OUTPUT of run 4 of at most 100:
```
Key-path matching: tr(QK_l @ OV_0)/(||QK_l||F ||OV_0||F)
L1H0 [0.0018 0.0341 0.0502 0.0058 0.0292 0.0479]
L1H1 [0.0004 0.0065 0.025  0.0141 0.0042 0.0212]
L1H2 [-0.0018  0.0296  0.0418  0.0121  0.0312  0.0467]
L1H3 [0.0012 0.0283 0.0453 0.0051 0.0242 0.0429]
L1H4 [0.0007 0.0167 0.0297 0.001  0.0233 0.0368]
L1H5 [-0.0012  0.0195  0.0396  0.0112  0.0194  0.0307]
L2H0 [-0.0004  0.0385  0.0584  0.0038  0.0229  0.0667]
L2H1 [0.0014 0.021  0.0334 0.0005 0.0129 0.0425]
L2H2 [0.0025 0.0257 0.0391 0.0053 0.0136 0.0461]
L2H3 [0.0031 0.0164 0.0226 0.0047 0.0122 0.0247]
L2H4 [-0.0014  0.0299  0.0451  0.0088  0.022   0.0512]
L2H5 [0.0008 0.0211 0.0362 0.0006 0.0135 0.0475]
L3H0 [-0.0025  0.0081  0.0128 -0.0104  0.0049  0.013 ]
L3H1 [ 0.0024  0.0098  0.0134 -0.0017  0.0037  0.0139]
L3H2 [-0.0004  0.0069  0.0119  0.0018  0.0006  0.0036]
L3H3 [-0.008   0.0019  0.0021  0.0626  0.0024 -0.0016]
L3H4 [0.0052 0.0093 0.0092 0.0064 0.0104 0.0089]
L3H5 [-0.0004  0.0024  0.0073 -0.0032  0.      0.0088]

Largest matching compositions across all preceding heads:
L1H0 [('L0H2', np.float64(0.0502)), ('L0H5', np.float64(0.0479)), ('L0H1', np.float64(0.0341)), ('L0H4', np.float64(0.0292))]
L1H1 [('L0H2', np.float64(0.025)), ('L0H5', np.float64(0.0212)), ('L0H3', np.float64(0.0141)), ('L0H1', np.float64(0.0065))]
L1H2 [('L0H5', np.float64(0.0467)), ('L0H2', np.float64(0.0418)), ('L0H4', np.float64(0.0312)), ('L0H1', np.float64(0.0296))]
L1H3 [('L0H2', np.float64(0.0453)), ('L0H5', np.float64(0.0429)), ('L0H1', np.float64(0.0283)), ('L0H4', np.float64(0.0242))]
L1H4 [('L0H5', np.float64(0.0368)), ('L0H2', np.float64(0.0297)), ('L0H4', np.float64(0.0233)), ('L0H1', np.float64(0.0167))]
L1H5 [('L0H2', np.float64(0.0396)), ('L0H5', np.float64(0.0307)), ('L0H1', np.float64(0.0195)), ('L0H4', np.float64(0.0194))]
L2H0 [('L0H5', np.float64(0.0667)), ('L0H2', np.float64(0.0584)), ('L0H1', np.float64(0.0385)), ('L0H4', np.float64(0.0229))]
L2H1 [('L0H5', np.float64(0.0425)), ('L0H2', np.float64(0.0334)), ('L0H1', np.float64(0.021)), ('L0H4', np.float64(0.0129))]
L2H2 [('L0H5', np.float64(0.0461)), ('L0H2', np.float64(0.0391)), ('L0H1', np.float64(0.0257)), ('L0H4', np.float64(0.0136))]
L2H3 [('L0H5', np.float64(0.0247)), ('L0H2', np.float64(0.0226)), ('L0H1', np.float64(0.0164)), ('L0H4', np.float64(0.0122))]
L2H4 [('L0H5', np.float64(0.0512)), ('L0H2', np.float64(0.0451)), ('L0H1', np.float64(0.0299)), ('L0H4', np.float64(0.022))]
L2H5 [('L0H5', np.float64(0.0475)), ('L0H2', np.float64(0.0362)), ('L0H1', np.float64(0.0211)), ('L0H4', np.float64(0.0135))]
L3H0 [('L0H5', np.float64(0.013)), ('L0H2', np.float64(0.0128)), ('L0H1', np.float64(0.0081)), ('L0H4', np.float64(0.0049))]
L3H1 [('L0H5', np.float64(0.0139)), ('L0H2', np.float64(0.0134)), ('L0H1', np.float64(0.0098)), ('L0H4', np.float64(0.0037))]
L3H2 [('L0H2', np.float64(0.0119)), ('L0H1', np.float64(0.0069)), ('L0H5', np.float64(0.0036)), ('L0H3', np.float64(0.0018))]
L3H3 [('L0H3', np.float64(0.0626)), ('L2H5', np.float64(0.0482)), ('L2H1', np.float64(0.0328)), ('L1H0', np.float64(0.0305))]
