Gear inspection missing-area bound

A gear inspection missing-area bound limits the whole-target pitting fraction using the area actually observed and explicit constraints on the unseen area. It formalizes what a gear inspection coverage audit can establish without assuming that excluded regions resemble visible ones.

Let $c=A_{\mathrm{seen}}/A_{\mathrm{target}}$, let $p$ be the observed gear pitting area fraction, and let $u$ be the unseen fraction affected by pitting. For a consistent surface-area convention,With only $0\le u\le 1$ known,The interval width is exactly $1-c$. Improving classification precision within the visible area cannot by itself remove uncertainty from a completely unseen region.

For an original example with $c=0.9$ and $p=0.02$, the possible whole-target fraction ranges from $0.018$ to $0.118$. The observed $2\%$ therefore does not prove that the whole surface is below an illustrative $5\%$ criterion. That chosen criterion is not a gear safety or acceptance standard.

The worst-case upper bound meets a target $\theta <1$ only iffor $p<1$. At $p=0.02$ and $\theta =0.05$, at least about $96.94\%$ coverage is required for this particular worst-case guarantee.

Additional justified information can narrow the range. If a separate observation establishes $0\le u\le 0.1$, then $cp\le \Phi \le cp+0.1(1-c)$. At $c=0.8$ and $p=0.02$, this gives $0.016\le \Phi \le 0.036$. The tighter bound depends on the new evidence about the unseen region, not on a mathematical preference for a smaller interval.

These deterministic bounds assume the observed classification and represented areas are correct. Image registration, image segmentation and geometric uncertainties can require additional allowances. They are not confidence intervals and do not describe pit depth or remaining fatigue life. The primary NASA endurance study combined inspection with vibration and debris monitoring; these different observations constrain different aspects of damage.^[ref]