E0 + E1 · Γ_ε = I − ε K²
Richardson contraction.
The signed selector is squared so the filter is monotone. Minimax step ε* = 2/(μmin+μmax) = 1/99144 equalizes the slowest and fastest complement modes at |λ| = 15/17. Q4.28 is applied to the normalized multipliers, not to unscaled K².
μ⁺_min
11,664
μ_max
186,624
ε*
1/99144
ρ*
0.882353
Engine
cycle 0float r⊥ 0.833364Q4.28 r⊥ 0.833364‖Δx‖ 1.206834e-8
Per-sector multipliers
- J=0 κ²=32,4000.67320261
- J=1 κ²=12,5440.87347696
- J=2 κ²=01.00000000 1
- J=3 κ²=11,6640.88235294
- J=4 κ²=19,6000.80230775
- J=5 κ²=01.00000000 1
- J=6 κ²=186,624-0.88235294
Stability window 0 < ε < 1/93312. Normalized path uses Ķ = K/432 and α = 32/17 ≈ 1.882353 so Q4.28 never has to store 186624.
Q4.28 monograph table
Published residual trajectory. LSB = 2⁻²⁸ ≈ 3.7253e-9. Hardware floor ≈ 17.6 LSB.
| n | float r⊥ | Q4.28 r⊥ | ‖Δx‖₂ |
|---|---|---|---|
| 1 | 6.1798e-1 | 6.1798e-1 | 1.5290e-8 |
| 5 | 3.3946e-1 | 3.3946e-1 | 2.7261e-8 |
| 10 | 1.7022e-1 | 1.7022e-1 | 3.5095e-8 |
| 20 | 4.6491e-2 | 4.6491e-2 | 6.0421e-8 |
| 50 | 1.0544e-3 | 1.0544e-3 | 1.3045e-7 |
| 100 | 1.9791e-6 | 1.9770e-6 | 2.5222e-7 |
| 150 | 3.7615e-9 | 6.5548e-8 | 3.1251e-7 |
| 200 | 7.1828e-12 | 6.5548e-8 | 3.1308e-7 |
| 220 | 5.8742e-13 | 6.5548e-8 | 3.1308e-7 |
| 250 | 1.3880e-14 | 6.5548e-8 | 3.1308e-7 |
Float continues to ~10⁻¹⁴. Q4.28 saturates at 6.554775×10⁻⁸ with no limit cycle because ρ* < 1.