Bell et al. (1997) reported the first detection of the cyanopolyyne HC$_{11}$N toward the cold dark cloud TMC-1; no subsequent detections have been reported toward any source. Additional observations of cyanopolyynes and other carbon-chain molecules toward TMC-1 have shown a log-linear trend between molecule size and column density, and in an effort to further explore the underlying chemical processes driving this trend, we have analyzed GBT observations of HC$_9$N and HC$_{11}$N toward TMC-1. Although we find an HC$_9$N column density consistent with previous values, HC$_{11}$N is not detected and we derive an upper limit column density significantly below that reported in Bell et al. Using a state-of-the-art chemical model, we have investigated possible explanations of non-linearity in the column density trend. Despite updating the chemical model to better account for ion-dipole interactions, we are not able to explain the non-detection of HC$_{11}$N, and we interpret this as evidence of previously unknown carbon-chain chemistry. We propose that cyclization reactions may be responsible for the depleted HC$_{11}$N abundance, and that products of these cyclization reactions should be investigated as candidate interstellar molecules.
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