Improvement arithmetic: four checks

Four improvement arithmetic checks. The operators that look free usually are not; these are the four that decide whether an improvement is real.

1. Cutting sample volume 10× costs SNR

Signal falls 10×, shot noise falls √10, so SNR falls by √10 = 3.16×. The improvement is real only if the original SNR had 3.16× to spare. Rule: state the SNR margin before claiming a volume reduction.

2. Removing a calibration trades accuracy for stability

Route	Error
With calibration	1.0%
Without calibration (stability-limited)	3.0%

Removing the step triples the error but removes the standard. Whether that is an improvement depends on which resource is scarce — it is a trade, not a gain.

3. Ratiometric measurements cancel common-mode drift

A 5% common-mode drift becomes 0.05% residual in a ratio. This is the one operator in the list that is close to free, and it is the same trick as the reference-ratio work in project 1.

4. Parallelisation adds bias while keeping variance

Channels	Throughput	Added bias
2	2×	2.8%
4	4×	4.0%
8	8×	5.7%
16	16×	8.0%

Averaging across independent channels reduces variance but cannot remove a bias that differs between channels. Throughput scales linearly; accuracy degrades as √n.

5. What this says about the 20,000

Of the twenty operators, one is close to free (ratiometric), several are trades (volume, calibration, cheaper detector), and some are arithmetic traps (parallelisation, proxy). The catalogue's scores rank expected gain; these computations say where the gain is paid back.

6. Limits

The four checks use illustrative constants; the scalings are the content.
No improvement was implemented or benchmarked.
"Close to free" for ratiometric assumes the second channel is available at no cost.

Incoprea Research · study 9, method improvements · 3 November 2026. Composition and computation. No experiment performed.