The lens is a line · the truth is a curve · the gap is the error

The gap.

Move a prompt's activation along a direction δ, away from the point the lens was fit on. The true transport (curve) bends away from the lens's linear reading (tangent). The field between them — opening ⟨⟩ on both sides of the fit point — is exactly how wrong the lens is, and it widens as κt².

What it signifies: the tangent is the Jacobian lens — a straight line, exact only at the fit point. The curve is reality. Their gap is the lens's reading error for an off-average prompt; κ is the trust radius — how far you can move before the line lies. Narrow gap → trust the readout far out. Wide gap → the lens is only locally true.

direction δ:

measure gap at t = 0.80

true transport (curve) lens (tangent line) the gap field ⟨⟩

—

distance t from fit point

—

gap width (lens error)

—

κt² (predicted gap)

—

κ (trust radius⁻¹)

the field opens ⟨⟩ from zero at the fit point (t=0, where the lens is exact) and widens on both sides — quadratically, gap ≈ κt². at t=0 the line and curve touch; that is the only place the linear lens is exactly right.
compare directions: δ₀ curves away fastest (widest gap at the edges), δ₂ slowest — the lens is more trustworthy along δ₂. same lens, different reliability by direction.
real data: the curve is the actual transport of a from-scratch transformer, projected on the curvature axis. far out, the gap exceeds κt² as third-order terms bend it further — honest, not a clean parabola forever.