The finite-difference curve · derived, measured, placed

‖r(ε)‖ = κε²

STEP 3 sampled this at four points and called it "second-order." Here is the full curve across nine decades of ε — the theoretical κε² line, the real measured residuals placed on it, and the precision floor where measurement peels off theory. Two independent computations of κ, one curve.

r(ε) = [F(x₀+εδ) − F(x₀)] − ε·Jδ = (ε²/2)·D²F[δ,δ] + O(ε³) ⟹ ‖r(ε)‖ → κ·ε², κ = ½‖D²F[δ,δ]‖ LIT

direction δ:

theory κε² (slope 2) measured ‖r(ε)‖ precision floor

the clean band (ε ≈ 1e-2…1e-6): measured points sit exactly on the slope-2 theory line — ‖r‖/κε² = 1.000. the ε² law, confirmed.
the floor (ε ≲ 1e-7): the FD subtraction F(x₀+εδ)−F(x₀) loses all significant digits to catastrophic cancellation; measured residual flatlines at machine-ε (~1e-14) while theory keeps dropping. this is a limit of float64 arithmetic, NOT of the identity.
the point: the constant κ isn't fit to the data — it's computed independently by second-order autograd, and the measured curve lands on it. derived, measured, matched.