Which projection depths n support the 4π lock? n = 10 → 50 sweep at L=8
The 4π holonomy at the canonical ring (r=5.30, N=32) is not exclusive to n=31. It appears for a discrete family of Fibonacci depths: n∈{11, 13, 15, 17, 18, 23, 31, 34}. The governing variable is σint (mean perpendicular-space norm of ring sites, r=−0.88 with gap), not the Diophantine residual |ε(n)|. Note XXI’s “n=31 unique” result was correct within the tested set {n=31, 36, 41, 46} — but 31 is one member of a larger compatible family. The absolute best lock (gap=0.0021) occurs at n=20 and n=30 at r=5.45.
The projection parameter n controls the Fibonacci approximant
φapprox = F(n)/F(n−1) used to build the
QR-decomposed projection basis. The signed residual
ε(n) = φapprox − φ
alternates in sign (odd n: positive overestimate; even n: negative underestimate)
and shrinks exponentially: |ε(n)| ≈ 1/(√5·F(n−1)²).
From Note XXI (thermodynamic limit), a targeted comparison of n∈{31, 36, 41, 46} at L=8 found n=31 uniquely near −2.0 while n=36, 41, 46 all gave ΔΘ/2π = −2.235. This Note tests the full range n=10→50 to determine whether n=31 is genuinely singular or one member of a family.
The first structural finding is a hard boundary at n ≈ 39. For n≥39, the ratio F(n)/F(n−1) equals φ to all 15 significant digits available in IEEE 754 double precision. All n≥39 produce identical projections, identical 521-site physical point clouds, and identical holonomies (ΔΘ/2π = −1.967 at r=5.55, gap=0.033).
The effective dynamic range of the projection is n=10…38 (29 distinct values). Any n≥39 is a repeated experiment. Claims about convergence of the holonomy as n→∞ cannot be tested with float64 arithmetic beyond this threshold.
At the canonical ring r=5.30 (N=32, Iaz=1.96), only a subset of n values produce near-integer holonomy. The full sweep reveals two sharply distinct populations:
The gap is bimodal: values cluster near 0.02 (family) or near 0.46 (incompatible). There is almost no middle ground. This is a discrete topological switch, not a gradual convergence.
A sharp result: n=31 and n=18 give exactly the same holonomy at r=5.30 (−1.9865, gap=0.0135), the same ring count (N=32), and the same azimuthal imbalance (I=1.96). Their Diophantine residuals differ by nine orders of magnitude (ε(18)=−1.75×10−7 vs ε(31)=+6.46×10−13), yet the holonomy is identical.
If the quality of the Fibonacci approximation determined the holonomy, n=31 (much better approximant) should differ from n=18. It does not. The holonomy depends on a property that saturates early in the Fibonacci sequence.
The phason strain σint(n) is the mean Euclidean norm of the perpendicular-space coordinates (pts_int = lat @ perp.T) for ring sites at the best radius. It measures how deeply the ring sites are embedded in the acceptance window — a direct proxy for their phason displacement from the ideal quasiperiodic tiling.
The smallest gap found in the entire sweep is gap=0.0021, occurring at both n=20 and n=30, at r=5.45 (Nring=34, Iaz=2.56). This is more than 6× smaller than n=31’s gap at r=5.30 (0.0135).
n=20 and n=30 fall at r=5.45 — a slightly different ring from n=31’s canonical r=5.30. The 34-site ring at r=5.45 has higher azimuthal imbalance (I=2.56 vs 1.96 for the 32-site ring). From Note XVII, I≥2.0 flags potential artefacts. These near-exact results at r=5.45 should be treated as candidates pending azimuthal-balance verification.
At its own canonical ring (r=5.30, I=1.96), n=31 gives gap=0.0135 — a well-balanced, confirmed result. n=20 and n=30 achieve smaller gaps at r=5.45 but on a ring with I=2.56, borderline for artefact classification. n=31’s strength is its combination of small gap AND low azimuthal imbalance.
