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A second-order reaction has overall order of reaction two: a rate law may depend on $[\mathrm{A}{]}^2$, or on $[\mathrm{A}][\mathrm{B}]$ with partial orders of reaction one and one. Doubling all relevant reactant concentrations multiplies the reaction rate by four at fixed conditions. This experimentally determined dependence does not establish molecularity or a unique reaction mechanism. For an elementary reaction obeying the law of mass action, molecularity and overall order of reaction agree; a net chemical equation alone cannot establish that the conversion is elementary.^[ref]

For the single-concentration model, explicitly choose the disappearance rate conventionHere $k_{\mathrm{obs}}$ is the coefficient for loss of A, held fixed at the specified temperature and other conditions. Its subscript labels the chosen observed disappearance rate; it does not imply a pseudo-first-order reaction. Assume a uniformly mixed closed system, fixed volume, and no appreciable reverse or competing chemical reactions. The integrated rate law isThe reciprocal-concentration plot is a straight line with slope $k_{\mathrm{obs}}$ and intercept $1/[\mathrm{A}{]}_0$. A brief derivation uses the chain rule: $d(1/[\mathrm{A}])/dt=-[\mathrm{A}{]}^{-2}d[\mathrm{A}]/dt=k_{\mathrm{obs}}$, then applies the initial condition. This is a hyperbolic decay, rather than the exponential decay of a first-order reaction.^[ref]

As an original prescribed example, choose $[\mathrm{A}{]}_0=0.020\mathrm{mol}{\mathrm{L}}^{-1}$ and $k_{\mathrm{obs}}=0.50\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. At $t=100\mathrm{s}$, the reciprocal-concentration calculation is $50+50=100\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}$, giving $[\mathrm{A}]=0.010\mathrm{mol}{\mathrm{L}}^{-1}$. The initial disappearance rate is $0.50(0.020{)}^2=2.0\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$; after $100\mathrm{s}$ it is $5.0\times 10^{-5}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$. Halving the concentration quarters the disappearance rate.

Dimensional analysis checks every formula: $k_{\mathrm{obs}}[\mathrm{A}{]}^2$ has units $\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$, $k_{\mathrm{obs}}t$ has units $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}$, and $k_{\mathrm{obs}}[\mathrm{A}{]}_{0}t$ is a dimensionless quantity. Changing concentration units changes the numerical rate constant: the same example is $[\mathrm{A}{]}_0=20\mathrm{mol}{\mathrm{m}}^{-3}$, $k_{\mathrm{obs}}=5.0\times 10^{-4}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$, giving $[\mathrm{A}](100\mathrm{s})=10\mathrm{mol}{\mathrm{m}}^{-3}$.

For an elementary reaction $2\mathrm{A}⟶\mathrm{P}$, instead define the stoichiometrically normalized reaction rate $r=k[\mathrm{A}{]}^2$. Two A entities disappear per P formed, soConsequently the reciprocal-concentration slope is $2k$, not $k$. In the prescribed example, the normalized rate constant would be $k=0.25\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$; initially $r=1.0\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ while the disappearance rate is twice that. With $[\mathrm{P}](0)=0$, stoichiometry gives $[\mathrm{P}](100\mathrm{s})=([\mathrm{A}{]}_0-[\mathrm{A}])/2=0.0050\mathrm{mol}{\mathrm{L}}^{-1}$. Both rate constants have the same dimensions, so a units check alone cannot catch their factor-of-two difference. The symbol $r$ denotes the same normalization as $v$ in reaction rate.

