Protected p-Division and the Extended Agrachev–Gamkrelidze Construction for Finite Pre-Lie Rings

Let p > 3, and let A be a finite pre-Lie ring whose additive group has p-power order and which satisfies Property 3 of Smoktunowicz. We give a two-part answer to Kourovka Notebook Problem 21.123, distinguishing the typographically literal formulas from an explicit factorial-compatible corrected prescription.First, the formulas displayed around the proposed modified group-of-flows construction do not, when read typographically literally, define a total operation. Several local indices do not align, an arbitrary set-theoretic right inverse of multiplication by p need not be iterable, and the movement identity printed in Remark 63 of the source need not hold with its stated shielding exponent. Second, after the factorial-compatible local adjustments and replacement of that identity by the block-shielding law proved here, the corrected partial prescription has a unique conservative totalization. Here conservative means that the totalized value agrees with every corrected literal section iterate at every point where that iterate is defined. The result covers both natural readings of the pullback notation: a direct set-theoretic \( p^{s_j} \)-pullback and a repeated p-section wherever the latter is defined.We construct this totalization through protected divided operator words. Every p-pullback is separated from the next pullback and from both endpoints by enough operator factors for Property 3 to annihilate all root-choice and additivity defects. This gives a choice-independent noncommutative functional calculus on finite abelian p-groups, stable under products and substitution and supporting protected exponential, logarithm, and Baker-Campbell-Hausdorff series.A faithful affine representation of the pre-Lie ring yields the decisive closure statement: the protected BCH series returns to the affine image, so the protected exponentials are closed under actual composition. Their orbit map is bijective, and transport of composition defines a group law on A whose \( \lambda \)-maps are additive automorphisms. Thus the conservative totalization is a left brace. If one fixed section has all required literal iterates defined, its literal construction is exactly this group law. Consequently, the typographically literal reading does not yield a total operation, whereas the explicitly corrected reading formalized in this paper gives an affirmative answer to the mathematical question underlying Problem 21.123.