= Wire-address position model
{title2=$\sigma_x=p/\sqrt{12}$}
Consider an original idealized <track reconstruction> model. Let equally spaced wires have positions $x_k=kp$ and assign each event to the nearest wire, reporting $\widehat x=x_k$. Assume \b[exactly one correct wire address], perfect <detection efficiency>, exact wire positions and a <uniform distribution> of true positions within each cell. These assumptions explicitly exclude missed pulses, false addresses and several unresolved tracks.
Writing $u=x-x_k$, the <measurement error> is $e=\widehat x-x=-u$, with $u$ uniform on $[-p/2,p/2]$. Symmetry gives zero <expected value>. Its <variance> and <standard deviation> follow by integration:
$$
\mathbb E[e]=0,\qquad
\sigma_x^2=\mathbb E[e^2]=\frac{1}{p}\int_{-p/2}^{p/2}u^2\,du=\frac{p^2}{12},\qquad
\sigma_x=\frac{p}{\sqrt{12}}.
$$
For chosen <wire pitches> $p=2\,\mathrm{mm}$ and $1\,\mathrm{mm}$ this gives $0.577\,\mathrm{mm}$ and $0.289\,\mathrm{mm}$ respectively. These are model <standard deviations>, not measured chamber <spatial resolutions>. The largest possible absolute error in this model is $p/2$, which is a different quantity.
A useful subtlety is that the formula depends on the illumination. For an unresolved pencil beam fixed at $x=x_k+0.4p$, every reconstructed position is $x_k$. The repeated estimates have zero <variance>, but every error is $-0.4p$. Small repeat-to-repeat spread can coexist with a biased position. The uniform-cell result therefore cannot be transferred to an arbitrary beam profile or detector selection.
The wire address supplies one transverse coordinate in this model. Combining several separated measurements supports <track reconstruction>; a single address does not specify a complete <trajectory>. If two <charged particles> enter the same cell during one accepted interval, an address-only record can be identical to that from one particle. No reconstruction of that record can recover the lost multiplicity without additional information.
\Image[/wire-pitch-and-calibrated-charge-centroid.png]
{title=Chosen wire-address errors and a calibrated two-wire response model}
{description=Original teaching calculations, not measured data. Left: uniform-cell address error. Right: stipulated linear charge response with independent small channel noise; the noise estimate is a first-order approximation, not a chamber performance claim.}
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