Calibrated increments along a parameter path

For a fixed recorded increment r and positive packet duration t, let b(s)=b₀+s db and g(s)=g₀+s dg. Write a₀=r−t b₀, c₀=1+g₀ and D=c₀+s dg. The native calibration expression is

\[ q(s)=\frac{a_0-st\,db}{D}. \]

After cancellation, its first derivative has numerator N=−t db c₀−dg a₀, independent of s. Therefore

\[ q'=N/D^2,\qquad q''=-2dg\,N/D^3,\qquad q'''=6(dg)^2N/D^4. \]

The implementation evaluates exact rational interval enclosures for supplied direction boxes and a subinterval of [0,1]. It requires the complete denominator range to stay positive. It cancels the common product before bounding it.

On one stable packet branch the partial fraction is affine, u(s)=u₀+u₁s. For the partial increment, the derivative of order n is

\[ (uq)^{(n)}=u q^{(n)}+n u_1q^{(n-1)},\qquad n=1,2,3. \]

The partial duration has derivatives tu, tu₁, 0, 0. A range for u extending outside [0,1] is rejected, rather than clipped or silently moved to another packet. Exact endpoints are retained and binary outputs are rounded outward.

These formulas bound packet calibration/partial timing only. Rotation, velocity, position, frame transforms, derivative jumps and whole-program roundoff still require a qualified native implementation.