Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

Cancellation in the packet formula

Reported diagnostics and arithmetic-aware interpretation › Cancellation in the packet formula · lines 2210–2240

The report evaluates $h = q + (1 - q) p_{N}$ close to one. The expression $1 - h^{2}$ then loses relative accuracy.

Applying the stable identities (eq:stabledefect)–(eq:stabledistance) to the supplied scalar values gives approximately

d_{N} (2 - d_{N}) \approx 2.52823210 \times 1 0^{- 6} .

This agrees with the reported operator-norm discrepancy at its displayed scale. The archived unstable formula differs from the norm value by approximately $3.59 \times 1 0^{- 11}$ .

This is an algebraic re-evaluation of supplied numbers, not a new execution or diamond-norm optimization. The two distances are not exactly equal in exact arithmetic:

W - W_{\infty} = 2 d_{N}, d_{⋄} (Ad_{W}, Ad_{W}) = 2 d_{N} - d_{N}^{2} .

At this small $d_{N}$ , their exact difference is much smaller than the cancellation error.

The original manifest is preserved. A future script should use the stable formula and distinguish arithmetic certification from tolerance checks.

Displayed formulas in this passage

unlabeled · line 2218
$d_{N} (2 - d_{N}) \approx 2.52823210 \times 1 0^{- 6} .$
unlabeled · line 2230
$W - W_{\infty} = 2 d_{N}, d_{⋄} (Ad_{W}, Ad_{W}) = 2 d_{N} - d_{N}^{2} .$