Symmetric relative cohomology for birth cochains

Background: The paper establishes an asymmetry where birth cochains relate to the preimage under $q^{\#}$ while death cochains relate to the image under $\lambda^{\#}$ in the long exact sequence of relative cohomology.

Question / Future Work: Develop a direct analogue of the results connecting death cochains to relative cohomology (like Proposition 4.3 for death cochains) for birth cochains, closing the established asymmetry between the two concepts.

Why It Matters: Closing this theoretical asymmetry would provide a more complete and unified algebraic framework for both birth and death events in persistent cohomology.

Evidence: At present, we are not aware of any direct analogues of the results in this section and Section 4.3 for birth cochains. This asymmetry is not surprising in the light of the assymetry between Proposition 4.1 and Proposition 4.3…