Where the 1/N Hypothesis Lives and Dies — Azimuthal Imbalance as the Governing Variable
The hypothesis ΔΘ/2π = −2.0 + 1/N was tested across 51 ring radii. It fails globally (R² = 0.17). Within the N = 32 family alone, the gap varies from 0.014 to 0.030 depending on the ring's azimuthal imbalance — conclusively ruling out a pure counting correction. The local coincidence at r = 5.517 (gap = 0.030 ≈ 1/32 = 0.031) is accidental. The governing variable is azimuthal site distribution, not N. The cleanest near-4π result found is at r = 5.30–5.35 (gap = 0.014, imbalance I = 1.964), not the canonical probe radius.
Note XIX confirmed the −0.031 gap at r = 5.517 (N = 32) is independent of α — it is structural. The leading candidate explanation was a discrete counting correction:
ΔΘ/2π = −2.0 + 1/Nring
The intuition: with N discrete edge-transport steps around the ring, one step's contribution is "consumed" by the Cantor boundary topology, leaving (N−1)/N of the full 4π winding. For N = 32, the prediction is gap = 1/32 = 0.031 — matching the observed 0.030 to within 3%.
To test this, the CBH probe (α = 0.18, λ = 0.145) was run across 51 radii in [4.5, 7.0] and a broader scan of [1.0, 9.5]. For each 4π-plateau result, the gap and 1/N were recorded.
| r | N | ΔΘ/2π | gap | 1/N | gap − 1/N | Imbalance I | Note |
|---|---|---|---|---|---|---|---|
| 4.900 | 22 | −2.2353 | −0.2353 | 0.04545 | −0.281 | 2.748 | overshoot |
| 5.250 | 36 | −1.9257 | +0.0743 | 0.02778 | +0.047 | 2.210 | 4π plateau |
| 5.300 | 32 | −1.9865 | +0.0135 | 0.03125 | −0.018 | 1.964 | best ring |
| 5.350 | 32 | −1.9865 | +0.0135 | 0.03125 | −0.018 | 1.964 | best ring |
| 5.450 | 34 | −2.0319 | −0.0319 | 0.02941 | −0.061 | 2.561 | overshoot |
| 5.500 | 32 | −1.9703 | +0.0297 | 0.03125 | −0.002 | 2.411 | canonical probe |
| 6.700 | 37 | −2.1572 | −0.1572 | 0.02703 | −0.184 | 1.690 | overshoot |
The critical test: three rings all have N = 32 sites. Under the 1/N hypothesis, all three should have gap ≈ 0.031. They do not:
| r | N | gap | 1/N | Imbalance I | gap/I |
|---|---|---|---|---|---|
| 5.300 | 32 | 0.0135 | 0.03125 | 1.964 | 0.0069 |
| 5.350 | 32 | 0.0135 | 0.03125 | 1.964 | 0.0069 |
| 5.500 | 32 | 0.0297 | 0.03125 | 2.411 | 0.0123 |
The rings at r = 5.30 and r = 5.35 contain the same 32 physical sites (the ring half-width dr = 0.326 overlaps these radii). Azimuthal imbalance I = 1.964. Gap = 0.014 — the cleanest near-4π result found in the entire dataset.
The ring at r = 5.50 (the canonical probe radius from Note XVIII) contains a different set of 32 sites with higher imbalance I = 2.411. Gap = 0.030 — 2.2× larger.
Conclusion: For the same N, higher azimuthal imbalance produces a larger gap. The gap is not a counting correction — it is a geometric property of how the ring sites are distributed around the center.
From Note XVII, azimuthal imbalance I = max_gap / mean_gap measures how unevenly ring sites are spaced in angle. A uniform ring has I = 1.0; clustered rings have I >> 1.
In the per-edge differential transport computation, each edge contributes:
Δαij = αrot,ij − αbase,ij, wrapped to (−π, π]
When ring sites are clustered, some angular sectors are over-represented and others empty. The empty sectors produce large inter-site angles at the cluster boundaries. These large angular jumps are likely where the wrapping truncation (mod 2π) is lossy — an edge spanning a large azimuthal gap will have its frame-rotation angle wrapped into (−π, π] before summing, potentially losing winding. Higher imbalance → more such boundary edges → larger truncation loss → larger gap from −2.0.
This mechanism predicts: gap ∝ f(I) where f is some increasing function. The N = 32 data shows gap(I=1.964) = 0.014 and gap(I=2.411) = 0.030 — consistent but insufficient to fit a specific functional form with only two points.
Gap is determined by ring geometry, not N alone. Higher azimuthal imbalance → larger gap. The cleanest 4π ring (lowest gap) has I ≈ 1.96, not I = 1.0. Even perfectly uniform ring would have some wrapping loss.
The exact functional form gap(I, N, r) is unknown. Whether gap → 0 as I → 1.0 (uniform ring) needs testing. Whether the gap can be derived analytically from the azimuthal Fourier spectrum of ring site positions.
Across the full scan, r = 5.30–5.35 with N = 32, I = 1.964 gives the closest approach to exact −2.0: gap = 0.014 → ΔΘ/2π = −1.986.
This is 7× closer to −2.0 than the canonical probe radius of Note XVIII (gap = 0.030 at r = 5.517 → ΔΘ/2π = −1.970). The result at r = 5.30 would meet a relaxed |ΔΘ/2π + 2| < 0.02 criterion, though not the exact integer.
Whether the r = 5.30 ring represents a more "resonant" configuration or simply a more symmetric one is unresolved. Notably, 5.30 ≈ 5.517 × φ⁻¹ ≈ 5.517 / 1.618 ≈ 3.41 — no obvious φ-relation. And 5.30 ≈ ln(2)/λ × 1.11 — not a clean multiple either. The specific ring geometry, not a deep resonance, likely governs this result.
Three sequential notes have progressively tightened the description of the near-4π lock:
| Note | Finding | Status |
|---|---|---|
| XVIII | CBH probe gives ΔΘ/2π = −1.970 at n=31, r=5.517. n=31 uniquely close to −2.0. | confirmed |
| XIX | Hard snap at α*≈0.091. 4π plateau [0.093, 0.36] — gap is not α-dependent. | confirmed |
| XX | 1/N hypothesis refuted. Gap governed by azimuthal imbalance. Best ring: r=5.30, gap=0.014. | confirmed |
| C (open) | Derive gap analytically from ring site Fourier spectrum / wrapping-loss model. | open |
The near-4π lock is structural (Notes XVIII–XIX), not parameter-tuned. Its deviation from exactly −2.0 is geometric (Note XX), not a counting artifact. A fully analytical description of the gap requires modeling the discrete ring's azimuthal edge-transport sum — still open but well-posed.
sovereign-lattice/data/gap_scaling_probe.json — coarse + fine scan, all plateau resultssovereign-lattice/tools/gap_scaling_probe.py — Avenue C scaling probe