ContinuumSurvivalOpen
plain-language theorem explainer
Continuum survival of the certificate residual is the open claim that some shape-regular refinement family keeps a strictly positive liminf of normalized separation from the metric image. Gravity analysts on the G4/G5 continuum-promotion campaign cite it as the survive terminal of the frozen trichotomy (survive / metric collapse / lattice washout). The declaration is a bare existential Prop, not a proved theorem; inhabiting it needs the Arc 2 geometric mesh Tendsto baseline.
Claim. There exists a shape-regular refinement family $F$ such that continuum survival holds for $F$: some $\varepsilon > 0$ eventually lower-bounds the normalized separation of $F$ along the refinement levels (positive liminf distance to the metric image).
background
Campaign G4/G5 replaces finite Boolean non-membership in the metric edge image by a normalized-separation trichotomy under a shape-regular refinement family. At refinement level $n$, with response difference $\Delta_n$ and metric image $M_n$, the three frozen terminals are: survive (positive liminf of normalized distance to the metric image), metric collapse (normalized distance tends to 0 while the response norm does not), and lattice washout (response norm tends to 0).
A refinement family packages a response difference, a metric projection, and a positive level norm at every $n$ (shape regularity). Survival for such a family means there is $\varepsilon > 0$ so that, eventually in $n$, the normalized separation stays at least $\varepsilon$. The finite residual already sits outside the metric edge image; that Boolean fact is imported and is not continuum novelty.
The residual itself is the signed genesis-versus-CODATA gap on $\alpha^{-1}$, confined to a certified band. Continuum terminals for the certificate pair remain open Props until a geometric mesh Tendsto baseline is in place.
proof idea
Definitional abbreviation, not a proof. The Prop is exactly the existential statement that some refinement family satisfies the continuum survival terminal (positive eventual lower bound on normalized separation). No tactics, no lemmas discharged.
why it matters
Records the survive arm of the G4/G5 continuum trichotomy as an explicit OPEN residual. Module honesty forbids promoting the finite outside-image theorem to continuum novelty; this Prop is the named place where a genuine continuum survive claim would land once the Arc 2 step-9 mesh Tendsto baseline exists. Sibling open residuals cover metric collapse and lattice washout; the continuum-promotion-earned flag stays false until a terminal is inhabited. No downstream theorems currently consume it (used_by empty), so its role is bookkeeping and campaign gating rather than a proved forcing-chain step. It does not touch T0–T8, RCL, or the mass ladder directly; it sits in the gravity analysis layer that polices residual claims about order-sensitive history response in 4D.
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