Outward arithmetic and the complete event sum

26 Outward arithmetic and the complete event sum

The preceding arguments identify each error as an event or a nonnegative current functional under the same initial coupling. The preparation, old-wave, new-prefix and copy/hold appendices specify their coefficient formulae and evaluation bounds. No numerical certificate is used in place of an analytic inequality.

For reproducible square-root enclosures, given a nonnegative rational $x$ and $n=10^{60}$, choose the least integer $k$ with $k^2\ge n^2x$; then $k/n$ is an upper bound. Polynomial integrations are exact rational operations, and positive Taylor remainders or explicit derivative bounds control the nonpolynomial terms as detailed in the appendices. The following current ceilings have been rounded outward before combining:

The rank row below is evaluated as $2C\sqrt{2N(I_{\rm old}+\Delta I_S+I_Z)}$, with $C=270000/67499$ and $N=256001$. Every displayed table entry is itself a rational upper bound; unlike a shortened diagnostic display, these entries may be added directly. The harmless rounding of the pointer-inner event is deliberately included in the sum.

The first seven rows belong to earlier-label transfer, the next four to copying and retention, and the last three to both arguments. In both proofs the last three are the same clock-core, entire-clock-deviation and actual-radial-exit events of the new wave. Their sum is therefore charged once in the union. Direct addition of the displayed rational ceilings gives

The reference-label ceiling $.002207585289$ already includes both original $10^{-4}$ allowances. Applying Lemma 2.1 proves the advertised joint-record bound without counting either allowance again. The explicit outward sums also show that the last displayed headline digits do not depend on substituting rounded intermediate values into an unreported computation.