Forward- and reverse-mode AD apply the same Jacobian in dual directions — the VJP's linear part is the transpose of the JVP's (pushforward vs pullback)

The two AD modes are not sibling algorithms but one linear object used two ways. Stated at Tier 1 from both vocabularies:

The classical framing (Baydin et al., read directly). Forward mode "can be viewed as evaluating a function using dual numbers" — truncated Taylor series v + v̇ε with ε² = 0, whose product rule carries derivatives. Seeding the tangent ẋ = r yields the directional derivative ∇f·r in one pass. Reverse mode is the two-phase sweep: record forward, then propagate adjoints v̄ᵢ = ∂yⱼ/∂vᵢ backward (claim-autograd-three-reifications-of-the-tape).

The geometric framing (JAX docs, fetched directly). JVP: (x, v) ↦ ∂f(x)v — the pushforward, building Jacobians "one column at a time." VJP: (x, v) ↦ vᵀ∂f(x) — the pullback, "one row at a time," and "the linear part of a VJP [is] the transpose (or adjoint conjugate) of the linear part of a JVP."

The cost duality follows from the shape, not from benchmarks. A gradient of f: ℝⁿ→ℝ needs n forward passes but one reverse pass — why training is reverse-mode (claim-backpropagation-special-case-of-reverse-mode-ad, claim-cheap-gradient-bound-two-figures). The memory mirror: forward mode's footprint is depth-independent; reverse mode's scales with the recorded computation. (JAX's "~3x FLOPs per JVP" is its own implementation's self-characterization — kept scoped, not a universal AD constant.)

Vocabulary caution carried from the capture: JAX never uses a literal dual-number type — dual numbers and per-primitive JVP rules are the same computation in two technical vocabularies, not two computations. Cluster: moc-backpropagation-origins.