Notes on the 3n + a family: a stratum invariant, exact anchor densities, and the component structure of the exceptional set

An independent note on the classical Collatz map and the wider 3n + a family. Three items, each labelled by the status of its evidence: (1) a proved stratum invariant — along any orbit of f_a, the 3-free part of gcd(n, a) never changes, so the multiples of a form a sealed stratum conjugate to the classical map, and a = 1 is the unique single-stratum member; (2) a computed result whose trapping mechanism is proved — for a in {5, 7, 13, 17, 19}, exhaustive counts of starting values to 10^6 give the anchor cycle's basin as exactly 1/a; (3) a conditional structural observation — the exceptional set of the classical map, if nonempty, is a union of full connected components (complete reverse trees) of the Collatz graph, so density results bound its size without constraining its shape. The note closes with two literature questions: whether the component structure of the hypothetical exceptional set has been exploited anywhere, and whether any Terras- or Tao-level almost-all result is known for prime starting values, or any result constraining the prime content of a hypothetical exceptional component. No proof of the Collatz conjecture is claimed.