Reposted by Damien Robert
“Dear mathematician, can you explain to me what mass is?”
“Very simple, dear: the mass of a dynamical system is the cohomology class of the Galilean group representing lack of equivariance of the moment map on the symplectic manifold that is the phase space of the system.”
![First page of the paper titled “An order-reversing embedding of Turing degrees into Arthur–Nimue–Merlin degrees” by Jean Abou Samra & David Alexander Madore, arXiv:2603.19946v1 [math.LO] 20 Mar 2026. Abstract reads as follows: «The Arthur–Nimue–Merlin degrees are a generalization of the Turing degrees introduced by Kihara as a tangible description of the partially ordered set of Lawvere–Tierney topologies on the effective topos (equivalently, subtoposes of the effective topos). They are defined in terms of a three-player game that introduces both angelic and demonic non-determinism into oracle queries. We construct an order embedding of the Turing degrees with their order reversed into the Arthur–Nimue–Merlin degrees, whose image we call the “co-Turing degrees”; we then study the order relationship of these co-Turing degrees with the (naturally embedded) Turing degrees within the Arthur–Nimue–Merlin degrees.» The opening paragraph of the introduction reads: «We begin with a riddle. King Arthur is desperate to know whether the Grail is in a particular French castle. He can ask arbitrary questions to the all-knowing mage Merlin, who promises to answer truthfully, but with a twist. Instead of merely answering “yes” or “no” in plain English, the mischievous Merlin provides a Turing machine, which halts if and only if the answer is “yes”. Can Arthur, a mere mortal limited by the Church–Turing thesis, find the Grail’s location before the end of days?»](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tnde52rcbqxotp7a2cl7bmxu/bafkreiei2hrbhkmwi3brv3ti2lx777t55fuiwj2lhxp733kg34iilx23vy)
