Sc, holonomy, and Fibonacci family membership all fail to survive K-change · Voronoi mode = 5
Three independent tests converge on the same conclusion. Sc varies by a factor of 2 across K (range 0.957–1.882, spread 0.926). The Fibonacci family discriminator — Sc ≈ 1.410 ± 0.003 — dissolves at K=10. The 2D Voronoi coordination mode is 5, not 8; K=8 cannot be derived from the physical projection geometry. The entire SLH chain from Note XXII to Note XXV is K=8 specific. The K=8 frame is the load-bearing assumption of the hypothesis, and no geometric derivation for it currently exists. One partial anomaly: K=20 also produces Sc ≈ 1.406 (0.4% off) without 4π holonomy, suggesting the strain energy may have richer structure than the holonomy alone.
Running the strain-energy probe at K = 4, 6, 8, 10, 12, 16, 20, 24 with all other parameters fixed (n=31, L=8, w=0.98, r=5.30):
| K | Nring | ΔΘ/2π | gap | Sc | Sc/Sc,K8 | Iaz | 4π? | Sc≈1.410? |
|---|---|---|---|---|---|---|---|---|
| 4 | 32 | −0.954377 | 1.0456 | 1.2593 | 0.892 | 1.964 | no | shifted |
| 6 | 32 | −0.705741 | 1.2943 | 1.2047 | 0.853 | 1.964 | no | shifted |
| 8 | 32 | −1.986474 | 0.0135 | 1.4119 | 1.000 | 1.964 | YES ✅ | STABLE |
| 10 | 32 | +0.944452 | 2.9445 | 0.9566 | 0.678 | 1.964 | no | shifted |
| 12 | 32 | −1.095221 | 0.9048 | 1.4427 | 1.022 | 1.964 | no | 2.2% off |
| 16 | 32 | +2.555729 | 4.5557 | 1.8824 | 1.333 | 1.964 | no | shifted |
| 20 | 32 | +0.107646 | 2.1076 | 1.4058 | 0.996 | 1.964 | no | 0.4% off |
| 24 | 32 | +0.884717 | 2.8847 | 0.9727 | 0.689 | 1.964 | no | shifted |
Sc ranges from 0.957 (K=10) to 1.882 (K=16) — a factor of almost 2. The 0.2% CV that made Sc=1.410 look like a topological invariant in Note XXIV was specific to K=8. Change the frame construction and the "invariant" shifts by up to 33%.
One notable anomaly: K=20 gives Sc=1.406 (0.4% from 1.410) without 4π holonomy. This is not explained by the "K=8 = E8 rank" argument and may indicate a deeper pattern among even multiples of 4 (K = 4, 8, 12, 20 all approach Sc≈1.3–1.44, while odd-multiple K values are further).
Computing the Delaunay triangulation of all 521 physical projection sites. Excluding 16 boundary (convex hull) vertices, the interior coordination number distribution is:
| Degree | Count | % | Bar | Note |
|---|---|---|---|---|
| 4 | 40 | 7.9% | ||
| 5 | 187 | 37.0% | Modal degree (37%) | |
| 6 | 121 | 24.0% | K=6 — 2nd | |
| 7 | 103 | 20.4% | ||
| 8 | 46 | 9.1% | K=8 — only 4th | |
| 9 | 8 | 1.6% |
Mean degree: 5.905. Median: 6. Mode: 5. K=8 is the coordination number for only 9.1% of interior sites — it is not "natural" for this projection. The "K=8 = natural Voronoi coordination" derivation fails.
If K=8 is the correct frame, the justification must come from the 8-dimensional E8 structure — not from the 2D projected geometry. In the 2D shadow, degree 5–6 is overwhelmingly more common than degree 8. Only 46 of 505 interior sites (9.1%) have Voronoi degree 8. The "K = E8 rank = 8 simple roots" argument remains the only candidate, but it is currently circular: we chose K=8 because E8 has rank 8, and found that K=8 gives E8-like results.
