Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

A conservative report-width bound

Exact algebraic and resource checks › A conservative report-width bound · lines 2371–2397

By lem:bounds,

r < \frac{7}{2}, B < \frac{200}{129}, 2 C < \frac{29}{10} .

For $0 \leq s \leq 1$ ,

ℓ (e_{i}) < \frac{7}{2}, ℓ (c_{i}) < 7, ℓ (t_{i}) < 4, ℓ (o_{i}) < 8, ℓ (u_{i}) < \frac{200}{129} .

An induced spacelike length is no larger than its projected spatial displacement, so

ℓ (v_{i}) \leq ∣ C_{i} ∣ < \frac{29}{10},

and

ℓ (w_{i}^{\pm}) \leq ∣ B_{i} ∣ + ∣ C_{i} ∣ < \frac{200}{129} + \frac{29}{10} < 8.

The same conservative bound covers incoming active values. Thus all represented finite particle lengths are below $8$ , and in particular below $16$ . Unsigned $b + 4$ -bit reports suffice.

Displayed formulas in this passage

unlabeled · line 2374
$r < \frac{7}{2}, B < \frac{200}{129}, 2 C < \frac{29}{10} .$
unlabeled · line 2378
$ℓ (e_{i}) < \frac{7}{2}, ℓ (c_{i}) < 7, ℓ (t_{i}) < 4, ℓ (o_{i}) < 8, ℓ (u_{i}) < \frac{200}{129} .$
unlabeled · line 2385
$ℓ (v_{i}) \leq ∣ C_{i} ∣ < \frac{29}{10},$
unlabeled · line 2389
$ℓ (w_{i}^{\pm}) \leq ∣ B_{i} ∣ + ∣ C_{i} ∣ < \frac{200}{129} + \frac{29}{10} < 8.$