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Frame-count switching in Raman acquisition-budget sensitivity occurs when a different integer number of exposure pairs becomes optimal as the available time changes. For each feasible pair count $n$, let $V_n(W)$ be the minimum predicted ratio variance after optimizing its live exposures. The complete design isThe marginal value of unused time below a Raman frame cap gives slopes for a fixed $n$. At a branch switch, those slopes need not be the derivative of the lower envelope on both sides.

A simple original example uses equal shot-noise coefficients $\alpha =\beta =a$, readout duration $d$ per frame and cap $c$ per exposure. Equal on/off totals are optimal, givingTake $a=10^{-6}\mathrm{s}$, $c=0.10\mathrm{s}$ and $d=0.010\mathrm{s}$. Two pairs saturate their $0.40\mathrm{s}$ live-time capacity once $W\ge 0.44\mathrm{s}$. Three pairs have live time $W-0.06\mathrm{s}$ near this budget. Their variances agree atJust below this value the two-pair variance is the constant $10\times 10^{-6}$. Just above it the three-pair variance is $4\times 10^{-6}/(W-0.06\mathrm{s})$ and begins decreasing. One pair has lower exposure capacity, while four or more pairs incur more readout and have less useful live time near the switch, so these two branches determine the global optimum there.

The one-sided derivatives at an optimal Raman frame-count switch are zero on the left and $-25\times 10^{-6}{\mathrm{s}}^{-1}$ on the right. The optimal variance is continuous at this switch, but has a kink. At $W=0.45\mathrm{s}$ two pairs give $10\times 10^{-6}$; at $0.47\mathrm{s}$ three pairs give about $9.7561\times 10^{-6}$. Extra time becomes useful again after the discrete design changes, despite the previous fixed-count plateau.

The primary Raman experiment's saturation-limited exposures and shuttered on/off readout motivate these competing resources.^[ref] The example uses chosen rounded timings. Read noise, drift and switching requirements can move the branch crossings and must be included when choosing an actual acquisition protocol.

Joint time-and-readout perturbations at a Raman design tie describe the directional response of the best acquisition variance when both available wall time and per-frame readout cost change. A tie generally has different derivatives on opposite sides, so there need not be one gradient for the optimum. The active branch derivatives must be combined along the actual perturbation direction.

Use the original balanced-noise model in one-sided derivatives at an optimal Raman frame-count switch. With $a=10^{-6}\mathrm{s}$, per-exposure cap $c=0.080\mathrm{s}$ and readout $d$, the locally competing branches areThey tie along $W=6c+8d$. At $(W_{\ast },d_{\ast })=(0.60,0.015)\mathrm{s}$, define $K=4a/(6c{)}^2\approx 17.3611\times 10^{-6}{\mathrm{s}}^{-1}$. The three-pair branch is locally constant, while the four-pair gradient is $(-K,8K)$.

For a path $W=W_{\ast }+ϵu_W$, $d=d_{\ast }+ϵu_d$ with $ϵ\downarrow 0$, the optimum has directional derivativeThe sign of the net live-time change $u_W-8u_d$ selects the improving branch. This conclusion assumes the branches remain feasible and nonactive designs have a positive value gap nearby.

For original finite examples, adding8ms of budget while adding1ms to each readout leaves $W-8d$ exactly unchanged, so these two branches remain tied. Adding10ms while adding1ms per readout increases that live time by2ms: the four-pair variance becomes $4\times 10^{-6}/0.482\approx 8.2988\times 10^{-6}$, below the three-pair value $8.3333\times 10^{-6}$. The first-order improvement is $-K(0.002)\approx -3.4722\times 10^{-8}$. Adding only6ms with the same readout increase instead makes the four-pair branch worse; the three-pair optimum remains unchanged locally.

The resource distinction has a physical motivation. The primary stimulated-Raman experiment used alternating pump states and shuttered detector readout; its exposure and readout intervals were separate parts of the acquisition sequence.^[ref] The rounded timings and noise parameter above are a mathematical planning example, not that instrument's measured values. Changing exposure caps, state-switching costs or noise terms requires the corresponding branch model to be recomputed.