For every prime-power dimension, a complete set of d+1 mutually unbiased bases (MUBs) is known, but in non-prime-power composite dimensions, the maximum number N(d) remains open. The tensor-product construction supplies mini(piai)+1 MUBs for d=∏ipiai, and this is the best lower bound currently known for every non-prime-power composite d≤100. We do not attempt to determine N(d); we ask whether these specific tensor-product sets admit one additional basis. We build and verify the sets for all 64 such dimensions, with pairwise overlap deviations below 10−15; run a construction-free joint search for d≤7; compare twelve optimizers; and test construction, convergence, and success thresholds. For a specified protocol A—algorithm, initialization distribution, success criterion, and stopping rule—let qA(d) denote the probability that one descent recovers a provably existing extension. Among the successful local-descent protocols tested, the inferred probabilities have comparable order, whereas the number of descents per computational budget differs much more strongly. With maxfun unbounded, exact gradients and optimization on U(d) recover guaranteed extensions at d=10,12, and 16. Raising the Riemannian CG iteration limit from 6000 to 150,000 changes none of the success counts or extension-search summaries. In d=6, the numerical failure to extend the tensor-product-basis triple reproduces a known analytic unextendibility theorem; it does not resolve the general four-MUB problem. Moreover, the actual extension target is maximally entangled. Haar-random initialization has zero probability of lying exactly on that structured submanifold, and the unrestricted search has not been validated for convergence to it, so recovery probabilities from the tensor-basis validation task cannot be transferred directly to the extension problem. The constructed bases, numerical summaries, per-restart arrays for the optimizer and threshold studies, complete per-descent arrays, and all code are publicly archived.
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