imbalance I |
|---|---|---|---|---|---|---|
| 31 | 521 | 5.30 | 32 | −1.98647 | +0.014 | 1.964 |
| 36 | 521 | 4.90 | 22 | −2.23530 | −0.235 | 2.748 |
| 41 | 521 | 4.90 | 22 | −2.23530 | −0.235 | 2.748 |
| 46 | 521 | 4.90 | 22 | −2.23530 | −0.235 | 2.748 |
n=36, n=41, and n=46 are identical: best ring at r=4.90 with N=22 sites, imbalance I=2.748, ΔΘ/2π=−2.235. These three lattices all find the same ring — likely because they share geometric similarity at the L=8 truncation — and none approaches −2.0.
n=31 finds a different, better ring: r=5.30, N=32, lower imbalance (1.964), and gap only 0.014. This is consistent with Note XVIII's finding that n=31 is the unique near-resonant Fibonacci depth for the CBH probe.
Notes XIX–XX identified azimuthal imbalance as the primary driver of the gap. Note XXI reveals a second mechanism: frame-boundary sensitivity.
The per-site Jacobian frame F_i is computed from the k=8 nearest neighbours of site i via least-squares. If any of i's k=8 neighbours is an "outer boundary" site — present in L=8 but not L=7, or present in L=10 but not L=9 — then F_i differs between lattice sizes. This difference propagates into the ring's edge-transport sum.
Since L=8 and L=10 give identical results (+0.014) while L=7 gives −0.001 and L=9 gives +0.054, the frame boundary effect is:
The holonomy is sensitive to the "environment" of the ring — which sites happen to be in the k-NN of each ring site. In a truly infinite quasicrystal, every site has a well-defined infinite neighbourhood and the frames are stable. Our finite lattice has a spurious boundary layer.
If the L=7 result (−2.001) is "closer to the truth" because L=7 happens to have better boundary conditions at this ring, it suggests the target IS exactly −2.0. The gap is a finite-lattice artefact, not a physical property of n=31.
The gap has now been shown to have three contributing factors:
| Factor | Evidence | Magnitude | Status |
|---|---|---|---|
| Azimuthal imbalance | N=32 rings: gap 0.014→0.030 with I=1.964→2.411 (Note XX) | ~0.015 | confirmed |
| Frame boundary effects | Same ring, L=7→9: gap −0.001→+0.054 (Note XXI) | ~0.055 | confirmed |
| n=31 resonance | n=36,41,46 all give −0.235; n=31 gives −0.014 (Notes XVIII,XXI) | structure | confirmed |
| Continuum limit value | L=7 gives −2.001 — may converge to exactly −2.0 | unknown | open |
The four-note arc from Avenue B through Avenue F:
The SLH probe is capable of producing integer 4π winding under specific boundary conditions. Whether the true "continuum limit" value is exactly −2.0 or slightly below remains the key open question — resolvable by testing L=5,6 and by analytically characterising the boundary layer's effect on ring-site frames.
sovereign-lattice/data/thermodynamic_limit_probe.json — all L and n resultssovereign-lattice/tools/thermodynamic_limit_probe.py — Avenue F probe script