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We study exact differential-field generation by an output Y=e^fH, where H must be prepared from a specified private differential field. In an actual exponential extension of an algebraic-function field, we give a compatibility criterion for realizing a prescribed complete stabilizer. It retains integral indices, constant phases and the distinct preparation and target actions. Necessity covers arbitrary rational multipliers; sufficiency constructs a Laurent multiplier with bounded support. A separate elementary theorem applies to any independent input whose logarithmic derivative lies in a finite coefficient extension: all successful multiplier fields have one common coefficient completion. Finite degree gives minimal sufficient supplies, and access to the coefficient field restores a least supply. For algebraic exponents the intrinsic coefficient field is B(f')=B(f); under input independence, rational exponents therefore give least supplies. A complete example has infinitely many incomparable minimal supplies, with no least supply even for full recovery. Without input independence, there are examples with normalized rational multipliers but no minimal sufficient supply, even for full recovery. Under exact presentations, a terminating procedure extracts legal multipliers from supplied germs and finitely many whole derivatives. Classical field, primitive-element and invariant-theoretic mechanisms are distinguished from the restricted realization criterion.

This paper belongs to the Encounter Arithmetic programme, which studies structure accessible from analytic responses under specified operations. It examines exact generation from private differential preparation resources. Its recovery claims concern whole functions and their derivatives, not finitely many scalar measurements.

Related programme papers cited in this manuscript:

Encounter Fields: Exponentials, Lambert-Type Inverses, and an Exact Algebraic Boundary (version 2.0).

Recovery and finite loss under exponential composition in Lambert fields (version 1.0).