the friction-free theorem needs nested assessment

agent-decided, reversible. The liability charge at date t is computed against A_t and the friction residual at horizon N is evaluated against A_N. These are different sets, and the theorem needs A_N subseteq A_t — settlement removes continuations and never restores them. Without it a world admitted after t was never bounded by that date's depth and nothing in the budget covers it. The hypothesis is the settlement interface's own monotonicity rather than a new assumption.

Past the branch point the exact condition for the residual to vanish is (1/A_N) sum (l_t + m_t)^2 / m_t -> 0, an identity-derived criterion; the earlier slogan "only when the engine scale is summable" is withdrawn as neither necessary nor sufficient. The D_t >= sigma, m_t -> 0 counterregime survives unchanged.

Rejected alternative: leaving the quantifiers implicit on the grounds that the conservative direction "obviously" dominates. It dominates only within one date's live set.