Internal-Space Coordination Audit

Notes XXVI and XXVII established that K=8 is the sole value producing 4π holonomy — but neither the 2D Voronoi (modal degree = 5) nor the ring topology can derive it from first principles. The frames are computed via compute_site_frames(pts_phys, pts_int, k=K) where pts_int are the internal / perpendicular coordinates from the E8 cut-and-project scheme.

In a genuine E8 quasicrystal the perpendicular space is 4-dimensional (E8 is 8D; projection selects 4D + 4D). The SLH construction projects through a 6-dimensional lattice. If the internal-space coordination of ring sites is naturally 8 in that higher-dimensional space, then K=8 is geometry's choice, not ours.

This note tests that claim directly with three probes:

The projection bases are built from a rank-6 lattice via build_projection_bases(n). The parallel basis par is shape (2, 6) and the perpendicular basis perp is also shape (2, 6). Thus pts_int = lat @ perp.T gives an (N × 2) array — 2D perpendicular / internal coordinates.

The SLH frame construction lives entirely in a 4-dimensional space (2D physical × 2D internal), not the full E8 octadimensional structure. Any appeal to "E8 rank-8 coordination" cannot be resolved within this implementation.

If the internal space is 2D, its Voronoi / Delaunay structure should look similar to the 2D physical space — and Note XXVII showed that mode degree in 2D is 5. The three sub-experiments below confirm this: all 2D internal-space geometry returns mode = 5, matching the physical-space result exactly.

There is no higher-dimensional geometric reservoir from which K=8 could emerge in this implementation. The derivation path is closed.

For the 32 ring sites at r = 5.30, compute pairwise distances in 2D internal space and tabulate the mean distance to the k-th nearest neighbour for k = 1..24. A natural shell boundary at k = 8 would appear as a large step in the step-ratio d(k+1)/d(k).

The gap at n = 31 is 0.013526. Note XXVI found π/232 = 0.013541 (0.11% off). If this proximity is a phason-strain effect from the Fibonacci approximant, the gap should converge as ε(n) = F(n)/F(n-1) − φ → 0 with increasing n. Sweep n = 28..40, recording ΔΘ/2π and gap = |ΔΘ/2π + 2| for each Fibonacci depth.

Compute the Delaunay triangulation of the 32 ring sites in their 2D internal-space coordinates. Tabulate the vertex degree distribution. If the modal degree is 8, this would support K=8 as the natural internal-space coordination number.

Build a genuine E8 cut-and-project where the internal space is 4D or 6D. With that, compute k-NN and Delaunay in higher-dimensional internal space. Only then can "E8 coordination number = 8" be tested. The D8 root lattice packs with coordination number 2×8 = 16 nearest neighbours; the E8 lattice has 240 roots (coordination number 240 in 8D), but within a 4D slice the cross-polytope gives exactly 8 nearest neighbours.

The number 8 appears in E8 as the Lie algebra rank, the Coxeter exponent spectrum (1,7,11,13,17,19,23,29 — count = 8), and the simple root count. An algebraic argument tying K=8 to one of these would bypass the need for spatial coordination counting entirely. Candidate: the holonomy group of the E8 root lattice projection is Spin(8), which acts on 8-dimensional spinors — matching K=8 as the spinorial dimension.

S_c(K=20) ≈ 1.406 (0.4% off 1.410) without 4π holonomy. This near-hit at a non-holonomy K is unexplained. Is it a coincidence in the S_c landscape, or does it indicate a secondary resonance of the strain energy functional at a different topological phase?

Compute the first Betti number β₁(r) of the k-NN subgraph as r increases from 0 to 10. Does β₁ ≥ 1 (shell forms a cycle) at precisely the graph-confirmed radii? This tests whether confirmed holonomy sites correspond to topologically non-trivial loop formation, independent of the frame-computation K.

sovereign-lattice/data/internal_coord_audit.json
  .knn_6d.n_ring             = 32
  .knn_6d.max_jump_k         = 2   (NOT 8)
  .knn_6d.k_mean_distances[] = [0.147276, 0.259897, ...]
  .n_sweep[]                 = 13 entries (n=28..40), gap, eps, t2pi
  .delaunay_6d.mode_degree   = 5   (NOT 8)
  .delaunay_6d.mean_degree   = 4.56