Borel-code, measure, category, and perfect-set absoluteness

Induction on the well-founded code gives $B^N=B^W\cap {\mathbb{R}}^N$: basic rational intervals are absolute, and complement and countable union commute with intersection with the shared reals. If the two models have the same reals, evaluations are identical.

A coded open or closed set has the same rational basis/tree description. If a Borel set is null, F2 says that for each $j$ there is a countable elementary-set cover of cost below $2^{-j}$; Countable Choice selects these covers, and pairing their indices and coding their real endpoints gives one real witness. Conversely that displayed witness proves outer measure zero. Thus such a witness remains valid in an outer transitive model, and when the two models have the same reals it transfers in both directions. A displayed sequence of closed nowhere-dense Borel codes behaves identically: closedness and the rational-basis test for empty interior are absolute on shared reals, and the same sequence witnesses meagreness. This is the coded content used below; no claim that the two bare definitions in F2 manufacture witnesses by themselves is made.

A simultaneous induction on a Borel code produces an open-mod-meagre pair together with an explicit sequence of closed nowhere-dense codes covering the error. A basic open code uses itself and the empty sequence; for complements, replace the complement of the current open set by its interior and append its closed nowhere-dense boundary; for countable unions, union the open representatives and pair the two natural indices of all exception sequences. The construction is recursive from the Borel code, so the witness lies in every model containing that code. Countable Choice from F4 closes the meagre ideal under the displayed union, while F4 supplies completeness and countable closure for the null ideal in $M$; the ground and forcing models have ambient Choice.

For a coded closed set in two models with the same reals, nonemptiness is absolute and absence of isolated points is equivalent to the rational splitting test: every basic interval meeting the set contains two disjoint smaller basic intervals meeting it. The real witnesses transfer in both directions. Alternatively, an explicit pruned binary tree whose successor cylinders are disjoint certifies nonemptiness and supplies branches by F4 inside $M$ (and by Choice in the ambient forcing models). These are exactly the two perfect-set transfers used below. A closed code can acquire a new branch in an outer model with new reals, so no such blanket downward absoluteness, and no absoluteness for arbitrary uncoded sets, is claimed.