[there will be a pre-seminar for students at 13:30. For zoom link please email s.nagy@qmul.ac.uk] Following a proposal by H. Sati, we have recently stated a hypothesis about the mathematical home of the quantum charges in M-theory. This “Hypothesis H” refines the traditional proposal for quantization of D-brane charge from K-theory to the non-abelian cohomology theory known as 4-Cohomotopy, whose classifying space is the 4-sphere. Besides its motivation from homotopy-theoretic re-analysis of 11d supergravity and of the old brane scan, Hypothesis H is currently justified by its rigorous implication of a list of long-conjectured M-theoretic consistency conditions on C-field flux and M-brane charges – such as shifted C-field flux quantization, dual Page charge quantization and M2/M5-brane tadpole cancellation. But if Hypothesis H is a correct assumption about the nature of M-theory, this suggests that quantum states of full M-theory should be reflected in the positive cohomology of the moduli space of Cohomotopy cocycles, much like quantum states of non-perturbative Chern-Simons theory are in the Dolbeault cohomology of moduli spaces of (flat) connections. In this talk I discuss how, in the topological sector of D6/D8-brane intersections, such quantum states according to Hypothesis H are identified with *weight systems* on *horizontal chord diagrams*, and how these do reflect a range of phenomena expected from the traditional approaches to understanding these brane intersections, such as non-abelian DBI-theory, the BMN matrix model, Rozansky-Witten theory and Hanany-Witten theory. Specifically, we have proven that the sl(2,C)-weight system satisfies the positivity condition that characterizes physical (i.e. non-ghost) quantum states. Under the above identification, this quantum state corresponds to an elementary fuzzy funnel configuration and/or to the elementary transverse M5-brane state in the BMN matrix model – both as expected for D6/D8-brane intersections. Besides possible implications for the elusive formulation of M-theory, this result may provide a unifying explanation for the plethora of unexpected appearances that chord diagrams are recently making in fundamental high energy physics, notably in discussion of holographic entanglement entropy. Slides and further pointers available at: https://ncatlab.org/schreiber/show/Some+Quantum+States+of+M-Branes+under+Hypothesis+H