The pit-smoking technique is documented among Jamaicas Taino inhabitants before European contact, and Maroon communities in the same region are separately credited with developing the spice marinade and much of the technique as it survives today; oral tradition and food-history writing attribute the fusion differently depending on which strand a given account emphasizes.
What would resolve this Colonial-era written accounts or archaeological evidence dating the spice marinade specifically, rather than the smoking pit alone, would help separate the two strands; much of the record is oral rather than documentary, which may keep this genuinely unresolved.
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