The half-life, meaning the time to reach half a specified starting concentration, follows by setting $[\mathrm{A}]=[\mathrm{A}{]}_0/2$:For $r=k[\mathrm{A}{]}^2$ in $2\mathrm{A}⟶\mathrm{P}$, this becomes $t_{1/2}=1/(2k[\mathrm{A}{]}_0)$. Its units are seconds because $k_{\mathrm{obs}}[\mathrm{A}{]}_0$ has units ${\mathrm{s}}^{-1}$. Doubling the starting concentration halves the first half-life; a quoted half-life needs its starting concentration. This differs from a first-order reaction, whose half-life is independent of starting concentration.^[ref]

Within the prescribed run, $[\mathrm{A}]$ reaches $0.010$, $0.0050$, and $0.0025\mathrm{mol}{\mathrm{L}}^{-1}$ at $100$, $300$, and $700\mathrm{s}$. The successive halving intervals are $100$, $200$, and $400\mathrm{s}$: each begins at a smaller concentration. More generally, reaching fraction $f$ of the initial concentration, with $0<f<1$, requires $t=(f^{-1}-1)/(k_{\mathrm{obs}}[\mathrm{A}{]}_0)$, so $90\%$ consumption takes nine initial half-lives, rather than a fixed sequence of equal waiting times.

A straight reciprocal-concentration plot tests this particular single-concentration rate equation. It does not uniquely identify a reaction mechanism, nor does curvature rule out every second-order reaction: two independently changing reactant concentrations require their own integrated rate law. At low concentration, a small measurement error $\delta [\mathrm{A}]$ becomes approximately $-\delta [\mathrm{A}]/[\mathrm{A}{]}^2$ in the reciprocal, magnifying late-time errors. Use the line as a diagnostic; a nonlinear regression of the original concentrations can retain their measurement scale and explicit error model.

Select units with first reading $X>1$, keep that membership fixed, and measure the same units again. The first reading now has a truncated normal distribution. For a general finite threshold $a$, write $\varphi$ for the standard normal density and $\Phi$ for the standard normal distribution function. Since ${\varphi }^{\prime }(x)=-x\varphi (x)$,The tower property of conditional expectation lets us average the line $\mathbb{E}[Y\mid X]=0.6X$ over the selected group.^[ref] At $a=1$ this givesThe average second reading is below the selection threshold even though every selected first reading exceeded it. This is a group expectation; any finite generated group's sample mean fluctuates around it. Selecting again on $Y>1$ would define a different group and answer a different question.

The latent variable decomposition makes the apparent drop especially clear. Gaussian conditioning gives $\mathbb{E}[T\mid X=x]=0.6x$ and $\mathbb{E}[{\epsilon }_1\mid X=x]=0.4x$. ThusThe first measurement error has zero unconditional mean, but positive selected mean. The second measurement error stays centered because it is independent of selection on $X$. The persistent signal $T$ never changed.

Stellar parallax is the apparent change in a star's viewing direction caused by the observer's changing position. Nearby stars shift more against a distant reference background. In an ideal circular Earth orbit, the conventional annual stellar parallax $p$ uses a baseline of one astronomical unit, approximately the Earth-Sun distance. Two opposite observing positions, about six months apart, are separated by approximately two astronomical units. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. ^[ref].

For the perpendicular baseline in the diagram, a right triangle giveswhere $p$ in the final expression is in radians. Combining the small-angle approximation with the parsec convention givesThus $p=0.05$ arcseconds implies $d\simeq 20$ parsecs, while the ideal six-month excursion is $0.10$ arcseconds. Geometry supplies this distance without assuming a luminosity or a particular stellar atmosphere.

In practical astrometry, many observations separate the annual stellar parallax signal from proper motion, the star's own changing sky position. The Gaia DR3 catalogue reports stellar parallax and its uncertainty in milliarcseconds: $50\mathrm{mas}=0.05$ arcseconds. A catalogue angle is a fitted measurement with measurement error. ^[ref].

For a positive stellar parallax with small standard deviation ${\sigma }_p$, first-order propagation gives ${\sigma }_d/d\simeq {\sigma }_p/p$. Thus $p=50\mathrm{mas}$ with ${\sigma }_p=1\mathrm{mas}$ gives $d\simeq 20$ parsecs with ${\sigma }_d\simeq 0.4$ parsecs, assuming systematic offsets are negligible. This linear estimate requires ${\sigma }_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.

