Differentiate rules for linear additive vs. linear multiplicative connectives

Determine how the proof-theoretic rules in the proposed "very linear logic" should distinguish the linear additive connectives (arithmetic sum and harmonic sum) from the linear multiplicative connectives (multiplication and its de Morgan dual) so that the two families of connectives are formally and coherently separated within a single system.

Background

The paper introduces a quantitative semantics over the extended positive reals supporting three generations of connectives: non-linear (min/max), linear additive (sum and harmonic sum), and linear multiplicative (product and its dual). These operations interact via various distributivity properties, suggesting a novel substructural logic the authors call "very linear logic."

While the semantic behavior of these connectives is described, the authors note uncertainty about the corresponding proof-theoretic rules that would differentiate the linear additive from the linear multiplicative fragment, and they are not aware of prior formulations of such a logic.