P(r)=r^-s / Zeta(s) with s > 1
Ça fait longtemps que j'avais pas sorti Maria
Elle voudrait que j'y passe la bague à la manita
J'ai loué la 458 Italia
Moi, j'suis comme maman, des fois, j'écoute Dalida v -> v + u
without u ⟂ v
|v+u|^2 = |v|^2 + 2 v•u + |u|^2
K(v+u) = 1/2 m|v|^2 + m v•u + 1/2 m |u|^2
K(v+u) - K(v) = m v•u + 1/2 m|u|^2
Euclidean rotation becomes hyperbolic rotation
Euclidean norm becomes Minkowski norm
The energy-momentum invariant replaces kinetic energy taken in isolation
Newton reappears as a term in the expansion
E = γmc² = mc² + 1/2 mv² + 3/8 mv⁴/c² + ...
S_rel = ∫ [ -mc² + 1/2 mv² + 1/8 mv⁴/c² + ... ] dt
- two functors between the theories:
I: T0 -> T1
R: T1 -> T0
- that compose into an identity:
R ∘ I = Id_T0
I(h) = b ∘ a
R(b ∘ a) = R(b) ∘ R(a) must hold too
Z(b ∘ a) = Z(b) + Z(a).
f(√(a^2 + b^2)) = f(a) + f(b)
Hey I think know what you're attempting to do here. I ... suspect you may be some kind of trisector (https://www.ufv.ca/media/faculty/gregschlitt/information/Wha...). A very successful one at that! Please don't be offended by the snarky remarks in this PDF nor by the comparison itself. I think I'm a kind of trisector myself, except that I'm more interested in trisectors themselves than in whatever is being trisected. That is in fact the only reason I think I understand what EchoKey is about.
You're not the only person I've seen pursuing a project like this. We're living through strange times in which people independently arrive at theories of composition, that quite don't fit together because you can compose as hard as you want, the gap between the real and the symbolic is conserved and is free to vary from one individual's imagination to another. At least this is my take. Composition may be a larger concept than applied categoricians tend to assume, and one question that could be asked in EchoKey is: can composition be expressed symbolically in a closed form, or is it essentially semiotic, i.e. it can't stabilize on a definition because it has to be iteratively revised?
The people I have in mind and that I think we're part of all have a certain definition, a specific theory of the same thing that is somehow always incomplete, often isolating in that it's a kind of rock we push with us with the perspective however diffuse of reaching some place in the relational or even affective domain. I'm not sure how to put it. My latest idea is "energy as the epistemic invariant of theoretical extension". I don't think we'll be able to figure out anything without what we end up with also amounting to composing with each other, because deep down energy is what we observe to remain when decomposable objects are routed through decomposable processes that once rerouted and recomposed preserve the same thing: the intactness of being as that which never changes through the processes it takes part in, and which allows us to stitch, without gaps or overlaps, the ever-extensible fabric of the world as the frontier containing all processes.
Is this realizable? Or just a dream? I'm not sure, it's a beautiful reality. Reality, not idea, what we are, not what we think. I'm hopeful this can happen and if I'm not mistaken you and I may be the only two people in this thread who understand what we're talking about—until some third person shows up, maybe? If so, then it's quite remarkable.
P.S.: what's the link between the Boltzmann generators papers and your pipeline? Why is energy involved?
> Our approach combines augmented coupling flows with graph neural networks to exploit local environments, enabling energy-based training and rapid inference.
Does this relate to the Jacobian conjecture?