A technical replicate repeats a measurement or analytical procedure on the same biological preparation. For example, read the length of the same leaf three times to assess measurement error. Measuring three different leaves within one pot instead samples variation within the plant; it should be described explicitly rather than silently equated with repeated readings of one leaf. Neither procedure supplies another pot-level treatment replicate.

An authored teaching experiment uses an adjustable steady-voltage supply, a digital multimeter, fixed resistors, and a $10\mathrm{k\Omega }$ potentiometer or resistance decade box. Its original circuit uses a $10\mathrm{V}$ supply and $3.3$, $6.8$, and $4.7\mathrm{k\Omega }$ resistors; the student compares $8.2$ and $2.2\mathrm{k\Omega }$ electrical loads. The readout is load voltage, recorded against prediction and deviation. The student then removes the electrical load to measure the open-circuit voltage, disconnects the supply and substitutes a wire to measure the Thévenin resistance with the multimeter's resistance mode, and restores the supply for an independent method: adjust the variable electrical load until the voltmeter reads half the unloaded voltage. Finally the student builds the equivalent and repeats both load measurements. Its tables are blank recording forms, not reported observations. ^[ref]

The half-voltage inference assumes negligible voltmeter loading and positive $R_{\mathrm{th}}$. A finite voltmeter input electrical resistance $R_m$ actually gives an apparent unloaded readingWhen the same voltmeter remains connected across a variable electrical load $R_L$, the network sees $R_p=R_L{R}_m/(R_L+R_m)$. Setting its reading to $V_m/2$, rather than to the true $V_{\mathrm{oc}}/2$, givesThus the uncorrected half-reading method underestimates positive $R_{\mathrm{th}}$. In the original $12\mathrm{V}$ example, a stipulated $R_m=10\mathrm{k\Omega }$ gives $V_m=120/17\mathrm{V}\approx 7.059\mathrm{V}$ and half-reading load $R_L=20/17\mathrm{k\Omega }\approx 1.176\mathrm{k\Omega }$, instead of $\frac{4}{3}\mathrm{k\Omega }$. These are model predictions for a deliberately loading meter, not the specification or observed readout of the source experiment's instrument.

Finite readings can also estimate the terminal line without a physical short. For two distinct load settings with corrected delivered electric currents $I_1,I_2$ and terminal voltages $V_1,V_2$,If the voltmeter remains across each electrical load, the delivered electric current is $I_k=V_k(1/R_{L,k}+1/R_m)$, including the meter branch. Omitting it biases the inferred slope.

Useful additional controls are to measure each resistor separately before assembly, verify terminal polarity, repeat an earlier load setting to detect supply drift, and compare a third setting with the inferred Thévenin terminal response. These are proposed controls, not reported outcomes of the source experiment. Closely spaced electric currents make the slope estimate sensitive to voltage resolution and measurement error; widely separated settings may trigger source current limiting or Joule heating that changes electrical resistance. Lead and contact electrical resistance, meter input uncertainty and rounding limit the inference. An ohmmeter injects its own test electric current; its reading on a powered network is not the suppressed-network ratio. Measurements with independent sources physically removed do not characterize an active dependent electrical source whose control or supply has also been disconnected.

A torsion balance measures a weak torque by suspending a body from a thin fiber and reading its twist angle. The fiber supports the body's gravitational weight while permitting rotation about the suspension axis. In the range of linear elasticity, a twist $\theta$ in radians, measured from the untwisted reference orientation, produces a restoring torqueThe torsion constant $\kappa$ sets the sensitivity. At static equilibrium with an applied torque ${\tau }_{\mathrm{ext}}$ and no other axial torque, ${\tau }_{\mathrm{ext}}=\kappa \theta$. A perpendicular force $F$ at lever arm $a$ therefore gives $\theta =aF/\kappa$. The torsion balance measures a turning effect: a force whose line of action passes through the suspension axis gives no axial torque. The Cavendish torsion balance is the gravitational application; Coulomb's law supplies an electrical one. ^[ref]

