Foundations of AbuGhuwaleh Jet-Transport Geometry Cocycles, Rigidity, and Singular Fingerprints

Let f be holomorphic on a domain, and let \ ₙᶠ (z, ): =f^ (n) (z+) f^{ (n) (z) } \ whenever f^ (n) (z) 0 and both points z, z+ lie in. We develop a transport-cocycle viewpoint on the derivative ladder \f^{ (n) \}₍₀ and call the resulting framework AbuGhuwaleh Jet-Transport Geometry. The basic object ₙᶠ simultaneously couples the derivative order n, the base point z, and the translation parameter. We prove five foundational results. First, ₙᶠ is an exact holomorphic cocycle for the local translation action, and every nonvanishing holomorphic translation cocycle is of the form g (z+) /g (z) for a unique zero-free holomorphic gauge g, up to a multiplicative constant. Second, with qₙ=f^ (n+1) /f^ (n), the transport law \ ₍+₁ᶠ (z, ) =ₙᶠ (z, ) qₙ (z+) qₙ (z) \ holds exactly, yielding the discrete Riccati-type recurrence q₍+₁=qₙ+ (qₙ) ' wherever qₙ0. Third, the renormalized cocycle \ ₙᶠ (z, ): =e^-qₙ (z) ₙᶠ (z, ) \ has logarithm \ ₙᶠ (z, ) =₌₂qₙ^ (m-1) (z) m!\, ᵐ, \ which defines a canonical curvature hierarchy attached to the transport. Fourth, we classify the finite-degree rigid models: ₙᶠ has finite transport degree d if and only if f^ (n) =Ce^P on the relevant zero-free patch, where P is a polynomial of degree at most d. Finally, for a meromorphic level g with local factorization g (z) = (z-a) ᵐ h (z), we show that g contains a universal singular packet \ m\![\! (1+z-a) -z-a, \] which detects the location and multiplicity of zeros and poles. These results establish jet transport as a coherent geometric object rather than a single derivative ratio, and they provide a precise starting point for higher-order asymptotics, inverse recovery, spectral transport, and multivariate extensions.