theorem
Hex.GraphIso.Nauty.Max.SweepInput.restore_scope
{n k : Nat}
{G : Colored n k}
{ctx : Ctx n}
{tcLevel fuel cfuel : Nat}
{first : Bool}
{level numcells tc tv1 tv index : Nat}
{cell : VSet n}
{st : Search n}
{l : Loop n}
{bs bs' fs : List Nat}
{parents : Parents n}
(h : SweepInput G ctx tcLevel fuel cfuel first level numcells tc tv1 (some tv) cell index st l bs fs parents)
(hgsz : ctx.g.size = n)
(hsymm : ∀ (u v : Nat), u < n → v < n → ctx.g[u]!.mem v = ctx.g[v]!.mem u)
(hloop : ∀ (v : Nat), v < n → ctx.g[v]!.mem v = false)
:
have raw :=
(Nauty.node (first && tv == tv1) ctx (n + 2) tcLevel fuel (level + 1) (numcells + 1)
(child first level tc tv st)).snd;
have middle := if (first && tv == tv1) = true then afterChildFirst level tv1 raw else raw;
have left :=
{ lab := middle.lab, ptn := middle.ptn, active := middle.active, orbits := middle.orbits,
fixedpts := middle.fixedpts.erase tv, autos := middle.autos, wsCap := middle.wsCap, firstcode := middle.firstcode,
canoncode := middle.canoncode, firsttc := middle.firsttc, firstlab := middle.firstlab, canonlab := middle.canonlab,
canong := middle.canong, samerows := middle.samerows, compCanon := middle.compCanon,
eqlevFirst := middle.eqlevFirst, eqlevCanon := middle.eqlevCanon, gcaFirst := middle.gcaFirst,
gcaCanon := middle.gcaCanon, canonlevel := middle.canonlevel, noncheaplevel := middle.noncheaplevel,
allsamelevel := middle.allsamelevel, cosetindex := middle.cosetindex, stabvertex := middle.stabvertex,
numnodes := middle.numnodes, tctotal := middle.tctotal, canupdates := middle.canupdates,
numorbits := middle.numorbits, numgenerators := middle.numgenerators, numbadleaves := middle.numbadleaves,
maxlevel := middle.maxlevel, order := middle.order, genTrace := middle.genTrace, workperm := middle.workperm };
have ready := recover (n + 2) level left;
Generic.Grows (SearchState.key ctx bs st) (SearchState.key ctx bs' ready) →
(∀ (t : Nat) (p : Parent n),
parents t = some p →
p.loop.first = true → ∀ (γ : Array Nat), γ ∈ ready.genTrace → CellStab p.state.ptn t p.state.lab γ) →
Scope G ctx tcLevel level (Loop.codes ctx l) bs' ready parents
A received child preserves every saved ancestor field except for the explicit generator-stabilization obligation. Incumbent growth comes from the child's maximum result and settled-code receipt.