The original schematic separates two ways of reading the same torque. In deflection operation the top attachment stays fixed and the body rotates. In null operation the experimenter rotates the top attachment by $\alpha$ while returning the body's laboratory angle $\beta$ to its original value. If the initially untwisted configuration has $\alpha =\beta =0$, the fiber twist is $\theta =\beta -\alpha$, not $\beta$ alone. ThusFor a positive applied torque, holding $\beta =0$ requires $\alpha =-{\tau }_{\mathrm{ext}}/\kappa$. Keeping the body at the same angle keeps the lever arm and source geometry fixed, making comparison of different applied forces easier. Either readout still requires a calibrated torsion constant and a stable reference orientation.

An illustrative calculation shows the amplification: with $a=0.050\mathrm{m}$, $F=2.0\times 10^{-9}\mathrm{N}$, and $\kappa =1.0\times 10^{-8}\mathrm{N}m/rad$, the applied torque is $1.0\times 10^{-10}\mathrm{N}\mathrm{m}$ and the equilibrium twist is $0.010\mathrm{rad}\simeq 0.573^{\circ }$. These are stipulated values, not measured data. Reducing the torsion constant tenfold increases this twist tenfold, provided the fiber remains in its calibrated linear elasticity range. The cost is a slower response, explained by the oscillation period in torsion constant.

A concrete laboratory experiment is PASCO's Coulomb balance: level its counterbalanced rod, discharge both conducting spheres, and set the unloaded zero. Charge the spheres at maximum separation; switch off the charging supply. Move the second sphere to a known center separation $R$, then turn the suspension dial until the rod returns to its index mark. Record the dial change and $R$, recharge, and repeat at several separations. The manual recommends a draft-free room, clean insulating supports, and recharging before each reading: leakage, especially in humid conditions, otherwise changes the electric charges during the sequence. At close separations the conducting spheres redistribute charge, so the point charge approximation needs correction. ^[ref]

For constant electric charges and a fixed perpendicular lever arm, the Coulomb's law prediction is $|\theta |\propto R^{-2}$. Plotting the logarithm of twist magnitude against the logarithm of separation gives $\log |\theta |=\text{constant}-2\log R$, so the predicted slope is $-2$. Absolute force measurement additionally needs the torsion constant; comparing ratios can eliminate it if it remains stable. Repeating a reference separation between other settings distinguishes a separation effect from a changing signal. A declining reading at the reference separation would invalidate the assumption of unchanged electric charges, even if a distance plot looks smooth.

The historical electrical example is Coulomb's first memoir of 1785. His apparatus enclosed a suspended needle with a small ball and a second ball; an adjustable head twisted the suspension. After charging the balls by contact, he reported angular separations $36^{\circ }$, $18^{\circ }$, and $8.5^{\circ }$ with head settings $0^{\circ }$, $126^{\circ }$, and $567^{\circ }$. The total twists were approximately $36^{\circ }$, $144^{\circ }$, and $576^{\circ }$: fourfold restoring torque accompanied approximately halved separation. These are his reported observations, not data from our diagrams. ^[ref] A precise reconstruction must use center distances and the changing lever arm, rather than identify angular separation exactly with distance; Alberto A. Martínez's replication explicitly analyzes that geometry. ^[ref]

Sensitivity also exposes changes of the apparatus itself. Fleischer and colleagues' cryogenic torsion balance used a temperature-controlled enclosure to reduce vacuum-vessel top-plate temperature fluctuations to approximately $\pm 0.01\mathrm{K}$, limiting changes in suspension geometry from thermal expansion. They used a mild bake-out near $40^{\circ }\mathrm{C}$ to speed initial fiber relaxation and reduce slow zero motion. Their optical readout compared the pendulum mirror with an internal reference mirror to reject shared optical motion. Room-temperature thermal radiation entering through the optical opening heated the pendulum; a cold filter reduced that load. The authors could not directly measure the pendulum's temperature without disturbing it, and explicitly distinguished the cold shield's temperature from the body's. Cooling therefore requires checking the actual noise and thermal environment, rather than assuming that the fiber and body share the shield's temperature. ^[ref]

