Proposition: Conditional necessary conditions

Let ${\mathcal{P}}_{\mathrm{int}}$, $\mathsf{Rec}$, $O$, $m^{\circ }$, $V$, $r_O$, $Y$ be as in XV-B's linear interface XVB. Assume:

1. a preliminary lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to V$ with $r_O{F}_0=m^{\circ }\mathsf{Rec}$ exists;

2. the permitted candidate family is $F_0+\mathcal{A}$ for a vector space $\mathcal{A}\subseteq \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ with $a({\mathcal{P}}_{\mathrm{int}})\subseteq {\mathcal{C}}_{\mathrm{part}}$ for all $a\in \mathcal{A}$;

3. the lift of the collision closure to $W$ satisfies $C_{\mathrm{coll}}\subseteq W\setminus O$, with $C_{\mathrm{coll}}$ closed.

Then for every $n\in \ker \mathsf{Rec}$:

1. $F_0(n)\in Y$, and a member of $F_0+\mathcal{A}$ descends only if $F_0(n)\in \{a(n):a\in \mathcal{A}\}\subseteq {\mathcal{C}}_{\mathrm{part}}$; in particular $\mathrm{supp}{F}_0(n)\subseteq C_{\mathrm{coll}}$, and $F_0(n)$ has every property shared by all members of ${\mathcal{C}}_{\mathrm{part}}$;

2. if there is an open $G\subset W\setminus C_{\mathrm{coll}}$ with $F_0(n){|}_G\ne 0$, no member of $F_0+\mathcal{A}$ descends;

3. if a functional $\ell$ on $Y$ satisfies $\ell (a(n))=0$ for all $a\in \mathcal{A}$ and $\ell (F_0(n))\ne 0$, no member descends.

For $n=(qi\theta +1)M_{\mathrm{R}}$ the descent equation is

Proof. (i) $r_O{F}_0(n)=m^{\circ }\mathsf{Rec}(n)=0$, so $F_0(n)\in Y$. If $F_0+a$ annihilates $n$ then $F_0(n)=(-a)(n)$ with $-a\in \mathcal{A}$, so $F_0(n)\in {\mathcal{C}}_{\mathrm{part}}$ by (H2). (ii) is XV-B's Theorem thm:contact with $C=C_{\mathrm{coll}}$, legitimate by (H3) and Proposition prop:classstatus(i). (iii): for all $a\in \mathcal{A}$, $\ell ((F_0+a)(n))=\ell (F_0(n))\ne 0$, so $(F_0+a)(n)\ne 0$ and XV-B's Proposition prop:descent excludes descent. The display is linearity.