Manuscript XII-F@1 · XII-F.tex · lemma · 54cb3169

Lemma: No ramp contribution

F2: companion support and the off-condition diagnostic · lines 619–635 · proof included or linked

XII-F@1#lem:no-ramp

For the handed companion,

X_{1} [K ρ_{ξ}] = ρ_{ξ} .

Proof. The tensor $K ρ_{ξ}$ is supported above the ramp, so $σ_{ε} K ρ_{ξ} = K ρ_{ξ}$ . Using $D K = □$ gives

X_{1} [K ρ_{ξ}] = E_{0}^{R} D K ρ_{ξ} = E_{0}^{R} □ ρ_{ξ} = ρ_{ξ},

where the last equality follows from zero past data.

Displayed formulas in this passage

unlabeled · line 622
$X_{1} [K ρ_{ξ}] = ρ_{ξ} .$
unlabeled · line 631
$X_{1} [K ρ_{ξ}] = E_{0}^{R} D K ρ_{ξ} = E_{0}^{R} □ ρ_{ξ} = ρ_{ξ},$