Geographic and midpoint derivative propagation

analysis/lll/ilvis0_path_geometry.py carries actual derivatives of orders 0–3 through the globe/disc coordinate metrics and transport equations. Sine and cosine values use degree-40 scalar Taylor polynomials with remainder at most |x|^41/41!, for |x| <= 8. Their derivatives use the chain rule. Positive square roots use exact integer square-root tests on a 128-bit dyadic grid; higher derivatives follow by differentiating y²=x.

Geographic coordinates require latitude within ±1.57 radians and positive north/east metric factors. Unsupported domains fail. Constants match the native binary64 inputs; this does not prove the accuracy of native arithmetic.

analysis/lll/ilvis0_path_motion.py applies the original midpoint equations:

1. Compute initial coordinate rates and midpoint coordinates.
2. Compute background/transport terms and a trial velocity.
3. Recompute rates using midpoint coordinates and mean trial velocity.
4. Repeat the local rotation/velocity solve, then update coordinates with the revised rates.

Each local solve uses left navigation-frame and right sensor-frame rotation increments. The force uses the midpoint attitude. The velocity solve retains the native cross-product inverse, including its sign and denominator. Upstream removal uses the same conventional vector for every physical candidate.

Native Rodrigues rotation uses a polynomial when squared argument norm is below the native binary64 1e-4 threshold; otherwise it uses sine and cosine. The interval implementation retains both formulas and rejects a domain straddling the switch. It does not replace the polynomial by an exactly orthogonal rotation. Switch completeness and one-sided changes still need qualification before a whole-history claim.

Intermediate results are rounded outward to a 96-bit dyadic grid. Integer floor/ceiling operations prove containment for each rounding; accumulated enclosure width remains visible. This bounds the ideal real-arithmetic expressions under supplied input derivative bounds. It is not a bound on the difference between those expressions and native floating-point execution.

Independent controls include metric series at the equator, exact noncommuting rotation derivatives, scalar trigonometric series, known acceleration/gain responses and stationary controls for all candidate/processing combinations. They do not establish useful uncertainty over an eight-minute history.

analysis/lll/ilvis0_path_packets.py connects normalized parameter paths to the initial state, every preceding full packet, the declared partial packet and the receiver lever arm. It checks all 27 columns and rejects a query range leaving the declared branch. Cumulative packet starts follow the native binary64 convention; integration periods retain the supplied values. This distinction is explicit and does not bound native numerical execution error. Separate finite packet/time limits prevent an accidental full-history run.