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Bell et al. reported the first detection of the cyanopolyyne HC11N towards the cold dark cloud TMC-1; no subsequent detections have been reported towards any source. Additional observations of cyanopolyynes and other carbon-chain molecules towards 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 analysed Green Bank Telescope observations of HC9N and HC11N towards TMC-1. Although we find an HC9N column density consistent with previous values, HC11N 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 HC11N, and we interpret this as evidence of previously unknown carbon-chain chemistry. We propose that cyclization reactions may be responsible for the depleted HC11N abundance, and that products of these cyclization reactions should be investigated as candidate interstellar molecules. [ABSTRACT FROM AUTHOR] more...