| n | ε(n) | rbest | ΔΘ/2π | gap | Nring | Iaz | σint | r=5.30 gap |
|---|---|---|---|---|---|---|---|---|
| 10 | −3.9e−4 | 4.6 | −1.566 | 0.434 | 16 | 1.91 | 0.692 | 0.476 |
| 11 | +1.5e−4 | 5.3 | −1.974 | 0.026 | 32 | 1.96 | 0.794 | 0.026 |
| 12 | −5.6e−5 | 5.55 | −2.051 | 0.051 | 38 | 2.73 | 0.773 | 0.417 |
| 13 | +2.2e−5 | 5.25 | −1.993 | 0.008 | 36 | 2.21 | 0.790 | 0.056 |
| 14 | −8.2e−6 | 5.55 | −2.127 | 0.127 | 38 | 2.73 | 0.773 | 0.449 |
| 15 | +3.1e−6 | 5.25 | −2.018 | 0.018 | 36 | 2.21 | 0.790 | 0.041 |
| 16 | −1.2e−6 | 5.55 | −1.930 | 0.070 | 38 | 2.73 | 0.773 | 0.475 |
| 17 | +4.6e−7 | 5.5 | −2.007 | 0.007 | 32 | 2.41 | 0.783 | 0.021 |
| 18 | −1.8e−7 | 5.3 | −1.987 | 0.013 | 32 | 1.96 | 0.794 | 0.014 |
| 19 | +6.7e−8 | 5.55 | −2.054 | 0.054 | 38 | 2.73 | 0.773 | 0.459 |
| 20 | −2.6e−8 | 5.45 | −1.998 | 0.002 ✓ | 34 | 2.56 | 0.789 | 0.039 |
| 21 | +9.8e−9 | 5.55 | −1.930 | 0.070 | 38 | 2.73 | 0.773 | 0.475 |
| 22 | −3.7e−9 | 5.55 | −2.006 | 0.006 | 38 | 2.73 | 0.773 | 0.486 |
| 23 | +1.4e−9 | 5.3 | −1.978 | 0.023 | 32 | 1.96 | 0.794 | 0.023 |
| 24 | −5.4e−10 | 5.55 | −2.054 | 0.054 | 38 | 2.73 | 0.773 | 0.459 |
| 25–29 | … | 5.55 | … | >0.06 | 38 | 2.73 | 0.773 | >0.4 |
| 30 | −1.7e−12 | 5.45 | −1.998 | 0.002 ✓ | 34 | 2.56 | 0.789 | 0.039 |
| 31 | +6.5e−13 | 5.3 | −1.987 | 0.014 | 32 | 1.96 | 0.794 | 0.014 |
| 32 | −2.5e−13 | 5.55 | −1.933 | 0.067 | 38 | 2.73 | 0.773 | 0.487 |
| 33 | +9.4e−14 | 5.55 | −1.969 | 0.031 | 38 | 2.73 | 0.773 | 0.501 |
| 34 | −3.6e−14 | 5.3 | −1.982 | 0.018 | 32 | 1.96 | 0.794 | 0.018 |
| 35 | +1.4e−14 | 5.55 | −2.060 | 0.060 | 38 | 2.73 | 0.773 | 0.450 |
| 36 | −5.0e−15 | 5.55 | −1.857 | 0.143 | 38 | 2.73 | 0.773 | 0.477 |
| 37 | +2.0e−15 | 5.25 | −2.017 | 0.017 | 36 | 2.21 | 0.790 | 0.120 |
| 38 | −1.0e−15 | 5.55 | −2.054 | 0.054 | 38 | 2.73 | 0.773 | 0.459 |
| 39–50 | ε=0 | 5.55 | −1.967 | 0.033 | 38 | 2.73 | 0.773 | 0.460 |
Blue rows: Population A (family, near-2.0 at r=5.30). Green rows: n=20,30 (absolute best gap=0.002 at r=5.45). Greyed rows: machine-precision frozen (n≥39 all identical).
Note XXI tested n∈{31, 36, 41, 46}. n=36, 41, 46 all gave ΔΘ/2π=−2.235
at the r=5.30 probe. Conclusion: “n=31 uniquely near −2.0.”
This Note shows that conclusion was accidentally correct within the tested set,
but incomplete. n=36 belongs to Population B (incompatible at r=5.30);
n=41 and n=46 are both machine-frozen (identical to n=39).
The correct statement is: n=31 belongs to a Fibonacci family
{11, 13, 15, 17, 18, 23, 31, 34} that supports 4π holonomy at r=5.30
with N=32, I=1.96.
The family {11, 13, 15, 17, 18, 23, 31, 34} does not follow a simple arithmetic or parity rule. The non-family members that achieve near-2.0 at other radii (n=13, 17, 20, 22, 30, 37) complicate a clean geometric derivation. The σint correlation (−0.88) points to phason-space geometry as the mechanism, but the specific alignment condition remains to be derived.
The gaps in the sequence 11, 13, 15, 17, 18, 23, 31, 34 hint at a sub-Fibonacci recurrence. Note that 18−17=1, 23−18=5, 31−23=8, 34−31=3. The differences {1, 5, 8, 3} are themselves Fibonacci-adjacent numbers. A modular arithmetic analysis of F(n) mod some resonance period may reveal a cleaner pattern.
The 34-site ring at r=5.45 achieves gap=0.0021 at both n=20 and n=30, with Iaz=2.56. Since I>2.0 is the artefact threshold (Note XVII), this result requires dedicated azimuthal verification before being classified as a confirmed detection.
for n in range(10, 51):
par, perp, meta = build_projection_bases(n) # φ_approx = F(n)/F(n-1)
lat = generate_hyperlattice(L=8)
pts_phys = lat @ par.T # physical 2D coordinates
pts_int = lat @ perp.T # perpendicular-space (phason) coords
acc = acceptance_mask(pts_int, golden_cantor)
pts_rot, _ = apply_boundary_helix(pts_phys, pts_int, ..., α=0.18, λ=0.145)
frames_base = compute_site_frames(pts_phys, pts_int, k=8) # phys→internal
frames_rot = compute_site_frames(pts_phys, pts_rot, k=8) # phys→rotated
for r in np.arange(4.5, 6.05, 0.05):
ΔΘ/2π = ring_holonomy(frames_base, frames_rot, center, r, dr)
# Select r with min |ΔΘ/2π + 2|