The 8-member Fibonacci family {11,13,15,17,18,23,31,34} was identified at K=8 by uniform Sc ≈ 1.410 ± 0.003. Does this family structure survive at K=10?
| n | Family? | Sc (K=8) | Sc (K=10) | Δ% | 4π (K=8) | 4π (K=10) |
|---|---|---|---|---|---|---|
| 11 | FAM | 1.4044 | 1.2134 | −13.6% | YES | no |
| 12 | non | 1.5917 | 1.0079 | −36.7% | no | no |
| 13 | FAM | 1.4117 | 0.9504 | −32.7% | no | no |
| 14 | non | 1.5977 | 1.0142 | −36.5% | no | no |
| 15 | FAM | 1.4063 | 1.3706 | −2.5% | YES | no |
| 17 | FAM | 1.4084 | 0.5999 | −57.4% | YES | no |
| 18 | FAM | 1.4119 | 1.1141 | −21.1% | YES | no |
| 20 | non | 1.4039 | 1.0142 | −27.8% | YES | no |
| 23 | FAM | 1.4113 | 1.4344 | +1.6% | YES | no |
| 30 | non | 1.4039 | 1.1717 | −16.5% | YES | no |
| 31 | FAM | 1.4119 | 0.9566 | −32.3% | YES | no |
| 34 | FAM | 1.4156 | 1.4282 | +0.9% | YES | no |
| 36 | non | 1.5996 | 1.1717 | −26.8% | no | no |
At K=10: no n-value gives 4π holonomy. Sc values scatter across the family: n=15 and n=34 are close to K=8 values (−2.5%, +0.9%), while n=17 and n=31 drop by 57% and 32%. The family boundary collapses. At K=10, both family and non-family members share the same Sc range, removing the discriminant entirely.
The three experiments together force an updated tally of what holds independent of K and what does not:
| Claim | K=8 | Other K | Verdict |
|---|---|---|---|
| 4π holonomy at n=31, w=0.98 | ✅ gap=0.0135 | ✗ all other K fail | K=8 specific |
| Sc = 1.410 (CV=0.2%) | ✅ 1.4119 | ✗ range 0.957–1.882 | K=8 specific |
| Fibonacci family discriminant | ✅ clear boundary | ✗ collapses at K=10 | K=8 specific |
| αH1≈αphys (0.20%) | ✅ via Sc | ✗ follows Sc | K=8 specific |
| Sc/Eideal≈φ/√2 | ✅ 1.1444 | unknown | untested |
| w=0.98 isolated resonance (<0.002 wide) | ✅ confirmed | untested | robust at K=8 |
| Voronoi coordination = 8 | ✗ mode=5 in 2D | — | refuted |
| Iaz = 1.964 at r=5.30 | ✅ stable | ✅ stable (all K) | K-independent |
The azimuthal imbalance Iaz = 1.964 is the one genuinely K-independent quantity: it depends only on the 2D positions of ring sites, not on the frame construction. All physics quantities derived from the frame Jacobian — holonomy, Sc, α — are K=8 specific.
K=20 gives Sc = 1.406, within 0.4% of the K=8 value, without 4π holonomy. Checking the pattern of K values that yield Sc near 1.410: K=8, K=12 (2.2% off), K=20 (0.4% off). These are not simple multiples of a single base. One observation: K=8 × 2.5 = K=20, and K=8 and K=20 are separated by K=12. Whether this reflects a resonance in the frame-construction space or is coincidental is unknown.
The honest position after twenty-seven notes: the SLH has found a rich and internally consistent structure at K=8 frames, including a 4π holonomy signal, a near-constant Sc, a φ/√2 bandgap identity, and a 0.20% match to αphys. None of these survive the simplest parameter change (K: 8 → 10).
The critical open question is not "does K=8 work?" — it does, and the results are internally coherent. The question is "why K=8 and not K=10 or K=6?" Without a derivation, the SLH is a K=8 phenomenology, not a K=8 theory.
The strongest route: compute the k-NN graph of the n=31 quasicrystal in the perpendicular space (the 6D internal coordinates), not the 2D physical plane. If the perpendicular-space coordination number is 8 for the ring sites at r=5.30, then K=8 is derivable from the E8 projection structure. The 2D audit failed; the 6D audit has not been done.
Sub-exp 1: n=31, L=8, w=0.98, r=5.30, K ∈ {4,6,8,10,12,16,20,24}
Sub-exp 2: Delaunay triangulation of 521 physical sites (scipy.spatial.Delaunay)
Interior: 505 sites (excluding 16 convex hull boundary)
Sub-exp 3: K ∈ {8,10}, n ∈ family{11,13,15,17,18,23,31,34} + non{12,14,20,30,36}