A useful control strategy is to record unloaded readings and temperature before and after each signal sequence, recalibrate the torsion constant under the same conditions, and compare forward and reverse settings. The algebra of torsion balance zero drift shows precisely which offsets such comparisons can cancel. A fine angular scale alone does not establish a small measurement error.

Oberlin College's documented demonstration uses a track, two collision carts, two force probes set to the $50\mathrm{N}$ range, added $500\mathrm{g}$ bars, and a laptop with a Vernier interface. Preparation consists of entering the measurement calibration, securing the force probes to the carts, and zeroing them. Rubber stoppers serve for elastic collisions, clay for inelastic collisions, and hooks with string for pulling. Comparing the two force records tests the mutual interaction; adding mass to one cart tests whether unequal masses change the equality. ^[ref]

Vernier's force probe uses a bending beam with strain gauges; changing electrical resistance produces a voltage readout calibrated as force. Its manual recommends zeroing in the working orientation. If recalibration is needed, use an unloaded reading and a known suspended gravitational weight; for the $50\mathrm{N}$ range its example is $1\mathrm{kg}$ giving $9.8\mathrm{N}$. Re-zero after changing orientation. The manual quotes calibrated accuracy $\pm 0.2\mathrm{N}$, distinct from the $0.05\mathrm{N}$ resolution of that range, and warns that the interface's data-collection rate can limit collision measurements. ^[ref]

The following sampling and control choices are proposed additions to that documented experiment. Level the track and check unloaded motion in both directions for slope and friction. Use a soft bumper and one interface for simultaneous records; start before contact and stop after separation. A trial sampling rate of $1000{\mathrm{s}}^{-1}$, if supported, would put about 100 samples across a $0.10\mathrm{s}$ interaction. Increase the sampling rate and repeat until the peak and integrated impulse comparison are stable; a large sample count alone does not establish adequate response. Measure each total moving mass, including the force probe and added bars.

Convert both channels to the same right-positive force convention. Oppositely facing force probes can have matching raw signs because their local positive directions differ; check each sign by a gentle push of known direction before the run. Overlay $F_{B\mathrm{on}A}(t)$ and $-F_{A\mathrm{on}B}(t)$, and also examine their signed sum. Integrate both over the same contact interval to compare impulses. Repeat with added mass on one cart, exchange the probes between carts, and check unloaded zeros before and after. These controls help distinguish a physical mass change from a channel offset or gain error.

Agreement within measurement error supports the pair-interaction model over the resolved interval; it does not establish exact equality at arbitrarily short time scales. Misalignment, a loose mounting, sensor-beam dynamics, saturation, unequal timing, and zero drift can spoil the comparison. Track friction and cable pulls are additional external forces; a probe measuring bumper contact need not measure the entire net force on its cart. If a massive connector carries appreciable changing momentum, compare the full cart–connector collection rather than assuming its endpoint forces are an instantaneous cart-to-cart pair. The constructed pulse diagram above illustrates what to compare, and supplies no measurements from this apparatus.

Choose two point-like astronomical sources with $({\theta }_1,I_1)=(-0.00625,1)$ and $({\theta }_2,I_2)=(0.009375,0.65)$, where the angles are in radians and strengths use arbitrary common units. Their separation is $\Delta \theta =0.015625$ radians. These are invented model parameters, not historical measurements. Their complex visibility follows immediately from the preceding derivation, and its squared magnitude isAt $u=0$, $V=1.65$; at $u=400$, $V=-1+0.65i$; at $u=800$, $V=0.35$. The unequal strengths prevent the complex visibility magnitude from reaching zero. Its modulation contains the source separation, while a common displacement of both sources changes wave phase without changing this magnitude. Amplitude alone loses that location information.

