The sweet potato and the Manila galleon: the camote line
Sweet potato left the Americas three separate times and genetics can tell the lines apart: a pre-Columbian Polynesian crossing, dated in southern New Zealand to 1430-1460 CE; a Portuguese line west from the Caribbean to Africa, India and the Moluccas; and the camote line, Mexican clones carried into Asia on the Acapulco-Manila galleons. The Spanish leg is a hypothesis stated in conditional verbs by the geneticists who proposed it, and this object keeps it in those terms.
Why this wins its question: English-language coverage of the sweet potato collapses into one of two stories: the pre-Columbian contact debate, or a vague line about Europeans spreading American crops. The genetics has since separated three crossings by cultivar line, and this object states each with the evidence behind it - kumara to eastern Polynesia roughly a millennium ago and in southern New Zealand at 1430-1460 CE at 95% probability from 24 radiocarbon determinations and starch granules; batata west from the Caribbean with the Portuguese, leaving West African diversity 18% below American; camote from Mexico to the Philippines on the Acapulco-Manila galleons. It also does the thing the corpus exists to do and prices its own best claim down: the Spanish leg is quoted in the authors' conditional ("could have spread ... very soon after Spanish conquest of these islands in 1522"), the 1522 date is flagged as too early for any Acapulco-Manila sailing, and the claim sits at 0.55 rather than being written up as a Spanish achievement. The counter-evidence that Polynesian voyagers got there first is not a footnote here but a claim at higher confidence than the Spanish one.
Claims
Every assertion below is bound to registered sources and carries its own confidence. Weight them; do not treat the page as uniformly authoritative.
The crop's diffusion out of America is modelled as three distinct cultivar lineages, and the names people use for the plant follow the lines. The kumara lineage, from the southern gene pool of north-western South America, was "introduced to Eastern Polynesia by voyagers approximately 1000 to 1100 y ago". The camote lineage, from the northern Mesoamerican pool, was "introduced by Spanish galleons sailing between the Philippines and Mexico at approximately A.D. 1500". The batata lineage, northern and Caribbean, was "circumlocuitously introduced by Portuguese traders via Europe in the 16th century".
The Spanish leg is a hypothesis about a possibility, not a documented shipment, and the authors' own wording says so: Spanish "Acapulco-Manila" galleons "could have spread sweet potato clones from Mexico to the Philippines very soon after Spanish conquest of these islands in 1522". The conditional verb is theirs, and the date attached to it is loose - 1522 is the year after Magellan's landfall, not the year of a Spanish settlement or of any sailing between Acapulco and Manila. Read the claim as the opening of a route hypothesis, not as a dated cargo.
Europeans were not the first to move the plant across the Pacific, and the counter-evidence is dated to the decade. Bayesian modelling of 24 radiocarbon determinations places sweet potato in southern New Zealand at "1430-1460 CE at 95% probability" - about 150 years after Polynesians settled Te Waipounamu - supported by eight starch granules over 13.2 micrometres with Ipomoea batatas characteristics and by four pit structures resembling rua kumara storage. Any account that begins the crop's Pacific history with a Spanish galleon is out by a century and a half.
What the lines rest on is genetic and herbarium evidence rather than archives: chloroplast and nuclear microsatellites run on modern material and on historically collected herbarium samples, including specimens from "Captain Cook's first voyage of 1769". The New Guinea study genotyped 417 landraces with six chloroplast and 11 nuclear microsatellites. Documents are what this reconstruction does not have; the herbarium sheet is doing the work a ship's manifest would otherwise do.
Where the introduced lines landed, they were made into something new by the people who received them. New Guinea holds an estimated 5,000 cultivars and is treated as a secondary centre of diversity; the crop is proposed to have reached the highlands "around 1700, where the plant was then relatively rapidly adopted and diffused", and the nuclear data indicate that New Guinea landraces derive principally from the northern neotropical gene pool - the camote and batata lines. The diversity is Melanesian work done on an American plant delivered by European shipping.
The other European line ran to Africa and was Portuguese: sweet potato "was introduced to Africa by the Portuguese in the 16th century", probably first as a tuber in Tanzania and dispersing westward, and West African diversity measured across 132 varieties - 112 from Togo in 2012 and 20 from Senegal in 2014 - is 18% lower than in America. The study does not name which cultivar line arrived and draws no link to the Atlantic slave trade; neither inference is made here.
The distinction that makes the whole reconstruction possible is that the pre-Columbian and the European-era introductions are separable in the genome: "distinctive genetic markers enable the differentiation of the kumara lineage derived from the Southern gene pool and the batata and camote lineages derived from the Northern gene pool". Without that split the Spanish, Portuguese and Polynesian crossings would be one undifferentiated arrival, which is how popular accounts still treat them.