(a) original model, rings of 2N sites, N=400 cells, 40 samples per lambda
  lam=  2.01: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [4017, 4017, 7966, 7966, 4017, 4017]
  lam=   2.5: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [4017, 4017, 7966, 7966, 4017, 4017]
  lam=   3.0: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [3987, 3987, 8026, 8026, 3987, 3987]
  lam=   5.0: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [3959, 3959, 8082, 8082, 3959, 3959]
  lam=  10.0: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [4005, 4005, 7990, 7990, 4005, 4005]
  lam=  -3.0: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [3968, 3968, 8064, 8064, 3968, 3968]
  lam=   1.5: eigenvalues=32000; x strictly in gaps: 0; |x|>sqrt(lam^2+4): 0; eigenvalue counts per x-band (lower/upper E branch): [4528, 4528, 9054, 9054, 4528, 4528]
(b) Lemma: q in {-L,0,L}, no two adjacent +L (random, not alternating); ring of 1000 sites
  L=2.2: eigenvalues strictly inside (2,L): 0
  L=3.0: eigenvalues strictly inside (2,L): 0
  L=6.0: eigenvalues strictly inside (2,L): 0
(c) control: allow adjacent +L sites (i.i.d. q in {-L,0,L})
  L=3.0: eigenvalues strictly inside (2,L): 603  (nonzero expected)
  L=6.0: eigenvalues strictly inside (2,L): 845  (nonzero expected)