The sparse case samples $u=\pm 12.5k$ for $k\in \{2,5,9,14,20,28,36,44,52,60,64\}$; the denser case samples every $k=1,\dots ,64$, also with both signs. All weights are one. Both reach $|u|=800$, so their shortest fringe period is the same $0.00125$ radians. Their dirty beams and false subsidiary features differ because the intermediate spatial frequencies differ. Neither case includes $u=0$ or frequencies above that maximum. The figure uses a periodic angular interval of length $0.08$ radians; this artificial boundary makes a discrete Fourier transform calculation exact for the chosen grid, rather than modelling an all-sky observation.

The plotted dirty image is the sum of two shifted dirty beams, not a fitted or deconvolved estimate of the source. Adding samples in this example reduces prominent spurious peaks; it does not establish a universal monotonic improvement for every weighting or sky. A missing zero baseline leaves the periodic dirty image with zero mean, so it cannot by itself recover the total positive brightness. Missing short spacings also weaken constraints on extended smooth emission.

Reconstruction is an inverse problem. A perturbation whose Fourier transform vanishes at every sampled coordinate can be added without changing the measurements: incomplete coverage leaves a null space. Nonnegative brightness, a restricted source region, or a model with a small number of components can reduce ambiguity, but each is additional information. Removing dirty beam artefacts means estimating a sky consistent with both data and such assumptions; a narrow central peak alone does not certify a unique image. Measurement error, uncertain calibration and source variability add further limitations.

A real voltmeter has finite input electrical resistance $R_d$; a galvanometer also has a finite winding electrical resistance. Away from balance, its electric current changes both intermediate electric potentials. This is voltmeter loading when the detector is a voltmeter. Model the detector as a passive resistor, with no offset or injected electric current. Thévenin's theorem supplies the full response without pretending the connected bridge remains two independent voltage dividers.

With the detector removed, the equivalent source is $V_{\mathrm{oc}}$. Replace the ideal supply by a wire to calculate the source electrical resistance seen between $B,C$: nodes $A,D$ merge, making $R_1,R_2$ parallel and $R_3,R_x$ parallel. ConsequentlySince the attenuation factor is strictly positive for $R_d>0$, the null remains $R_2{R}_3=R_1{R}_x$. A smaller detector electrical resistance reduces the measured electric potential difference off balance but does not shift the ideal null. Its electric current sensitivity instead depends on $1/(R_{\mathrm{th}}+R_d)$; detector gain and resolution also matter in deciding how closely a null can be located.

In the plotted example, $R_x=1.100\mathrm{k\Omega }$ gives $V_{\mathrm{oc}}=-119.048\mathrm{mV}$ and $R_{\mathrm{th}}=1.02381\mathrm{k\Omega }$. A detector with $R_d=1.000\mathrm{k\Omega }$ instead reads $V_d=-58.824\mathrm{mV}$. Using that loaded reading in the unloaded voltage divider formula would underestimate the deviation of $R_x$ from $1\mathrm{k\Omega }$. Balancing the bridge avoids this particular systematic measurement error.

The local derivative makes the sensitivity explicit. For $R_1=R_2=R_3=R$ and $R_x=R+\delta R$, the first-order Taylor expansion isAt the plotted null, the open-circuit slope is $-1.25mV/\Omega$, compared with $-0.625mV/\Omega$ for $R_d=1\mathrm{k\Omega }$. Increasing the supply improves this ideal sensitivity but also increases Joule heating. A finite source electrical resistance requires recomputing $R_{\mathrm{th}}$; the displayed parallel-pair expression assumes an ideal supply.

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 $\hat{x}=x_k$. Assume 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 $e=\hat{x}-x=-u$, with $u$ uniform on $[-p/2,p/2]$. Symmetry gives zero expected value. Its variance and standard deviation follow by integration: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.