Covariant Differentiation from the AASC Kernel

AASC is a role-constraint formalism for the necessity conditions of non-degenerate determinate construction. It is not an interpretive overlay placed on independently complete physics. The canonical kernel theorem establishes, by fixeddomain exhaustion, that determinate same-domain use requires Admissibility, Standing, Reference, and Irreversibility/Fixation. This paper asks a narrower question: once a governed domain occupies the covariant-differentiation target, what smooth local first-order realization can faithfully represent the standingpreserving transport-comparison role? The proof separates role necessity, realization existence, and representative classification. First, target occupation forces a pre-differential transport-comparison role: standing-bearing carrier content cannot be compared across admitted continuation by raw representative comparison, static transition bookkeeping, privileged representatives, witnessless assignment, same-act repair, or undeclared scope enrichment. Second, smooth realization is a branch condition: the kernel does not numerically select a connection, a gauge group, a global bundle topology, or a dynamical action. Third, in the standard finite linear carrier branch, an admissible smooth realization is not an arbitrary first-order operator. It is the canonical infinitesimalization of finite standing-preserving linear transport. Identity, restriction, reversal, weak composition, smooth first-jet dependence, lawful-redescription equivariance, and preservation of the already fixed finite linear carrier operations force additivity, direction linearity, and the identity principal symbol σ∇(ξ) = ξ ⊗ idE . Consequently the difference from the raw representative derivative is C∞-linear in both the section and direction arguments, and locally ∇ = d + A, A ∈ Ω1(EndE). Gauge, tangent, and tensor covariant derivatives are typed specializations. Metricity remains downstream from governance. In the minimal first-order natural metric branch, metric comparison preservation gives ∇g = 0, and a natural difference-tensor theorem excludes every nonzero first-order natural metriccompatible contorsion, whether primitive or derived. Hence ∇ = ∇LC in that branch. Torsionful, metric-affine, higher-jet, nonlocal, and modified carrier theories are not denied; they are declared richer-scope or higher-response branches. The novelty is not the familiar local formula. It is the dependency theorem showing that connection type is the forced smooth first-jet realization of kernelforced standing-preserving transport once the covariant-differentiation target, standard finite linear carrier branch, and admissible smooth transport realization are occupied.