Research Proposal: Towards a Non-Degenerate Cohesive Linear ∞-Topo

This research proposal outlines a systematic programme to construct a non‑degenerate cohesive linear ∞‑topos, overcoming the degeneracy of the centre comonad model presented in [Woo 2026, DOI: https://doi.org/10.5281/zenodo.20443143]. After identifying the structural limitations of precomposition modalities and explaining why the derived centre (Hochschild cohomology) is unsuitable, the document explores four alternative directions: an abelianisation modality, a maximal commutative subalgebra site, the CPM construction, and a loop‑space approach via Hochschild homology. Each direction is analysed with precise mathematical hypotheses, open problems, and concrete first steps. A list of immediate mathematical tasks is provided. The proposal is intended as a research blueprint, inviting collaboration and further development.