Clock cover and residual jumps

For a fixed affine path, the ideal query is q(s)=t+c+s d, with 0≤s≤1. The packet endpoints are the native sequential binary64 sums. For nonzero d, each interior crossing is s=(end−t−c)/d. Sorting those exact rational crossings provides a complete partition for this ideal affine query. Open segment interiors and exact node ownership are recorded separately. The native ≤ test assigns an exact tie to the earlier packet, even when only an isolated endpoint of the path belongs to that packet.

The partial fraction is (q−start)/dt. Because the rounded cumulative end need not equal start+dt in real arithmetic, a branch can require a fraction just above one. The report preserves that overshoot, the cumulative discrepancy and whether the native fraction guard supports it. This does not prove coverage of rounding in the native query multiplication, addition or predicate.

For a piecewise smooth residual r, let h=r′ on each open segment. Then

\[ r(1)=r(0)+\int_0^1 h(s)\,ds+\sum_c [r]_c +[r]_{0+}+[r]_{1-}. \]

Here the endpoint terms mean r(0+)−r(0) and r(1)−r(1−). The integral uses the previously checked piecewise trapezoid allowance, with the interior one-sided derivative values at 0 and 1. All signed residual-jump intervals are added outward. Bounds on jumps in h and h′ remain required independently; they are not substitutes for jumps in r. Missing bounds, including endpoint zeros, fail.

The jump component verifies this arithmetic under supplied piecewise smoothness, partition and jump assumptions. It does not establish those assumptions for native trajectories or qualify any observational confidence level.