"Is the orientation of ℂ derived or chosen?" This is the foundational question that complex analysis fails to ask in the standard form. The question of the orientation of ℂ might seem already trivially defined. Orthogonal is orthogonal. What else could need specification? But consider ℍ with respect to the real line and identity of ℂ within ℍ with respect to ℝ is ambiguous. AC doesn't fix a reason for a choice, it only makes one. This produces a proliferation of choices to accommodate this ambiguity: Stiefel-Whitney classes manage the ℤ/2 explosion at ℂ. Pontryagin and Chern classes manage the S² explosion at ℍ. Twisted cohomology and local systems manage the path-dependent explosion at 𝕆. Stable homotopy manages the zero-divisor explosion beyond. The Universal Coefficient Theorem sits underneath all of them as the accounting system that tracks how much structure is lost when you project from one level's explosion to another's. Tor tracks distortion relative to ℤ homology. But why ℤ? The free resolutions underlying Tor computations are covering maps of S¹, so the correction terms already implicitly reference connection structure on S¹ , pointing toward a family of connections ω = (1/n)dθ without singling out which is canonical. So then we ask, what is the connection form that establishes orientation canonically before ambiguities proliferate? Orientation is binary, and for canonicity needs to be both discrete and minimal. Thus ω = ½dθ emerges as the canonical connection form that fixes orientation throughout. Tor then naturally emerges with perfect interpretability from knowing the canonical form first, and the distortion that all other coefficient rings induce from it

One problem I have while doing something similar is that ChatGPT inserts standard stuff too early, without caring that learning needs steps along a path

honestly fuck pythagoras for "solving problems" therefore preventing me from gaining new mathematical knowledge

For decades, Crackpot Physics has loudly announced itself upon entry, as it parlays a largely invented - or at the very least viciously bastardized - mother tongue of eclectic physics vocabulary: It’s fuckin gobbledygook. But I am increasingly worried about the extremely dangerous new form of scientific crankery in the post AI era: An increasing number of moderately trained people seem to be operating in a dangerous middle zone: not obviously ignorant of technical jargon, but apparently missing the discipline that prevents real concepts from being rearranged into grandiose conclusions. They’re leaning extremely heavily on a “silent coauthor”, spending hours and hours querying it and arranging the ideas (incorrectly) in their own deficient mental model and presenting it, confidently, as if the product of their own insights because they prompted it. That can be FAR more persuasive to lay audiences than outright crankery because the vocabulary is recognizably legitimate. But the mathematical control is not there. What makes these “papers” troubling is that they use real technical nouns from physics but the relations between those nouns are scrambled. That can be more misleading than obvious nonsense because it is recognized as domain language even to educated non-specialists. It’s going to mislead a higher caliber of intellect than old school crackpottery ever did.

The Möbius tower traces directly through the quaternions, but imposes a unique structure on it, with direct relevance for physics. So the Möbius Octonionic sector is slightly different from standard Octonions, since there are distinguished directions. It’s an oriented fano plane, and the symmetries there are literally SU(3) color confinement from first principles. I don’t know much about coquaternions, or other split algebras. The Möbius connection didn’t run through those, so I haven’t explored it. My progression has been a fairly linear exploration of the tower structure, as evidenced by my paper series: 1. Foundational connection form, canonical orientation 2. RH, generation of C and rigidity thesis 3. Set theory, 3 valued logic classifier 4. Tower generation- central mechanism, RCHO progression 5. Physics bridge: Topos structure and continuation into post NDA structure 6. Group symmetry derivation, structural assignments 7. (Still writing) Constraint geometry derivation of nuclear magic numbers, other properties. 1-> 6 are already present on my GitHub.