L3H4 [('L2H2', np.float64(0.0143)), ('L2H4', np.float64(0.0122)), ('L0H4', np.float64(0.0104)), ('L2H5', np.float64(0.0094))]
L3H5 [('L0H5', np.float64(0.0088)), ('L0H2', np.float64(0.0073)), ('L0H1', np.float64(0.0024)), ('L0H4', np.float64(0.0))]

L0H3 OV spectrum after LN scale/centering
[1.126  0.9554 0.9495 0.8895 0.8632 0.8537 0.8418 0.7799 0.7586 0.7228 0.7078 0.6727 0.6691 0.6516 0.6339 0.6167 0.5652 0.528  0.4384 0.3824 0.3491
 0.3158 0.2968 0.2947 0.2815 0.2755 0.2646 0.246  0.2403 0.233  0.2243 0.2131 0.2093 0.1972 0.1892 0.1722 0.1586 0.1483 0.1368 0.1327 0.1111 0.0978
 0.083  0.0703 0.0613 0.0516 0.0467 0.0461 0.0386 0.0335 0.0318 0.0305 0.0265 0.0261 0.0251 0.0226 0.0221 0.0213 0.0206 0.0184 0.0178 0.0146 0.0127
 0.0108]
L0H0 OV spectrum
[0.9574 0.898  0.8182 0.6413 0.5544 0.4742 0.4388 0.4058 0.3721 0.3476 0.3294 0.3082 0.2579 0.2439 0.2052 0.1953 0.1708 0.1431 0.1304 0.1084 0.096
 0.0926 0.0869 0.0828 0.0698 0.0674 0.0635 0.0576 0.0526 0.0519 0.0457 0.0442 0.0425 0.0417 0.0408 0.039  0.0365 0.0354 0.034  0.0323 0.0303 0.0287
 0.0279 0.0267 0.0253 0.0246 0.0235 0.0224]

MLP0 input-weight Gram coupling between current-token and H3-written subspaces
rank 12 eigs [39.741+0.j     7.056+0.j     4.765-3.776j  4.765+3.776j -2.433+4.831j -2.433-4.831j -5.141+0.689j -5.141-0.689j  5.113+0.j     1.648-2.403j]
rank 16 eigs [39.852+0.j     7.234-2.059j  7.234+2.059j -0.98 -6.718j -0.98 +6.718j  5.039-4.178j  5.039+4.178j -5.491+0.j    -3.633+1.446j -3.633-1.446j]
rank 20 eigs [41.786+0.j     7.33 -6.542j  7.33 +6.542j -8.043+0.j     6.207-4.055j  6.207+4.055j -5.59 -3.353j -5.59 +3.353j  0.14 +6.279j  0.14 -6.279j]
rank 24 eigs [41.757+0.j     7.518-6.519j  7.518+6.519j -8.102+0.j     6.02 +4.113j  6.02 -4.113j -5.638-3.474j -5.638+3.474j  0.077-6.596j  0.077+6.596j]
rank 28 eigs [41.366+0.j     7.666-6.504j  7.666+6.504j -8.123+0.j     6.198+4.074j  6.198-4.074j -5.748-3.661j -5.748+3.661j  0.223+6.696j  0.223-6.696j]
rank 32 eigs [41.282+0.j     7.748-6.285j  7.748+6.285j -7.918+0.j     6.278+3.815j  6.278-3.815j -5.778+3.556j -5.778-3.556j  0.61 -6.066j  0.61 +6.066j]

Later-head QK on H3's read-to-write circuit (r=24)
L1H0 [ 0.128   0.0389  0.5584  0.0466 -0.042 ]
L1H1 [ 0.1472  0.0892  0.5631  0.0608 -0.0361]
L1H2 [ 0.1417  0.0771  0.5887  0.0544 -0.0481]
L1H3 [ 0.1321  0.0271  0.5715  0.0402 -0.051 ]
L1H4 [ 0.1014  0.007   0.5075  0.0277 -0.0313]
L1H5 [ 0.1394  0.0696  0.5988  0.0558 -0.0541]
L2H0 [ 0.0662  0.0166  0.5246  0.0219 -0.0148]
L2H1 [ 0.1033  0.0017  0.5632  0.032  -0.0293]
L2H2 [ 0.0711  0.0257  0.5064  0.021  -0.016 ]
L2H3 [ 0.0729  0.0241  0.5548  0.025  -0.0186]
L2H4 [ 0.1015  0.0754  0.5898  0.0356 -0.0282]
L2H5 [ 0.1057  0.0023  0.5316  0.031  -0.0299]
L3H0 [ 0.1062 -0.0634  0.4957  0.0332 -0.0332]
L3H1 [ 0.0521 -0.0047  0.5157  0.0165 -0.0141]
L3H2 [ 0.0577  0.0106  0.5564  0.0192 -0.0143]
L3H3 [ 0.152   0.4151  0.7693  0.0613 -0.0104]
L3H4 [ 0.0252  0.0139  0.5185  0.0096 -0.0056]
L3H5 [ 0.0711 -0.0108  0.517   0.0202 -0.0214]

```