# Characterization of the FTLE success/failure separation

Describes the separation (shape, spectrum, time-scaling, effect size). Mechanism = open.

## Char 1 — full spectrum vs lambda1 (one mode or whole-spectrum shift?)

### TRM official @58590 (n=2048)
- per-exponent AUC(-λ_i→correct), i=1..8: 0.993, 0.994, 0.994, 0.993, 0.992, 0.987, 0.984, 0.977
- median spectrum | success: +0.011, +0.009, +0.007, +0.006, +0.005, +0.004, +0.003, +0.002
- median spectrum | failure: +0.103, +0.074, +0.057, +0.046, +0.036, +0.029, +0.024, +0.019
- AUC λ1=0.993 | AUC spectral-mean=0.995 | AUC KS-proxy(Σλ⁺)=0.995
- shift uniformity: success−failure median gap per exponent: -0.091, -0.065, -0.050, -0.040, -0.031, -0.026, -0.021, -0.017

### HRM joint @26040 (n=2048)
- per-exponent AUC(-λ_i→correct), i=1..8: 0.987, 0.990, 0.989, 0.990, 0.991, 0.992, 0.992, 0.993
- median spectrum | success: -0.150, -0.163, -0.182, -0.191, -0.203, -0.212, -0.220, -0.227
- median spectrum | failure: +0.031, +0.002, -0.018, -0.030, -0.041, -0.050, -0.056, -0.064
- AUC λ1=0.987 | AUC spectral-mean=0.992 | AUC KS-proxy(Σλ⁺)=0.877
- shift uniformity: success−failure median gap per exponent: -0.181, -0.165, -0.164, -0.161, -0.161, -0.162, -0.163, -0.163

### HRM @26040 (n=8192, est-2)
- per-exponent AUC(-λ_i→correct), i=1..8: 0.984, 0.989, 0.990, 0.991, 0.991, 0.991, 0.991, 0.991
- median spectrum | success: -0.858, -0.887, -0.908, -0.927, -0.941, -0.952, -0.964, -0.972
- median spectrum | failure: -0.600, -0.635, -0.656, -0.671, -0.684, -0.696, -0.706, -0.714
- AUC λ1=0.984 | AUC spectral-mean=0.991 | AUC KS-proxy(Σλ⁺)=0.500
- shift uniformity: success−failure median gap per exponent: -0.259, -0.252, -0.252, -0.255, -0.256, -0.257, -0.258, -0.258

## Char 2 — bimodality vs threshold-on-continuum

BC>0.555 hints bimodal. outcome-rate-vs-λ1 sharpness: width of 25→75% outcome transition,
in λ1 units, divided by the 10–90 inter-percentile spread of λ1 (small = sharp threshold).

- TRM official @58590 (n=2048): BC(all)=0.855, BC(succ)=0.780, BC(fail)=0.303; outcome transition width/spread=0.298 (λ1@75%=+0.068, @25%=+0.094)
- HRM joint @26040 (n=2048): BC(all)=0.555, BC(succ)=0.331, BC(fail)=0.287; outcome transition width/spread=0.124 (λ1@75%=-0.069, @25%=-0.037)
- HRM @26040 (n=8192, est-2): BC(all)=0.554, BC(succ)=0.401, BC(fail)=0.258; outcome transition width/spread=0.150 (λ1@75%=-0.749, @25%=-0.693)

## Char 3 — separation magnitude vs integration window H (all examples, final outcome)

Distinct from E5 (which restricted to undecided@H for *prediction*). Here: how does the
two-class λ1 separation BUILD as the integration window grows?

### TRM: H, AUC(-λ1→final correct), Cohen's d(fail−succ)
- H= 2: AUC=0.782, d=+1.059
- H= 4: AUC=0.891, d=+1.759
- H= 6: AUC=0.929, d=+2.250
- H= 8: AUC=0.947, d=+2.680
- H=10: AUC=0.964, d=+3.118
- H=12: AUC=0.975, d=+3.577
- H=16: AUC=0.993, d=+4.840

### HRM: H, AUC(-λ1→final correct), Cohen's d(fail−succ)
- H= 2: AUC=0.428, d=+0.028
- H= 4: AUC=0.728, d=+0.932
- H= 6: AUC=0.871, d=+1.691
- H= 8: AUC=0.918, d=+2.154
- H=10: AUC=0.940, d=+2.497
- H=12: AUC=0.960, d=+2.821
- H=16: AUC=0.987, d=+3.448

## Char 4 — effect size / overlap beyond AUC (full-window λ1 and KS-proxy)

- TRM official @58590 (n=2048):
    λ1: AUC=0.993, Cohen's d(fail−succ)=+4.84, Bhattacharyya(gauss)=0.042, hist-overlap=0.049
    KS-proxy: AUC=0.995, Cohen's d=+5.55, hist-overlap=0.039
- HRM joint @26040 (n=2048):
    λ1: AUC=0.987, Cohen's d(fail−succ)=+3.45, Bhattacharyya(gauss)=0.227, hist-overlap=0.074
    KS-proxy: AUC=0.877, Cohen's d=+1.14, hist-overlap=0.330
- HRM @26040 (n=8192, est-2):
    λ1: AUC=0.984, Cohen's d(fail−succ)=+3.34, Bhattacharyya(gauss)=0.248, hist-overlap=0.101
    KS-proxy: AUC=0.500, Cohen's d=+nan, hist-overlap=1.000