= Solution
\b[A housekeeping Ct is unnecessary for the within-sample short:long ratio], because a common reference cancels. With matched doubling efficiencies,
$$
2^{-[(C_{t,S}-C_{t,\mathrm{TBP}})-(C_{t,L}-C_{t,\mathrm{TBP}})]}=2^{C_{t,L}-C_{t,S}}.
$$
However, a validated reference such as <TATA-binding protein> transcript is needed when comparing the abundance of either isoform separately between samples while correcting for input, <RNA> recovery and <reverse transcription>. Reference stability must be tested under the differentiation conditions. Neither reference normalization nor cancellation removes the need to validate the two isoform assays' efficiencies and response calibration.
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