title	dif
tags	cyber, language
alias	Dif, differential language, differential geometry language
crystal-type	entity
crystal-domain	cyber

differential [[geometry]]. Riemannian manifolds, tangent spaces, geodesics, Laplace-Beltrami operator. the [[geometry]] of continuous curved space

Op	Action
chart(M, coords)	Define coordinate patch on manifold
metric(g_ij)	Specify Riemannian metric tensor
christoffel(g)	Compute connection coefficients
geodesic(p, v, t)	Trace geodesic from point p with velocity v
covariant_deriv(T, v)	Parallel transport / covariant derivative
curvature(g)	Riemann curvature tensor
laplacian(f, g)	Laplace-Beltrami operator on manifold

required for: latent space embeddings, [[tri-kernel]] diffusion formalized as heat flow on manifolds, physics simulation. programming model: coordinate charts, metric tensors, covariant derivatives — none of which exist in [[Ren]]. [[proof]]-hard over finite [[field]]s. research horizon

see [[cyb/languages]] for the complete language set. see [[cyb/multiproof]] for the proving architecture